THE DYNAMICAL DECOMPOSITION OF GROWTH AND CYCLES

Luís Francisco Aguiar Tese de Mestrado em Economia

Orientação

Francisco Louçã

Faculdade de Economia Universidade do Porto 2000

Nota Biográfica Formação Académica Pós-graduação em Economia ⎯ especialização em Métodos Quantitativos Aplicados à Economia ⎯ na Faculdade de Economia da Universidade do Porto em 11 de Outubro de 2000, com média final de 17 valores. Licenciado em Economia na Faculdade de Economia da Universidade de Coimbra em 2 de Setembro de 1997, com a classificação final de Bom com Distinção — 16 valores. Experiência Profissional Desde 15 de Abril de 1998 é Assistente Estagiário na Escola de Economia e Gestão da Universidade do Minho, onde já leccionou a cadeira de Teoria do Comércio Internacional, a cadeira de Inovação e Transferência de Tecnologias e a cadeira de Economia Política II. De 10 de Novembro de 1997 a 1 de Abril de 1998 realizou um estágio no Núcleo de Mercados e no Núcleo de Estudos e Planeamento do Instituto de Gestão do Crédito Público (IGCP).

Agradecimentos Sendo um trabalho deste género por natureza individual ⎯ dir-se-ia até solitário ⎯, não teria sido possível concretizá-lo sem o apoio e colaboração de algumas pessoas. Sem pretender elaborar uma lista exaustiva, gostaria de, em primeiro lugar, agradecer ao Prof. Doutor Francisco Louçã, que sendo uma pessoa obviamente ocupada, nunca deixou que a orientação desta tese passasse para segundo plano, tendo os seus comentários críticos sido marcantes na evolução deste trabalho. Redigido numa segunda língua, foi importante o apoio que recebi na sua escrita. Agradeço, por isso, a meu pai, Cristóvão de Aguiar, a extrema paciência com que foi lendo e relendo o trabalho, procurando sempre a melhor forma de me expressar. Sublinho, porém, que quaisquer erros gramaticais ou de outra natureza são da minha exclusiva responsabilidade. Num registo mais íntimo, gostaria também de agradecer à minha família mais chegada, tanto pelo apoio incondicional que dela recebi como pelas oportunidades que me foram proporcionadas. Assim sendo uma referência/reverência a minha mãe, Otília Meirinho, a meu pai e a meu irmão mais velho, Artur João. Um agradecimento muito especial, e particularmente carinhoso, à minha companheira desta e de outras ocasiões, Sandra, pelo incentivo, pelo apoio e pela presença.

Contents 1 Introduction

5

2 Some Reference Growth Models

9

2.1 The Harrod-Domar Model — An AK Model . . . . . . . . . . . . . . .

9

2.2 The Basic Neoclassical Growth Model . . . . . . . . . . . . . . . . . . 10 2.3 The Arrow-Frankel-Romer Model of Endogenous Growth . . . . . . . 12 2.4 A Model of Growth Through Vertical Innovations . . . . . . . . . . . 15 2.5 Consequences of Endogenous Growth Theories . . . . . . . . . . . . . 18 3 Some Reference Business Cycle Models

20

3.1 Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1.1

The Slutzky-Frisch decomposition . . . . . . . . . . . . . . . . 20

3.1.2

Equilibrium Business Cycles (EBC)– Lucas’ Model . . . . . . 23

3.2 Deterministic Linear Models – Samuelson’s Multiplier-Accelerator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Deterministic Nonlinear Models . . . . . . . . . . . . . . . . . . . . . 27 3.3.1

The Kaldor Trade Cycle . . . . . . . . . . . . . . . . . . . . . 30

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4 Growth and Cycles: an Actual Distinction or a Theoretic Abstraction?

36

4.1 Decomposition Between Trend and Cycles in Nonstationary Time Series 38 4.1.1

Empirical findings and consequences . . . . . . . . . . . . . . 41

4.2 Models of Growth and Cycles . . . . . . . . . . . . . . . . . . . . . . 43 4.2.1

Real Business Cycle Models . . . . . . . . . . . . . . . . . . . 43 1

4.2.2

Business Cycle Models with Endogenous Technology . . . . . 48

4.2.3

Growth Based on General Purpose Technologies . . . . . . . . 53

4.2.4

Goodwin’s Predator-Prey Growth Cycle . . . . . . . . . . . . 60

4.2.5

Attraction for Chaos in Goodwin’s Model . . . . . . . . . . . 72

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5 Chaotic Cycles

77

5.1 Chaos, Attractors and SDIC . . . . . . . . . . . . . . . . . . . . . . . 78 5.1.1

Transition to Chaos . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Some Economic Applications of Chaos . . . . . . . . . . . . . . . . . 83 5.2.1

The Neoclassical and the Endogenous Growth Models . . . . . 83

5.2.2

Kaldor’s Trade Cycle in Discrete Time . . . . . . . . . . . . . 84

5.2.3

A Model of Growth and Cycles . . . . . . . . . . . . . . . . . 88

5.3 The Adequacy of the Traditional Econometric Approach to Nonlinear Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3.1

The Augmented Dickey-Fuller (ADF) Test – A Unit Root Test 92

5.3.2

An Econometric Application to Our Artificial Model . . . . . 94

5.3.3

Problems with Heteroskedasticity . . . . . . . . . . . . . . . . 96

5.3.4

The BDS Statistic . . . . . . . . . . . . . . . . . . . . . . . . 100

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6 Some Conclusions

107

2

List of Figures 3-1 English Business Cycles and Stochastic Moving Summations . . . . . 21 3-2 A Frischian Version of Samuelson’s Multiplier-Accelerator Model . . . 27 3-3 Goodwin’s Multiplier-Accelerator Model Dynamics . . . . . . . . . . 29 3-4 Kaldor’s Nonlinear Investment and Saving Functions . . . . . . . . . 31 3-5 A Phase Diagram of Kaldor’s Trade Cycle Model . . . . . . . . . . . 33 4-1 Diffusion of the New GPT . . . . . . . . . . . . . . . . . . . . . . . . 57 4-2 Time Evolution of Aggregate Output . . . . . . . . . . . . . . . . . . 59 4-3 Predator-Prey Growth Cycle . . . . . . . . . . . . . . . . . . . . . . . 66 4-4 Goodwin’s Model with Anti-Cyclical Productivity Growth . . . . . . 68 4-5 Limit Cycle in Goodwin’s Model with Pro-Cyclical Productivity Growth 69 4-6 Period Doubling (g = 90) . . . . . . . . . . . . . . . . . . . . . . . . . 74 4-7 Second Period Doubling (g = 125) . . . . . . . . . . . . . . . . . . . . 74 4-8 Rossler Strange Attractor (g = 150) . . . . . . . . . . . . . . . . . . . 75 5-1 Kaldor Investment and Savings Function . . . . . . . . . . . . . . . . 86 5-2 Kaldor Business Cycle Attractor (Y) . . . . . . . . . . . . . . . . . . 86 5-3 Kaldor Business Cycle Attractor (K) . . . . . . . . . . . . . . . . . . 87 5-4 SDIC in Kaldor’s Model . . . . . . . . . . . . . . . . . . . . . . . . . 87 5-5 Capital Evolution for 50 years . . . . . . . . . . . . . . . . . . . . . . 89 5-6 Output Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 90 5-7 The Estimated Model Dynamics . . . . . . . . . . . . . . . . . . . . . 96 5-8 Two Simulated Time Series . . . . . . . . . . . . . . . . . . . . . . . 97 5-9 Dimension of Stochastic Processes vs Deterministic Processes . . . . . 100

3

List of Tables 5.1 ADF Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 ARCH LM Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3 ARCH LM Test to EGARCH standard residuals . . . . . . . . . . . . 99 5.4 BDS Test to the AR(4) residuals . . . . . . . . . . . . . . . . . . . . 102 5.5 BDS Test to the EGARCH(2,4) residuals . . . . . . . . . . . . . . . . 103 5.6 BDS Test to Deterministic Chaotic Time Series . . . . . . . . . . . . 105

4

Chapter 1 Introduction The main objective of this work is to question the traditional decomposition between growth and cycles. This approach is common both in theoretical and applied works. This main objective may be divided into three sub-objectives: first is to introduce the dominant paradigm – the Slutzky-Frisch approach, which supports the decomposition – and the leading alternative – a nonlinear approach, in which the separation between trend and cycles is rejected. The second sub-objective is to indicate some fragilities of the theoretical construction that supports the decomposition of growth and cycles. The last sub-objective is to explore one extension of the nonlinear approach: chaos theory, not only at a theoretical level but also at an econometric level. We will show not only the inadequacy of the traditional econometric technique to deal with chaotic time series, but also the limitation of some specific procedures developed so far. The decomposition between trend and cycles was first rationalized by Slutzky and Frisch in the late twenties and early thirties. Since then, this approach has been subject to various criticisms. Authors who have not accepted this approach are innumerous (Schumpeter, Harrod, Goodwin, Kaldor, Kalecki, etc.). The main problem with the critics is the absence of valid alternatives. As Goodwin once recognized, it is far more difficult to treat growth and cycles jointly. However, these difficulties should not refrain our efforts to find better ways to deal with growth and cycles together. In this work we begin with a quick revision of some growth models. The models 5

chosen were models which form the basis for some attempts to unify growth and cycles theory. The neoclassical (Solow) growth model is the basic growth model needed to understand the theory of Real Business Cycles. Theories of endogenous growth are essential to understand some more successful attempts to integrate cycles and growth in the same theory. Here we can distinguish two main approaches: the Romer endogenous growth model is the basis for Stadler analysis where the usual decomposition between short run (cycles) and long run (growth) is questioned. Stadler teaches us that we should not forget long run consequences of short run shocks. The other approach to endogenous growth is a neo-Schumpeterian model developed by Aghion and Howitt, which is the basis for the theory of growth based on General Purpose Technologies, where the reverse causation is shown. Namely, short run effects of long run growth. After a quick revision of some growth models the same revision must be made for business cycles models. Here we find again two main approaches. The first one, rationalized by Slutzky and Frisch, consists of considering linear stable economic systems disturbed by a stream of exogenous shocks. So a distinction is made between the impulse (a stream of exogenous shocks) and the propagation (a linear oscillating stable system) mechanism. The ability of this approach to provide an explanation of business cycles was called in question: to rely on exogenous shocks amounts to a confession of ignorance. In the other approach, nonlinear systems are considered and business cycles are endogenous. This approach was popularized by Goodwin and Kaldor. The basic idea of these authors was that instability and fluctuations are a consequence of market failures and thus must be endogenous to the economic system. This class of models has two main advantages: first, more realistic models may be created; second, we do not need to rely on exogenous shocks anymore to explain cycles1 . Before the analysis of some of the attempts to integrate both growth and cycles, we begin by studying some of the criticisms to the decomposition between them. It 1

Obviously, these kind of models do not exclude exogenous shocks. Any manageable model must exclude some relevant variables, so there will always be shocks which are exogenous to the system. What these models try to show is that even in the absence of those shocks fluctuations would occur.

6

is interesting to note that one of the assaults against the traditional approach came from the inside of the paradigm. Nelson and Plosser showed some of the problems involved with a wrong econometric specification of the trend when analyzing time series data. More specifically they showed that if the trend of an economic time series is stochastic and extracted as deterministic, spurious short run cycles will be generated. This idea opened the way to Real Business Cycles theory and to Stadler’s model which extended the approach to endogenous growth models. A more radical critique is made by authors who, denying the linear structure of the economic system, cannot accept an additive decomposition of the trend (be it stochastic or deterministic) and cycles. One of the most celebrated nonlinear models of growth cycles (Goodwin’s predator-prey model) is analysed at some length in this work. We extend the model in order to accommodate the possibility of some endogenous labour productivity growth and we also relax the assumption of a fixed coefficients technology by considering a general CES production function. This way, a more realistic growth model is achieved. When that extension is made it is not hard to point out some of its limitations, namely its structural instability. The above limitation rather than being a big flaw is an opportunity to conceive a much richer model. The introduction of a control parameter turns Goodwin’s model dynamically unstable, but structurally stable. With this model, we can also learn that from nonlinear models to deterministic chaos takes only one step. The consequences are severe. Generally nonlinear models could generate endogenous cycles of a wide range of cyclical behaviour. The aperiodic (erratic) behaviour of economic time series could not be explained by those models, unless we admitted exogenous shocks. The possibility that, in a chaotic model, a seemingly stochastic behaviour can be generated deterministically forces us to adopt the hypothesis that there are two distinct sources of irregularity in economic statistics: exogenous and endogenous. In the last chapter, after some basic definitions and theorems needed to understand chaotic dynamics, we show how chaos theory may be applied to some of the models studied in previous chapters. We use an extension of Goodwin’s model to generate artificial data. Following an approach initiated by Blatt, with the help of 7

those data we show that the traditional econometric approach is not able to deal with the possibility of deterministic chaos. We show that, with standard procedures, a linear econometric model can be estimated. That econometric model shows no evidence of misspecification with some traditional econometric tests. The estimated properties are qualitatively wrong. We conclude then that to accommodate the possibility of deterministic chaos specific procedures have to be developed. We present one of them: the BDS statistics, which was developed to detect low dimensional deterministic chaos (although it has power to detect other nonlinearities as well). We also show what can be an explanation for little evidence of deterministic chaos in macroeconomic time series: aggregation can hide evidence of chaos.

8

Chapter 2 Some Reference Growth Models 2.1

The Harrod-Domar Model — An AK Model

To our work, the most relevant feature of this model is its ability to generate endogenous growth. We may call it an early variant of the AK model. Although not in an explicitly way, Harrod (1939) and Domar (1946, 1947) assumed a production function with fixed coefficients. Consider the following production function1 : Y = min Obviously if

K a

≤

L b

½

K L , a b

¾

(2.1)

we have: 1 Y = K a

(2.2)

So, assuming a constant saving rate2 : K0 1 1 =s K0 = s K ⇔ a K a Where s is marginal/average propensity to save and Since Y = AK (with A ≡ a1 ) we have only if g ≡

K0 K

≤

L0 L

Y0 Y

=

K0 . K

(2.3)

d() dt

= ()0 is the time derivative.

This growth is self-sustained

≡ n, otherwise labour will become the limitational factor.

Naturally, this is only a crude model of endogenous growth since it requires 1

We are closely following the analysis of the Harrod-Domar model made by Sollow (1956). We are assuming away the possibility of depreciation of capital, hence there is no distinction between gross and net investment. 2

9

an increasing rate of unemployment (being unable to explain a steady increase in output per capita) and is based on a non-consensual assumption, namely that there is no substitutability between labour and capital.

2.2

The Basic Neoclassical Growth Model

The most relevant feature of the neoclassical growth theory is that in order to have a positive growth rate of output per capita we need exogenous technological progress. This can be illustrated with Solow’s (1956) basic growth model. To understand Solow’s idea, a simple Cobb-Douglas production function may be used: Y = K α L1−α

(2.4)

Considering part of the output is consumed and the other saved and invested (and assuming no depreciation) we have the basic identity: K 0 = sY = sK α L1−α

(2.5)

To keep the Harrod-Domar assumptions Solow admitted an exogenous constant rate of population growth n, so that L (t) = L0 ent

(2.6)

Hence ¡ ¢1−α K 0 = sK α L0 ent

Introducing a new variable k ≡

K L

so that

k0 k

k 0 = sk α − nk

=

K0 K

(2.7) −

L0 , L

we have (2.8)

This nonlinear first-order equation may be transformed into a linear equation by

10

a suitable transformation. Consider

K Y

= r = k 1−α . Then equation 2.8 becomes

r0 + n (1 − α) r = s (1 − α)

(2.9)

whose equilibrium value is r0 = 0 ⇔ r =

s n

(2.10)

and the general solution ³ s ´ −n(1−α)t s e + r (t) = r0 − n n

(2.11)

The first term of the right side of the equation vanishes as t gets larger, so we conclude that the Capital/Output ratio tends asymptotically to its equilibrium value ¡ ¢ 1 1 s . Since k = r 1−α we have, in equilibrium K = ns 1−α . Thus, capital and labour n L

will tend to grow at the same rate n. Obviously, output will tend to grow at the same rate, i.e., output per capita will tend to be constant.

In order to have a positive growth rate of output per capita, we can introduce technological change. Consider a simple case in which productivity raises constantly at rate g: Y = egt K α L1−α ³ ´1−α g ⇔ Y = K α L0 e(n+ 1−α )t

(2.12)

Proceeding the analysis as before we would conclude that the rate of growth of output becomes n + rate

g . 1−α

This way, output per capita would grow at the constant

g . 1−α

The main problem with this approach is that the determinants of economic growth are exogenous and are left unexplained by the model. Thus, the model is only descriptive being unable to give a real explanation of the causes of economic growth.

11

2.3

The Arrow-Frankel-Romer Model of Endogenous Growth

Frankel (1962) argued that a production function of the form Y = AK was more suitable for growth problems than the usual neoclassical production function but, since it was a function of only one factor was wrong at a microeconomic level. However, if consideration

is

taken

for

the

external

effects

of

investment,

an

aggregated production function of the type Y = AK can arise. Frankel considers a Cobb-Douglas production function Y = AK α L1−α , in which ¡ ¢γ A stands for an index of the development of the economy. Choosing A = A K , L the aggregated production function will become: Y = AK α+γ L1−α−γ . The exponent α + γ captures both the private (α) and the external (γ) effects. To have the AK model, he focuses his attention in the case α + γ = 1. Arrow (1962), in his paper on learning by doing, argued that the productivity of an industry is an increasing function of the cumulative aggregate investment for the industry: new knowledge is discovered as investment takes place. Arrow had to face a technical problem related to the increasing returns. In an optimizing growth model, which maximizes a discounted integral over an infinite horizon problem, the presence of increasing returns may eliminate the possibility of achieving a maximum. To overcome this problem, Arrow assumed that, while the production function exhibits increasing returns in capital and labour, for a fixed amount of labour it presents diminishing returns (i.e., α + γ < 1) . The consequence of this assumption was that the rate of growth of output per capita was an increasing function of the rate of population growth, being zero in the case of a fixed labour force. Unlikely did Arrow have this latter result in mind when he tried to create a model of endogenous growth. Romer (1986) borrowed Arrow-Frankel’s idea but generalized the model to the case in which the aggregate elasticity of output with respect to capital and knowledge (which can be seen as a composite good) is not less than one (as in the Arrow model) or one as in the Frankel Model (i.e., α + γ > 1). To understand the Arrow-Frankel-Romer model we can consider the usual Cobb12

Douglas production function of a representative firm Y = AK α L1−α

(2.13)

where, as before, A is the state of knowledge. Romer assumed the number of workers was equal to the number of firms, so that we have a representative one-worker firm. Equation 2.13 becomes Y = AK α

(2.14)

Romer also assumed no depreciation of capital, so K 0 = AK α − c

(2.15)

where c stands for consumption per capita. The state of knowledge is an increasing function of the total stock of capital. So, having N firms A = A (NK)β

(2.16)

Romer considered an isoelastic utility function. The instantaneous utility is: u (c) =

c1−θ − 1 1−θ

(2.17)

This way, the dynamic optimization problem the owner of a representative firm faces with is R∞ max 0 u (ct ) e−ρt dt

s.t. K 0 = AK α − c and K 0 ≥ 0 The time path of A is exogenously given to the representative agent. This way

we are considering that the increasing returns are external to the firm so that a competitive equilibrium exists, A being a public good with no remuneration. The Hamiltonian of this maximization problem is ¡ ¢ H = e−ρt u (c) + λ AK α − c 13

(2.18)

or, considering µ = λeρt ¢¤ £ ¡ H = e−ρt u (c) + µ AK α − c

(2.19)

being the canonical equations K 0 = AK α − c, K (t0 ) = K0 ∂H , lim λ (t) = 0 λ0 = − t→∞ ∂K

(2.20) (2.21)

λ is what we may call the dynamic Lagrange multiplier. To have an interior maximum of the Hamiltonian

∂H ∂c

= 0 ⇔ u0 (c) = µ, this way

we have c−θ = µ

(2.22)

From 2.21 we have λ0 = −λαAK α−1 , so λ0 = −αAK α−1 λ Now, since λ = µe−ρt , we have

λ0 λ

=

µ0 e−ρt −ρµe−ρt µ−ρt

(2.23) , so

¡ ¢ µ0 = − αAK α−1 − ρ µ But from 2.22 we know that

µ0 µ

(2.24)

0

= −θ cc , introducing in equation 2.24 we have

¢ 1¡ c0 = αAK α−1 − ρ c θ

(2.25)

and from 2.20 c K0 = AK α−1 − K K

(2.26)

Assuming rational expectations, agents will correctly anticipate the level of capital which other firms will choose, hence A = A (N K)β . Equations 2.25 and 2.26

14

become ¢ 1¡ c0 = αAN β K α+β−1 − ρ c θ 0 K c = AN β K α+β−1 − K K

(2.27) (2.28)

It is easy to check that if α + β < 1 (as in the case studied by Arrow) the diminishing returns on capital will lead to an equilibrium level of capital which means that growth in consumption per capita will tend to be zero. If α + β = 1 (the Frankel case) the steady-state consumption growth rate is c0 c

=

αAN β −ρ , θ

so 1

ct = c0 e θ (αAN

)t

β −ρ

(2.29)

which means consumption per capita grows without bounds. Romer also showed how α + β > 1 could be accommodated, in which case the rate of growth would become monotonically increasing. It is interesting to note that since firms do not internalize the side effect of capital accumulation the equilibrium growth rate is less than the optimal growth rate. To an omniscient social planner the equation of motion of the optimization problem would not be K 0 = AK α − c. Internalizing the external effect of capital accumulation would change the equation of motion to K 0 = AN β K α+β − c. If α + β = 1 the optimal growth rate would be

2.4

(α+β)AN β −ρ θ

>

αAN β −ρ . θ

A Model of Growth Through Vertical Innovations

Romer’s main contribution to growth theory was to endogenize the source of sustained growth in per capita income: knowledge accumulation. Aghion and Howitt (1992) made a different approach to endogenize growth: they considered industrial innovations which improve the quality of products – vertical innovations. Admit an economy populated with L workers. The production function of the

15

consumption good depends only on the input of the intermediate good X: Y = AX α

(2.30)

with 0 < α < 1. A is a technological parameter. When an innovation occurs A is raised by a constant factor γ > 13 . The labour force can be used in two different sectors: it can produce intermediate goods (to simplify admit the production is one for one), or can be employed in the research sector L = X + LR&D

(2.31)

Obviously LR&D is the amount of labour devoted to research and X to manufacturing. Aghion and Howitt admit that innovations arrive randomly with a Poisson arrival rate of λLR&D , with λ being a productivity indicator of the research sector. The firm well succeeded in the innovation will monopolize the intermediate sector until the next innovation occurs. The amount of labour devoted to research is determined by wτ = λVτ +1

(2.32)

where wτ is the wage and Vτ +1 is the discounted payoff of the (τ + 1)th innovation: Vτ +1 =

π τ +1 r + λLR&D τ +1

(2.33)

where π τ +1 is the profit flow that the (τ + 1) monopolist will achieve, and r is the ¡ ¢ interest rate. r + λLR&D can be interpreted as the obsolescence adjusted interest τ +1 rate4 , since λLR&D τ +1 is the flow probability that the innovation becomes obsolete.

