Market distortions and local indeterminacy: a general approach∗ Teresa Lloyd-Braga1, Leonor Modesto2 and Thomas Seegmuller3 1 2 3

Católica Lisbon School of Business and Economics

Católica Lisbon School of Business and Economics and IZA

Aix-Marseille University (Aix-Marseille School of Economics), CNRS-GREQAM and EHESS

August 7, 2013

Abstract We provide a methodology to study the role of market distortions on the emergence of indeterminacy and bifurcations. It consists in introducing general specifications for the elasticities of the crucial functions defining the aggregate equilibrium dynamics of the model. This allows us to study how market distortions influence the range of values for the elasticity of input substitution under which local indeterminacy and bifurcations occur, highlighting the main channels and classes of distortions responsible for indeterminacy. Most of the specific market imperfections considered in the related literature are particular cases of our framework. Comparing them we obtain several equivalence results in terms of local dynamic properties. Applying this methodology to the Woodford (1986) framework we find that distortions in the capital market, per se, do not play a major role. We further show that, for empirically plausible values of elasticity of substitution between inputs, indeterminacy requires a minimal degree of distortions. This degree seems to be high under output market distortions, while with labor market distortions the required degree is empirically plausible.

JEL classification: C62, E32. Keywords: Indeterminacy, endogenous fluctuations, market distortions, externalities, imperfect competition, taxation. ∗

Financial support from FCT under the PTDC/EGE-ECO/103468/2008 is gratefully acknowledged. We thank J.M. Grandmont for his supporting and enlightening comments and R. Dos Santos Ferreira for useful remarks.

1

1

Introduction

Several papers have studied the effects of specific market distortions (linked to externalities, imperfectly competitive markets, or government intervention) on local dynamics.1 However, a systematic analysis within a general unified framework, able to compare the importance of different distortions for the emergence of indeterminacy and bifurcations, is still missing. In order to fill this gap, we develop a methodology to study and fully characterize the role of market distortions on the occurrence of local indeterminacy and bifurcations, introducing a general framework that accounts for market distortions without determining a priori their specific source. Market distortions play a role on the local stability properties of the steady state because they modify the elasticities of the crucial functions characterizing the general equilibrium dynamic equations of the model. Our approach consists in generalizing these elasticities, introducing new parameters that represent distortions. We then study how our distortion parameters influence the values of the elasticities of input substitution in production and of labor supply under which indeterminacy and bifurcations occur. Most of the usual specific market distortions, and different combinations among them, can be recovered as particular specifications of our distortion parameters and, thereby, our results can be used to have an immediate idea of their potential role on the emergence of indeterminacy. We also show that several different specific market distortions have equivalent representations in terms of our distortion parameters, and therefore influence local stability properties in the same way, sharing the same indeterminacy mechanisms.2 Even if our approach can be applied to any dynamic general equilibrium model, we consider here a dynamic framework based on the Woodford (1986) perfectly competitive one sector model of a segmented asset market economy with heterogeneous agents (capitalists and workers).3 In accordance with empirical evidence, we assume that inputs are not weak substitutes, a case where indeterminacy and bifurcations would not occur in the absence of distortions, as shown in Grandmont et al. (1998). In contrast, with distortions, local indeterminacy and bifurcations (Hopf, transcritical and/or flip) may occur in 1

See the references in Section 2.3. As shown in Grandmont et al. (1998) we may construct stochastic (sunspot) equilibria, i.e. expectation-driven fluctuations, along indeterminacy and/or flip/Hopf bifurcations. However stochastic (sunspot) endogenous local dynamics obtained for equivalent models should only be expected to be the same if we consider the linearized version of the respective models with additive sunspot shocks that do not influence the distortion parameters. 3 This is a suitable framework for our purpose, as several papers have introduced specific market distortions in this model, providing examples for our approach. See Section 2.3. 2

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the presence of sufficiently high capital-labor substitution. Our approach allows us to highlight the main channels through which local indeterminacy and bifurcations emerge. One of the main results is that distortions affecting the real interest rate do not play a major role on local dynamics. In contrast, indeterminacy easily occurs when distortions modify the real wage and/or consumption and labor supply decisions. However, when these distortions are arbitrarily small, indeterminacy requires arbitrarily large elasticities of input substitution (and of labor supply). Hence, indeterminacy can only prevail for values of the elasticity of capital-labor substitution around one (those considered empirically plausible) under a minimal degree of distortions. To illustrate these results we consider examples of specific distortions on output, capital and labor markets, that can be represented as particular cases of our framework. Our major findings are that: (i) indeterminacy does not occur with capital market distortions (such as capital income taxation); (ii) indeterminacy requires implausible high output market distortions (such as positive productive externalities or countercyclical market power); and (iii) on the contrary, under labor market distortions (such as unemployment benefits with efficiency wages or unions) indeterminacy and bifurcations occur for empirically relevant values of the parameters. Since indeterminacy and bifurcations are linked to the emergence of endogenous fluctuations, driven by volatile self-fulfilling expectations, our results suggest that labor market imperfections are the most likely cause for this type of business cycles. Hence, our paper fully answers the research question raised in Grandmont et al. (1998) on whether "features such as increasing returns to scale, imperfect competition, and/or sluggish adjustment of wages or prices, alter the dynamics and may or may not improve the range of parameters that give rise to endogenous fluctuations". The rest of the paper is organized as follows. In Section 2, we present our general framework, applying it to some examples and showing equivalence results. We study the role of our distortion parameters on local dynamics in Section 3, and apply our results to examples with specific market distortions. In Section 4, we further illustrate the usefulness of our general approach applying it to a new example. Section 5 provides concluding remarks. Proofs and technical details are relegated to the Appendix.

2

The model

The dynamic model here considered is based on the perfectly competitive Woodford (1986)/Grandmont et al. (1998) framework. In order to ease the presentation we begin with a brief exposition of this model. 2

2.1

The perfectly competitive economy

In each period t = 1, 2, ..., ∞, output is produced using a constant returns to scale technology AF (Kt−1 , Lt ), where A > 0 is a scaling parameter, F (K, L) is a strictly increasing concave function, homogeneous of degree one in capital, K > 0, and labor, L > 0. From profit maximization, the real interest rate ρt and the real wage ω t are respectively equal to the marginal productivities of capital and labor, i.e. ρt = AFK (Kt−1 , Lt ) ≡ Aρ(Kt−1 /Lt ) and ω t = AFL (Kt−1 , Lt ) ≡ Aω(Kt−1 /Lt ). There are two types of infinitely-lived consumers, workers and capitalists. Both can save through two assets, productive capital and money, the latter being given by a fixed amount M at the economy level, constant over time. Capitalists are less impatient than workers and do not supply labor, whereas workers face a finance constraint which prevents them from borrowing against their wage earnings. Focusing on equilibria where the finance constraint is binding and capital is the asset with the greatest return, capitalists hold the entire stock of capital and only workers hold money, consuming out of their past wage income (saved in money holdings, Mt ). The behavior of the representative worker can be summarized by the maximization of w w U Ct+1 /B − V (Lt ) subject to the budget constraint Pt+1 Ct+1 = wt Lt = Mt , where Pt is the price of the final good and wt the nominal wage at period t, w Ct+1 ≥ 0 the worker’s consumption at period t + 1, B > 0 a scaling parameter, V (L) the desutility of labor and U (C w /B) the utility of consumption. The solution of this problem is given by the intertemporal trade-off between future consumption and leisure: ω t+1 Lt+1 /B = γ t

(1)

where γ t ≡ γ(Lt ) is the usual offer curve with εγ (L) ≡ γ ′ (L)L/γ(L) ≥ 1,4 w and Ct+1 = ω t+1 Lt+1 at the monetary equilibrium, because wt Lt = M in every period t. The representative capitalist maximizes the log-linear lifetime utility func∞ t c c tion t=1 β ln Ct subject to the budget constraint Ct + Kt = (1 − δ + rt /Pt )Kt−1 , where Ctc represents his consumption at period t, β ∈ (0, 1) his subjective discount factor, rt the nominal interest rate and δ ∈ (0, 1) the depreciation rate of capital. Solving the capitalist’s problem we obtain the 4

V (L) is a continuous function for L ∈ [0, L∗ ], and C r with r high enough, V ′ > 0, V ′′ ≥ 0 for L ∈ (0, L∗ ), and limL→L∗ V ′ (L) = +∞, where the worker’s time endowment L∗ may w w be infinite. Also, U Ct+1 /B is a continuous function for Ct+1 ≥ 0, and C r , with r high w enough, U ′ > 0, U ′′ ≤ 0 for Ct+1 > 0, and −xU ′′ (x)/U ′ (x) < 1. These assumptions imply that εγ (L) ≥ 1.

3

capital accumulation equation Kt = β [1 − δ + ρt ] Kt−1

(2)

A perfectly competitive intertemporal equilibrium with perfect foresight is a sequence (Kt−1 , Lt ) ∈ R2++ , t = 1, 2, ..., ∞, that, for a given K0 > 0, satisfies (1) and (2), with γ t ≡ γ(Lt ), ω t ≡ Aω(Kt−1 /Lt ) and ρt ≡ Aρ(Kt−1 /Lt ). ′

(K/L) For further reference, let us define s(K/L) ≡ (K/L)f ∈ (0, 1) as f (K/L) the (private) elasticity of output with respect to capital and σ(K/L) ≡ ′

− f (K/L)[1−s(K/L)] as the (private) elasticity of capital-labor substitution. In (K/L)f ′′ (K/L) the rest of the paper, we denote by εGX the elasticity of the function G with respect to the variable X, evaluated at the steady state. Accordingly, the elasticities of the real wage, the real interest rate and the offer curve with respect to capital and labor, evaluated at the steady state, are given by: 1−s 1−s s , ερL = , εωK = , σ σ σ s = − , εγK = 0, εγL = εγ , σ

ερK = − εωL

(3)

where εγ − 1 0 represents the inverse of the elasticity of (private or competitive) labor supply of the representative worker evaluated at the steady state and s ∈ (0, 1) and σ > 0 denote the steady state values of s(K/L) and σ(K/L).

2.2

The general framework

We now present our general framework with market distortions, explaining and motivating the main differences with respect to the perfectly competitive case. First, in many models characterized by market imperfections, the real interest rate and/or the real wage relevant to consumers’ decisions are no longer equal to the perfectly competitive marginal productivities of capital and labor. This will happen, for example, in the cases of productive externalities, imperfect competition in the product market or with consumption, labor or capital taxation. Second, with some other distortions, like consumption or government spending externalities on preferences, the relevant intertemporal choice of workers becomes a choice between future effective consumption5 (that no longer coincides with the wage bill) and leisure. Third, with leisure externalities or in the presence of labour market imperfections, such as unemployment benefits and efficiency wages or unions, the private offer curve 5

By effective consumption we mean the argument of the utility for consumption, which with consumption or public spending externalities on preferences includes them.

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derived for the perfectly competitive economy is no longer valid at the social level. Several examples are provided in Section 2.3. As market imperfections introduce a wedge between social (general equilibrium) and private (competitive) factor productivities, and/or labor supply, we propose a more general dynamic equilibrium system given by (4)-(5) in Definition 1 below. Our approach consists in replacing the functions ρ(K/L), ω(K/L) and γ(L), in the competitive dynamic system (1)-(2), by three more general functions: R(K, L), the real interest rate relevant to capitalists’ decisions, Ω(K, L) effective consumption per unit of labor and Γ(K, L) the generalized offer curve. However, the functions ρ(K/L), ω(K/L) and γ(L) are still in the background of our model as they still measure the competitive factor productivities and labor supply. Definition 1 A perfect foresight intertemporal equilibrium of our economy, which encompasses market distortions, is a sequence (Kt−1 , Lt ) ∈ R2++ , t = 1, 2, ..., ∞, that, for a given K0 > 0 satisfies: Kt = β [1 − δ + Rt ] Kt−1

(4)

(1/B)Ωt+1 Lt+1 = Γt

(5)

where Rt ≡ AR(Kt−1 , Lt ), Ωt ≡ AΩ(Kt−1 , Lt ) and Γt ≡ Γ(Kt−1 , Lt ). The functions R(K, L), Ω(K, L) and Γ(K, L) are positively valued and differentiable as many times as needed for (K, L) ∈ R2++ . As under perfect competition, the dynamics of the economy with market distortions are governed by a two dimensional system in capital and labor, where the first equation represents capital accumulation and the second one the intertemporal choice of workers. The perfectly competitive case is recovered from Definition 1 for R(K, L) = ρ(K/L), Ω(K, L) = ω(K/L) and Γ(K, L) = γ (L). The purpose of our analysis being to study local indeterminacy near a (normalized) steady state of (4)-(5),6 we must look at the eigenvalues of the following log-linearized system: Kt Lt+1

=

(1 + θεRK ) εΓK −εΩK (1+θεRK ) 1+εΩL

θεRL εΓL −θεRL εΩK 1+εΩL

Kt−1 Lt

≡ [J]

Kt−1 Lt

(6) where θ ≡ 1 − β(1 − δ), hat-variables denote deviation rates from the steady state and all elements of J are evaluated at the steady state. The trace T and the determinant D of J are given by: 6

We consider the normalized steady state (K, L) = (1, 1), whose existence is shown in Appendix 6.1.

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T =1+

εΓL + θ(εRK (1 + εΩL ) − εΩ,K εRL ) 1 + εΩL

D=

εΓL (1 + θεRK ) − θεΓK εRL 1 + εΩ,L

(7) (8)

We consider that: Definition 2 β KK 1−s β 1−s − , εRL = αKL + KL + σ σ σ σ β LK s β LL s εΩK = αLK + + , εΩL = αLL + − σ σ σ σ β ΓK β ΓL , εΓL = αΓL + + εγ , εΓK = αΓK + σ σ

εRK = αKK +

(9)

where αij ∈ R and β i,j ∈ R, for i = K, L, Γ and j = K, L, are parameters independent of εγ and σ,7 that represent a large class of specific market distortions. The local stability properties of the model, being determined by the eigenvalues of the Jacobian matrix J (or, equivalently, by its trace, T, and determinant, D, as explained below) depend on the values taken by εRj , εΩj , and εΓj . Therefore, distortions influence the local dynamics of the model by modifying these elasticities relatively to the perfectly competitive case, through the parameters αij and β ij . Indeed, the perfectly competitive case given in (3) is recovered from (9) with αij = β ij = 0 for all i and j. Hence, the term αij + β ij /σ = 0 represents market distortions, adding two new components to the different elasticities: αij , which provides a measure of market distortions when inputs are high substitutes in production (σ high), and β ij , which becomes more relevant when inputs are weak substitutes in production (σ low).

2.3

Examples and equivalence results

We now present several examples of specific market distortions that provide microeconomic foundations for the model developed above. All of them have already been studied in the literature, but not always in a finance constrained 7

Note that σ, s and 1/(εγ − 1) still denote, respectively, the competitive steady state elasticities of capital-labor substitution, of output with respect to capital and of labor supply.

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Woodford economy. For each example, we identify αij and β ij , for i = K, L, Γ and j = K, L, as functions of the parameters that represent the specific distortions considered.8 We also emphasize that there are classes of specific distortions which have equivalent representations in terms of our distortion parameters. Since market distortions influence local stability properties through the parameters αij and β ij , equivalent market distortions share the same local indeterminacy mechanisms, even if their economic interpretations are different. However, these equivalence results may not hold with sunspot shocks. Even when there are no stochastic shocks to the parameters αij , β ij ,9 introducing stochastic sunspot shocks to expectations leads to endogenous fluctuations that depend significantly on higher order derivatives (e.g. risk aversion, nonlinearities) that are purposely not considered here, so that the different equivalence results established in this paper about local indeterminacy for the local approximate log-linearized dynamics, may no longer hold for the resulting local endogenous sunspot equilibria. 2.3.1

Examples without distortions on the offer curve

In these examples, the generalized offer curve coincides with the competitive one, Γ = γ(L). However, effective consumption and the real interest rate depart from their competitive counterparts, ω(K/L) and ρ(K/L), due to wedge functions D1 and D2 that summarize the specific distortion considered, i.e. we have Ω = Aω(K/L)D1 (K, L) and R = Aρ(K/L)D2 (K, L). Computing the elasticities of Ω and R, and using (3) and (9), we have εD1 K = αLK + β LK , σ β LL β KL β KL εD1 L = αLL + σ , εD2 K = αKL + σ , εD2 L = αKL + σ . Using these relations we can easily identify the parameters αij and β ij . We start with two examples of output market distortions, where the real interest rate and the real wage are affected in the same way, i.e. D1 = D2 = D, proceeding then with two examples where D1 = D2 . Productive externalities Production is given by AF (K, L)ξ(K, L), where K (L) denote average levels of capital (labor) in the economy, taken as given by individual firms, and ξ(K, L) stands for externalities. We assume that 8

For the sake of conciseness, we present these models providing only their main economic features. For more details, the interested reader can look at the references herein. 9 Under some specific market failures with strategic interactions between agents not considered in this paper, distortions parameters may become stochastic in the presence of extrinsic uncertainty and, in this case, the results in Grandmont et al. (1998) about the existence of sunspot equilibria around indeterminacy and bifuractions may no longer be applicable.

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εξL + εξK < 1, i.e. that the degree of externalities is not too large.10 At a symmetric equilibrium with K = K and L = L, we have D(K, L) = ξ(K, L). Accordingly, we obtain: αLL = αKL = εξL , αLK = αKK = εξK , αΓi = β ji = 0, for i = K, L and j = K, L, Γ.

(10)

Output externalities can also be represented in this framework considering D(K, L) = Z F (K, L) , where F (K, L) is the average private level of output, taken as given by individual firms. Defining z ≡ εZF , we obtain αLL = αKL = (1 − s)z, αLK = αKK = sz. We conclude that output externalities are a particular case of productive externalities with εξL = z (1 − s) and εξK = zs. Imperfect competition in the product market We now emphasize that models with internal increasing returns, imperfect competition and free entry on the product market are equivalent to models with positive output externalities. Benhabib and Farmer (1994) and Cazzavillan et al. (1998) underlined that this is the case when marginal costs are decreasing, the production function being homogeneous of degree 1 + z > 1 implying a constant markup of prices over marginal costs. Here, we obtain a new result. The same type of equivalence exists when imperfect competition is associated with markup variability and a fixed cost, but constant marginal costs.11 In this case imperfect competition in the output market introduces a countercyclical markup factor m (yt ) > 1 which is decreasing in output yt , i.e. εmy < 0. This markup distorts the real wage (or effective consumption) and the real interest rate with respect to their competitive counterparts, leading to D(K, L) = 1/m(y). Defining ν ≡ −εmy > 0, αLL = αK,L = (1 − s)ν, αLK = αKK = sν and αΓi = β ji = 0, for i = K, L and j = K, L, Γ. A similar representation applies to models with taste for variety, but in this case m (yt ) corresponds to the ratio between the aggregate price and the price set by a single firm.12 Comparing with productive externalities we can state the following result.13 10

See Barinci and Chéron (2001), Benhabib and Farmer (1994), Cazzavillan et al. (1998). 11 We focus on markup variability linked to strategic interactions between producers and business formation. See Dos Santos Ferreira and Lloyd-Braga (2005), Seegmuller (2009). 12 See Jacobsen (1998) and Seegmuller (2008). 13 As long as capital and labor demands are derived from (static) profit maximization, this equivalence must also hold in several different types of macrodynamic models and not only in the Woodford framework.