To close the model we only need to specify the profit flow π τ and the demand 3

We assume here that when an innovation occurs, all sectors of the economy will imediately adopt it. A more realistic dynamics for the speed of adoption and for the costs of adjustment is considered when we study General Purpose Technologies in chapter 4.2.3. π τ +1 R&D 4 Note that Vτ +1 = r+λL ⇔ Vτ +1 = πτr+1 − Vτ +1 λL . So we are saying that the expected r R&D payoff of an innovation is equal to the present value of the flow of profits subtracted by the present value of the expected capital loss that will occur when the monopolist is replaced by a new innovator. We are implicitly saying the current monopolist does not invest in research. This is so since the payoff for the current monopolist to innovate is only (Vτ +1 − Vτ ), which is strictly less than the payoff of another researcher (Vτ +1 ) .

16

for manufacturing labour Xτ . To explore all the monopoly rents the monopolist will maximize: π τ = maxpτ (X) X − wτ X X

(2.34)

Assuming the final good sector to be competitive, pτ (X) = Aτ X α−1 α. Solving the above expression to X and π : Xτ∗ πτ where ω τ =

wτ Aτ

µ

¶ 1 α2 1−α = ωτ ¶ µ 1−α ω τ Xτ ≡ Aτ π ∗τ = Aτ α

(2.35) (2.36)

is the productivity adjusted wage rate. Solving 2.32 and 2.33 and

remembering that

Aτ +1 Aτ

= γ: γπ ∗τ +1 ωτ = λ r + λLR&D τ +1

(2.37)

From 2.31, and assuming a frictionless labour market: L − Xτ∗ = LR&D τ

(2.38)

, so the system characterizing the In the steady state, ω = ω τ and LR&D = LR&D τ stationary solution is:

⎧ ⎨ ω = λ γπ∗ r+λLR&D ⎩ LR&D = L − X ∗

(2.39)

´ ³ R&D ˜ is unique5 . So, It is possible to show that the steady state equilibrium ω ˜, L

in the steady state (assuming the τ th innovation is active) ³ ´ ˜ R&D Yτ = Aτ L − L

(2.40)

Yτ +1 = γYτ

(2.41)

and, obviously,

5

See Aghion and Howitt (1992).

17

We know that ln Y increases ln γ when an innovation occurs. But the time it takes between each innovation is random. The growth between two periods will be: ln Yt+1 − ln Yt = εt ln γ

(2.42)

where the subscript t stands for time and εt is the number of innovations that ˜ R&D we occurred between t and t + 1. Taking expectations, and since E (εt ) = λL have the average growth rate: ˜ R&D ln γ g = λL

(2.43)

It is interesting to note that in this model the stationary growth rate may be more or less than the socially optimal growth rate, while in Romer’s model the equilibrium growth rate is less than optimal. As in the previous model there are positive external effects that are not internalized by the market, namely that the benefit of the next innovation will continue forever. But, counteracting the external positive effect, firms when investing in research do not internalize the loss to the previous monopolist caused by a new innovation.

2.5

Consequences of Endogenous Growth Theories

When an inherently stable growth model is considered – like Solow growth model – the long run equilibrium path is well defined. The impact of exogenous shocks will be felt only in the short run, without consequences in the long run equilibrium. Thus we have a clear distinction between long run growth (essentially determined by the exogenous productivity growth rate) and short run cycles (explained by a stream of exogenous shocks). We will see how this vision of growth and cycles entirely respects the Slutzky-Frisch paradigm, which we will study next. The above distinction between growth and cycles theories is questioned when we consider learning by doing technologies as those in the Frankel-Arrow-Romer Model.

18

We will see how productivity shocks may lead to permanent changes in real income – Stadler (1990). The idea is quite simple: if there is a positive shock, with real temporary effects, we will have a higher level of economic activity. This will raise our technological level in a permanent basis. Thus, the neutrality of short run shocks as, for example, a monetary shock, is very questionable. This issue will be dealt with in chapter 4.2.2. Growth is no longer a continuous process in the model of Aghion and Howitt. When an innovation occurs output jumps to a higher level. Thus, although recessions are not explained, we have a model where growth takes place cyclically.

19

Chapter 3 Some Reference Business Cycle Models 3.1

Stochastic Models

3.1.1

The Slutzky-Frisch decomposition

“Thanks to the early work of Frisch (1933) and Slutsky (1937) (...) most macroeconomists now share the same general analytical approach, that based on the distinction between impulse and propagation mechanisms.” — in Blanchard and Fischer (1989) Actually, the usual business cycle models mainly differ about the source (the impulse) of the cycle. Obviously the specific propagation mechanism is not exactly the same from model to model. But the general framework is the same. Slutzky’s paper published in 1937 in Econometrica was a revised English version of an earlier paper published in Russia. His main objective was to explain “the undulatory character of the processes and the approximate regularity of the waves” using nothing but random series. To obtain such series (computer simulations were not in fashion that time) he used results from a Russian lottery loan. Then he computed a moving summation and compared the resulting series to an index of English business cycles. The results can be seen in figure 3-1

20

Figure 3-1: English Business Cycles and Stochastic Moving Summations The series are doubtless remarkably similar1 , thus, although Slutzky did not give a general law to explain what kind of cycles can be generated by this mechanism, he established that some sort of swings can be produced by the aggregation of erratic influences. When Frisch wrote his 1933 article, he was aware of Slutzky’s results. He constructed a model which, when disturbed, would generate damped oscillations (the propagation mechanism). To explain why the cycles are undamped and show some regularity, he considered the impact of exogenous random shocks which provide the necessary energy to feed the cycles. His basic idea was the rocking horse analogy of Wicksell: “if you hit a rocking-horse with a stick, the movement of the horse will be very different from that of the stick. The hits are the cause of the movement, but the system’s own equilibrium laws condition the form of the movements”2 . 1

The similarity between the two series is not innocent. According to Judy Klein (1997), Slutzky, to get a good graphic match, chose only the first 23 years of a database of 60 years and picked only the terms 20 to 145 of a series of 1009 terms. Besides that he plotted 92 observations of the English business cycle against 126 observations of his model considering a different scale for each series. 2 Wicksell, K. (1918), “Karl Petander: Goda och dårliga tider”, Ekonomisktidskrift, vol. 19, pp. 66-75, cit. in Thalberg (1990).

21

Frisch considered the following dynamic equations describing a simple economy: yt = mxt + µx0t Z ε D (τ ) yt−τ dτ zt =

(3.1) (3.2)

0

x0t = c − λ (rxt + szt )

(3.3)

To simplify Frisch assumed D (τ ) = 1ε , so from 3.2 we get zt0 =

(yt − yt−ε ) ε

(3.4)

with m, µ, ε, c, λ, r and s being positive parameters. 3.1 gives us the production of capital goods. The first term represents the level of production of capital just necessary to maintain the level of consumption xt . It is positive due to the depreciation of capital. Thus, it is the replacement investment. The second term is only the acceleration principle. 3.2 is what Frisch called the carry-on-activity. Dτ is “the amount of production activity needed at point of time t + τ in order to carry on the production of a unit of capital goods started at the point of time t”. Frisch assumed that D (τ ) =

1 ε

generating 3.4. Equation 3.3 is the consumption equation. The constant c expresses the tendency to expand consumption. The second term represents the negative effect generated during an upturn of x and z. (rx + sy) represents the cash needed for the transaction of consumer and production goods. Since money supply cannot be infinitely expanded the increasing need for cash will create a tension counteracting the expansion. Frisch calibrated the model in order to find explicit solutions. The values considered were: ε = 6, m = 0.5, µ = 10, λ = 0.05, r = 2, s = 1, c = 0.165. For example, the solution Frisch found for x is:

x (t) =

[1.32 − 0.32e−0.08045t ] + [0.6816e−0.371335t sin (0.73355t)] + + [0.27813e−0.5157t sin (1.79775t)] + [0.17524e−0.59105t sin (2.8533t)]

22

(3.5)

As we can see in equation 3.5 the resulting trend and cycles can be additively aggregated. With this model, Frisch rationalized the usual separation between the study of cycles from the study of the trend. In the first square brackets we have a damped exponential term (the upper limit is 1.32). In the second we have an ¢ ¡ 2π = 8.5654 . The third and fourth oscillation with a period of 8.57 years 0.73355 terms represent a cycle of 3.5 and 2.2 years respectively (actually Frisch found some

more cycles of minor importance). Once all e exponents are negative, all cycles generated on this model are damped cycles. Frisch considered that to understand the regularity of the economic cycles we only need to imagine “what would become the solution of a determinate dynamic system if it were exposed to a stream of erratic shocks that constantly upsets the continuous evolution, and by so doing introduces into the system the energy necessary to maintain the swings”. This idea will be graphically illustrated in the context of Samuelson’s model.

3.1.2

Equilibrium Business Cycles (EBC)– Lucas’ Model

The central idea of Lucas’ (1972, 1973) model is the existence of some rigidity which prevents the economy from being in a constant equilibrium state. Imperfect information is the source of the rigidity – Fischer (1977) and Taylor (1979) considered the existence of staggered contracts as the source of the rigidity, but the structure of their models is essentially the same. To rational agents (suffering from no monetary illusion) the relevant prices are relative prices; thus, the supply of a firm should not be influenced by the general price level. Following this idea Lucas (1973) assumed that supply of firm i is (y is the logarithm of output of the firm): c yi,t = α + βt + yi,t

(3.6)

α + βt is the “secular component, reflecting capital accumulation and population c is the cyclical component which is influenced by variations in the change” and yi,t

relative prices. It can be seen in equation 3.6 the decomposition between the trend 23

and the cycle Frisch defended. But Lucas assumed that a producer can only observe the evolution of his own price and could only have expectations about general prices. When a producer observes a variation in his product price there are two possibilities: it can be due to a shift in the demand function for his product (which leads to variations in relative prices) or may be the consequence of a general raise in prices. The rational response of the producer will be to attribute part of the variation to his relative price and the other part to a variation of the general level of prices. We can consider the cyclical component depends on the perceived relative price and on its own lagged value: c c = b (pi,t − E (pt )) + λyi,t−1 yi,t

(3.7)

pi is the log of the producers’ price and E (p) is his expectation of the log of the general price level. λ is assumed to be in the interval ]0, 1[. Averaging across producers gives the aggregate supply production function: c yt = α + βt + b (pt − E (pt )) + λyt−1

(3.8)

To simplify consider α = β = 0. In this case yt = ytc . So yt = b (pt − E (pt )) + λyt−1

(3.9)

To close the model we can consider a simple demand side pt = mt − yt

(3.10)

where m stands for the log of a generic variable affecting demand (it can be considered, for our purposes, money stock). Combining equations 3.9 and 3.10, we get ⎧ ⎨ y = t ⎩ p = t

b m 1+b t 1 m 1+b t

− +

b E 1+b

(pt ) +

λ y 1+b t−1

b E 1+b

(pt ) −

λ y 1+b t−1

24

(3.11)

Taking expectations of the two sides of the price equation, we obtain E (pt ) = E (mt ) − λyt−1

(3.12)

Introducing this result in the output equation we get yt =

b (mt − E (mt )) + λyt−1 1+b

(3.13)

Since 0 < λ < 1 the model is dynamically stable, being the fluctuations explained by exogenous and unpredictable shocks in money supply (= mt − E (mt )). So we found here a model completely inserted in Slutzky-Frisch’s paradigm: the economic system is inherently stable and cycles are explained by a continuum of stochastic shocks.

3.2

Deterministic Linear Models – Samuelson’s Multiplier-Accelerator Model

Samuelson (1939a) develops a very simple model capable of generating endogenous cycles. Consider a simple closed economy. In moment t national income will be: Yt = Ct + It + Gt

(3.14)

Consumption depends on one period lagged national income: Ct = αYt−1

(3.15)

Investment depends on the variation of consumption (accelerator principle): It = β (Ct − Ct−1 ) = αβYt−1 − αβYt−2

25

(3.16)

To our aim government expenditure can be considered exogenous, so: Gt = G

(3.17)

Then equation 3.14 can be rewritten as Yt = α (1 + β) Yt−1 − αβYt−2 + G

(3.18)

In order to have cycles with this model, the roots of the characteristic equation 3.19 must be complex: λ2 − α (1 + β) λ + αβ = 0 √ 2 α(1+β)± [α(1+β)]2 −4αβ λ= 2

(3.19)

To have complex roots the discriminant must be negative [α (1 + β)]2 − 4αβ < 0 α<

4β (1+β)2

(3.20)

To have constant amplitude oscillations a much more stringent restriction is needed: αβ = 1. If αβ > 1 the model will generate explosive oscillations. On the other hand, if αβ < 1 the oscillations will be damped, the model being unable to explain the persistence of cycles without the help of exogenous shocks. This is the problem with this model: there is no reason for a real economy to obey to such restrictive conditions. This model is not capable of giving us a reasonable explanation for the persistence of cycles in a real economy. Stable solutions make the system tend to an equilibrium value, thus eliminating cycles and unstable solutions will make explosive oscillations which are economically implausible3 . It is possible to illustrate Frisch’s idea with this model. According to Frisch’s model, presented earlier, the deterministic part (propagation mechanism) of the economic system is inherently stable. This way, αβ < 1. Introducing a stochastic 3 One way to overcome this problem would be to admit explosive oscillations and to consider the existence of a ceiling (e.g. output capacity) and a floor (e.g. a minimum investment level). This approach is used by Hicks (1950) in his model of the trade cycle.

26

156 154 152 150 148 146 144 142 100

110

120

130

140

150

160

170

180

190

200

Y

Figure 3-2: A Frischian Version of Samuelson’s Multiplier-Accelerator Model shock (ut ) (the impulse) in equation 3.18, we have Yt = α (1 + β) Yt−1 − αβYt−2 + G + ut

(3.21)

Calibrating the model expressed in the above equation with the values G = 30.0, α = 0.8, β = 1 the equilibrium value for Y is 150. Considering Y0 = Y1 = 150 and ut ∼ N (0, 1)4 , we can see in figure 3-2 that there is no tendency for the shocks to disappear because the system is continually fed with exogenous shocks.

3.3

Deterministic Nonlinear Models ”I first met Richard Goodwin 15 years ago: I was 25 and he was 60 years old. [...] Now, this year, he has reached the venerable age of 75 and I am 15 years older. [...] He has in the intervening years, been growing at the rate of 1.25 and I at 1.6. At this rate, it is to see, I could even become older than Richard Goodwin. This is the sort of linear nonsense Richard Goodwin has been trying to expose for over 40 years.” — in Velupillai (1990)

4

Generated with the econometric program Eviews, version 3.1.

27

The first interest in exploring nonlinear models is to escape the highly unrealistic assumptions which are necessary in linear models to generate oscillations in economic models. As we could see in Samuelson’s model, oscillations in a linear model will tend to die away or to explode. A way to overcome this problem is to introduce exogenous shocks. But this does not seem to be the best way if we want to be able to explain as much as possible with a mathematical model. By definition, random shocks are left unexplained by the model. Another unpleasant feature of linear models is that upswings and downswings are symmetrical. With nonlinear models it is easy to avoid this characteristic. Samuelson was quite aware of these defects and suggested – Samuelson (1939b) – his multiplier-accelerator model could be improved with the introduction of nonlinear relations. For example, he defended that marginal propensity to consume should decrease as income increases and that one should impose a ceiling and a floor to the investment function – ideas developed by Kaldor (1940). We can see how the introduction of a simple nonlinearity is sufficient to generate cycles with Goodwin’s Multiplier-Accelerator model – Goodwin (1951). He used, as the basic framework, Samuelson’s model without any kind of lags and in continuous time. Y =C +I +G

(3.22)

Consumption is proportional to income C = αY

(3.23)

Government can be considered exogenous and constant G=G

(3.24)

The nonlinearity is introduced in the investment function. Goodwin assumed there is a desired level of capital which is proportional to income. K ∗ = AY

28

(3.25)

Figure 3-3: Goodwin’s Multiplier-Accelerator Model Dynamics When actual capital is below the desired capital, investment is carried out at the ¢ ¡ + = I . But when the stock of capital is above the maximum rate possible I = dK dt

desired then net investment will be negative, with a floor defined by the depreciation

rate (I = I − ). Investment will be zero if actual capital stock equals its desired level. Solving 3.22, 3.23, 3.24, and 3.25 for K ∗ we have ⎧ ⎪ ⎪ K∗ = ⎪ ⎨ K∗ = ⎪ ⎪ ⎪ ⎩ K∗ =

A I+ 1−α

+

AG , 1−α

AG , 1−α A I− 1−α

K∗ > K K∗ = K

+

AG , 1−α

(3.26)

K∗ < K

There is only one rest point which is unstable. Any deviation from equilibrium will lead to self-maintained oscillations. Consider figure 3-3, the system is in the equilibrium point E. Since actual capital equals desired capital, net investment is zero and income will be

G . 1−α

If the

economy is disturbed from this equilibrium, for example, if there is any kind of shock that raises income, then the desired level of capital will change. This change will induce a raise in net investment from zero to I + . Income will increase to

I+ 1−α

G + 1−α .

Since net investment is positive capital stock is increasing. The economy will travel continuously in segment [AB] until it reaches B. At that point, desired capital equals 29

actual capital, net investment will cease, so income decreases and net investment diminishes to I − . The economy jumps from B to C and then goes continuously to D, where it jumps again to A, an so on. With this extremely simple and simplified model we can see some of the basic features of nonlinear models: the oscillations are self-sustained (it is not necessary to have random shocks), equilibrium is unstable and the economy will oscillate around its equilibrium, and no lags need to be introduced. Goodwin, in the same paper, considered the effects of introducing exogenous technical progress. He concluded in that case there would not even exist an equilibrium state and, thanks to the stimulus effect of technical progress, depressions would be shorter than booms.

3.3.1

The Kaldor Trade Cycle

Kaldor (1940) presented an extremely ingenious keynesian model. The introduction of simple nonlinearities in the investment and saving functions was sufficient for the model to generate endogenous cycles. He considered that in equilibrium ex-ante saving is equal to ex-ante investment. If ex-ante investment exceeds ex-ante saving this will lead to an economic expansion. Kaldor considered the employment rate (x) as an indicator of the economic activity level5 . Now, according to Kaldor,

dI dx

will be small for low and high levels of x. It will be

small for low levels because the excess of installed capacity is not a good stimulus to investment. For high levels of activity the rising costs of investments and the lack of new opportunities will refrain investment plans. Kaldor for

dS dx

had precisely the opposite view. He argued that, when incomes are

unusually low, then savings are cut drastically. For high income levels, people will save a more and more proportion of their income. To have a situation of multiple equilibria he also assumed that for “normal” values 5

dI dx

>

dS . dx

The mathematical development made here follows Gandolfo (1997). The main difference is that Gandolfo used current output as an indicator of the activity level, while we consider the employment rate, which, in our opinion, is closer to Kaldor’s spirit.

30

Figure 3-4: Kaldor’s Nonlinear Investment and Saving Functions In figure 3-4 we have a graphical interpretation of these functions and, as can be seen, there are three equilibrium points. Only the two extreme intersection points are stable equilibrium points, so the economy will tend to stay in a low activity equilibrium or in an abnormally high activity equilibrium point. Kaldor argued that the economy would fluctuate between these two equilibria. According to him the above described functions are short run functions. In the long run, if the economy stays in the high activity equilibrium, the savings curve will shift upwards. This is so because the high level of investment will lead to an increasing stock of capital which permits, for a given level of employment, to raise the total production and, consequently, raise both the consumption and savings amount. On the other hand, the accumulation of capital will restrict the investment opportunities leading to a downward shift of the investment function. Likely, if the economy stays in the low equilibrium level, the savings function will shift downwards and the investment curve upwards. Mathematically, Kaldor implicitly assumed that ∂S ∂K

> 0 and

∂I ∂K

< 0 (K stands for the stock of capital).

This model basic principles are: The activity level varies positively with the difference between ex-ante investment

31

and ex-ante savings x0 = F [I (x, K) − S (x, K)] with F (0) = 0 and

dF d(I−S)

(3.27)

> 0.

Assuming ex − ante investment = ex − post investment – this means it is ex-post savings that will adjust so that we would have the macroeconomic identity guaranteed (ex − post investment = ex − post savings); and also assuming no depreciation, we have K 0 = I (x, K)

(3.28)

With these equations we will only have a rest point if x0 = K 0 = 0. For x0 = 0 we need to have I (x, K) − S (x, K) = 0. Differentiating µ

∂S ∂I − ∂x ∂x

¶

dx +

this way dK = dx

µ

∂I ∂S − ∂K ∂K

∂S ∂x ∂I ∂K

¶

(3.29)

dK = 0

∂I − ∂x ∂S − ∂K

The denominator of the right side of equation 3.30 is negative. For

(3.30) ∂S ∂x

−

∂I ∂x

we have negative values for a “normal” range of activity levels and positive values elsewhere. Thus, for “normal” activity levels the curve x0 = 0 will be positively sloped, and for both low and high values of x it will have a negative slope. Differentiating K 0 = 0 we have ∂I dK = − ∂x >0 ∂I dx ∂K

(3.31)

We can see this in the phase diagram represented in figure 3-5. Obviously, since ∂x0 dK

=

∂I ∂K

−

∂S ∂K

< 0 for points above x0 = 0 we must have x0 < 0 (for points below

x0 = 0 we have x0 > 0). When we look for points outside K 0 = 0 we can conclude that with

∂K 0 ∂K

< 0 for points above/below K 0 = 0 we have K 0 < 0 / K 0 > 0.

Looking at the arrows in the diagram it is easy to conclude that this system generates forces that impede the economy from leaving the rectangle [ABCD]. So if the economy is initially in the rest point (x∗ , K ∗ ) we can be sure that no endogenous

32

Figure 3-5: A Phase Diagram of Kaldor’s Trade Cycle Model forces will pull the system out of [ABCD]. If the rest point is locally unstable, then we can intuitively imagine that the economy will move in a closed orbit between the rest point and the polygon [ABCD]. This intuitive idea is assured by the following theorem: Theorem 1 (See Gandolfo (1997)) Poincaré-Bendixson criterion — in a phase plane consider two closed curves C1 and C2 , with C1 ⊂ C2 . Then if no rest point exists between C1 and C2 and the integral curves passing through the points of C1 and C2 penetrate in C1c ∩ C2 (the index

c

means the complement of), then C1c ∩ C2

contains at least one limit cycle. We have already found a candidate for C2 (the rectangle [ABCD]). For C1 , if we show that the rest point is locally unstable, we can consider any curve in its neighbourhood. To study the local behaviour of the rest point we can consider a linear approximation 3.27 and 3.28 around the rest point: ⎧ of the system of equations £¡ ¡ ¤ ¢ ¢ ∂I ∂S ⎨ x0 = F (0) + F 0 (0) − (x − x∗ ) + ∂I − ∂S (K − K ∗ ) ∂x

∂x

∂K

∂K

⎩ K 0 = I (x∗ , K ∗ ) + ∂I (x − x∗ ) + ∂I (K − K ∗ ) ∂x ∂K Since F (0) = I (x∗ , K ∗ ) = 0, and transforming the coordinates of the system,

33

we have

⎧ ⎨ x0 = F 0 (0) £¡ ∂I − ∂S ¢ (x) + ¡ ∂I − ∂x ∂x ∂K ⎩ K 0 = ∂I (x) + ∂I ¡K ¢ ∂x

∂K

∂S ∂K

¢ ¡ ¢¤ K

(3.32)

where () stands for deviations from the equilibrium.

The characteristic equation of the system of differential equations 3.32 is £ ¡ ∂I ∂S ¢ ∂I ¤ λ2 − F 0 (0) ∂x − ∂x + ∂K λ+ £¡ ¡ ∂I ¢ ∂I ¢ ¤ ∂I ∂S ∂I − ∂S − ∂K − ∂K =0 +F 0 (0) ∂x ∂x ∂K ∂x

(3.33)

With our assumptions about the signs of the partial derivatives we can be sure £¡ ∂I ∂S ¢ ∂I ¡ ∂I ¢ ¤ ∂S ∂I − ∂x ∂K − ∂K − ∂K > 0. So the necessary and sufficient that F 0 (0) ∂x ∂x

condition to have local instability is ¡ ∂I ∂S ¢ ∂I ¤ ¡ ∂I £ − ∂x + ∂K < 0 ⇔ F 0 (0) ∂x − − F 0 (0) ∂x

∂S ∂x

¢

+

∂I ∂K

>0

This means that movements along the curves in figure 3-4 are quicker than

movements of the investment curve. This condition is implicit in Kaldor’s model because he considers that investment and savings depend on the level of activity in the short run; while, on the other hand, only in the long run they do depend on accumulated stock of capital after significant variations occur.

3.4

Conclusion

In this chapter we were able to distinguish the two main approaches to cycles modeling. The stochastic approach, based on the work of Slutzky and Frisch, considers a dynamically stable model which is ruffled by exogenous shocks. As a paradigmatic example of this approach we chose Lucas (1972, 1973) model, where exogenous shocks are related with unexpected variations on the stock of money. On the contrary, the defenders of a different approach consider the economic system inherently unstable. Even in the absence of exogenous shocks, the economic system fluctuates cyclically. This deterministic approach demands the consideration of a nonlinear model – to work with a linear model we would have to make highly restrictive assumptions. These deterministic cycles have the unpleasant feature of 34

being easily predictable being incompatible with optimizing agents. In chapter 5 we will see that this problem may be overcome with possibility of deterministic chaos.

35

Chapter 4 Growth and Cycles: an Actual Distinction or a Theoretic Abstraction? “[...] it is possible [...] that the trend of growth may itself generate forces making for oscillation. This, if so, would not impair the importance of the study of the effects of lags. But it may me that the attempt to explain the trade cycle by exclusive reference to them is an unnecessary tour de force. The study of the operation of the forces maintaining a trend of increase and the study of lags should go together.” — in Harrod (1939) Although most of authors have studied growth and cycles separately, this occurred essentially for technical reasons. Goodwin in his 1955’s paper declared that his primary concern was “with the simultaneous existence and mutual conditioning of economic growth and economic cycles [...] [But] it may be argued that this is too ambitious an aim and, in the light of difficulties that I have encountered, I am inclined to agree” (our emphasis). Actually, it is rather difficult to integrate both growth and cycles in one model and most of the efforts done until now have proved to be unsuccessful. Harrod (1939) tried to tackle this issue with a model which was inherently un36

stable. Harrod welcomed this instability because he believed it to be an explanation of fluctuations in real world. His analysis failed because, mathematically, his unstable system did not generate fluctuations, it simply broke down. Any point away from the equilibrium path would tend, self-sustainedly, to go further and further away from the equilibrium – not to generate oscillations. As we could see in the previous chapters it is possible to create models of economic growth without any concern with cycles, and vice-versa. The question is whether it is desirable. We saw that Fischer and Slutzky provided an analytical justification for this decomposition. The problem with their analysis is that it may be valid only if we are talking about linear relationships. And we say may because, as we will see, this decomposition is valid only if economic time-series are trend-stationary and not difference stationary. Or, in other words, if we are talking about a deterministic trend and not about a stochastic trend. This distinction between short and long run (cycles and growth) can make sense if we consider Solow’s growth model (specially with exogenous technical progress), which is the basis for real business cycle models. But, when we look to modern theories of endogenous economic growth it is impossible to abstract from long run effects of short run cycles, this is the basic idea of Stadler’s article (1990). When we consider pure cycle models only, we face an unrealistic characteristic feature common to these models which was expressed by Kaldor (1954): “Each depression phase lasts precisely as long as is required for the capital stock to fall to the same extent as it had risen in the previous boom; the capital stock, as well as the output, in the corresponding phases of successive cycles are identical. As a pure cyclical model it has therefore little resemblance to the fluctuations in the real world, the most characteristic feature of which has been that successive booms carried production to successively higher levels; while the creation of capital in boom periods has exceeded many times the net capital depletion in depression periods”. It is easy, of course, to introduce an exogenous trend in a pure model cycle, but, naturally, from the analytical point of view, this is not satisfactory. The trend is left unexplained, therefore this kind of models cannot be the basis for a theory of 37

economic growth. Schumpeter was probably the author who saw more deeply the connection between growth and cycles. To him booms and depressions were just the form which progress takes in our economy. According to Schumpeter even if there is a stable flow of new ideas and of technological progress major innovations have to wait for a favorable economic climate. This will lead to the adoption of these technologies by a number of imitators, leading to an increase of investment. When this new technology is fully adopted the economic growth will cease creating a favorable climate to the adoption of new innovations. The main problem with this theory is that it is very descriptive and hard to deal with analytically, but, as we shall see there are some models which try to capture Schumpeter’s ideas. Goodwin (1955) tried to couple Schumpeter’s theory with Keynes’ ideas of effective demand. Unfortunately his model cannot be accepted as good explanation of economic growth because the latter depends on an exogenous parameter representing technical progress. This way we can consider that Goodwin’s model consist of a superposition of a [linear] trend into an otherwise trendless cycle model. Actually Goodwin’s most celebrated model – his predator-prey model of cyclical growth, Goodwin 1967 – suffers from the same problem. Economic growth in this model is caused by an exogenous increase in labour supply and in labour productivity, while cycles are essentially unaffected by the existence or inexistence of growth. We will deal with this model with some detail trying to endogenize technical progress.

4.1

Decomposition Between Trend and Cycles in Nonstationary Time Series

Although not denying the decomposition between trend and cycles, Nelson and Plosser (1982) – see also Nelson and Kang (1984) – showed some of the risks involved with this procedure. As we know a stationary time series is one that fluctuates around a mean value. More precisely, it has a mean value, variance and autocovariances that are constant (and finite) through time. Obviously, this is not the case of real output (at least 38

for developed countries) which has been increasing for the few last centuries. This way we say that the real output time series exhibit a trend. Typically the secular component (which is in the domain of growth theory) is considered to be a [linear] function of time, while the cyclical component (which is in the domain of business cycle analysis) is assumed to be stationary – this is because cyclical fluctuations are assumed to dissipate over time, so any permanent movement (nonstationary) is attributed to the secular component. Thus it is usual to detrend time series by regression on time, being the residuals considered the cyclical component (or, which is the same, include time as an explanatory variable – see, for example, equation 3.6 in Lucas’ model). The main problem with this approach is that if the secular component is not a deterministic trend then the resulting model will be misspecified. Nelson and Plosser analyzed two possibilities: — a trend-stationary process (TS) — a difference-stationary process (DS) The first one can be represented as: (4.1)

Yt = α + βt + ct ct

¡ ¢ = φ (L) vt ; vt ∼ i.i.d 0, σ 2v

(4.2)

where L is the lag operator and φ (L) is a polynomial in L satisfying the conditions for stationarity and invertibility. Obviously, in this case, the long run expectations are (4.3)

E (Yt ) = α + βt

so the cyclical component has no effect in the long run. This idea can also be seen by considering the long run effect of a innovation in period t:

∂Yt+n ∂vt

= 0, for n big

enough. A DS process may be represented as: Yt = β + Yt−1 + dt ¡ ¢ dt = δ (L) ut ; ut ∼ i.i.d 0, σ 2u 39

(4.4) (4.5)

where L is the lag operator and δ (L) is a polynomial in L that satisfies the conditions for stationarity and invertibility. Equation 4.4 can be rewritten as

Yt = Y0 + βt +

t X

dj

(4.6)

j=1

So, although this process can also be written as a function of time plus deviations from the trend, the deviations from the trend are an accumulation of stationary changes, which will not be stationary since its variance grow without bounds, making impossible to make long run forecasts. For example, if δ (L) = 1 then

V ar

à t X j=1

dj

!

= tV ar (ut ) = tσ 2u

(4.7)

this way, we can conclude that the inclusion of a time trend is not enough to capture the nonstationarity of the time series. In this case innovations in period t have permanent effects:

∂Yt+n ∂ut

= 1.

The distinction between TS and DS time series is decisive in the study of business cycles. For example, if we believe aggregate output is composed by a nonstationary stochastic growth component (Yt ) plus a stationary cyclical component (ct ), then: (4.8)

Yt = Yt + ct Yt ct

¡ ¢ = α + Yt−1 + δ (L) ut ; ut ∼ i.i.d 0, σ 2u ¢ ¡ = φ (L) vt ; vt ∼ i.i.d 0, σ 2v

(4.9) (4.10)

with δ (L) and φ (L) being polynomials in L that satisfy the conditions for stationarity and invertibility. In this case, if we regress Yt as a function of time and consider the residuals as being solely explained by the cyclical component, we are confounding the two sources of variation, “greatly overstating the magnitude and duration of the cyclical component and understating the importance of the growth component”1 . 1

In Nelson and Plosser (1982).

40

It is interesting to note that, although first differentiating is sufficient to “stationarize” the series, the stochastic growth component will always be present: Yt − Yt−1 = α + δ (L) ut + (1 − L) φ (L) vt

(4.11)

We can see in the above equation that the innovation in the permanent component ut is still present. This leads us to another problem: unless we consider severe restrictions we cannot infer the behaviour of each unobserved component. In general, if we assume δ (L) = 1 and that ut and vt are independent then it will be possible to identify the parameters of the unobserved components of 4.11.

4.1.1

Empirical findings and consequences

Nelson and Plosser, after showing the dangerous of confounding the sources of the nonstationarity, proceeded to the time series analysis of some important American macroeconomic variables (namely output, interest rates, wages, etc). They tested the null hypothesis of DS time series against the alternative hypothesis that the series are TS or simply stationary. The formal test used was the Augmented Dickey Fuller (ADF)2 test and they concluded that, except for the unemployment rate, it was impossible to reject the null hypothesis of a unit root. They also concluded that innovations in the permanent components are likely to have a larger variance than innovations in the cyclical components. The main implication Nelson and Plosser found for business cycle theory was the rejection of EBC models. Since these models assume that disturbances (generally monetary disturbances) have only transitory impact they cannot account for DS time series, in which, by definition, there are disturbances that have permanent effects, and which, according to Nelson and Plosser, have greater variance than the former. This line of thought led them to say that most of output fluctuations should be 2

We will explain the mechanics of the ADF test later, when testing the stationarity for simulated time-series.

41

explained by real disturbances and not by monetary shocks. This argument opened the way to Real Business Cycle (RBC) we will study in next section. Campbell and Mankiw (1987) followed another strategy to measure the persistency of disturbances. After estimating general ARIMA models (of various orders) for real US GDP growth they tried to answer the question: if real output falls 1 percent lower than expected “how much should one change one’s forecast of GNP for five or ten years ahead?” Their answer was quite surprising: “a 1 percent innovation in real GNP should change one’s forecast of real GNP by over 1 percent over a long horizon.” So they also concluded in favor of the highly persistence of the effects of disturbances. But, in spite of this common conclusion with Nelson and Plosser, they did not point to RBC as the correct path for economic investigation. Campbell and Mankiw agreed that business cycles traditional theories have two premises which are inconsistent with observed data: first, fluctuations in output are driven primarily by aggregate demand shocks, and, second, that demand shocks only have temporary effects. Nelson and Plosser defended the abandonment of the first premise3 . But Campbell and Mankiw pointed to the abandonment of the second premise: “Shocks to aggregate demand could have permanent effects if technological innovation is affected by the business cycle”4 . It should be stressed that not all authors agree with the above approaches to test the persistency of shocks. Harvey (1997) although not denying the historical importance of the study of Nelson and Plosser – “[Their] attack on the so-called trend-stationary model needed to be made because of its wide acceptance amongst applied economists”5 – totally rejects ARIMA models and unit roots tests based on autoregressive approximations (like the ADF test) in favour of structural time series models (like the Kalman filter). According to Harvey there is a large class of ARIMA models and parameter values which have no sensible economic interpretation. He also alerted to the difficulty of identifying higher order ARIMA models. For example 3

Which, by itself, is not equivalent to considering RBC models that are basically an intertemporal Walrasian model where the allocation of resources is always Pareto efficient (as we shall see next). It is possible to consider that real shocks affect the economy through some Keynesian model. 4 In Campbell and Mankiw (1987), p. 977. 5 In Harvey (1997), p. 196.

42

Harvey and Jaeger (1993) fitted a stochastic trend plus cycle model to US GNP, with an ARIMA (2,2,3) reduced form; but standard Box-Jenkins methodology would lead to the identification of an ARIMA(1,1,0), which is not able to generate the cyclical movements of the series. Harvey has also warned that when a series needs to be differenced to become stationary there is the risk that the corresponding model be close to non-invertibility. In this situation the autoregression will become a very poor approximation. Under these conditions, unit root tests are incapable of detecting two unit roots in the process. According to Harvey this is “why most economists apparently believe that real macroeconomic time series are integrated of order one and construct models with this basis”6 . The use of autoregressive models usually imply a higher level of persistence than structural models. “The point is that a structural model of a real series such as GNP will typically contain one or two unit roots, but will be perfectly consistent with a cycle about a more or less stable trend path”7 .

4.2 4.2.1

Models of Growth and Cycles Real Business Cycle Models

Kydland and Prescott (1982), launched the idea that business cycles were mainly explained by technology shocks (i.e., disturbances in the aggregate production function). This seminal article is a paradigm of RBC models. The point of departure of their analysis is the neoclassical equilibrium growth model (illustrated in the first chapter). They intended (how well, we will discuss later) to use an “approach [which] integrates growth and business cycles theory”. They also assumed the heritage of Frisch’s approach (see chapter 2), making a clear distinction between the impulse and the propagation mechanism – Kydland and Prescott (1991). 6 7

In Harvey (1997), p. 197. In Harvey (1997), p. 197.