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Proposition 1 Models with markup variability or taste for variety (characterized by ν > 0) have an equivalent representation, in terms of our general framework, to models with positive output externalities or to models with constant markup and increasing returns (characterized by z > 0) when ν = z. Public spending financed by variable taxation In this example, we consider public expenditures, Gt ≥ 0, financed by variable tax rates on capital or labor income, under a balanced budget rule. Accordingly, Gt = τ L (ω t Lt ) ω t Lt + τ K (ρt Kt−1 ) ρt Kt−1 . The tax rates on labor and capital incomes are determined respectively by τ L (ω t Lt ) ≡ zL (ω t Lt /ωL)φL and τ K (ρt Kt−1 ) ≡ zK (ρt Kt−1 /ρK)φK , with zi ∈ [0, 1) the stationary level of the tax rate and φi ∈ R its degree of pro(counter)-cyclicality with respect to the tax base, for i = L, K, and ωL and ρK respectively the wage bill and capital income, evaluated at the steady state.14 Assuming wasteful public expenditures and that households take tax rates as given, D1 (Kt−1 , Lt ) = 1 − zL (ω t Lt /ωL)φL and D2 (Kt−1 , Lt ) = 1 − zK (ρt Kt−1 /ρK)φK . We will address separately the role of each type of taxation on indeterminacy. In the case of capital taxation only, we have D1 (K, L) = 1 and: αji = β ji = 0 for i = K, L and j = L, Γ, (11) zK αKK = −φK , αKL = 0, β KK = −αKK (1 − s) = −β KL 1 − zK In the case of labor income taxation, we have D2 (K, L) = 1 and obtain: αij = β ij = 0 for i = K, Γ and j = K, L, zL αLL = −φL , αLK = 0, β LL = −αLL s = −β LK 1 − zL

(12)

w

w Consumption externalities Workers’ utility is U(Ct+1 ϕ(C t+1 )/B)−V (Lt ), w w where ϕ(C t+1 ) is the externality function and C t+1 denotes workers’ average consumption, taken as given by individual workers.15 At a symmetric equilibrium we have C¯ w = C w and D1 (K, L) = ϕ(C w ). Defining χ ≡ εϕC¯ w we have αLL = χ, αLK = 0, β LL = −αLL s = −β LK . Comparing with the example on labor taxation, we have the following result. 14

This example follows closely Lloyd-Braga et al. (2008), and covers as particular cases the fiscal rules in Dromel and Pintus (2008), Guo and Lansing (1998), Gokan (2005), Schmitt-Grohé and Uribe (1997), among others. 15 Consumption externalities have been considered, for instance, in Alonso-Carrera et al. (2008), Gali (1994), Ljungqvist and Uhlig (2000).

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Proposition 2 Models with consumption externalities in preferences where individual workers compare their own consumption to that of the average worker (characterized by χ) and models with labor income taxation (characterized by (φL , zL )), have equivalent representations in terms of our general zL 16 framework when χ = −φL 1−z . L 2.3.2

Examples with distortions on the generalized offer curve

In these examples the generalized offer curve departs from its competitive counterpart γ(L) due to a wedge function D3 (K, L) ≡ Γ(K, L)/γ(L), so that, using (3) and (9), εD3 K = αΓK + β ΓK , εD3 L = αΓL + βσΓL . We start with σ two examples (leisure externalities and efficiency wages with unemployment) where distortions only modify the offer curve and proceed with an example (unemployment and unions) where distortions on effective consumption, D1 , are also relevant. Note that distortions involving unemployment (with efficiency wages or unions) consider, typically, the case of indivisible labor with a constant real reservation wage at the private level, implying an infinitely elastic private labor supply (εγ = 1). w Leisure externalities Let the utility function of a worker be Ct+1 /B − η εγ Lt Lt , where Lt denotes aggregate labor, taken as given by individual workers, η > −1 is the constant degree of leisure externalities17 and εγ ≥ 1. In the absence of leisure externalities, η = 0, we recover the perfectly competitive ε case with γ(L) = Lt γ . Since at a symmetric equilibrium Lt = Lt , we obtain D3 (K, L) = Lη and:

αΓL = η, αij = 0 for i = Γ, j = L and β ij = 0

(13)

Unemployment insurance and efficiency wages This example follows Grandmont (2008), who introduced unemployment insurance in a Woodford economy with efficiency wages and indivisible labor. The efficiency wage contract involves a level of effort, x∗ , and a level of consumption of employed workers, C ∗ , both constant over time. They only depend on h ∈ (0, 1) , the constant percentage of the wage received by unemployed workers, which is financed by a uniform tax rate on income of all workers, taken as given by individuals. The generalized offer curve Γ, being identical to aggregate consumption of employed and unemployed workers, becomes Γ = C ∗ n + 16

Note that, when individual workers compare their own consumption to average consumption of both capitalists and workers, we have an equivalence with models with consumption taxation of the type proposed in Lloyd-Braga et al. (2008). 17 See Benhabib and Farmer (2000) and Weder (2004).

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C ∗ (1 − n)h, where n = L/x∗ denotes the steady state employment rate, so ∗ ∗ (1−n)h −h that D3 (K, L) = C n+C . Since εγ = 1, we have αΓL = h+(1−h)n ∈ γ(L) (−1, 0), αij = 0 for i = Γ, j = L, and β ij = 0. Comparing this economy with the one with leisure externalities, we can state the following: Proposition 3 For εγ = 1, the model with externalities in leisure (characterized by −1 < η < 0) and the model with unemployment benefits and efficiency wages (characterized by h ∈ (0, 1), n > 0) have equivalent repre−h sentations in terms of our general framework when η = h+(1−h)n . Unions and unemployment benefits This example, developed by Dufourt et al. (2008), introduces unions and unemployment benefits in the Woodford framework, considering indivisible labor. A constant real benefit b > 0 is paid to each unemployed worker, financed by a real tax τ > 0 on each employed worker, both b and τ taken as given by individuals. Unions, under efficient bargaining, are able to set real wages above the marginal ≥ 1, productivity of labor, with a markup factor µ(K/L) = 1−αs(K/L) 1−s(K/L) where (1 − α) ∈ [0, 1) represents unions bargaining power. Hence, the real wage (or effective consumption) departs from the perfectly competitive marginal productivity of labor and becomes Ω = µ(K/L)Aω(K/L), i.e. D1 (K, L) = µ(K/L), while the real interest rate is lower than the marginal productivity of capital, R = αAρ(K/L). Aggregate consumption of employed and unemployed workers is bµ(K/L) so that Γ = bµ(K/L) and (σ−1) D3 (K, L) = bµ(K/L)/γ(L). Since εγ = 1 and as εµK = −εµL = s(1−α) , 1−αs σ we get: αΓK = αLK = −αLL = −β ΓK = β LL = −β LK = β ΓL = αΓL = αLL − 1, αKj = β Kj = 0 for j = K, L

3

s(1 − α) ∈ (0, s), 1 − αs (14)

Local stability properties

We now characterize the role of market distortions on local stability properties. We use the geometrical method developed in Grandmont et al. (1998), analyzing how T and D, given in (7)-(9), change in the space (T, D) according to the different values taken by the parameters of the model (see Figures 1 to 4). Note that T and D correspond, respectively, to the sum and product of the two eigenvalues of J, i.e. the two roots of the associated characteristic polynomial P (λ) ≡ λ2 − λT + D = 0. Hence, along the line (AB), one eigenvalue is equal to −1, i.e. P (−1) ≡ 1 + T + D = 0. Also, 11

on the line (AC) one eigenvalue is equal to 1, i.e. P (1) ≡ 1 − T + D = 0, and on the interior of the segment [BC], the two eigenvalues are complex conjugates with a unit modulus, i.e. D = 1 and |T | < 2. It can be deduced that the steady state is a sink, with both absolute eigenvalues lower than 1, when D < 1 and |T | < 1 + D, i.e., when (T, D) is inside the triangle ABC. It is a saddle-point, when |1 + D| < |T |. Otherwise, it is a source (locally unstable). Since K is a predetermined variable, the steady state is locally indeterminate when it is a sink.18 This geometrical method is also useful to study the occurrence of bifurcations. Considering that a parameter of the model is made to continuously vary in its admissible range (for instance εγ ∈ [1, +∞)),19 a Hopf bifurcation generically occurs when (T, D) crosses the segment [BC] in its interior (εγ crossing the critical value εγ H ). When (T, D) crosses the line (AC) a transcritical bifurcation generically occurs (εγ crossing the critical value εγ T ).20 When (T, D) crosses (AB) (εγ crossing the critical value εγ F ), a flip bifurcation generically occurs.21 Let us start by recalling what happens in the competitive case (αij = β ij = 0). Using (7), (8) and (9), we have D = σ−θ(1−s) εγ and T = 1 + σ−s σεγ −θ(1−s) . Take fixed values of θ and s such that 0 < θ (1 − s) < s and σ−s consider σ > s. Then, as the bifurcation parameter εγ varies from 1 to +∞, the point (T, D) describes a half line ∆ (σ) for a fixed σ, with an origin (T1 (σ), D1 (σ)) for εγ = 1 that belongs to the line (AC) and lies above point C. This half line points upwards as εγ increases, with a slope S (σ) = σ−θ(1−s) ∈ σ (0, 1) (see Figure 1). Also, as σ moves from s to +∞, the origin (T1 (σ), D1 (σ)) moves down along (AC) from infinity to the limit point (T1 (+∞), D1 (+∞)) = C, describing in this way another half line ∆1 that coincides with the part of (AC) above C, with a slope S1 = 1. At the same time, the slope of ∆(σ), S (σ), goes up and tends to 1 when σ increases to +∞. Therefore, whenever σ > s, the whole half line lies in the saddle point region below (AC) and above (BC), with D > 1. Local indeterminacy is impossible, a result already established in Grandmont et al. (1998). With distortions, because they may alter the half lines ∆(σ) and ∆1 , this result may be reversed and several paths to indeterminacy emerge. When εγ varies, one possibility is that ∆ (σ) crosses the interior of the triangle ABC 18