43

A Simplified Model The model we will present is essentially the same put on view by Prescott (1986), with the exception that we will consider total depreciation of capital in order to achieve an explicit solution – see McCallum (1989) and Romer (1996). First, we need to consider a neoclassical constant returns production function (generally a Cobb-Douglas production function) with a multiplicative technological parameter A: Yt = At Ktα L1−α t

(4.12)

where variables have the usual meaning. Capital stock in period t + 1, considering a depreciating parameter of δ, will be: Kt+1 = (1 − δ) Kt + It

(4.13)

Kt+1 = (1 − δ) Kt + st Yt In order to obtain an explicit solution, we need to consider the case of complete depreciation (δ = 1): Kt+1 = st Yt

(4.14)

Since these models assume perfect competition, labour and capital will be paid according to their marginal productivity: µ

¶α

Yt = (1 − α) wt = (1 − α) At Lt µ ¶1−α Yt Lt =α 1 + rt = αAt Kt Kt Kt Lt

(4.15) (4.16)

A characteristic feature of these kinds of models is the consideration of a representative household, thus ignoring the problem of aggregating heterogenous agents. So the representative household maximizes the expected value of

U=

+∞ X t=0

e−ρt u (Ct , 1 − Lt )

(4.17)

where u (.) is the instantaneous utility function, ρ is the discount factor, Ct is con-

44

sumption in period t, and normalizing time to one 1 − Lt is the time allocated to leisure. According to Prescott (1986) leisure per capita has maintained constant while the real wage has increased, so the elasticity of substitution between consumption and leisure should be one. Thus we can consider an instantaneous utility function of the type: ut = (1 − θ) ln Ct + θ ln (1 − Lt )

(4.18)

The exogenous technological parameter is assumed to follow an autoregressive process: At+1 = Aλt eεt

(4.19)

where λ is the autocorrelation parameter. We are assuming competitive markets, no externalities, an infinite lived representative household. These assumptions imply that model’s equilibrium corresponds to a Pareto optimum. Thus, it is the same to solve the maximization problem (which becomes a dynamic programming problem) or to find the competitive equilibria. Standard results of dynamic optimization state that the solution is unique – see Lucas and Stokey (1989). To find the competitive equilibrium we have to consider the two conditions for household optimization. The first one represents the inter-temporal efficiency condition: the marginal cost of investing in more capital (in terms of utility) must be the same as the expected marginal utility gain (1−θ)e−ρt Ct

=E

1 Ct

−ρ

⇔

h

=e E

(1−θ)e−ρ(t+1) Ct+1

h

1+rt+1 Ct+1

i

i (1 + rt+1 )

(4.20)

The second condition represents the intratemporal efficiency condition (the labour-leisure trade-off): the marginal rate of substitution between labour and consumption must equal the marginal product of labour (1−θ)e−ρt wt Ct (1−θ) Ct = θ wt 1−Lt

θe−ρt 1−Lt

⇔

=

45

(4.21)

We are now able to find the equilibrium paths of labour supply and for the marginal/average propensity to save. Yt , we have From 4.20 and knowing that Ct = (1 − st ) Yt and (1 + rt ) = α K t 1 (1−st )Yt

⇔ Et

h

∙

−ρ

= e Et i

1 (1−st+1 )

=

Y

α Kt+1

t+1

(1−st+1 )Yt+1 eρ st α(1−st )

¸

Since in the right hand side all variables are known in period t, there is no uncertainty. The fixed-point is found equating st+1 = st = s: s=

α eρ

(4.22)

From 4.21 and knowing wt = (1 − α) LYtt (1−s)Yt 1−Lt

t = (1 − α) (1−θ)Y θLt

⇔L=

(4.23)

(1−α)(1−θ) (1−α)+(α−s)θ

Thus, in this simplified model, the saving rate and labour supply are constant. So the specific form of output fluctuations will be mainly determined by the dynamics of the technology. From 4.12 we have ln Yt = ln At +α ln Kt +(1 − α) ln Lt . Knowing that Kt = sYt−1 : ln Yt = α ln s + (1 − α) ln L + ln At + α ln Yt−1 ³ α ln s+(1−α) ln L ⇔ ln Yt − = ln At + α ln Yt−1 − 1−α

α ln s+(1−α) ln L 1−α

⇔ yt = at + αyt−1 8

´

(4.24)

From 4.19 we know at = λat−1 + εt , so we can rewrite 4.24 as yt = (λ + α) yt−1 − αλyt−2 + εt [1 − (λ + α) L + αλL2 ] yt = εt 9

ln L Where yt = ln Yt − α ln s+(1−α) and at = ln At . 1−α 9 In this equation L represents the lag operator and not labour supply. 8

46

(4.25)

Solving the polynomial equation 1 − (λ + α) L + αλL2 = 0

(4.26)

we have L=

1 1 ∨L= α λ

(4.27)

Obviously, since 0 < α < 1 the first root lies outside the unit circle, so in order to have a nonstationary process we must have λ ≥ 1. If λ = 1 we will have a DS time series, if λ > 1 the time series will be explosive. Thus in this model, there is no mechanism capable of transposing transitory technology shocks in long-lasting output movements. In order to have this latter feature we need to have highly persistent technology shocks (so λ must be close to unity). Naturally this crude model cannot catch all business cycles characteristics. Namely, our extremely unrealistic assumption of complete depreciation of capital led to a constant saving rate and a constant labour supply. With a constant saving rate the variation of consumption will equal the variation of the investment. Once one relaxes that assumption we lost the possibility of finding an explicit solution, but it is easy to show numerically that the variability of investment will be much higher than that of consumption – see Hansen (1985). Are RBC Models Real Models of Growth AND Cycles? “Given the finding that business cycle fluctuations are quantitatively just what neoclassical growth theory predicts, the resulting deviations from trend are nothing more than well-defined statistics. We emphasize that given the way the theory has developed, these statistics measure nothing. Business cycle theory treats growth and cycles as being integrated, not as a sum of two components driven by different factors.” — in Kydland and Prescott (1996) In the following discussion we do not worry about critics to this theory that are

47

not related to the problem of the decomposition between trend and cycles10 . Kydland and Prescott claimed that in their model there is no distinction between growth an cycles. But they were the first to admit (implicitly) this is not that quite so. In 1991 they point to Frisch’s (1933) article as a precursor of their RBC model. We have already seen that Frisch rationalized not only the decomposition between trend and cycles but as well between cycles of various magnitudes. This idea is also implicit when they analyze the statistical properties of the United States business cycles. They only study detrended time series refusing the accusation that they are generating spurious cycles11 . The implicit assumption is that the extracted trend is a good approximation of the steady state growth, which is not explained. A model which is concerned with deviations from the trend without explaining it can hardly be seen as an integrated approach of the trend and cycle. Even the use of the neoclassical production function to model long run growth is quite unfortunate since the steady state growth rate is completely explained with exogenous factors in the neoclassical growth model (see chapter 2.2). For the reasons illustrated above we are forced to say that, although RBC theory treats cycles as a moving equilibrium of a neoclassical growth model subject to technological shocks, it does not present an integrated approach of the trend and cycle. This idea was perfectly explicit in Hansen (1985): “Detrending was necessary because the models studied abstract from growth”.

4.2.2

Business Cycle Models with Endogenous Technology

Stadler (1990) and Stiglitz (1993) studied the main long run implications of business cycles. They followed different approaches to endogenize technology. While Stadler considered a process of learning-by-doing, similar to the Arrow-Frankel-Romer studied earlier, Stiglitz explicitly introduces Research and Development (R&D) investments, which vary pro-cyclically with the business cycle. However the main conclu10 For example, questions may be raised about the validity of aggregation procedures implicit in representative agent models, the irrelevance of monetary shocks (since we are in a walrasian equilibrium), recessions explained by questionable technological recessions, etc. 11 As shown by Cogley and Nason (1995) and by Harvey and Jaeger (1993).

48

sion reached is the same: there is long run nonneutrality of “temporary” short run shocks, even if we are talking of pure monetary shocks. A positive monetary shock with real effects will induce a higher level of economic activity in the short run, which through a process of learning-by-doing, or due to an increase of R&D investments, will shift the production function allowing a permanent higher level of the real output. Since Stadler’s model considers a process of learning-by-doing already studied and uses the same framework of EBC and RBC we will discuss Stadler’s model instead of Stiglitz’ model. Stadler Model Stadler considered a closed economy, with a large number of firms, producing an identical good, and behaving competitively. There is only one input (labour). The representative firm i maximizes the present value of expected profits: µ ¶# W t+j V = max β j Yt+j − Lt+j {Lt } P t+j j=0 "∞ X

(4.28)

where Lt is employment of labour by the representative firm, Yt is output of the representative firm, Wt is the wage, P is the price level, and 0 < β < 1 is the discount factor. The production function is a Cobb-Douglas which depends on labour and the technological level (At ): Ft Yt = Lαt A1−α t

(4.29)

where Ft is a strictly positive stochastic shock that contains a permanent component ξ t and a temporary component η t : Ft = Ft eηt Ft = Ft−1 eξt

(4.30)

The technological level At depends on its previous value and varies positively

49

with the activity level: λ At = At−1 Yt−1

(4.31)

with 0 < λ < 1. Thus, a high activity level will induce a higher technological level in the subsequent periods12 . Considering an exogenous evolution of knowledge, with the concavity of the production function, we can be sure that a competitive equilibrium exists. So, taking the first-order condition of equation 4.28, after introducing equation 4.29 we get labour demand:

   P α ln αFt Wt 1−α

Pt = αAt Ft e Wt

Ldt

t

(4.32)

or, in logarithms (small letters stand for logarithms, e.g. pt = ln Pt ): ltd =

pt − wt ln α ft + + at + 1−α 1−α 1−α

(4.33)

For labour supply consider a log-linear function: lts = φ1 + φ2 (wt − pt )

(4.34)

with 0 < φ2 < 1. At the beginning of the period the money wage is set at the expected market clearing level. Taking expectations of equations 4.33 and 4.34: ⎧ ⎨ E (ls ) = φ + φ (w − pe ) t−1 t t 1 2 t ¡ ¢ e ⎩ E ld = ln α + pt −wt + a + t−1

t

1−α

1−α

t

ft−1 1−α

(4.35)

where Et−1 pt ≡ pet , and Et−1 ft = ft−1 because Et−1 η t = Et−1 ξ t = 0. Solving 4.35 in order to wt : wt = pet +

12

ft−1 ln α − φ1 (1 − α) (1 − α) at + + 1 + φ2 (1 − α) 1 + φ2 (1 − α) 1 + φ2 (1 − α)

Actually Stadler considered At = At−1

³

Yt−1 Lt−1

´λ

Lθt−1 , thus we are imposing the restriction that

λ = θ. The main conlusions are not affected by this simplification.

50

(4.36)

Substituting the above equation into equation 4.33 gives the employment per firm: lt =

(ln α)φ2 +φ1 (1+(1−α)φ2 )

+

(pt −pet ) (1−α)

ft + (1−α) −

(1−α)φ2 a (1+(1−α)φ2 ) t ft−1 (1+(1−α)φ2 )(1−α)

+

(4.37)

Now introducing equation 4.37 in the production function will allow us to get the supply function, aggregating all the N firms will give us the aggregate production function: φ2 ln α+φ1 yts = n + α (1+(1−α)φ + ) 2

α (1−α)

(1−α)(1+φ2 ) a+ 1+φ2 (1−α) t 2 η t ) + 1+φ1+φ ft−1 2 (1−α)

(pt − pet ) +

1 (ξ t + + (1−α)

(4.38)

yts = b0 + b1 (pt − pet ) + b2 at + b3 (ξ t + η t ) + b4 ft−1 where n is the logarithm of the number of firms. Now that we have already found the aggregate supply function, in order to close the model we need to assume an aggregate demand function: Ytd =

Mt Pt

ytd = mt − pt

(4.39)

where M is the money stock. The money supply policy is known and follows a simple rule: mt = µ + mt−1 + εt

(4.40)

Taking expectations from 4.39 and subtracting from 4.39 and considering 4.40 ¡ ¢ pt − pet = εt − ytd − ytde

(4.41)

Since we assume rational agents expected market clearing implies ytde = ytse , so from 4.38 we have ytde = b0 + b2 at + b4 ft−1

(4.42)

thus subtracting 4.42 from 4.38, and taking in consideration that market clearing implies ytd = yts gives us ytd − ytde = b1 (pt − pet ) + b3 (ξ t + η t ) 51

(4.43)

Introducing this expression in 4.41 pt − pet =

1 [εt − b3 (ξ t + η t )] 1 + b1

(4.44)

Finally to have the output process we only need substitute equation 4.44 in the aggregate supply function yt = b0 + b2 at + b4 ft−1 + αεt + b3 (1 − α) (ξ t + η t )

(4.45)

The Effects of Monetary Shocks With this model no longer is necessary to rely on technology shocks to explain the nonstationarity of aggregate output time series. Considering ft = 0 for all periods (i.e., ξ t = η t = 0) yt = b0 + b2 at + αεt

(4.46)

and having in consideration that at = at−1 + λyt−1 yt = (1 + b2 λ) yt−1 + α (εt − εt−1 )

(4.47)

[1 − (1 + b2 λ) L] yt = α (εt − εt−1 ) where L is the lag operator. Solving the equation 1 − (1 + b2 λ) Z = 0 we get Z =

1 1+b2 λ

(4.48)

< 1. The root of the equation lies inside the unit circle so the

output process is nonstationary without being DS: we have a greater than unit root process13 . We have here an explosive process. Even in the absence of external shocks (εt = 0 for all periods) the economy will be growing at a higher and higher growth rate (yt − yt−1 = (b2 λ) yt−1 ). Thus, this model captures one of the stylized growth 13 There might by some semantic confusion here. There are two usual ways of solving the homogeneous equation of equation 4.47. One is using the lag operator as it was done in the text. The other way is to solve it as an usual difference equation: yt − (1 + b2 λ) yt−1 = 0. In this case we would solve the characteristic equation: W − (1 + b2 λ) = 0 ⇔ W = (1 + b2 λ) = Z1 > 1. In this case we have an explosive process and we say that the model has a greater than unit root. See Hamilton (1994).

52

facts pointed by Romer (1986). Another interesting feature of this model is that money supply shocks (or, more generally demand shocks) do have permanent, and increasing, effects in output. For example, if εt = 1, the impact on yt is α. In the next period output falls but not to its previous level. The impact on yt+1 is b2 λα. And, from there on, the impact will grow at rate b2 λ, so in period t + n the effect of the shock occurred in period t will be (1 + b2 λ)n−1 b2 λα. It is easy to understand why this happens. If there is a positive shock output will raise. In the next period the higher level of output will induce a higher productivity which permits output to be bigger than would be in the absence of shock. The same argument can be extended to the following periods, leading to a higher and higher output. Stadler also studied the effect of pure real shocks (with εt = 0 for all periods) and concludes that the two main differences between his results and previous RBC models (which we have already studied) are: — the model has a greater than unit root (which is common to the model with pure monetary shocks), and — both permanent and temporary technology shocks have permanent effects while in a typical RBC only persistent technology shocks have long run effects. Stadler with this model questioned the usual decomposition between trend and cycles. Within this framework all “temporary” shocks have permanent effects, so the long run path of an economy will not be independent of short run fluctuations. Naturally in this model the causation works in only one way, from cycles to growth. There is no reverse implication: cycles do not depend of growth. So this model does not fully integrate growth and cycles, but is a serious contribution to link both theories.

4.2.3

Growth Based on General Purpose Technologies

“Industrial change is never an harmonious advance with all elements of the system actually moving, or trending to move in step. At any giving time, some industries move on and others stay behind; and the discrepancies arising from this are an essential element in the situations 53

that develop. (...) In every span of historic time it is easy to locate the ignition of the process and to associate it with certain industries and, within these industries, with certain firms, from which the disturbances then spread over the system”. — in Schumpeter (1964) In the previous section we saw how economic fluctuations can influence long-term growth. In this section, with the introduction of the concept of General Purpose Technologies (GPT), we will try to explore the reverse causality. According to Helpman and Trajtenberg (1998a) in “any given “era” there typically exists a handful of technologies that plays a far reaching role in widely fostering technical change and thereby bringing about sustained and pervasive productivity gains”. Common examples of such technologies are the steam engine (first industrial revolution), the electric dynamo (first part of this century) and microelectronics and computers in the last decades. As soon as a new GPT (which is more productive than the previous one) is discovered there will be sectors interested in adopting this new technology. But the adoption of a new technology requires a costly process and the adjustment takes time. Thus the adoption of a new GPT requires a deflection of resources from the production activity to invest in the implementation of such technology. This way a positive technology shock can cause a temporary recession. In RBC models to explain a recession we would need to appeal to a technological regression. In the model presented in this section, the adjustment costs for adopting a new technological paradigm may be enough to explain a “slump” before the definitive take-off. Diffusion of General Purpose Technologies and Cyclical Growth: a Simple Model In this model – presented by Aghion and Howitt (1998a and 1998b) – we will consider the existence of N different sectors, each one producing a different intermediate good in a monopoly situation. 54

We assume there exists a perfectly widespread GPT and that a new, more productive GPT, exogenously arrives. We will also assume that the probability of the arrival of a third GPT is negligeable at least before almost all sectors adopt the new GPT. We consider that each sector is in one of three stages: — stage 0, if it uses the old technology and is making no effort in order to use the new GPT, — stage 1, if it is still using the old GPT but is investing in order to be able to use the new GPT, and — stage 2, if it is already working with the new technology. Let n0 , n1 , n2 be the proportion of sectors in stage 0, 1, and 2 respectively. The aggregate production function depends only on the amount of labour used in the production of the N intermediate goods. We consider a Constant Elasticity of Substitution (CES) production function with constant returns to scale14 :

Y =

ÃN X

(Ai )α (Li )α

i=1

! α1

(4.49)

with Ai = 1, if the sector uses the old GPT, and Ai = γ > 1 if it uses the new technology. Li is the amount of productive labour used by sector i. Diffusion of the New GPT At least since the pioneer work of Mansfield (1961) it is usual to model the diffusion of innovations in a similar fashion to biological and, more precisely, to epidemiological models of spread of diseases. Thus, to describe the probability of a sector to move from stage 0 to stage 1, we will consider not only the independent determination of investing in the new GPT but also a contagion effect. The independent effect is considered to follow a Poisson distribution with an 14

The only relevant difference between the model exposed here and the original model – Aghion and Howitt (1998a and 1998b) – is that while we consider a discrete number of sectors (N ), Aghion and Howitt considered a continuum of sectors (uniformly distributed on the unit interval).

55

arrival rate of λ0 . To model the “epidemic” effect it is considered that at each moment a firm i can observe m other firms and if at least k of them are in stage 2 (i.e., they are already working with the new GPT) then firm i will be compelled to invest in the new GPT, thus passing from stage 0 to stage 1. The probability of a firm watch m other firms and at least k of them use the new GPT, follows a Binomial Distribution: B (k; m, n2 ) =

m X j=k

m! (n2 )j (1 − n2 )m−j j! (m − j)!

(4.50)

Thus, the flux of firms moving from stage 0 to stage 1 is given by (λ0 + B (k; m, n2 )) n0 N = (λ0 + B (k; m, n2 )) (1 − n1 − n2 ) N Now, we assume the costs suffered by a firm in stage 1 to implement the new GPT consist of hiring a fixed number of workers (= Lr&d ), who will not be linked with productive activities but will be working in research and development. The probability of success follows a Poisson process with an arrival rate of λ1 . Thus d(n2 N) dt

= λ1 (n1 N) ⇔

dn2 dt

= λ1 n1 . So we have a system of nonlinear differential

equations describing the diffusion of the new GPT: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

dn1 dt dn2 dt

= (λ0 + B (k; m, n2 )) (1 − n1 − n2 ) − λ1 n1 = λ1 n1

(4.51)

⎪ ⎪ n1 (0) = 0 ⎪ ⎪ ⎪ ⎪ ⎩ n (0) = 0 2

Calibrating the system of equations with the parameter values λ0 = 0.005, m = 10, k = 3, λ1 = 0.215 , we have the evolution of firms adopting the new GPT that is described in figure 4-1. In the dashed line, we can see the evolution of the proportion of firms that 15

The parameter values considered are very close to the values used by Aghion and Howitt (1998).

56

1 0.8 0.6 0.4 0.2

0

10

20

30

40

50

Figure 4-1: Diffusion of the New GPT had successfully implemented the new GPT. As we may observe, the pattern of the evolution is in perfect agreement to the technological diffusion models generally used in Economics and that were imported from Biology. Basically the evolution of firms adopting the new GPT follows a logistic curve. In the solid line we can see the evolution of firms in stage 1. i.e., firms that are investing in the new technology but have not been able to implement it. We can observe a sharp increase between the years 19 and 21. This means that a large proportion of firms is diverting workers from productive activities to non-productive activities. Thus we can expect an output depression between these years. Time Evolution of Aggregate Output To determine the evolution of aggregate output we only need to determine the evolution of the labour market and introduce it in equation 4.49. Since we are assuming symmetrical firms, firms using the same GPT will have the same demand for productive workers. This way, equation 4.49 can be simplified: ⎛

(1−n2 )N

Y ≈⎝

X

(Li )α + γ α

i=1

N X

i=(1−n2 )N

⎞ α1

(Li )α ⎠

(4.52)

Firms producing with the old GPT demand labour according to the following

57

maximization problem: M ax (pi Li − wLi )

(4.53)

Li

with pi =

∂Y ∂Li

=

³P (1−n2 )N i=1

(Li )α + γ α

PN

α i=(1−n2 )N (Li )

expression for pi in 4.53 we get the following problem Max Li

∙³ P(1−n2 )N i=1

α

(Li ) + γ

= Max (Y 1−α Lαi − wLi )

PN α

α

i=(1−n2 )N

(Li )

´ α1 −1

´ 1−α α

Lα−1 . Introducing this i

Lαi − wLi

¸

(4.54)

Li

which results in the demand for productive labour by each firm using the old GPT: L0,1 =

1 ³ w ´ α−1

α

Y

(4.55)

where the subscript 0,1 represents demand of firms in stage 0 and in stage 1. The same line of thought would lead us to productive labour demand by firms using the new GPT: L2 =

µ

w αγ α

1 ¶ α−1

Y

(4.56)

Assuming that labour market is always in equilibrium: (1 − n2 ) NL0,1 + n2 NL2 + n1 N Lr&d = L

(4.57)

Introducing equations 4.55 and 4.56 in the above equation and solving it in order to

w α

we get ³w´ α

³ ´1−α α = (L − n1 NLr&d )α−1 (NY )1−α (1 − n2 ) + n2 γ 1−α

(4.58)

Inserting this expression in equations 4.55 and 4.56 L0,1 = L2 =

(L−n1 N Lr&d )   α N (1−n2 )+n2 γ 1−α

α (L−n1 NLr&d )  1−α  γ α N (1−n2 )+n2 γ 1−α

(4.59)

Substituting these expressions for L0,1 and L2 in 4.52 we find the value for ag-

58

1.3

1.2

1.1

1

0.9

0

10

20

30

40

50

Figure 4-2: Time Evolution of Aggregate Output gregate output: ´ 1−α ³ α α 1−α (L − n1 NLr&d ) Y = (1 − n2 ) N + γ n2 N

(4.60)

Calibrating the above expression with the parameter values N = 500, α = 0.5, γ = 1.3, L = 5000, Lr&d = 4 we find the time path for aggregate output (as a proportion of output in the year 0) described in figure 4-2. As expected there is a major slump around the year 20 before the definitive take-off to a higher productivity level. Safety Clause and Conclusion Obviously, the parameter values chosen to calibrate the model were not innocent. In fact the slump observed is not insensitive to the choice of the parameter values. For example, we considered γ = 1.3, saying the new technology is 30% more productive than the old one. If we considered a much higher value, e.g. γ = 3, then the much higher productivity of the sectors using the new GPT would compensate the adjustment costs of the sectors in stage 1 avoiding the recession. Even more important than the previous example is the value chosen for Lr&d . We assumed the existence of 5000 workers and 500 firms, thus the average number of workers per firm is 10. It seems a bit unnatural to consider that a firm, to implement 59

the new GPT, needs 4 workers devoted to research when generally no more than 2.5% of the labour force is allocated to R&D activities. If we considered Lr&d = 1 then we would have a monotonic increase in output. Naturally the model could be improved. We could admit, for example, the existence of unemployment and skill differentials. But these changes would only accentuate the slump verified in the basic model. Workers that were not able to learn to work with the new GPT would be fired in the transition process worsening the recession and delaying the recovery. This model has a Schumpeterian flavour. It reflects the idea that recessions are just the form growth takes. It illustrates the notion of “creative destruction”. The recession reflects only the adjustment costs of changing the technological paradigm. Once again the decomposition between growth and cycles is questioned. Here the causality goes from growth to cycles. It is the dynamic of growth that generates cycles. We can even say that cycles are, in this context, a deterministic consequence of the growth process. What remains to be explained is the origin of growth. To explain growth we have a model that relies, once more, on an exogenous technical change, here in the form of new General Purpose Technologies. Thus, although questioning the trend/cycles decomposition it can hardly be seen as a perfect growth model. It is, however, a major advance when compared with RBC models. In those models cycles were also explained by productivity shocks, but to accept recessions we would have to consider technological regressions. In GPT models (temporary) recessions occur even with positive productivity shocks.

4.2.4

Goodwin’s Predator-Prey Growth Cycle

In 1967 Goodwin presented what would become his most celebrated model. It was his intention to analyze growth and cycles simultaneously. Or better, he wanted to model cycles of growth. Actually according to him, economists should study growth rates and not levels. Goodwin’s model shows how an antagonist relationship between workers and capital owners can lead to cycles. The mechanism is quite easy to understand. In 60

a situation of rising profitability, investment will be raised, thus creating more jobs and destroying the reserve army of labour. This will give more bargaining power to labour which can demand higher wages. Then, in Marx words, “accumulation slackens in consequence of the rise in the price of labour, because the stimulus of gain is blunted. The rate of accumulation lessens; but with its lessening, the primary cause of that lessening vanishes, i.e., the disproportion between capital and exploitable labour-power. The mechanism of the process of capitalist production removes the very obstacles that it temporarily creates. The price of labour falls again to a level corresponding with the needs of the self-expansion of capital”16 : a new cycle begins. The similarities between this class struggle and the antagonist relationship between two species (a predator and a prey) are obvious. This fact was not unnoticed by Goodwin. In his words: “It has long seemed to me that Volterra’s problem of the symbiosis of two populations — partly complementary, partly hostile — is helpful in the understanding of the dynamical contradictions of capitalism, especially when stated in a more or less Marxian form”. Interestingly, and, probably, not so Marxian, it is the workers who are predators in Goodwin’s model and capitalists the prey, as Solow (1990) pointed out. Here we will not present Goodwin’s model exactly. Instead of that we will try to put his model in a more general framework. His results may be analyzed as a special case of our model. The advantage of doing so is that we will be able to understand the consequences of relaxing some of his assumptions, allowing the evaluation of their robustness. The Model Goodwin made five assumptions for convenience (in his words) and two assumptions of disputable sort. The first five assumptions were: (a) productivity of labour growing exogenously at rate β, (b) steady growth of labour force, (c) two factors of production, both homogenous, 16

In Marx, K. (1887/1974), Capital, Vol. 1, Lawrence&Wishart, p.580, cit in Harvie (2000).

61

(d) all quantities real and net, (e) all wages consumed and all profits invested. We will lose nothing essential by considering the exogenous growth of labour force to be zero. In this model full-employment is not guaranteed, so we have to make a distinction between labour force (= N) and employed workers (= L). We will normalize the labour force to unity (N = 1), so the rate of employment is given by L =

L . N

Contrary to Goodwin’s wishes, the first assumption is not so innocuous. First of all, a positive trend is exogenously imposed, so the model may describe the growth process but does not explain it. Another important point is that the belief of Aghion and Howitt (1998b), and of the generality of the economists, that this was the first model in which cycles are a deterministic consequence of the growth process, is wrong. Even in the absence of growth the fluctuations would still be there, so they are not an implication of the growth process (at least in this model with no further changes). Relaxing this assumption, we will try to expose the implications of introducing an endogenous component in labour productivity. More specifically, we will admit some process of learning-by-doing à la Arrow-Frankel-Romer. So we consider that the bigger the accumulated net investment (which corresponds to the stock of capital) the bigger labour productivity will be. So labour productivity is given by Y = a = eβt K γ , with 0 < γ < 1 L

(4.61)

so the productivity growth rate will be a0 K0 =β+γ a K

(4.62)

We can also see the implications of considering an anti-cyclical productivity by assuming −1 < γ < 0. Goodwin’s case will be found considering γ = 0. The other two assumptions were: (f) constant capital-output ratio, (g) real wage rises in the neighbourhood of full employment. 62

We will accept the last assumption by considering a Phillips curve to explain the behaviour of wages: w0 = f (L) w

(4.63)

We will admit, like Goodwin did, that as L approaches 1 the function will become indefinitely large. This function will be negative for low values of L. We will establish some downward rigidity of wages by imposing a floor to f (L). So f (0) < 0 but not “too” negative. We will not accept assumption (f). Or better, we will relax that assumption by considering a general CES production function: ¤− 1δ £ Y = A αK −δ + (1 − α) L−δ ef

(4.64)

where Lef is the effective employed labour force: Lef = Leβt K γ As we know the Leontief production function, which is Goodwin’s implicit assumption in (f), is a particular case of the above function. Namely, ³£ ¤ 1´ −δ − δ −δ = min (AK, AL) lim AαK + (1 − α) Lef

δ→+∞

And with δ → 0 it will become a Cobb-Douglas: Y = AK α L1−α ef . With this new function we are forced to assume a profit-maximizing assumption: ∂Y ∂Lef ∂Y ∂L

= we−βt K −γ

(4.65)

=w

so firms will hire workers until their marginal productivity equals the real wage. Ploeg (1985) also made these two latter assumptions He did consider, however, a different bargaining equation. From equation 4.65 we can determine the optimal factor demand ratio17 (in 17   −1 δ ∂ A[αK −δ +(1−α)L−δ ef ] ∂Lef

we−βt K −γ ⇔

i −1−δ h δ = we−βt K −γ ⇔ − 1δ A αK −δ + (1 − α) L−δ (−δ) (1 − α) L−δ−1 = ef ef

A[αK −δ +(1−α)L−δ ef ]

(

−1−δ δ

F K,Lef Lef

−δ−1 (1−α)Lef

)

63

=

w a

⇔

(1−α)L−δ ef

[αK −δ +(1−α)L−δ ef ]

=

w a

⇔

αK −δ (1−α)L−δ ef

=

effective terms):

µ

K Lef

¶

(u) =

µ

(1 − α) (1 − u) αu

¶− 1δ

(4.66)

where u = wa , represents worker’s proportion of national income. The optimal capital-output ratio will be18 1 σ (u) = A

µ

¶ 1δ

(4.67)

eβt K γ

(4.68)

α (1 − u)

Labour’s productivity will be given by19 µ

u a (u) = A 1−α

¶ 1δ

After obtaining the above relations we are in good conditions to describe the model in the usual form of two differential equations representing the evolution of labour’s share of national income and of the employment rate: ⎧ ⎨ From 4.63 we know

w0 w

⎩

u0 u

=

w0 w

L0 L

=

K0 K

−

a0 a

−

(4.69)

0

( KL ) K L

= f (L), from 4.68 we can derive

0

a0 a

=

1 u0 δ u

+β+

0

Y = γ KK . Since we assumed that all profits are invested we have KK = (1 − u) K ´ 1δ ³ 0 K 0 ( ) 1 u0 . From 4.66 we have LK = γ KK + β + 1δ 1−u . Putting all these A (1 − u) (1−u) α u L

together we have

⎧ ⎨ ⎩

1

f (L)−β−γAα− δ (1−u) 1+ 1δ

1+δ δ

u0 u

=

L0 L

= (1 − γ) Aα− δ (1 − u)

1

a w

−1⇔

³

K Lef

´

(u) =

³

1+δ δ

−

1 u0 δ(1−u) u

(1−α)(1−u) αu

(4.70) −β

´− 1δ

∙ ¸ δ1 ³ ´− 1δ (1−u) K −δ 1 α + (1 − α) ⇔ σ (u) = −1 A α L−δ δ ef A[αK −δ +(1−α)L−δ ef ] ¸ ∙ 1 ³ ´−δ − δ1 − δ A[αK −δ +(1−α)L−δ F (K,Lef ) Lef ef ] K 19 a (u) = = = A α + (1 − α) = L L L K ³ ´− δ1 h i− 1δ ´ 1δ ³ αu u βt γ e K ⇔ a (u) = A eβt K γ α + A (1−α)(1−u) αu (1−u) 1−α 18

σ (u) =

K F (K,Lef )

=

K

=

K ALef

64

We are now able to understand Goodwin’s model and extend some of his conclusions. The Model with Leontief Technology and Exogenous Productivity Growth In this particular case we discover a model that is formally equivalent to the Lotka-Volterra predator-prey model. As we saw earlier the Leontief production function may be approximated by a CES production function by considering δ → +∞. If we do not admit endogenous productivity growth then γ = 0. The bargaining function is approximated by a linear function (as Goodwin did):f (L) = −φ + ρL, with large φ and ρ. With these simplifications system 4.70 becomes: ⎧ ⎨ u0 = (−φ − β + ρL) u ⎩ L0 = (A (1 − u) − β) L

(4.71)

The properties of this model are perfectly known. Namely that it has an equilibrium point that is not stable or unstable. We follow Blatt (1983) by considering it a neutral equilibrium. If the system is in equilibrium there will be no force pushing it off the equilibrium, so it cannot be considered an unstable equilibrium; on the other hand, if the system is in disequilibrium, there will be no endogenous force pulling it to the equilibrium state, so it cannot be considered a stable equilibrium. ´ ³ φ+β . If the system is placed , The equilibrium point is given by (u∗ , L∗ ) = A−β A ρ

out of this point, it will evolve in a closed cycle. There is, however, no limit cycle. The closed orbit, which the system will follow repeatedly, depends on the initial

conditions. An interesting property of this model is that even if the system is not in the rest point the average values of u and L will be the equilibrium values. Another point, which has already been indicated before, is that even in the absence of an exogenous productivity growth the system maintains its formal properties. Thus we still have a cyclical motion if the system is not in the stationary point. To illustrate this we can simply observe the phase portrait generated by system 4.71 in figure 4-3, with values β = 0, A = 0.25, φ = 9, ρ = 10. The initial values were (L, u) = (0.9, 0.98). With these values the complete cycle takes a little bit more than four years. In this model the possibility of having, for some periods, u > 1 65

Figure 4-3: Predator-Prey Growth Cycle means that total consumption is higher than total output. This is possible since we admit an homogenous output and allow the possibility of disinvestment. Solow (1990) used annual data (1947 to 1986) of the Unites States economy to plot the phase diagram of Goodwin’s model and compared its dynamics with the one described by figure 4-3. He observed that predictions of Goodwin’s model were basically correct, but only in three separate sub-periods. However, he considered the displacements so large that could not accept Goodwin’s model as the only mechanism ruling the relation between wage share and the employment rate. Solow finishes his article suggesting that “it would be enlightening to try the model out of similar data for some European countries”. Using data for a similar period (1956-1994) Harvie (2000) follows Solow’s suggestion and makes a similar analysis for ten OECD countries (Australia, Canada, Finland, France, Germany, Greece, Italy, Norway, the United Kingdom and the United States). Interestingly, Goodwin’s model worked extremely well for all countries except for the United States and for the United Kingdom. One possible explanation for these divergent results, at least for the United States, was given by Solow (1990): “[...] I should point out that the US may not be an appropriate trial horse for this model. Part of the folklore is that the US has a nominal66

wage Phillips curve, whereas the main European countries do indeed have a real-wage Phillips curve. The difference is very important for the interpretation of the model. In an economy with a nominal-wage Phillips curve, the wage share will be significantly affected by such forces as the speed and strength with which nominal prices respond to the facts of supply and demand”. Without having a real-wage Phillips curve, one of the main assumptions of model is violated and the Goodwinian mechanism may be seriously hurt, because it was the evolution of real wages that determined the evolution of labour’s share of national income, and it was the evolution of labour’s share that determined the level of investment. The Model with Leontief Technology and Cyclical Productivity Growth By analyzing this model specification we are able to understand the consequences to the stability of Goodwin’s model of introducing an endogenous element to labour productivity growth. Thus we still keep the assumption that δ → +∞ but we now consider γ 6= 0. More specifically we assume that 0 < γ < 1. If we admit that the productivity growth is anti-cyclical and want to study that situation we only have to consider −1 < γ < 0. With these assumptions system 4.70 becomes: ⎧ ⎨ u0 = [f (L) − γA (1 − u) − β] u ⎩ L0 = [A (1 − γ) (1 − u) − β] L

The rest point of this system is given by (u∗ , L∗ ) =

h

(4.72)

A(1−γ)−β , f −1 A(1−γ)

³

β 1−γ

´i . To

analyze the stability of the system in the neighbourhood of the equilibrium we

can take a linear approximation around the stationary point (u∗ , L∗ ). The system becomes:

⎧ ⎨ u0 = γAu∗ (u − u∗ ) + f 0 (L∗ ) u∗ (L − L∗ ) ⎩ L0 = A (γ − 1) L∗ (u − u∗ )

(4.73)

The characteristic equation of the system of differential equations 4.73 is λ2 − γAu∗ λ + A (1 − γ) f 0 (L∗ ) u∗ L∗ = 0 67

(4.74)

Figure 4-4: Goodwin’s Model with Anti-Cyclical Productivity Growth Since A (1 − γ) f 0 (L∗ ) u∗ L∗ > 0 the stability of the system depends on the sign of −γAu∗ . If −1 < γ < 0 then −γAu∗ > 0 and the system is stable — it will approach the rest point in an oscillating fashion if the value of f 0 (L∗ ) is high enough. If 0 < γ < 1 then the system is unstable, generating explosive cycles. These drastic changes in Goodwin’s model stability properties should not surprise us. The Lotka-Volterra equations are known by their structural instability, which means that small differences in the model (for example γ is in the neighbourhood of zero but is not exactly zero) can lead to significant changes in the properties of the model. In figure 4-4 we can see a phase portrait of the system 4.72 in the case of anti-cyclical productivity growth. The values considered were β = 0.02, A = 0.25, γ = −0.3, f (L) = −0.040064 +

0.000064 20 . (1−L)2

As expected, the dynamics of the

system corresponds to a stable spiral. As mentioned in the analysis of the characteristic equation 4.74, if productivity growth is pro-cyclical then the system is (locally) unstable. Thus we would have explosive cycles. This would be a truly Marxist model where the internal contradictions of the capitalist society would lead to its destruction. But if we adopt a more 20 By assuming this formulation to the Phillips curve we guarantee a lower bound to the growth of wages (-4%). For L = 0.96 wages growth rate become zero. This curve becomes indefinitely large as the employment rate approaches 1, as assumed by Goodwin before making the linear approximation.

68

Figure 4-5: Limit Cycle in Goodwin’s Model with Pro-Cyclical Productivity Growth realistic approach to the model it is well known how to transform a globally unstable model into a stable one. The usual way is to impose a ceiling, or a floor, to one of the variables impossibilititating the occurrence of an explosion in the evolution of the system. A simple way to transform our model is to consider that if the labour share is above some limit then workers will no longer demand a rise in their wage rates. For example, we can consider that ⎧ ⎨ −0.040064 + f (L) = ⎩ 0

0.000064 (1−L)2

if u < 1

(4.75)

if u ≥ 1

By doing this we guarantee that u will not be higher than one, so we are imposing a ceiling in labour’s share and consequently in consumption (do not forget that in this model all wages are consumed). As a result we are also imposing a floor in investment — it cannot become negative. So we guarantee the system will not leave a bounded region; since the rest point is unstable, we know through Poincaré-Bendixson theorem that we will have a limit cycle. So, unlike Goodwin’s model, the cycle the system tends to will be independent of the initial conditions. In figure 4-5 we can see the evolution of the system from the rest point — (u∗ , L∗ ) = (0.885 71, 0.969 46) — until it reaches the limit cycle (the

69

parameter values are the same, except, obviously, the value of γ, which will be 0.3). The limit cycle generated lasts approximately eight and a half years. This modified model has, in our opinion, some advantages relatively to the original model. First of all, we have, in this model, an unstable equilibrium point, so there are forces impeding the stationary equilibrium. Secondly, the existence of limit cycle rules out the possibility of having cycles of unrealistic dimensions. Finally, labour productivity is no longer constant. The Model with a CES Production Function and no Cyclical Productivity Growth This is the case studied by Ploeg (1985)21 . We now consider γ = 0 and 0 < δ < +∞. We are studying the properties of Goodwin’s model, relaxing the assumption of a constant capital-output ratio. In this case system 4.70 becomes: ⎧ i h ⎨ u0 = f (L)−β u 1 h 1+ δ 1 ⎩ L0 = Aα− δ (1 − u) 1+δ δ −

1 u0 δ(1−u) u

i

(4.76)

−β L

∙ ¸ δ ¡ β ¢ 1+δ 1 −1 1+δ The rest point of this system is given by (u , L ) = 1 − A α , f (β) . ∗

∗

By linearizing the system around this point we get: ⎧ ⎨ u0 = δ u∗ f 0 (L∗ ) (L − L∗ ) δ+1 ´ ³ ⎩ v 0 = − 1+δ A α− 1δ (1 − u∗ ) 1δ L∗ (u − u∗ ) − δ

f 0 (L∗ ) L∗ (δ+1)(1−u∗ )

(L − L∗ )

(4.77)

whose characteristic equation is given by λ2 +

Since

´ ³ 1 1 f 0 (L∗ ) ∗ −δ ∗ δ f 0 (L∗ ) L∗ u∗ = 0 L λ + A α (1 − u ) (δ + 1) (1 − u∗ )

f 0 (L∗ ) L∗ (δ+1)(1−u∗ )

(4.78)

´ ³ 1 1 > 0 and A α− δ (1 − u∗ ) δ f 0 (L∗ ) L∗ u∗ > 0 we can be sure that

the system will converge to the equilibrium point — once again if f 0 (L∗ ) is high enough the system will oscillate towards the equilibrium. In this analysis we can see a re-edition of the debate between Solow and Harrod21

Although he considers a more general bargaining equation the conclusions reached are the same.