We only deal with the role of market distortions on local indeterminacy linked to the sink property, i.e. we do not address the cases of static or global dynamic indeterminacy and bifurcations, which may also appear in the presence of some market imperfections. 19 In examples with unemployment, where εγ = 1, we use σ as the bifurcation parameter. 20 The case of a saddle node bifurcation is ruled out, since we apply our analysis to (K, L) = (1, 1) whose existence is persistent, under the usual scaling procedure. For the sake of simplicity we also disregard pitchfork bifurcations. See Cazzavillan et al. (1998). 21 The expressions of εγ T , εγ F and εγ H are given in Appendix 6.4.4.

12

for σ large. For this to be possible, the limit point (T1 (+∞), D1 (+∞)), if it still belongs to (AC), needs to be in the interior of the segment [AC], and the limit slope S (σ) of ∆ (σ) when σ goes to infinity must exceed 1. As we shall see, this becomes possible when αKK > 0. Another possibility (which is the only one open for models with unemployment where εγ is fixed at one) is to move the half line ∆1 from its position in the competitive case (the part of the line (AC) above C) so that it crosses the interior of the triangle ABC, where (T1 (σ), D1 (σ)) is above (AC). This involves again that the limit point ((T1 (+∞), D1 (+∞)) belongs to the interior of the segment [AC] and appropriate conditions on the slope S1 of ∆1 that, as shown below, can be ranked in terms of the indexing parameter αΓL .

3.1

The role of market distortions on local dynamics

As a preliminary result, by direct inspection of (7)-(8), we may immediately deduce that distortions affecting the R function do not play a major role. Indeed, εRi appears in T and D always multiplied by θ, a parameter that takes rather small values when the period of time considered is short, β being close to 1 and δ close to 0. However, distortions affecting Γ and/or Ω can significantly influence the dynamic behavior of our system. To see this, take as a first approximation θ arbitrarily close to zero, so that from (8) D ≈ εΓL /(1 + εΩL ). Using (9), εΓL may become lower than 1 (αΓL < 0 under σ large) and 1 + εΩL may take values greater than 1 (αLL > 0 under σ large), so that the required condition for indeterminacy D < 1 can easily be obtained. In fact, as we shall see and in contrast to the perfectly competitive case,22 indeterminacy is then possible even with arbitrarily small distortions affecting the Γ and/or the Ω functions. We now study in more detail how market distortions influence the occurrence of indeterminacy, focusing on empirically relevant parameterizations. We consider that σ is large, i.e., that inputs are not weak substitutes. This is a plausible assumption. Empirical studies show that the wage bill is increasing in labor, which, in the absence of market distortions, means that σ > s and that consumption is increasing in labor.23 We extend this assumption to our economy with distortions, assuming that effective consumption (ΩL) is increasing in labor, i.e. 1 + ǫΩL > 0. From s−β LL (9), this is equivalent to have αLL > −1 and σ > 1+α . Also, since LL calibrated values for the capital share of output (which, in the absence of distortions, is represented by s) are usually lower than 1/2, and since 22

Under perfect competition (see (3)) this is not possible for σ large since εΓL = εγ > 1 and 0 < 1 + εΩL = 1 − s/σ < 1 for σ > s > θ(1 − s). 23 See Hamermesh (1996).

13

values for θ are rather small when the period is short, we assume that 1/2 > s > θ(1 − s), as typically done in Woodford economies. We further extend the latter assumption to our economy with distortions assuming that θ [(1 + αLL ) (1 − s − β KK ) − (s − β LL ) αKK ] < s − β LL and β i,j < s. Given the previous assumptions, both are satisfied in the absence of distortions, or when they are small enough. Moreover, these inequalities imply that s−β LL KK ) αKK > − 1θ and σ > 1+α > θ(1−s−β , i.e. 1 + θǫRK > 0 (see (9)). Hence, 1+θαKK LL capital income is increasing with capital, as suggested by empirical works. All these assumptions are summarized in Assumption 1 and we consider them satisfied in the rest of the paper. Assumption 1 1. Small distortions, short period and capital share of output small a. β i,j < s, with i = K, L, Γ, j = K, L b. 0 < θ(1 − s) < s < 1/2

c. θ [(1 + αLL ) (1 − s − β KK ) − (s − β LL ) αKK ] < s − β LL

2. αLL > −1, so that effective consumption is increasing in L, i.e. 1 + s−β LL εΩL > 0 for σ > 1+α . LL Our main interest is to analyze how distortions, measured by αij and β ij , influence the values of εγ and σ under which indeterminacy occurs, for fixed values of s and θ. Hence, using the geometrical method, we examine how they affect the half lines ∆(σ) and ∆1 . The half line ∆(σ) From (7)-(9), as εγ varies in [1, +∞), the point (T, D) still describes a half line ∆ (σ) in the plane (T, D), starting at (T1 (σ), D1 (σ)) γ RK for εγ = 1. Its slope S (σ) ≡ ∂D/∂ε with ∂D/∂εγ = 1+θǫ and ∂T /∂εγ = ∂T/∂εγ 1+ǫΩL 1 , is given by: 1+ǫΩL S (σ) = 1 + θαKK − θ

1 − s − β KK . σ

(15)

Under Assumption 1, both 1+ εΩL and 1+ θεRK are positive, so that, as s−β LL under perfect competition, T and D increase with εγ . Hence, for σ > 1+α , LL the half line ∆ (σ) points upwards as εγ increases. See Figures 2-4. The slope S (σ) > 0 increases towards 1 + θαKK when σ goes up to +∞. Therefore, for αKK positive, and in contrast to the perfectly competitive case, the slope of the half line ∆ (σ) can now become higher than 1. To analyze how the 14

half line ∆(σ) moves in the space (T, D) as σ runs its admissible interval s−β LL , +∞ , we must also study how its origin (T1 (σ), D1 (σ)) changes with 1+αLL σ.24 The half line ∆1 Since our general approach involves 16 distortion parameters, we will impose simplifying Assumptions on T1 (σ) and D1 (σ), which are verified for most distortions considered in the literature and in all the examples presented in Section 2.3. In what follows, and as under perfect competition, we assume that: Assumption 2 1. The locus of points (T1 (σ), D1 (σ)) describes a half line ∆1 as σ moves s−β LL from 1+α > 0 to +∞. LL 2. The limit point (T1 (+∞), D1 (+∞)) of the half line ∆1 belongs to the (1+αΓL )−αΓK αKL ] . line (AC), i.e. D1 (+∞) = T1 (+∞)−1, where D1 (+∞) = 1+αΓL +θ[αKK1+α LL 3. D1 (σ) is decreasing in σ >

s−β LL . 1+αLL

Conditions on parameters implied by these assumptions are given in Appendix 6.3. Assumption 2.1 means that the slope of the half line ∆1 , 1 /∂σ S1 ≡ ∂D whose expression is given in Appendix 6.4.2, does not de∂T1 /∂σ pend on σ, while Assumption 2.3 means that αΓL > α0ΓL , with S1 = 0 for αΓL = α0ΓL (see (19) and (23)). As under perfect competition, where the point (T1 (+∞), D1 (+∞)) coincides with point C, the origin of the half line ∆1 is still on the line (AC). See Assumption 2.2. However, in contrast to the case of perfect competition, we assume that: Assumption 3 (T1 (+∞), D1 (+∞)) is in the interior of the segment [AC]: ΓK αKL −αKK ) 1. D1 (+∞) > −1, i.e. αΓL > αLΓL ≡ − 2+αLL −θ(α 1+θαKK

2. D1 (+∞) < 1, i.e. αΓL < αH ΓL ≡

αLL +θ(αΓK αKL −αKK ) 1+θαKK

The relevance of Assumption 3.2 for the emergence of indeterminacy was already briefly discussed above. In fact, given Assumptions 1 and 2, D1 (+∞) < 1 ends up being a necessary condition for indeterminacy. Because ∆ (σ) points upwards from (T1 (σ), D1 (σ)), we have D > D1 (σ). Also as D1 (σ) is decreasing in σ, we have D1 (σ) > D1 (+∞). Hence, D > D1 (σ) > 24

Expressions for T1 (σ) and D1 (σ) are given in Appendix 6.2.