70

Domar. Once again, when it is considered a production function with zero elasticity of substitution22 between factors, the system does not approach the equilibrium point, just like in the Harrod-Domar model – although in their model the disequilibrium is cumulative and self-sustained, while in Goodwin’s model the system has a perpetual cycle around the equilibrium point. If we admit some substitutability ¢ ¡ 1 > 0 the equilibrium will no longer be unstable and the system between factors 1+δ

will approach a steady state growth, as in Solow growth model.

The General Case In∙the general case the equilibrium is¸given by the rest point of δ ´ 1+δ ³ ´ ³ 1 β β system 4.70, (u∗ , L∗ ) = 1 − (1−γ)A α 1+δ , f −1 1−γ . The linearized version of the system will be

⎧ 1 1 δ ⎪ f 0 (L∗ ) u∗ (L − L∗ ) u0 = Aγα− δ (1 − u∗ ) δ u∗ (u − u∗ ) + δ+1 ⎪ ⎪ ⎨ ´³ ´ ³ 1 γ 1 − 1δ ∗ δ L∗ (u − u∗ ) − Aα (1 − u ) L0 = (1 + δ) (γ − 1) − (1−u ∗) δ ⎪ ⎪ ⎪ 0 ∗ ⎩ − f (L ) L∗ (L − L∗ )

(4.79)

(δ+1)(1−u∗ )

The characteristic equation will be:

h 1 1 λ2 − Aγα− δ (1 − u∗ ) δ u∗ −

f 0 (L∗ ) L∗ (δ+1)(1−u∗ )

1

1

i

λ+

(4.80)

+A (1 − γ) α− δ (1 − u∗ ) δ L∗ u∗ f 0 (L∗ ) = 0 1

1

with A (1 − γ) α− δ (1 − u∗ ) δ L∗ u∗ f 0 (L∗ ) > 0 the stability of the system will be i h 1 1 f 0 (L∗ ) ∗ . determined by the sign of −Aγα− δ (1 − u∗ ) δ u∗ + (δ+1)(1−u ∗) L

We have already seen the consideration of a pro-cyclical productivity growth

(0 < γ < 1) has a destabilizing effect while the consideration of a non-null substitutability between factors has a stabilizing effect. Which one prevails will depend on 1

their magnitudes. If γ >

αδ A

f 0 (L∗ ) 1+δ (δ+1)(1−u∗ ) δ

L∗ u∗

then the system will be locally unstable,

and, to avoid explosive oscillations, we may establish a floor level to net investment 1

similar to the one imposed by equation 4.75. If γ =

αδ A

f 0 (L∗ ) 1+δ (δ+1)(1−u∗ ) δ

L∗ u∗

the system

1 The elasticity of substitution of a CES production function is given by 1+δ . In the case of a Leontief technology (δ → +∞) it will be zero and, in the other extreme, it will be one in the case of a Cobb-Douglas production function. 22

71

is characterized by constant amplitude oscillations, and, naturally, if 1

L∗ αδ f 0 (L∗ ) γ< 1+δ A (δ + 1) (1 − u∗ ) δ u∗ the system is stable oscillating towards the steady state. By equation 4.67 we can see that the capital-output ratio is no longer constant. When u increases the capitaloutput ratio will also increase. To give an example: we can see what happens for the parameter values already considered: β = 0.02, A = 0.25, γ = 0.3, f (L) = −0.040064 +

0.000064 , (1−L)2

and α = 0.5.

With these values the system will be stable for δ < 573.03. This implies an elasticity of substitution between factors of

1 1+δ

> 0.0017. Even if we considered γ = 0.95

the system would be stable if the elasticity of substitution were higher than 0.0052. Thus, even for extremely low substitutability between factors, the system tends to be stable. The stabilizing effect of introducing some flexibility in the production function is much stronger than the destabilizing effect of endogenous productivity growth. Only when the production function is close to a Leontief technology does the system generate perpetual oscillations. Obviously even considering a stable system we could perpetuate oscillations if we introduced stochastic shocks following Frisch’s idea.

4.2.5

Attraction for Chaos in Goodwin’s Model

We saw earlier Goodwin’s model is structurally unstable. For example, when we introduced some pro-cyclical labour productivity behaviour we reached a model that generated explosive cycles. We “stabilized” the model by imposing a floor to investment. Then the system converged to a limit cycle. Yet there is another way to “stabilize” the model. We can introduce a control parameter which will counteract the evolution of the system when it is too far away from the equilibrium point. To understand how this control parameter works consider the linearized version of Goodwin’s model with cyclical labour productivity growth, system 4.73 expanded

72

with a third equation, here expressed in terms of deviations from the equilibrium: ⎧ ⎪ ⎪ u0 = γAu∗ u + f 0 (L∗ ) u∗ L ⎪ ⎨ 0

L = A (γ − 1) L∗ u − z ⎪ ⎪ ⎪ ⎩ z 0 = b + gz ¡L − c¢

(4.81)

z is a control parameter (for example government budget surplus) which provides a growing downward pressure beyond a given high and a growing upward pressure for low values. The behaviour of this system depends crucially on the value of the constant g. Lets consider γ = 0.3, A = 1, u∗ = 0.9, f 0 (L∗ ) = 3, L∗ = .9, b = 0.005, c = 0.04523 . So whenever the employment rate is above 0.945 the control parameter z restricts further increases in employment. With these values the above system becomes:

⎧ ⎪ ⎪ u0 = 0.27u + 3L ⎪ ⎨ 0

L = −.63u − z ⎪ ⎪ ⎪ ⎩ z 0 = 0.005 + gz ¡L − 0.045¢

(4.82)

If g < 83 we have a limit cycle. But g ≈ 83 is a bifurcation point. For higher values of g the cycle period doubles. We can see the effect of the period doubling in figure 4-6. In figure 4-7 we can observe the effect of another period doubling (which occurs when g ≈ 120).In this situation we no longer have a limit cycle and, as we raise the value of g, the cycle gets higher and higher until the trajectory is chaotic as can be seen in figure 4-8. We have already seen how the consideration of nonlinear models was enough to create endogenous cycles, but until now those cycles had little resemblance with the quite erratic and unpredictable behaviour of economic time series. With this model, with only one nonlinearity in the third equation, we notice that a quite erratic behaviour of time series can be completely deterministic. Another feature of this system is that in spite of its erratic behaviour the trajectories are bounded in a closed region. This region works as an attractor, as it will be seen in next chapter. 23

Considering u∗ = 0.9, corresponds in system 4.72 to assume that β = 0.0175. The assumption of L∗ = .9 and f 0 (L∗ ) = 3 corresponds to assume that f (L) can be approximated by f (L) ≈ −3 + 3.33L.

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0 .2

v

0 .1

0

u -0.1

-0.05

0

0 .05

0.1

-0 .1

-0 .2

Figure 4-6: Period Doubling (g = 90)

0 .2

v

0 .1

0

u -0.1

-0.05

0

0 .05

0.1

-0 .1

-0 .2

Figure 4-7: Second Period Doubling (g = 125)

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0 .2

v

0 .1

0

u -0.1

-0.05

0

0 .05

0.1

-0 .1

-0 .2

Figure 4-8: Rossler Strange Attractor (g = 150) Obviously the consideration of nonlinear system is not a sufficient condition to have chaos, but the possibility of its existence has severe consequences: the erratic movement of time series is generally attributed to stochastic shocks, but once we admit the possibility of deterministic chaos this other root has also to be investigated.

4.3

Conclusion

In the beginning of this chapter we surveyed some of the critics usually made to the traditional decomposition between growth and cycles. Some emphasis was given to an econometric critique born in the inside of the dominant paradigm, namely the danger of confounding a stochastic trend with a deterministic trend – Nelson and Plosser (1982). It must be stressed that we did not explore all possible processes. Harvey (1997) warned that not all processes can be reduced to TS or DS processes, and restricting the analysis to those processes may create some misspecification problems. The consideration of a unit root is not enough to solve all the problems in the decomposition between the trend and fluctuations. We have also studied some models that question the dissociation of growth and cycles. The first of these models (RBC models) tried to incorporate Nelson and Plosser criticism. But we saw that RBC models can hardly be considered well succeeded in the unification of both theories, since an atheoretic distinction between 75

trend (e.g. Hodrick-Prescott filter) and cycles is made. We saw the importance of endogenous technology models in questioning the separation of growth and cycles. We saw, with the help of two different models, how easy it is to establish causal relations in both ways, i.e., how cycles influence growth (Stadler model) – this way money is no longer neutral, being questioned one of the main characteristics of neoclassical growth models – and how cycles can be influenced by long run growth (models of growth based on GPT). Finally we revisited Goodwin’s Predator-Prey model. Since growth was achieved with the help of an exogenous trend, we concluded that the model, in the original formulation, could not be considered a model of growth and cycles. Van der Ploeg (1985) had already analysed the implications of using a more flexible production function. Here we extended his analysis in order to study the consequences of considering some sort of endogenous labour productivity. We examined the conditions needed in order to have self-sustained endogenous cycles. We also saw that the extension of Goodwin’s original model could easily take us to chaotic processes – something to be studied more deeply in the next chapter.

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Chapter 5 Chaotic Cycles In previous chapters we saw that simple nonlinear models were able to, endogenously, reproduce a cyclical behaviour. Those kind of models are usually criticized with the argument that they are not compatible with rational expectations. Since, generally, those models have periodic cycles it would not be hard for agents to understand the mechanics of the cycle and take advantage of unexplored profit possibilities, which would lead to the end of cycles. But what if the deterministic cycles were unpredictable? In the fifties a meteorologist named Lorenz was, with the help of a computer, generating data from a simple system of three nonlinear differential equations. Once he wanted to reproduce a particular lengthy simulation and decided to start half way. So he introduced the intermediate values as the initial conditions and run the model again. He was taken by surprise when he realized the time series generated were in complete disagreement to the previous solution. After a while he understood the problem. The printed values were rounded to three decimals while the computations were made at six decimals. So, when he introduced the intermediate values he was making the mistake of ignoring the last three decimals. This small error was sufficient to generate a completely different time series sequence. This problem is known as Sensitive Dependence on Initial Conditions (SDIC), and it is the basic characteristic of a chaotic model. The first consequence of a model exhibiting SDIC is the impossibility of making accurate long run forecasts, since this would imply perfect knowledge of the condi77

tions in a certain moment of time (which is virtually impossible), even if the underlying model was perfectly known. This may explain the poor accurate record of meteorological forecasts, and one is tempted to ask about the deplorable record of economic forecasts. After all, we have already seen that it takes only one step from a nonlinear system to a chaotic system.

5.1

Chaos, Attractors and SDIC

The technical definition of a chaotic motion lies heavily in three concepts: — Conservative and Dissipative Systems, — Sensitive Dependence on Initial Conditions, and — (Strange) Attractors.1 Conservative and Dissipative Systems The definitions of conservative and dissipative systems were imported from Physics. Definition 2 (See Medio (1992)) A conservative system is a system without any kind of friction, thus it conserves its total volume (for example total energy) along the phase space. On purely formal grounds a system is conservative if there is a continuous, single-valued, analytic function of the state variables, which is constant along the path. For example, Goodwin’s basic model is a conservative system. System 4.71 is reproduced for convenience: ⎧ ⎨ u0 = (−φ − β + ρL) u ⎩ L0 = (A (1 − u) − β) L

With some simple calculations we can verify that along the trajectories eρL eAu L−(φ+β) u−(A−β) = C te

(5.1)

1 The definitions which will be given next are valid for systems of differential equations only. Extensions to difference equations are straightforward.

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Since the volume is kept constant this system cannot have an attracting region (which could be an equilibrium point, a limit cycle or a strange attractor). Definition 3 (See Medio (1992)) A dissipative system is characterized by a contraction of phase space volumes with increasing time. Formally this can be defined as the Jacobian of the system of differential equations having a negative trace. Because of dissipation, an n-dimensional system will eventually become confined to a subset of Hausdorff dimension2 less than n (which can be a fractal dimension). Sensitive Dependence on Initial Conditions Definition 4 (See Medio (1992)) Consider φt : U → U , U ⊆ Rn , the solution of a system of ordinary differential equations. The system has SDIC if there exists δ > 0 such that, for any x ∈ U and any neighbourhood N of x, there exists y ∈ N and t ≥ 0 such that |φt (x) − φt (y)| > δ. The above definition only says that even small differences in the initial conditions will lead to completely different paths, turning impossible to make long run forecasts. Strange Attractors In the course of this work we have already come across with some attractors, namely stable equilibrium points and limit cycles, since all trajectories converge there. Definition 5 (See Medio (1992) and Gandolfo (1997)) A compact set A ⊂ U , invariant under φt is a strange attractor for system if there is a set B with the following properties: (a)B is an n-dimensional neighbourhood of A, (b) for any initial point x (0) ∈ B, the trajectory representing the solution x (t) remains in B for all t > 0, andx (t) → A, t→∞

(c) there is SDIC, (d) there is a dense orbit on A for the flow φt . 2

Let N (ε) be the number of squares with length ε needed to cover an object, then, according to Martin (1994), the Hausdorff dimension is given by D = lim lnlnN 1(ε) . ε→0

79

ε

The meaning of the first three conditions is obvious. The last one means the attractor cannot be split into two regions. To have a (strange) attractor it is important that the system is a dissipative system, since it is dissipation that makes transitory phenomenons disappear.

5.1.1

Transition to Chaos

Period-Doubling and Feigenbaum’s Universal Constant In the present state of art we are not able to list precisely the prerequisites of chaotic behaviour. Some of the already known forms are period-doubling, intermittency and saddle connection. In this work we will only explore the one which have received more attention in economic literature: period doubling. When there is a change in the forces that act in a dynamical system sometimes one observes a period duplication. A periodic orbit is replaced by another (close to the former), but in which it is necessary to take two rounds to return to the starting point. This means that the necessary time needed to return to the starting point has approximately doubled. What is interesting is that period doubling can occur repeatedly, originating a period four times as long, then eight, sixteen, etc. This period doubling cascade can be observed in figures 4-6 and 4-7. Admit A as the relevant parameter determining the qualitative features of the model. Admit also that the points in which a bifurcation occurs are A1 , A2 , A3 , ...A∞ . In 1978 Feigenbaum3 , a physicist, found a constant arising in many one-dimensional discrete maps: An − An−1 ∼ = 4.6692016091... An+1 − An

(5.2)

This is known as the Feigenbaum universal constant. It is universal in the sense that it is valid not only for a very large class of discrete maps, but also for some continuous multi-dimensional system which, thanks to strong dissipation, may be characterized by a one dimensional map. 3

Feigenbaum, M. (1978), “Quantitative Universality for a Class of Non-Linear Transformations”, Journal of Statiscal Physics, vol. 19, pp. 25—52, cit. in Creedy, J. and Martin, V. L. (1994).

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This universal constant gives us the way to determine A∞ , only with the know´ ³ An −An−1 . It has been shown that ledge of a few bifurcation points An+1 ∼ = An + 4.669202

beyond A∞ exists chaos.

For example, one of the most simple used applications of chaos theory in economics is the logistic function: yn+1 = Ayn (1 − yn )

(5.3)

If 0 < A < 3 the above equation has a stable fixed point, but when A = 3 a period doubling bifurcation happens and a stable two period cycle appears. Another √ period doubling occurs when A = 1 + 2 6, generating a four-period cycle. With the knowledge of these two bifurcation points, and with the help of Feigenbaum constant, it is easy to conclude that beyond A∞ ∼ = 3.569446, we have chaos. Period Three Implies Chaos A very interesting result which gives us sufficient conditions to have chaos was given by Sarkovskii4 : Theorem 6 (See Ruelle (1994)) Consider the following ordering of all positive integers: 3  5  7  ...  2 × 3  2 × 5  2 × 7  ...  22 × 3  22 × 5  22 × 7  ...  2n × 3  2n × 5  2n × 7  ...  2n  ...  22  2  1 If f is a continuous map of an interval into itself with a period a, and a  b in the above ordering, then f also has a periodic point of period b. This means: if we detect that f has a period point whose value is not a power of 2, then we can be sure that f has infinitely many period points. 4

Sarkovskii, A. N. (1964), “Coexistence de cycles d’une application continue de la droite dans elle-même”, Ukr. Mat. Zh., vol. 16, pp. 61-71, cit in Ruelle (1994).

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A very interesting application of Sarkovskii’s result was given by Li an Yorke in 19755 : Theorem 7 (See Gandolfo (1997)) Let f : I → I, be a continuous map. Con-

sider that the first three iterates of x0 are given by f (x0 ) = x1 , f 2 (x0 ) = f (x1 ) = x2 , f 3 (x0 ) = f (x2 ) = x3 satisfying the inequality x3 ≤ x0 < x1 < x2 or, alternatively, x3 ≥ x0 > x1 > x2

Then for every k = 1, 2, 3, ... there is a periodic point in I having period k and there is an uncountable subset Λ ⊂ I containing no periodic points with the following properties: (1) for every x, y ∈ Λ, x 6= y lim sup |f n (x) − f n (y)| > 0, and n→∞

lim inf |f n (x) − f n (y)| = 0 n→∞

(2) for every point x ∈ Λ and periodic point y ∈ I lim sup |f n (x) − f n (y)| > 0 n→∞

Property (1) of this theorem tells us that in the aperiodic set Λ, points starting from arbitrarily closed points will infinitely often become very closed to one another, and also become infinitely often finitely separated. This is equivalent to SDIC. On the other hand, property (2) tells us that there will be no point of contact between aperiodic orbits and periodic orbits. Note that if a periodic point with period three exists the hypotheses of the theorem are satisfied, thus the statement that period three implies chaos6 . This condition is not a necessary condition: it is possible to have chaos without having a three period cycle. 5

Li, T. and Yorke, J. A. (1975), “Period Three Implies Chaos”, American Mathematical Monthly, vol. 82. pp. 985-992, cit. in Ruelle (1994). 6 One should take this statement carefully since period three does not imply a chaotic attractor. So Li-Yorke definition of chaos does not coincide exactly with our previous definition. Chaos in the Li-Yorke sense is also known as topological chaos.

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5.2 5.2.1

Some Economic Applications of Chaos The Neoclassical and the Endogenous Growth Models

Day (1982) attracted considerable attention to chaos by alerting economists to the possibility of chaos in a quite known model: Solow’s growth model. It is interesting that the year of publication of this short paper coincided with the seminal paper of Kydland and Prescott, who also used the neoclassical production function in their explanation of cycles. The similarities end here. While, as we have already seen, in Kydland and Prescott model explanation of cycles was based in stochastic technology shocks (i.e., stochastic shifts in the production function), Day bases his explanation in simple nonlinearities. Day’s approach is purely deterministic. Day considered a discrete time version of Solow’s model, but changed the saving function. He considered saving propensity as an increasing function of the capitaloutput ratio and of the rate of interest. The model is fully characterized by the equations yt = Aktα ¶ µ b kt st = a 1 − r yt kt+1 = st yt

(5.4) (5.5) (5.6)

where yt and kt are respectively output and capital per capita, and r is the rate of interest. Note the system implies a purely circulating capital and a stationary labour force. Solving the above system in order to kt+1 , and having in mind that r = α ky , yields: kt+1

µ ¶ b 1−α k = akt 1 − αA t

(5.7)

This equation is a variant of the logistic equation 5.3 and it is not hard to prove that for certain parameter values it generates chaotic motions7 . Consider for example the parameter values a = 6.6, α = 0.5, Ab = 1 and the initial value k0 = 0.025. Then k1 = 0.1128, k2 = 0.2444, k3 = 0.0181. Thus, we have a sequence of points such that kt+3 < kt < kt+1 < kt+2 , so by Sarkovskii and Li-Yorke results we can be sure to have a 7

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An even simpler way to show the emergence of chaos is to admit an AK production function, but admit some negative effect caused by the accumulation of capital (e.g. pollution effect). In this case we can admit a constant savings ratio st = s. The model becomes: yt = Akt (m − kt )γ

(5.8)

st = s

(5.9)

kt+1 = syt

(5.10)

These equations can be reduced to: kt+1 = sAkt (m − kt )γ

(5.11)

which is another variant of the logistic equation capable of generating chaos8 .

5.2.2

Kaldor’s Trade Cycle in Discrete Time

In the previous example we saw an economic application of a one dimensional map. When we try to extend the analysis to higher dimensional maps we do not have many theorems to help us identify attractors. One possible useful theorem is given by Marotto9 , who generalizes Li-Yorke theorem to higher dimensional maps with the concept of snap-back repellers. Basically, if the eigenvalues of the system around the fixed point lie outside the unit-circle, and for state variables not close to the fixed point the eigenvalues lie inside the unit-circle, then the system has a snapback repeller and the process is chaotic. The problem with this theorem is that it is analytically difficult to identify a snap-back repeller and we often have to rely on numerical analysis. We have already studied Kaldor’s trade cycle model in continuous time, so we chaotic region in equation 5.7 8 For example, admit sA = 1.63, γ = 0.3, m = 2, k0 = 0.97. With these values we would have k1 = 1.595, k2 = 1.982, k3 = 0.962. Once more Li-Yorke and Sarkovskii sufficient condition for chaos are respected. 9 Marotto, F. R. (1978), “Snap-back Repellors Imply Chaos in Rn ”, Journal of Mathematical Analysis and Applications, vol.72, pp. 119-223, cit. in Creedy and Martin 1994.

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do not need to explain the rationality of the model now. We will just admit some functions that obey Kaldor’s hypothesis10 : Yt+1 − Yt = α (It − St )

(5.12)

Kt+1 − Kt = It − δKt St = sYt µ a+ It =

b 1 + ec−dYt

(5.13) ¶

(5.14) − eKt

(5.15)

The above equations speak for themselves. The first one tells us that changes in output are proportional to the gap between investment and savings. In the second one we admit that to have capital increases, capital depreciation must be exceeded by investment. In the third one we assumed a linear savings function (violating one of Kaldor’s assumptions). The nonlinearity is introduced in the fourth equation. The first term is a logistic function of the output, so marginal propensity to invest is extremely low for extreme values of Y . The second term represents the negative impact of the stock of capital in investment opportunities. The parameters are chosen in such a way that the investment and savings functions intercept in three different points. If we calibrate the model with the values a = 4, b = 30, c = 5, d = 0.1, e = 0.000001, s = 0.3, we have the graphic representation of savings and investment shown in figure 5-1. The system is governed mainly by the parameter α. For low values the system displays a unique stationary equilibrium, and as its value is raised bifurcations (period doubling) occur until chaos is reached. In figure 5-2 Yt+1 is plotted against Yt (α = 8.8). As can be seen, Y varies between 35 and 160. The stock of capital (figure 5-3) varies between 550 and 650. In figure 5-4 we can see the different evolution of the time series caused by a small variation in the initial value of Yt (K0 = 600 in both cases) for fifty iterations. 10

It should be stressed that Kaldor’s reasoning is clearly in continuous time. And, since replacing derivatives with differences is not an innocuous operation, we cannot be sure that Kaldor would agree with this representation of his model.

85

40

30

20

10

0

20

40

60

80 Y

100

120

140

Figure 5-1: Kaldor Investment and Savings Function

Yt+1

160

35 35

Yt

160

Figure 5-2: Kaldor Business Cycle Attractor (Y)

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K t+1

650

550

Kt

550

650

Figure 5-3: Kaldor Business Cycle Attractor (K)

180 160 140 120 100 80 60 40 20 0 1

4

7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 Yo=80.000001

Yo=80

Figure 5-4: SDIC in Kaldor’s Model

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It is obvious that this model suffers from SDIC.

5.2.3

A Model of Growth and Cycles

We have so far explored only chaotic systems in discrete time. When the analysis is made in continuous time we need at least a three dimensional system. This is so because in a continuous system variables cannot jump discretely. In a planar system a trajectory representing the solution cannot intercept itself, so it is not possible to have an irregular complex behaviour in a two-dimensional system. Goodwin (1991) extended his 1967 predator-prey model in order to accomplish growth and cycles. Specifically it generates a Kondratieff growth cycle, which also incorporates Juglar cycles. He tries to incorporate the Schumpeterian swarm of innovations, according to which, after a weak beginning, the path-breaking innovation proves its importance and more and more firms will adopt the innovation. At the end the rate of adoption will diminish since the majority of the firms have already adopted it. We have already seen that from Goodwin’s model to the possibility of chaos it takes only one step. For convenience Goodwin’s system of equations is reproduced here (u and L should be understood as deviations from equilibrium): ⎧ ⎪ ⎪ u0 = hL ⎪ ⎨

L0 = −du + f L − ez ⎪ ⎪ ⎪ ⎩ z 0 = b + gz (L − c)

(5.16)

The similarities between this system and system 4.81 are quite obvious, so there is no need for further rationalization. As we have already seen, for sufficiently high values of g the system is able to generate deterministic chaos. To include growth in the above model we have to consider the effects of the investment in labour productivity. We admit there is a cyclical component in labour productivity (γ in equation 4.61) and no exogenous productivity growth (β = 0 in the same equation). About investment we will admit an historically given fifty years

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30 25 20 15 10 5 0

10

20

30

40

50

Figure 5-5: Capital Evolution for 50 years Schumpeterian swarm of innovations. Specifically we will admit that: n−qt

K 0 = men−qt−e

11

(5.17)

For example, if we calibrate this equation with the values m = 4.5, n = 3, q = 0.15 and K0 = 1 the capital accumulation would be as represented in figure 5-5. The dynamics of output will be determined by the evolution of employment 0

(= L∗ + L) and by the evolution of labour productivity (= γ KK ): Y0 L0 K0 = ∗ +γ Y L +L K

(5.18)

This formulation has one problem. The investment function influences output, but should also be influenced by. To answer, at least partially, to this criticism we will admit that the employment level has its role in the dynamics of the investment, so: n−L−qt

K 0 = men−L−qt−e 11

(5.19)

This function is known as the Gompertz curve and it is a special case of the generalized logistic. An advantage of this formulation relative to the usual simple logistic is that it is more flexible; namely we are not restricted to a symmetrical curve.If we had used a simple logistic its main implication would have been a more variation of the series accumulated in early stages. For further information on this curve see Cadima (1999), who applies this function to a dynamical study of the cellular phones in Portugal. Stone (1990) has also used this function to describe the population growth dynamics.

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8

6

4

2

0 00

10

20

30

40

50

60

70

80

90

00

Figure 5-6: Output Time Evolution In this formulation the employment level enters directly in the investment function. It can be interpreted as an accelerator mechanism. Joining equations 5.16, 5.18, and 5.19, the complete model becomes: ⎧ ⎪ u0 = hL ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L0 = −du + f L − ez ⎪ ⎪ ⎨ z 0 = b + gz (L − c) ⎪ ⎪ ⎪ n−L−qt ⎪ ⎪ K 0 = men−L−qt−e ⎪ ⎪ ´ ³ ⎪ n−L−qt ⎪ L−ez men−L−qt−e ⎩ Y 0 = −du+f Y + γ L∗ +L K

(5.20)

To understand the kind of output dynamics generated by this model, the model was calibrated with the parameter values: b = 0.001, c = 0.048, d = 0.5, e = 0.8, f = 0.15, g = 85, L∗ = 0.9, n = 3, p = 35, q = 0.15, γ = 0.3. The data generated are for a hundred years. Since the parameters chosen to the investment equation imply a swarm of fifty years we had to introduce a second swarm of fifty years. This way in the first fifty years m takes the value m = 4.5. Beyond that m takes the value m = 135. The evolution of investment will be essentially the same in each half century.

The initial values were u = 0.02, L = 0.04, z = 0, k = 1, y = 1.

The results can be observed in figure 5-6. One interesting feature of the time series generated is that the cycles generated are not identical, even considering identical

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capital accumulation dynamics for both half centuries. Once again we observe that a chaotic deterministic system can generate a quite erratic behaviour. The possibility that an erratic behaviour can be purely deterministic raises an interesting question: to what extent are the traditional econometric techniques appropriate to deal with this new issue? We will try to sketch the answer in the next point.

5.3

The Adequacy of the Traditional Econometric Approach to Nonlinear Cycles

Blatt (1983) alerted to the dangerous consequences of an error in the identification of the stability properties of an economic system. He then asks if the traditional econometric tools are a good instrument to analyze the stability of an economic system. To answer this question he makes a simple test. He generates some economic time series with the help of a nonlinear, locally unstable, macro-model Hicks proposed. With these artificial data he tries to estimate the original model. The results are quite unpleasant: the estimated model did not identify (not even close) the inherent instability of the original model. Basically, a dynamically stable model was estimated and the endogenous cycles were attributed to stochastic shocks. Louçã (1997) made a similar approach. He considered a more general model to generate artificial data for output12 which was able to simulate growth and cycles endogenously. But, in his treatment of the time series output, he extracted a linear trend13 and then he modelled the residuals as a linear autoregressive process. The problem with Louçã’s approach is that when one tries to apply usual econometric procedures to time series data it is not possible to forget testing previously the stationarity of the series. An applied econometrist before extracting a (linear) time 12

The model he used was very similar to the system of equations 5.20. The main difference is that he represented the investment dynamics with a simple logistic and did not introduce an accelerator component in the investment function. 13 The trend was extracted from logaritmized time series, so it is an exponential trend relative to the original data.

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trend, would test the possibility of the series being difference stationary and not trend stationary. We propose to do that next. Before that we will explain the procedure used to test the stationarity of the series.

5.3.1

The Augmented Dickey-Fuller (ADF) Test – A Unit Root Test

If we admit that Y is an economic time series which can be modelled as an autoregressive process (AR) of order one then: Yt = α + βYt−1 + ut

(5.21)

In economic time series it is not too restrictive to consider that β > 0. We know Yt is stationary iff β < 1. So, it would seem normal to test stationarity using the ordinary t-test: H0 : β = 1 H1 : β < 1 Obviously, if one rejects H0 in favour of H1 , the possibility of β > 1 is also rejected. Unfortunately this procedure is not correct. First of all, with the presence of a lagged dependent variable as a regressor, βˆ 14 will be biased. Per si this would not be a tremendous problem since Mann and Wald15 showed that the limit distribution ´ √ ³ √ α − α) and n βˆ − β was a bivariate Normal, with mean value zero and of n (ˆ ³ ´ finite variances and covariances. This way, p lim βˆ = β. Unfortunately this result

depends crucially on the assumption that the series Yt is stationary. ´ √ ³ˆ d Dickey and Fuller (1979) proved that if β = 1 then n β − β → v, being v ³ ´ a random variable with finite variance and E βˆ < 1. Thus, the β estimator will 14 ˆ

β is the OLS estimate of β. Mann, H. and Wald, A. (1943), “On the Statistical Treatment of Linear Stochastic Difference Equations”, Econometrica, vol. 11, cit. in Greene (1997) 15

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be downward biased if β = 1, invalidating the usual significance tests. To overcome this problem Dickey and Fuller made some Monte-Carlo simulations to obtain the relevant critical values16 . To test the stationarity of Yt one should run the regression Yt = α + βYt−1 + ut ∆Yt = α + γYt−1 + ut

(5.22)

with γ = (β − 1). To test the hypothesis that γ = 0 against the alternative γ < 0, one should compute the usual t-statistic and compare it with the Mackinnon’s critical values. This procedure presents a failure: the fact that we are only considering an AR(1) process. If the real process is an AR(p), with p > 1, then the residuals will be serial correlated. To overcome this situation, we can add as a regressor the necessary number of lags of the dependent variable – this is the Augmented Dickey-Fuller test. Consider for example an AR(3) process: Yt = α + β 1 Yt−1 + β 2 Yt−2 + β 3 Yt−3 + ut

(5.23)

The polynomial associated lag equation is: 1 − β 1 L − β 2 L2 − β 3 L3 = 0

(5.24)

From equation 5.23 we can reach: ∆Yt = α + β ∗1 Yt−1 + β ∗2 ∆Yt−1 + β ∗3 ∆Yt−2 + ut 17

(5.25)

If there is a unit root then 5.24 can be written as (1 + ψL + θL2 ) (1 − L) = 0, by 16

The program Econometric Views (EViews), and most of other econometric packages has already incorporated the critical values to test the series stationarity. The critical values included were simulated by Mackinnon (1991) and are more robust than the originals Dickey and Fuller proposed. 17 with β ∗1 = β 1 + β 2 + β 3 − 1, β ∗2 = − (β 2 + β 3 ), and β ∗3 = −β 3 .

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which equation 5.23 can be written as: ¢ ¡ 1 + ψL + θL2 (1 − L) Yt = α + ut ¡ ¢ 1 + ψL + θL2 ∆Yt = α + ut

∆Yt = α − ψ∆Yt−1 − θ∆Yt−2 + ut

(5.26)

So, if there is an unit root, the coefficient of Yt−1 in equation 5.25 must be zero. Thus, testing β ∗1 = 0 in equation 5.25 is equivalent to testing the hypothesis of the series having a unit root, i.e., of being DS. A possible criterium to choose the number of lags is to estimate equation 5.22 and test the residuals serial correlation. If there is evidence of autocorrelation we add a new lag and test again the residuals. This procedure is repeated successively until we accept the hypothesis of no serial correlation. To attend to the possibility of the series being TS it is usual to also add a linear time trend to equation 5.22, and test the significance of its coefficient by the usual methods.

5.3.2

An Econometric Application to Our Artificial Model

We will use the time series represented in figure 5-6 to test its stationarity. The test used will be the ADF and will be applied after logarithmizing the series. The results obtained are illustrated in table 5.11819 . According to this test, we cannot reject the null hypothesis of nonstationarity for the log of the output (LY ), while for the growth rate we reject the null hypothesis accepting the growth rate20 to be stationary around a constant. With these results an applied econometrist would not extract a linear trend to stationarize the series. He would rather consider the first differences of LY (growth rate of Y ). 18

We first considered a constant and a linear trend as deterministic components. Since the latter was not significant we excluded it from the test regression. 19 To choose the number of lags we began by considering no lags. We then tested the residuals autocorrelation; the test used was the Breusch-Godfrey LM test. If the hypothesis of no serial correlation was rejected we introduced another lag and repeated the procedure. The level of significance considered was 5% and the order of serial correlation tested went from 1 to 4. 20 DLY is the first difference of LY . Since LY = log Y, DLY will be the growth rate of Y .

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Series Deterministic Number t-Stat Critical Value Component of Lags 1% 5% 10% LY Constant 4 -0.653931 -3.4648 -2.8762 -2.5745 DLY Constant 3 -5.448247 -3.4648 -2.8762 -2.5745 Table 5.1: ADF Test Results So, we will try to model DLY as an autoregressive process. The estimated results are: DLYt = 0.0009 + 1.9571DLYt−1 − 1.6237 DLYt−2 + (2.836)

(29.026)

(−11.511)

+0.9392DLYt−3 − 0.3552DLYt−4 21 (6.66)

(5.27)

(−5.273)

where the values in parenthesis are the t-statistics. The R-squared (and the adjusted R-squared) is about 96%. The residuals show no evidence of serial correlation. The number of lags chosen were based on the Akaike and Schwarz information criteria and were strengthened by the fact that higher lags were not statistically significant22 . It is interesting to note that equation 5.27 is a simple difference equation with an explicit analytic solution: DLYt = 0.0104 + 0.6627t (A1 cos (1.4073t) + A2 sin (1.4073t)) + +0.8994t (A3 cos (0.2534t) + A4 sin (0.2534t))

(5.28)

where A1 , A2 , A3 , and, A4 are arbitrary constants that can be determined with the help of four initial conditions. The similarities between the above equation and Frisch’s equation 3.5 are obvious. In both of them we have trigonometric functions determining the cycles and in both of them the system is stable so the generated cycles will die away. It is remarkable how well this completely wrong estimated model fits in Frisch’s paradigm: we have an exogenous trend (determined by the constant 0.01042) and two different growth cycles (one with 4. 5 years and the other with 24. 8 years) aggregated additively. In figure 5-7 we can see how this estimated model would work in the absence of stochastic shocks. 21

Since we have considered semi-annual data, we have the growth rate of period t depending on the previous four semesters. 22 For example, if we introduced a fifth lag, its P-value would be 0.39.

95

0.020 0.015 0.010 0.005 0.000 -0.005 -0.010 -0.015 00

10

20

30

40

50

60

70

80

90

00

Figure 5-7: The Estimated Model Dynamics To explain the persistence of cycles in this model, Frisch would suggest the addition of a stream of exogenous shocks. This is what we will do next. We add a stream of exogenous shocks with mean zero and variance 0.0001444 to equation 5.27 with the help of a normal random number generator23 . In figure 5-8 we compare the original artificial time series with the time series generated by the estimated model (augmented with the stochastic shocks). Although not reproduced here, we would have reached the same qualitative results if we had applied the Hodrick-Prescott filter instead and then fitted an autoregressive process to the residuals.

5.3.3

Problems with Heteroskedasticity

We have so far neglected the possibility of having heteroskedastic disturbances. In traditional time series analysis it was usual to consider homoskedastic processes (associating heteroskedasticity to cross-sectional data). But, at least since Engle (1982), one cannot put aside the possibility of having an Autoregressive Conditional Heteroskedasticity (ARCH) model or one of its extensions as we shall see. 23

The variance was chosen in such a way that the original time series and this new time series have the same variance.

96

0.10 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 00

10

20

30

40

50

60

Original Series

70

80

90

00

Frischian Series

Figure 5-8: Two Simulated Time Series Consider a pth order ARCH process: Yt = β 0 Xt + t v u q X u t = u + αi t t

(5.29) 2 t−i

(5.30)

i=1

where ut follows a standard normal. It easy to derive the conditional and unconditional variances of

t:

V ar [ t |

t−1 , ..., t−p ]

=

σ 2t

=

+

q X

αi

2 t−i

(5.31)

i=1

V ar [ t ] =

1−

Pq

24

i=1 αi

(5.32)

It is interesting to note that, in this situation, although the Ordinary Least Squares (OLS) estimator is BLUE it is not BUE, i.e., it is the best linear unbiased estimator, but there is a more efficient nonlinear estimator. Engle (1982) derived the likelihood function for this model and also presented a Lagrange Multiplier (LM) test for the ARCH process. First, we estimate equation 24

It was assumed that the unconditional variance remains unchanged over time.