15

D1 (+∞) and the half line ∆ (σ) can go inside ABC, where D < 1 and indeterminacy emerges, if and only if D1 (+∞) < 1. Finally, Assumption 3.1 is introduced to facilitate the analysis, being satisfied for not too large distortions and in all the examples. To locate the half line ∆1 with respect to the triangle ABC, we note that, since from Assumption 3 the point (T1 (+∞), D1 (+∞)) belongs to the interior of the segment [AC] and, since from Assumption 2.3, D1 (σ) is decreasing, one needs simply to assume that (T1 (σ), D1 (σ)) lies above the line (AC) in order to ensure local indeterminacy for σ significant but finite: that condition reads directly D1 (σ) > T1 (σ) − 1. Note that the slope S1 of ∆1 can take any value different from 1 (see (22) in Appendix 6.4.2), but the above statements mean in terms of that slope, that S1 > 1 or S1 0. This is summarized in the following assumption: Assumption 4 The half line ∆1 lies above the line (AC), i.e. D1 (σ) > T1 (σ) − 1, or equivalently: S1 0 or S1 > 1. Assumption 4 means that αΓL < α1ΓL , with S1 = 1 for αΓL = α1ΓL , i.e. the half-line ∆1 is over the line (AC) when αΓL = α1ΓL (see (18) and (24)). To analyze the variation of the slope S1 of ∆1 , we choose αΓL as the central indexing parameter, taking the other parameters as given. Note that this slope is a decreasing function of αΓL , so that the values of S1 can be ranked in terms of this parameter.25 When αΓL decreases from α1ΓL to α0ΓL (recall that S1 = 0 when αΓL = α0ΓL ), the half-line ∆1 makes a counterclockwise rotation around (T1 (+∞), D1 (+∞)).26 As shown in Appendix 6.5, as αΓL decreases from α1ΓL to the critical value α∞ ΓL , S1 increases from 1 to +∞, the half line ∆1 becoming vertical for αΓL = α∞ ΓL . Then S1 increases from −∞ to −1 −1 −1 when αΓL goes down from α∞ to α ΓL ΓL , with S1 = −1 for αΓL = αΓL . At that point, ∆1 becomes parallel to (AB). When αΓL decreases from α−1 ΓL , the slope S1 of the half line ∆1 becomes larger than −1 and when αΓL takes the value αSΓLB the half-line ∆1 goes through point B. Finally, as αΓL decreases B from αSΓL to α0ΓL , the half line ∆1 goes below point B, becoming flat when αΓL tends to α0ΓL . This allows us to define three relevant configurations for S1 . Lemma Let Assumptions 1-4 be verified. Then, we have: 25

The expressions for all the critical values of αΓL not provided in the main text are given in Appendix 6.4.1. 26 Under Assumption 1, D1 (+∞) is increasing in αΓL . Hence, following a decrease of αΓL , for given values of the other distortion parameters, (T1 (+∞), D1 (+∞)) moves down along (AC) but stays between A and C (see Assumptions 2.2 and 3).

16

• Configuration (i): |S1 | > 1, i.e. αΓL > α−1 ΓL ; • Configuration (ii): S1 ∈ (−1, SB ), where SB ∈ (−1, 0) is the slope S1 when ∆1 goes through point B, i.e. αΓL ∈ αSΓLB , α−1 ΓL ; B • Configuration (iii): S1 ∈ (SB , 0), i.e. αΓL < αSΓL .

Figures 2-4 illustrate these three different configurations for the half line ∆1 . In each configuration, the half line ∆ is also represented in the (T, D) plane for some critical values of σ. Note that in Figures 2a-4a, since αKK 0, the slope S of this half line ∆ is always smaller than 1. In contrast, in Figures 2b-4b where αKK > 0, the slope S, given in (15), becomes larger than 1 for σ large enough (i.e. for σ > σ T , with σ T given in (28)). Configuration (i) is the relevant one when distortions are arbitrarily small. As seen in beginning of Section 3, S1 = 1 in the perfectly competitive case where distortions are absent. Hence, introducing arbitrarily small distortions, we should, by continuity, obtain a value for S1 arbitrarily close to 1. As shown in the above Lemma, under our Assumptions, this is only possible in Configuration (i). Since θ is small, this is also the relevant configuration in the absence of distortions on the Γ function.27 When αΓL becomes more and more negative, Configurations (ii) and (iii) may also occur, which already suggests the crucial role played by labor market distortions. Before exploring the results on local dynamics, let us remark that we have proceeded in the discussion above as if, for given values of the other distortion parameters, αΓL could be moved independently, even though its value may be restricted by Assumption 2.2 (see Appendix 6.3.2). Note however that in all the cases considered so far in the literature, and used as illustrative examples in Section 3.3, Assumption 2.2 is always satisfied imposing no restrictions on αΓL . Indeed, the examples presented in Section 2.3 are either characterized by αKK = αKL = 0 or by αΓL = αΓK = 0 and αLL αKK = αKL αLK , so that Assumption 2.2 is naturally satisfied.28 However, even when Assumption 2.2 does indeed restrict αΓL , our general approach can still be used. If we get a non empty set of values for the distortion parameters satisfying all our Assumptions (including Assumption 2.2), we can always compute α−1 ΓL and αSΓLB and, using the above Lemma, check what is the relevant configuration in that situation and proceed with the corresponding analysis of the local dynamics as explained below. 27

In fact, in this case, we can easily define an upper bound for θ ensuring α−1 ΓL < 0, so that 0 = αΓL > α−1 (see Lemma and (21)). ΓL 28 In this last case without distortions on the Γ function (αΓL = αΓK = 0), Configuration (i) will be obtained, as seen above.

17

3.2

Results on local dynamics

When εγ = 1 the dynamics of the model are fully characterized by the half line ∆1 , along which σ continuously varies in its admissible range (i.e. σ becomes the bifurcation parameter). This half line, starting for σ = +∞ in the interior of the segment [AC], points upwards as σ decreases and crosses s−β LL , +∞ , such that D1 (σ H1 ) = always the line (BC) for a value σ H1 ∈ 1+α LL

s−β LL 1.29 Define also σ F as the value of σ ∈ 1+α , +∞ for which the half line LL ∆1 crosses the line (AB), i.e. 1 + D1 (σ F ) + T1 (σ F ) = 0. Note that the half line ∆1 never crosses the line (AC) and (T1 (σ), D1 (σ)) leaves the triangle ABC for σ < max {σ H1 , σ F }. From Figures 2-4, it is easy to see that in Configuration (i) the steady state is a source for σ < σ H1 , undergoes a Hopf bifurcation for σ = σ H1 and becomes a sink for σ > σ H1 . In Configuration (ii) the steady state is a saddle for σ < σ F , undergoes a flip bifurcation for σ = σ F , becomes a source for σ F < σ < σ H1 , undergoes a Hopf bifurcation for σ = σ H1 becoming a sink for σ > σ H1 . In Configuration (iii) the steady state is a saddle for σ < σ F , undergoes a flip bifurcation for σ = σ F and becomes a sink for σ > σ F . However, for εγ > 1 we have also to consider the position of the half line ∆(σ), that starts for εγ = 1 on ∆1 and points upwards as εγ increases. This half line also does not cross the line (AC) if its slope is higher than 1. From (15), this can only occur if αKK > 0 and σ > σ T , where σ T satisfies S(σ T ) = 1. As an exposition device we assume that:

Assumption 5 If εγ > 1 and αKK > 0, a half line ∆(σ) with a slope S(σ) ≥ 1 (σ ≥ σ T ) always starts inside the triangle ABC, i.e., σ T > Max {σ H1 , σ F }. Consequently, for σ < σ H1 , and also for σ < σ F , the slope of ∆(σ), S(σ), is lower than 1, since S(σ) is increasing in σ. Using simple geometrical arguments, we are now able to discuss the occurrence of indeterminacy and bifurcations in each configuration. In Configuration (i) (see Figure 2) the half line ∆1 with a slope |S1 | > 1 cannot cross (AB), but crosses for σ H1 the segment [BC] in its interior. For σ < σ H1 , the half line ∆(σ) starts above [BC] in the source region, while for σ > σ H1 , its origin is inside the triangle (ABC) in the sink region. Consider first αKK 0, so that the half line ∆(σ) has always a slope S (σ) ∈ (0, 1), crossing the line (AC) for εγ T . Hence, for σ < σ H1 the steady state is a source when εγ < εγ T , undergoes a transcritical at εγ = εγ T and becomes a saddle for εγ > εγ T . For σ > σ H1 we have to consider σ H2 ∈ (σ H1 , +∞) 29

Expressions for all critical values of σ are given in Appendix 6.4.3.