97

obs×R2 1 order 2.979 2nd order 15.199 3rd order 15.149 4th order 16.030 st

P-value 0.084 0.001 0.002 0.003

Table 5.2: ARCH LM Test 5.29 using OLS. Then, regress the square of estimated residuals on a constant and its past values. The number of observations times R2 follows asymptotically a Qui squared distribution with q degrees of freedom (with q being the number of lags considered in the test regression). In table 5.2 we can see the results of the ARCH LM test. With these results an applied econometrist would have to deal with the conditional heteroskedasticity problem. In our work we will begin by considering a Generalized ARCH (GARCH) model proposed by Bollerslev (1986)25 . The advantage of this approach is that it is usually more parsimonious with the number of lags needed. In a GARCH(p, q) model the conditional variance is given by: σ 2t

=

+

q X

αi 2t−i

i=1

+

p X

β j σ 2t−j

(5.33)

j=1

Bollerslev et al. (1994) show that this is equivalent to saying that

2 t

can be modelled

as an ARMA([max (p, q) , p]) model. Obviously, if equation 5.33 is correctly specified the residuals should not exhibit additional ARCH. After considering several GARCH models of different orders we concluded that the standardized residuals26 continued to exhibit ARCH, indicating that equation 5.33 was misspecified. Nelson (1991)27 proposed an Exponential GARCH (EGARCH) model. Equation 25

Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, vol.31, pp. 307-327, cit. in Bollerslev et al., (1994). 26 Standardized residuals are defined as the conventional mean equation residuals (obtained from equation 5.29) divided by the conditional standard deviation (from equation 5.29) – σtt . 27 Nelson, D. (1991), ”Conditional Heteroskedasticity in Asset Returns: A New Approach”, Econometrica, vol. 59, 347-370, cit. in Bollerslev et al (1994).

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obs×R2 1 order 0.009 2nd order 1.693 3rd order 1.936 4th order 2.096 st

P-value 0.924 0.429 0.586 0.718

Table 5.3: ARCH LM Test to EGARCH standard residuals 5.33 is replaced by: ¡ ¢ ln σ 2t =

¯ q µ X ¯ αi ¯¯ + i=1

¯ ¶ X p ¯ ¡ ¢ ¯ + γ i t−i + β j ln σ 2t−j ¯ σ t−i σ t−i t−i

(5.34)

j=1

When we re-estimated equation 5.27, admitting that the conditional heteroskedasticity followed an EGARCH(2, 4)28 the results were:

DLYt =

¡ ¢ ln σ 2t =

0.00003 + 3.0965 DLYt−1 − 3.6082 DLYt−2 (19.830)

(281.625)

(−135.434)

+1.8744DLYt−3 − 0.3679 DLYt−4 (84.615) (−55.842) ¯ ¯ ¯ ¯ − 7.2006 + 2.1347 ¯ σt−1 ¯ − 0.0650 σt−1 t−1 t−1 (−14.975) (17.355) ¯ ¯ (−0.816) ¯ ¯ +1.8102 ¯ σt−2 ¯ − 0.0242 σt−2 t−2 (9.183) (−0.264) t−2 ¡ 2 ¢ ¡ ¢ +0.0583 ln σ t−1 + 1.3331 ln σ 2t−2 (0.930) (36.398) ¡ 2 ¢ ¡ ¢ −0.1151 ln σ t−3 − 0.5251 ln σ 2t−4 (−2.611)

(5.35)

(5.36)

(−16.551)

where the values in parenthesis are the z-statistics. As it can be seen, the results of equation 5.35 do not differ substantially from the results obtained in equation 5.27. Specially, it is easy to verify that the stability properties do not change. When the ARCH LM test is applied to the standard residuals the results are conclusive. As we can see in table 5.3, no economist would reject the null hypothesis of conditional homoskedastic residuals. Even the Jarque-Bera normality test tends to accept the good specification of the model: the Jarque-Bera statistic29 has a value of 2.4 with 28

q = 2, p = 4 in equation 5.34. The order of the ARCH process was chosen, basically, with the help of the Akaike and Schwartz information criterion, and with significance tests. 29 This test is quite simple.With the estimated standard residuals, one calculates the ³ kurtosis and ´ 2 n , the skewness of the standard residuals distribution. Then one computes: JB = 6 S 2 + (K−3) 4 where S is the skewness, K is the kurtosis and n the number of obsrevations. Under the null

99

Figure 5-9: Dimension of Stochastic Processes vs Deterministic Processes a P -value of 0.3. So, we would even accept the normality of the standard residuals. Obviously we did not perform here an exhaustive battery of tests, so we cannot be sure that the estimated model would pass in all specification tests, but we can affirm that there is a tendency to accept this wrong model. We say wrong because equation 5.35 represents a linear stable model, when we know the true model is a nonlinear unstable one. Even equation 5.36 tells us that, although the conditional variance of the residuals will vary with time, it will stabilize, unless it is fed with exogenous shocks. The intrinsic instability of the model is not captured in any of the components of the EGARCH estimates.

5.3.4

The BDS Statistic

Dimension of Stochastic Processes Although nonlinear deterministic models can generate random processes, there is one important difference between deterministic and stochastic processes: while deterministic processes have finite dimension, stochastic processes have infinite dimension. We can see in figure 5-9 the phase portrait of chaotic deterministic series based on the logistic equation (Yn+1 = 4Yn (1 − Yn )) and of a stochastic series. It is easy to see that while the deterministic process is one-dimensional the stochastic series hypothesis of a normal distribution, the Jarque-Bera statistic is distributed as χ2 with 2 degrees of freedom.

100

fills out the entire area. Thus the stochastic process is at least two dimensional. If we had plotted a three-dimensional phase portrait we would conclude that the stochastic process would fill out the entire cube and so on for higher dimensions. So a stochastic process approaches an infinite dimension. The Correlation Dimension and the BDS Statistic Based on the above notion of dimension Brock et al.30 propose a statistical procedure to test departures from independently and identically distributed (i.i.d.) observations. Consider T observations of a time series (x1 , x2 , ..., xT ), after removing all non-stationary components. Define the m-histories of xt process as the vectors (x1 , ..., xm ) , (x2 , ..., xm+1 ) , ..., (xT −m+1 , ..., xT ). Now define the correlation integral as the fraction of the distinct pairs of m-histories lying within a distance ε in the sup norm31 :

Cε,m,T =

T T X X ¡ ¡ ¢¢ 1 H ε − sup norm xi , xj (T − m + 1) (T − m) i=1 j=1

(5.37)

i6=j

⎧ ⎨ 0 if x ≤ 0 . where xi = (xi , ..., xi+m−1 ), and H (x) = ⎩ 1 if x > 0 Under some assumptions Cε,m,T converges to a limit Cε,m . The true correlation

dimension is given by

d ln(Cε,m ) . d ln ε

It is possible to show that the correlation dimension d ln(Cε,m,T ) tends to a constant has the Hausdorff dimension as its upper bound. If d ln ε

as m, increases then data will be consistent with deterministic chaotic behaviour, d ln(Cε,m,T ) increases without bound with m then one will conclude that data is if d ln ε stochastic. Brock et al. employed the correlation dimension to obtain a statistical test of 30

Brock, W. Dechert, W. and Sheinkman, J. (1987), ”A Test for Independence Based on the Correlation Dimension”, University of Winsconsin, Madison, University of Houston, and University of Chicago, cit. in Brock et al. (1991). 31 Brock showed that the correlation dimension was independent of the choice of the norm, so it is not restrictive to consider the sup norm. Brock (1986), ”Distinguishing Random and Deterministic Systems: Abridged Version”, Journal of Economic Theory, vol. 40, pp. 168-195, cit. in Creedy and Martin (1994b).

101

m ε = 0.5σ ε=σ

2 3 4 5 13.24 15.83 18.56 21.82 (3.71)

(4.04)

(4.85)

11.47 12.13 12.21 (2.79)

(2.92)

(2.96)

(6.44)

12.3

(3.06)

Table 5.4: BDS Test to the AR(4) residuals nonlinearity: they proved that under the null (xt i.i.d.) ln (Cε,m ) = m ln (Cε,1 ), which is the basis for the BDS statistic: BDS =

Cε,m,T − (Cε,1,T )m σ ε,m,T

(5.38)

where σ ε,m,T is the standard deviation consistently estimated32 . Under the null BDS has a limiting standard normal distribution. The asymptotic distribution behaves reasonably well, if the sample size is not less than 500, but it is poor for smaller sample dimensions.. To implement the BDS test, Monte Carlo Simulations of Brock et al. (1991) suggest that ε should vary between 0.5 and 2 standard deviations of the data, and m between 2 and 5. BDS Applied to our Model We now apply the test to the residuals of our models. First, we apply it to the residuals of the model given by equation 5.2733 . This is a valid procedure since DLYt is stationary (as can be seen in table 4.1), and the statistic is robust to linear filtering (e.g. see Bollerslev et al. (1994)). Since our sample size is too small (about 200 observations), we cannot rely on the asymptotic distribution. Thus, we use the critical values given in Brock et al. (1991) (Table C.2 pp. 233), which were obtained by Monte Carlo Simulations after 5000 replications. The values in parenthesis are the critical values for 1% significance. The results (table 5.4) are extremely convincing, the null hypothesis is strongly rejected suggesting some implicit structure not captured by the AR(4) process. But, 32

See Brock et al. (1991) for details on how to estimate σ ε,m,T . The statistic values were obtained with the software program ”BDS Stats”, version 8.20, developed by W. Dechert. 33

102

m ε = 0.5σ ε=σ

2 3 4 5 13.85 18.01 25.44 41.47 (0.00)

7.16

(0.00)

(0.00)

7.14

(0.00)

(0.00)

7.37

(0.00)

(0.00)

7.80

(0.00)

Table 5.5: BDS Test to the EGARCH(2,4) residuals although the BDS test is based on the correlation dimension, it does not provide direct evidence of deterministic chaos, since the BDS statistics (as reported by Brock et al.) has high power for detecting other departures from the null. Thus, the BDS test for chaos is also good to detect other types of nonlinearities (for example, GARCH and EGARCH models). Since an ARCH model and its extensions typically assumes i.i.d. standard residuals, Bollerslev et al. (1994) suggest the use of the BDS test as a specification test applied to the standardized residuals of a model. We have already seen that the Jarque-Bera test applied to the standardized residuals of our EGARCH(2, 4) would not reject the normality of those residuals. We now apply the BDS test to the same residuals. Two difficulties need to be faced with. First, the small dimension of the sample; second, the asymptotic distribution of the test is strongly affected by the fitting of the EGARCH model and has not been derived yet. To overcome both problems, we follow a procedure suggested by Brock et al. (1991), also applied by Louçã (1997): after estimating the BDS statistic we shuffle randomly the time series sample and then estimate the statistic again. This procedure is repeated 100 times. If the process is purely random the dimension of the process will be unchanged and so will the estimated statistic. If the process is purely deterministic, then shuffling will destroy the correlation structure of the process. In table 5.5 we can see the results achieved. In parenthesis we have the proportion of the statistic values (obtained after reshuffling) that are higher (in absolute value) than the statistic applied to the original series. As we can see, the results of the statistic point, correctly, to a misspecification of the model.

103

Some Problems The above results suggest that it is easy to determine whether a time series follow a chaotic process or not. We must take this conclusion very carefully. First, the rejection of the null hypothesis does not tell us anything about the alternative. For example, the data generator process may be a stochastic nonlinear model and not a chaotic deterministic model. Second, there is no practical distinction between a high dimensional chaotic model and a pure stochastic model, so this test is only appropriate to detect low dimensional chaos. An interesting problem, particularly when we are analyzing macroeconomic time series, is the problem with aggregate data. One of the flaws Schumpeter found in Keynes’ work was the use of aggregate functions (consumption, investment, etc.). He argued aggregation could mask innovative processes which are specific to some industries. Goodwin (1991) agreed to this idea and defended the use of large multidimensional systems. Unfortunately, for simplicity sake, he presented an aggregated model (which we have already studied). This problem has never been taken seriously by the Economics mainstream, since its relevance in linear models is not very pertinent. In an interesting exercise, Louçã (1997) applied the BDS test to economic time series from Great Britain. More precisely, the time series considered were industrial production, exports, investment in non-residential structures, investment in machinery, and gross domestic product (GDP). The evidence of nonlinearities was sharp for all series except for GDP (Louçã estimates the BDS statistic for eight different combinations of m and ε, and only for (ε, m) = (σ, 2) the statistic was significant at a 5% level). Obviously, since GDP is a function of investment, exports, and industrial production, if the latter ones exhibit nonlinearities, so will GDP. Louçã concludes that the aggregation at a GDP level hides nonlinearities’ evidence of at least some of its components. To illustrate this problem we can see in table 4.6 the BDS test applied to five different series34 and to their average (ft =

at +bt +ct +dt +et ). 5

34

Since the sample has 2000

The series were generated according to the formula: xt = 4xt−1 (1 − xt−1 ). The initial values for series at , bt , ct , dt , and et were, respectively, 0.1, 0.2, 0.3, 0.4, and 0.49. 3000 observations were

104

m ε σ

at bt ct dt et ft

2 = 0.5 ε = σ 709 287 691 286 733 285 692 287 690 288 0.07 -1.08

ε σ

3 = 0.5 ε = σ 939 273 915 265 969 270 925 269 912 271 1.02 -1.36

ε σ

4 = 0.5 ε = σ 1239 262 1191 252 1265 259 1218 260 1190 259 2.18 -1.25

ε σ

5 = 0.5 ε = σ 1690 265 1613 244 1711 252 1657 254 1598 252 1.57 -1.18

Table 5.6: BDS Test to Deterministic Chaotic Time Series observations we can use the standard normal distribution to find the critical values. The results speak for themselves. While for any of the series obtained from a logistic chaotic equation there is overwhelming evidence of nonlinearities, for the average of five chaotic series that evidence has almost completely disappeared: it is impossible to reject the null hypothesis at a 5% significance level except for (ε, m) = (0.5σ, 4).

5.4

Conclusion

It is known that some important results were achieved in Physics and other Natural Sciences with the help of chaos theory. It is also known that economic theory has imported much of its methodology and techniques from Physics, e.g. see Louçã (1997), so the use of chaos theory in Economics should not surprise us. We believe that the examples given in the section 5.2 show that chaos theory in Economics is not a fad. We have also shown that the traditional econometric techniques are not able to deal with the possibility of chaos. Using the traditional econometric approach we will tend to accept that the source of the erratic movements is exogenous and that the system is dynamically stable, even though the model is known to be inherently stable. Another problem is that specific econometric techniques, designed to deal with the possibility of chaos, are not as powerful as one might wish. We saw that aggregation can hide evidence of nonlinearities, a problem that can arise in many macroeconomic time-series. Another flaw is that stochastic processes are practically generated, being the first 1000 thrown away.

105

indistinguishable from high dimensional chaos, unless we have real large economic time series.

106

Chapter 6 Some Conclusions The main objective of this work was to criticize the traditional decomposition of growth and cycles. We followed two different, inter-related, approaches to question the decomposition. First, we showed that the study of growth without taking in consideration cycles is incompatible with the new endogenous growth theory. If we accept the idea of endogenous growth we cannot abstract from the long run effects of short run “temporary” effects. We have to agree with Goodwin and Kaldor who defended that “the most characteristic feature of [cyclical fluctuations in the real world] has been that successive booms carried production to successively higher levels”1 . To materialize this idea we used two models of endogenous growth. With the help of Stadler’s model we saw how cycles can have important effects on growth. And we also saw how “the trend of growth may itself generate forces making for oscillation”2 when we studied the effects of technological jumps in General Purpose Technologies. The other approach to criticize the decomposition of growth and cycles is to question the Frischian hypothesis of linear relations. This hypothesis is essential to sustain the decomposition process. If we accept the economy as being ruled by a complex system of nonlinear relations the decomposition will be irrelevant and purely descriptive. This critique opens the way to the study of growth and cycles with the help of nonlinear tools, e.g. chaos theory. 1 2

In Kaldor (1954), p. 222. In Harrod (1939), p. 15.

107

We studied two different approaches to macrodynamics. In the first one, a linear stable economic system ruffled with exogenous shocks is admitted. In other one, a nonlinear system capable of generating deterministic oscillations is considered. Which one is correct? The correctness of the traditional decomposition between trend and cycles depends heavily on the answer to the above question. An additive (Frischian) decomposition between trend and cycles – and between cycles of various magnitudes – only makes sense in a linear system. But a linear system is just a particular case of a nonlinear system. Why do macroeconomists work so much with linear mathematical models (or, under suitable transformations, linearized versions)? Slutzky (1937) showed that even a simple linear non-oscillatory system, subject to exogenous shocks, could produce sequences similar to some macroeconomic time series. But Louçã (1997) has also shown that a low dimension chaotic system can equally generate a sequence that simulates the randomness of the residuals, the moving average of which exhibits the same effect. Does it make any sense to consider the economic system as being linear? If it is linear, a small deviation from the equilibrium will be essentially the same as a big deviation; only scale changes. In economic life this is not so: a large upward fluctuation must face some limitation (e.g. scarcity of labour, land, etc.), or a large downward fluctuation must have a floor (e.g. it does not make sense to consider negative quantities or negative prices)! If we accept that the economic system is nonlinear, to what extent can we admit linear representations of it? If the system has a locally stable equilibrium, then a linear approximation may be correct. But, when the system is locally unstable, a linear representation of it is not appropriated, since it would lead to the explosion of the system. Thus, when modelling the economic system with linear equations, the local stability of the system must be investigated. But that investigation must be made a priori and not a posteriori. Why? Because the traditional linear econometric approach is not adequate to deal with locally unstable nonlinear systems. With the help of an artificial nonlinear chaotic model we saw in the previous chapter how the properties of the estimated linear model were qualitatively wrong: the estimated model was stable. 108

A question arises: who should bear the burden of proof? Interestingly the economic science is not in syntony with other mathematical sciences. In Physics and Chemistry (from which much of our methodology has been imported) and in Biology, complex systems are the rule and not the exception. If we accept that economic life is nonlinear and that the oscillators are not independent but rather they interact between them, then chaos may come out. More precisely, in a continuous system we need at least three oscillators linked in order to produce chaos (e.g. Rossler band in Goodwin’s model). And the higher the number of oscillators, the higher the probability of chaos3 . It does not seem unrealistic to believe this is the case of the economic system with heterogenous agents, imperfect markets, monopoly power, commercial and political relations between countries with different economic and political systems, etc, etc. It is known that financial time series show evidence of chaos. Tests made to macroeconomic time series also show some, but not overwhelming, evidence of chaos. More concretely: although there is a large consensus that the data generating processes are characterized by a pattern of nonlinear dependence, there is no consensus about the presence of chaos in macroeconomic time series. We have already sketched some of the reasons why this may happen. First of all macroeconomic time series are rather short. Second, they probably include a substantial dose of noise (this should be particularly true for aggregate time series). Third, we have already shown how aggregation can hide evidence of chaotic processes. Thus an objective for future research will be to move towards new data sets at the most disaggregated level as possible. A different but also relevant problem is that, until now, tests of chaos in economic series do not test the source of chaos. This way, if chaos evidence is found we dot not know if it is the economic system which generates this chaotic behaviour or if it is the economic system that is subject to chaotic shocks from the outside. For example, if evidence of chaos is found in the prices of agriculture products this may 3

Dechert et al. (1999) report the result of a Monte Carlo study on the probability of chaos in large dynamical systems (they used neural networks as proxies for the equations that describe the dynamics of the system). Their results were quite impressive: “as the dimension of the system and the complexity of the network increase, the probability of chaotic dynamics increases to 100%”.

109

be due to the chaotic weather system that surrounds the planet. To determine if the economic system is the source of chaos, a model of the economy must be constructed and tested. This is another relevant question for future research. The consideration of chaotic models does not exclude exogenous shocks. Any manageable model must exclude some relevant variables. Thus, when compared to real data, a representation of the economic system has to include exogenous stochastic shocks. But, as we have been defending, exogenous shocks may not be the only source of the erratic behaviour of real systems. But only when a nonlinear approach is made, we understand that equilibrium states, or even periodic orbits, are nothing more than special cases of a lot more general dynamical solutions. Obviously chaos theory is not the answer to all questions in Economics. For example ordinary chaos theory usually works with recurrent time evolutions, i.e., the system returns to states near the ones already observed. But economic history does not repeat itself. We must not forget that cycles occur in a more general framework of economic growth. Cycles are ot only influenced by growth, but they have important effects on growth as well. Besides that, economic cycles have an historical interpretation. Each cycle is different from all the others and not a mere repetition of the same dynamic process. Louçã (1999) considers that purely chaotic models are inadequate to study social systems. In his opinion a wider approach to complexity is needed, since an extremely simple universe with deterministic trajectories is assumed in chaos theory. In the same paper Louçã suggests another route to explore. He considers that wide systems of dynamic and stochastic interaction, which characterize ecological models, can be applied to Economics. “Some of the most important outcomes of this interaction may well be the advent of a new generation of economic models. The ecological models take into consideration both the existing and the emerging properties in this type of interaction; they emphasize the coordenation in disequilibrium of interdependent elements and the various dynamics of conflict and cooperation – the institutions – that cannot be accounted for in the current economic models”4 .

4

In Louçã (1999), p. 17.

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