18

defined as the value of σ such that the half line ∆(σ) goes through point C. In Appendix 6.6 we prove uniqueness of σ H2 when αKK 0, and that the half line ∆(σ) crosses the line (AC) above C if σ H1 < σ < σ H2 , while the reverse happens if σ > σ H2 .30 Therefore, when σ H1 < σ < σ H2 , the steady state is a sink (indeterminate) for ε < εγ H , undergoes a Hopf bifurcation at ε = εγ H , is a source for εγ H < ε < εγ T , undergoes a transcritical bifurcation at ε = εγ T , becoming a saddle for ε > εγ T . When σ > σ H2 the half line ∆(σ) only crosses (AC) below C, so that the steady state is a sink when εγ < εγ T , undergoes a transcritical at εγ = εγ T and becomes a saddle for εγ > εγ T . Consider now αKK > 0, so that ∆(σ) becomes steeper than (AC) for σ > σ T . Hence, for σ < σ H1 , results are identical to those obtained for αKK ≤ 0. For σ H1 < σ < σ T , in order to simplify the exposition, we assume that: Assumption 6 If εγ > 1 and αKK > 0, the half line ∆(σ) always crosses the line (AC) above point C for any σ ∈ (σ H1 , σ T ), i.e. εγ H < εγ T .31 Therefore, for σ H1 < σ < σ T , the steady state is a sink when εγ < εγ H , undergoing a Hopf bifurcation at εγ = εγ H , becomes a source when εγ ∈ εγ H , εγ T , undergoes a transcritical bifurcation at εγ = εγ T and becomes a saddle when εγ > εγ T . In contrast, for σ ≥ σ T , the half line ∆(σ) only crosses [BC]. The steady state is a sink when εγ < εγ H , undergoes a Hopf bifurcation at εγ = εγ H and becomes a source when εγ > εγ H . Accordingly, indeterminacy emerges for σ > σ H1 and εγ sufficiently close to 1. All these results are summarized in Table 1. In Configuration (ii), the half line ∆1 points upwards to the left, crossing first [BC] and then (AB) above point B (see Figure 3). In the case where αKK 0, S (σ) ∈ (0, 1). Therefore, for σ < σ F , the half line ∆(σ) crosses first (AB) and then (AC). For σ F < σ < σ H1 , ∆1 is above (AB) and [BC], and ∆(σ) only crosses (AC). For σ > σ H1 , ∆1 enters the triangle (ABC) and we obtain the same results of the previous configuration. When αKK > 0, for σ < σ H1 , S (σ) is still smaller than 1 (Assumption 5) and the results obtained for αKK 0 apply. When σ > σ H1 , everything is as in the previous configuration. Again we will have indeterminacy for σ > σ H1 and εγ sufficiently close to one. In Configuration (iii), the half line ∆1 , that points upwards to the left, crosses the line (AB) below point B (see Figure 4). Let us define σ H3 ∈ 30

For αKK < 0, σ H2 exists and is unique, while for αKK = 0 it may not exist. This 1 will happen if and only if αH ΓL ≤ αΓL , as shown in Appendix 6.6. In this case the half line ∆(σ) crosses the line (AC) above C for all σ > σH1 . 31 As shown in Appendix 6.6, this means that for αKK > 0, σH2 does not exist. A 1 sufficient condition is that αH ΓL ≤ αΓL .

19

(σ H1 , σ F ), as the value of σ for which the half line ∆(σ) goes through point B. In Appendix 6.7 we prove existence and uniqueness of σ H3 and that, for σ H1 < σ < σ H3 , the half line ∆(σ) crosses (AB) above B, while for σ H3 < σ < σ F , it crosses first (AB) below B. Consider the case where αKK 0 and S (σ) ∈ (0, 1). In this case it is relevant to know whether σ H2 > σ F or not. To simplify the exposition we assume that: Assumption 7 In Configuration (iii), if εγ > 1, αKK ≤ 0 and σ = σ F , then ∆(σ) crosses (AC) above C, i.e., σ H2 > σ F .32 Therefore, for σ < σ H3 , the half line ∆(σ) starts on the left-hand side of (AB), crossing (AB) above B and (AC) above C. For σ H3 < σ < σ F , ∆(σ) crosses (AB) below B, the segment [BC], and (AC) above C. For σ F < σ < σ H2 , ∆(σ) crosses first [BC] and then (AC) above C. When σ > σ H2 the half line ∆(σ) only crosses (AC) below C. When αKK > 0, under Assumptions 5 and 6, for σ < σ F , results are identical to those obtained with αKK ≤ 0. For σ F ≤ σ < σ T , ∆(σ) starts inside (ABC) with a slope smaller than 1. Then, it crosses [BC] and then the line (AC) above C. For σ ≥ σ T , the slope S being greater than 1, ∆(σ) only crosses [BC]. Proposition 4 summarizes our results. Note that Assumptions 5, 6 and 7, here used as an exposition device, are verified in all the examples presented. Proposition 4 Consider our Lemma and let Assumptions 1-7 be verified. Then, the nature of the steady state, whether a saddle, a sink or a source, is indicated in Table 1.33 Also, a Hopf bifurcation (resp. a flip or transcritical bifurcation) generically occurs as εγ crosses εγ H (resp. εγ F or εγ T ). From Table 1, we see that indeterminacy and bifurcations may occur in the presence of market distortions. Indeterminacy (sink) requires values of the elasticity of substitution between capital and labor (σ) above a lower bound, which is higher or equal to σ H1 . Also, Hopf and/or transcritical bifurcations may occur in all configurations. There are however differences across configurations: (a) The lower bound on σ required for indeterminacy is equal to σ H1 in Configurations (i)−(ii), while it is identical to σ H3 (> σ H1 ) in Configuration (iii); (b) Flip bifurcations are only possible in Configurations (ii) and (iii); (c) However, indeterminacy can only occur through a flip bifurcation in Configuration (iii); (d) Finally, while an upper bound on εγ 32

Indeed, as shown in Appendix 6.6, when αKK 0, σ H2 is unique and ∆(σ) crosses (AC) above(below) C if σ < (>) σ H2 . (s−β LL ) 33 In Table 1, when αKK = 0, the lines with a * disappear if σ H2 ∈ (1+α , +∞ does LL ) not exist, the upper limit of σ in the preceding line becoming +∞. See Appendix 6.6.

20

σ Configuration (i) αΓL >α−1 ΓL

1) αK,K ≤ 0 * 2) αK,K > 0

Configuration (ii) αSΓLB <

αΓL <α−1 ΓL

1) αK,K ≤ 0

* 2) αK,K > 0

Configuration (iii) αΓL <αSΓLB

1) αK,K ≤ 0

s−β LL , σ H1 1+αLL

(σ H1 , σ H2 ) (σ H2 , ∞)

s−β LL , σ H1 1+αLL

(σ H1 , σ T ) (σ T , ∞)

2) αK,K > 0

(εγ T , ∞) (εγ T , ∞) (εγ T , ∞) (εγ T , ∞) (εγ T , ∞)

[σ F , σ H1 ] (σ H1 , σ H2 ) (σ H2 , ∞)

s−β LL , σF 1+αLL

[σ F , σ H1 ] (σ H1 , σ T ) [σ T , ∞)

s−β LL , σ H3 1+αLL

[σ F , σ H2 ) (σ H , ∞) 2

s−β LL , σ H3 1+αLL

(σ H3 , σ F ) [σ F , σ T ) [σ T , ∞)

εγ Sink -

1, εγ H 1, εγ T -

1, εγ H 1, εγ H

-

s−β LL , σF 1+αLL

(σ H3 , σ F )

*

Saddle

1, εγ F and (εγ T , ∞) (εγ T , ∞) (εγ T , ∞) (εγ T , ∞) 1, εγ F and (εγ T , ∞) (εγ T , ∞) (εγ T , ∞)

and (εγ T , ∞) 1, εγ F and (εγ T , ∞) (εγ T , ∞) (εγ T , ∞) 1, εγ F and (εγ T , ∞) 1, εγ F and (εγ T , ∞) (εγ T , ∞)

-

1, εγ H 1, εγ T

-

1, εγ T (εγ H , εγ T ) (εγ H , ∞)

1, εγ T (εγ H , εγ T ) -

-

(εγ F , εγ T )

-

1, εγ T (εγ H , εγ T ) (εγ H , ∞)

1, εγ H 1, εγ H

(εγ F , εγ T )

-

(εγ F , εγ H ) 1, εγ H 1, εγ T − (εγ F , εγ H ) 1, εγ H 1, εγ H

Table 1: Local stability properties and bifurcations with market distortions

21

1, εγ T (εγ H , εγ T )

(εγ F , εγ T )

-

-

1, εγ F

Source

(εγ H , εγ T ) (εγ H , εγ T ) − (εγ F , εγ T )

(εγ H , εγ T ) (εγ H , εγ T ) (εγ H , ∞)

is always required for indeterminacy, in Configuration (iii) a lower bound on εγ may be needed in some cases: when σ ∈ (σ H3 , σ F ), indeterminacy only emerges for εγ > εγ F > 1. We can now discuss the role of different distortions. As already explained at the end of Section 3.1, Configuration (i) is obtained when distortions are arbitrarily small. Also, as seen before, Configuration (i) is the relevant one in the absence of distortions on Γ since θ is small. 34 It follows that (non arbitrarily small) distortions on Γ play a crucial role for the occurrence of flip bifurcations, which are only possible in Configurations (ii) and (iii). Since flip bifurcations are frequently a route for chaos, distortions affecting the offer curve may be associated with complex chaotic behavior of capital and labor/employment trajectories. Since under arbitrarily small distortions we obtain Configuration (i), we can see from Table 1 that indeterminacy occurs if and only if σ exceeds σ H1 with εγ below εγ H or εγ T . However, when distortions become arbitrarily small (with αij and β ij close enough to 0), αLL − αΓL → 0 so that σ H1 → +∞ (see (27)) and, therefore, indeterminacy requires an arbitrarily large σ.35 This means that a significant difference between αLL and αΓL is needed to have indeterminacy for finite and reasonable values of σ. Hence, a minimal degree of distortions is required for indeterminacy to occur with plausible values of the elasticity of substitution between labor and capital. In Sections 3.3 and 4, we discuss this minimal degree of distortions for each example considered.

3.3

Discussing the examples: The minimal degree of distortions required for indeterminacy

In this section, we apply Figures 2-4 and the results in Proposition 4 and Table 1 to the examples presented in Section 2.3, discussing the minimal degree of distortions required for indeterminacy with plausible values of σ. We focus on the case of a Cobb Douglas technology (σ = 1). Our numerical 1 examples are obtained considering that β = (1/1.03) 4 and δ = 0.1/4, consistent with most calibrations used in the business cycle literature for quarterly data. Hence, θ = 0.03475, and we fix s = 0.35, so that θ (1 − s) < s < 0.5, as required by Assumption 1. In all the examples presented, Assumptions 2.1 and 2.2 are always satisfied and, under Assumption 1, Assumption 2.3 is also verified. It can be further checked that, under Assumptions 1 and 2.3, 34

This is indeed what we obtained in the calibrated examples without distortions on Γ that are discussed below. 35 In this case, it also requires an arbitrarily high elasticity of labor supply 1/(εγ − 1) as εγ < min{εγ H , εγ T } with εγ H → 1 and εγ T → 1 (see (29) and (31) in Appendix 6.4.4).

22

Assumption 3.1 is also satisfied in all the examples. Therefore, we mainly discuss Assumptions 1, 3.2 and 4.36 Recall that Assumption 3 is required for indeterminacy and note that Assumption 4 becomes also a necessary condition for indeterminacy when αKK ≤ 0.37 3.3.1

Examples without distortions on the offer curve

In the case of productive externalities (see (10)), Assumption 1 means εξL > −1, εξK > − 1θ , σ > 1+εs ξL and θ[(1+εξL )(1−s)−sεξK ] < s, while Assumption 3.2, required for indeterminacy, means that εξL > θεξK . In the standard case of positive externalities, εξj > 0, Assumption 4 (εξL + εξK > 0) is also s satisfied.38 Consider now that, as in Cazzavillan et al. (1998), εξL > 1−s εξK .39 Under positive externalities satisfying εξL +εξK < 1, the condition θ (3 − s) < s, verified under our calibration, is sufficient to guarantee that Configuration (i) applies, since in this case α−1 ΓL < 0 = αΓL (see Lemma and (21)). Figure 2b applies and, provided εγ is small enough, indeterminacy emerges for σ > σ H1 = (s − θ(1 − s))/(εξL − θεξK ). It follows that in the Cobb-Douglas case indeterminacy requires εξL +εξK > [s−θ(1−s)]+(1 + θ) εξK > s−θ(1−s) = 0.3274125, under our calibration. This is a high value, hardly reconcilable with empirical studies. Basu and Fernald (1997) find degrees of increasing returns between 0.03 and 0.18 for the U.S. economy, and Harrison (2003) finds no significant externalities for the U.S. industry at the 2 digits level. In the particular case of output externalities where εξL = (1 − s)z and εξK = sz, for σ = 1, indeterminacy requires z > s−θ(1−s) = 0.51 under our calibration. 1−s−θs With markup variability we apply the equivalence result of Proposition 1. Therefore, ν should be high enough to get indeterminacy with σ close to 1. In the case of Cournot competition (Dos Santos Ferreira and Lloyd-Braga (2005)), we have ν = 0.5 (m − 1), where m is the markup factor. We should have m > 2.02 under our quarterly calibration (or m > 1.85 with an annual calibration), which exceeds empirical estimates. See, for instance, Morrison (1993) where the average annual markups of U.S. industries are between 1.179 and 1.803. With capital taxation (see (11)), Assumption 3.2 means 36

Assumptions 5 and 6 are only relevant in the case of output market imperfections, when αKK > 0. Assumption 7 only applies when Configuration (iii) is possible. 37 Note that the slope S of the half line ∆ is lower than 1 when αKK ≤ 0 (see (15)). It is easy to see geometrically that, in this case, indeterminacy is not possible if Assumption 4 is not verified. 38 When externalities are negative (εξj < 0), where εξK = αKK < 0, indeterminacy is not possible since Assumption 4, which is necessary for indeterminacy when αKK < 0, is not satisfied. s 39 When εξ,j > 0 , α1ΓL > αH ΓL so that Assumption 6 is verified. When εξ,L > 1−s εξ,K > 0 we have σT > σ H1 so that Assumption 5 is satisfied.

23

φK > 0, while Assumption 4 can only be satisfied when φK < 0. Since both are required for indeterminacy, indeterminacy is not possible. We conclude that distortions on capital or output markets per se do not seem to be empirically plausible sources of local indeterminacy. In all the following examples distortions do not influence R.40 With labor zL income taxation (see (12)), Assumption 1 means σ > s and −1 < φL 1−z < 1. L Indeterminacy requires Assumptions 3.2 and 4, both only satisfied if φL < 0. Therefore, indeterminacy does not occur when tax rates are constant or vary positively with the tax base. Since, under our calibration, θ is sufficiently small and satisfies 3θ(1 − s) < 2s, which ensures that α−1 ΓL < 0 = αΓL (see (21)), we obtain Configuration (i) and Figure 2.a applies. Then indeL )[s−θ(1−s)] terminacy occurs for σ > σ H1 = −φL szL +(1−z , provided that the (−φL zL ) elasticity of labor supply is high enough. Consider, for instance, constant real government spending, i.e., φL = −1. Indeterminacy can emerge with σ ≥ 1 for zL > [s − θ(1 − s)]/[1 − θ(1 − s)] ≡ zL∗ = 0.33 under our calibration, provided zL < 0.5 so that Assumption 1 is satisfied. This range of values for zL is in accordance with average labor income tax rates for several European countries, as obtained in Mendoza et al. (1994) and in Volkerink et al (2002). Using the equivalence result of Proposition 2 between labor income taxation and consumption externalities, we see that in this last case Assumption 1 means χ < 1 and indeterminacy only occurs for χ > 0. Hence, indeterminacy is only possible when consumption externalities are of the "keeping-up with the Joneses" type (Gali (1994)), occurring for σ > σ H1 = [s(1+χ)−θ(1−s)]/χ. Under our calibration, indeterminacy with a Cobb-Douglas technology emerges for 1 > χ > [s−θ(1−s)]/(1−s) = 0.5037, which seems to be an excessive value. Indeed, Maurer and Meier (2008) found significant peer effects, but in any case lower than 0.44. However, the indeterminacy mechanism involved is important since it is equivalent to labor income taxation, a distortion that may lead to indeterminacy under empirically relevant parameterizations. 3.3.2

Examples with distortions on the offer curve

With leisure externalities only the parameter αΓL = η > −1 is different from 0 (see (13)). Assumption 1 means σ > s and indeterminacy requires Assumption 3.2 and Assumption 4, both satisfied when η = αΓL < 0, meaning that the (private marginal) desutility of labor is lower when oth40

As αKj = β Kj = 0, and since αLK = αΓK in all the following example, we have = α1ΓL . Hence σH2 becomes +∞ (see Appendix 6.6 for αKK = 0). As a finite σ H2 does not exist, and as αKK = 0, footnote of Proposition 4 applies. Also, when Configuration (iii) is possible Assumption 7 is verified. αH ΓL

24

θ(1−s) ers also work more. In this example α−1 ΓL = −1+ 2s−θ(1−s) = −0.967 and √ 2 s[s−θ(1−s)]] SB 2[s−θ(1−s)] αΓL = θ(1−s) − < −1 under our calibration. Accordingly, θ(1−s) we obtain Configurations (i) for η ∈ (−0.967, 0) and Configuration (ii) for η ∈ (−1, −0.967). Hence, from Figures 2a and 3a, indeterminacy emerges for , provided εγ is small enough. In the Cobb Douglas σ > σ H1 = s−θ(1−s)(1+η) −η case, indeterminacy is obtained for −1 < η < η1 ≡ −[s−θ(1−s)]/[1−θ(1−s)], 1 appearing in both Configurations (i) and (ii) since α−1 ΓL < η < 0. Under our 1 calibration, we obtain η ≃ −0.33, a value that is significantly closer to zero than the values used in related literature, as for instance the value η = −1.23 considered in Benhabib and Farmer (2000). Using the equivalence result of Proposition 3 between leisure externalities and unemployment insurance with efficiency wages,41 we see that in this last case indeterminacy oc−h curs with σ = 1 if h+(1−h)n < −0.33, i.e. indeterminacy occurs for empirically plausible values of n and h. For instance, if the employment rate is n = 0.95, indeterminacy occurs as soon as the replacement ratio h is higher than 0.32, as in most developed economies. In the example with unemployment benefits and unions (see (14)), Assumption 1 means θ (1 − s) < αs and σ > αs. Assumption 3.2 and Assumption 4, required for indeterminacy, are always satisfied in this example. We obtain Configurations (i) and (ii) for θ(1−s) = 0.065 < α < α∗ where α∗ ≡ 1 − θ(1−s) = 0.984 is the s s(4−θ) 1−s value of firm’s bargaining power, α, such that αΓL = 1−αs = αSΓLB (see Appendix 6.4.1). Configuration (iii) applies for α∗ < α ≤ 1. Since εγ = 1 only ∆1 is relevant and, from Figures 2a and 3a indeterminacy emerges for σ > σ H1 = s ∈ (0, 1) when α ∈ (0.065, 0.984), while from Figure 4a it emerges for σ > σ F = 1 − 2(1−s)(1−θ/2)(1−αs) ∈ (0, 1) when α ∈ (0.984, 1]. 2(1−2s)+2sα Therefore, indeterminacy prevails when σ = 1, independently of the degree of union power, 1 − α.42 This result shows that the existence of financed constrained workers together with unemployment benefits, a situation characterizing many developed economies, is likely to create indeterminacy and also complex employment fluctuations through the occurrence of Hopf and flip bifurcations. We conclude that with plausible labor market imperfections, either leisure externalities, unemployment benefits or labor income taxation, indeterminacy 41 In this example and in the next one, since εγ = 1, in Figures 2-4 only the half line ∆1 is relevant. Note also that in both examples αΓL < 0 so that εΓL < 1, a parameterization that helps indeterminacy as explained in the beginning of Section 3.1. (σ−1) 42 Note that when α = 1 or σ = 1 we have εµK = −εµL = s(1−α) = 0, so that the 1−αs σ variability of the mark up disappears. However, since unemployment benefits are still in place and have to be financed by taxation, the reservation wage, and therefore the offer curve, are still afected, i.e. αΓL = −1.

25

emerges under reasonable degrees of capital-labor substitution.

4

Applicability of the general approach: a new example

This new example, not found in the literature, combines leisure externalities and labor income taxation, and illustrates the applicability of our general η ε w approach. Workers maximize Ct+1 /B − Lt Lt γ , with εγ ≥ 1 and 0 > η > w −1, under the budget constraint Ct+1 = (1 − τ L (ω t Lt ))ω t Lt (pt /pt+1 ), taking η as given both the externality in leisure, Lt , and the tax rate, τ L (ω t Lt ) ≡ zL (ω t Lt /ωL)φL with zL ∈ [0, 1) and φL ∈ R. Government revenue is used to finance wasteful public expenditures, under a balanced budget at each period. Finally, capitalists behave as under perfect competition. At equilibrium, we get Ωt = D1 (Kt−1 , Lt )ω t where D1 (Kt−1 , Lt ) = 1 − zL (ω t Lt /ωL)φL , and Γt = ε D3 (Kt−1 , Lt )Lt γ with D3 (Kt−1 , Lt ) = Lηt , while Rt = ρt . In contrast to the last example, where the same specific distortion (the unions’ markup factor) affected Ω and Γ, here Ω and Γ are affected by different specific distortions. We easily deduce that αKi = β Ki = αLK = 0, αΓK = β Γi = 0, αΓL = η, zL αLL = − 1−z φ and β LK = −β LL = sαLL . L L zL Assumption 1 means −1 < 1−z φ < 1 and σ > s, while Assumptions L L zL 3.2 and 4, required for indeterminacy, mean η < −φL 1−z . Then, the other L −1 assumptions are fulfilled. Under our calibration 0 > αΓL > αSΓLB > −1, so that all configurations, including Configuration (iii), may emerge in contrast to the examples where leisure externalities and labor taxation were considered separately.43 Therefore, since Configuration (iii) is possible, indeterminacy can also occur through a flip bifurcation (see Figure 4a). This shows that economies facing several distortions may exhibit complex dynamic behavior. In Configurations (i) and (ii) (see Figures 2a and 3a), indeterminacy occurs z

for σ > σ H1 =

L φ )−θ(1−s)(1+η) s(1− 1−z L L

z

L φ −η− 1−z L

, provided εγ is sufficiently low. Let us

L

s−θ(1−s) = −0.33 assume that εγ = 1.01 and η = −0.35 < η 1 , with η 1 ≡ − 1−θ(1−s) in our calibration. Then, since εγ is sufficiently low, indeterminacy would emerge in the Cobb-Douglas case in the absence of labor income taxation (see Section 3.3.2). However, the introduction of a procylical tax rate policy zL (φL > 0) is able to eliminate indeterminacy. Indeed, for 0.35 > 1−z φ > L L θ(1−s)(1−

43

zL

φL )

1−zL From Appendix 6.4.1, we obtain α−1 ΓL = −1 + 2s−θ(1−s) √ zL (1− 1−z φ ) 2s 1− 1−θ(1−s)/s −θ(1−s) SB zL L L φ ), and α = −1 + ΓL 1−zL L θ(1−s) zL 1−zL φL ).

26

= −1 + 0.033(1 −

= −1 + 0.0167(1 −

[1−θ(1−s)](η 1 −η) (1−s) 44

= 0.0226 > 0, we obtain σ H1 > 1. Since Configuration (i) applies, indeterminacy is no longer possible with σ = 1. Moreover, for all zL combinations of 0 < zL < 1 and φL > 0 such that 0.34 < 1−z φ < 0.35 a L L 45 saddle stable steady state appears as εγ = 1.01 > εγ T . This shows that fiscal policy rules can be used to eliminate expectation-driven instability created by other distortions.

5

Concluding remarks

With our general approach we were able to emphasize several interesting results, some of them already latent in previous works, but which are here confirmed, generalized and highlighted. First, our work enabled us to find classes of specific distortions within which equivalence results, in terms of linearized dynamics, are obtained. Our equivalence results have strong implications. We will not be able to identify, when estimating the relevant parameters of our general formulation, a particular source of specific distortions among those, which belonging to the same class, are observational equivalent. Also, even if indeterminacy requires an empirically unreasonable degree of some specific distortion, the associated indeterminacy mechanism is not necessarily unimportant, since an equivalent empirically plausible model may exist. Another implication is that simulations of equivalent linearized versions of the model with additive shocks lead exactly to the same trajectories of aggregate capital and labor (in deviations from the steady state) and, thereby, equilibrium cyclical properties of variables that only depend on aggregate capital and labor are, up to the first order, identical in equivalent models. Second, the dynamic effects of one specific distortion may be compensated/eliminated by the existence of another specific distortion. This last result suggests some policy implications. Indeed, we saw that some forms of taxation eliminate local indeterminacy and endogenous fluctuations caused by the presence of leisure externalities, so that distortionary taxes may be defended on stability grounds. Third, capital market distortions per se do not seem to play a major role for the occurrence of indeterminacy. On the contrary, bifurcations and indeterminacy emerge under labor market rigidities, without imposing strange 44

Indeed, defining η2 ≡

−2[s−θ(1−s)] 2s−θ(1−s) zL 1−zL φL > Z

= −0.967 in our calibration and using (21), we

2 −η] obtain αΓL = η > ⇔ ≡ [2s−θ(1−s)][η , Z < 0 for η > η2 . Hence, for θ(1−s) −1 zL 1−zL φL > 0 and η = −0.35 > η 2 we have η > αΓL and Configuration (i) applies. zL 45 Note that, in this case, εγ T = 1.35 − 1−z φL under our calibration. L

α−1 ΓL

27

or implausible restrictions, whereas for output market distortions, indeterminacy requires conditions that might be considered less relevant from an empirical point of view. These findings suggest that the functioning of labor markets, which in the real world show significant deviations from the competitive paradigm, may be responsible for the persistency along business fluctuations and for the existence of expectation-driven cycles. For empirical support see Dufourt et al. (2007) and Chari et al. (2007). Further analysis on this issue is therefore important for future research. A possible explanation for these results may be linked to the fact that future expectations, which open the room for fluctuations driven by self-fulfilling expectations, only affect, as in the Ramsey and Overlapping Generations models, the current decisions of consumers/workers, rendering distortions that affect the intertemporal trade-off of consumers/workers more important than those affecting the capital accumulation equation. Strategic considerations by firms owning productive capital, which are usually disregarded, may render future expectations of capitalists/producers relevant, and change the results. Further research on this issue is welcome.

6

Appendix

6.1

Existence of a normalized steady state

Using (4)-(5), a steady state solution (K, L) satisfies A̺(K, L) = θ/β and (A/B)Ω(K, L)L = Γ(K, L). Then, (K, L) = (1, 1) satisfies both equations by fixing appropriately the scaling parameters A = θ/(β̺(1, 1) > 0 and B = [β̺(1, 1)Γ(1, 1)]−1 θΩ(1, 1) > 0.

6.2

Expressions for T1(σ) and D1 (σ)

T1 (σ)≡ 1 + {σ[ (1 + θαKK ) (1 + αΓL ) −θαΓK αKL ] + β ΓL −θ[(1 + αLL )(1 − s − β KK ) + αKK (s − β LL ) + αLK (1 − s + β KL ) (1 − s − β KK )(s − β LL ) +αKL ]}/{σ(1 + αLL ) − (s − β LL )} 1 − s + β KL (16) D1 (σ) ≡ {σ[(1 + θαKK )(1 + αΓL ) − θαΓK αKL ] + β ΓL −θ[(1 − s − β KK )(1 + αΓL ) + αΓK (1 − s + β KL ) 1 − s − β KK −αKL β ΓL −αKK β ΓL ]}/{σ(1 + αLL ) − (s − β LL )} 1 − s + β KL 28

(17)

Assumptions 2.1 and 2.2 have already been accounted for in the previous expressions.

6.3

Assumption 2 written in terms of the parameters

KK )(s−β LL ) KK 1. (β LK +s) = (1−s−β and β ΓK = − β ΓL 1−s−β (1−s+β ) 1−s+β KL

KL

2. αKK (αΓL −αLL ) = αKL (αΓK −αLK ) 3. αΓL >α0ΓL given in (19)

6.4 6.4.1

Expressions of critical values Definitions and expressions for critical values of αΓL 1 − s − β KK ] (s − β LL +β ΓL ) + (1 − s − β KK )αLL 1 − s + β KL + (αLK −αΓK ) (1 − s + β KL )}/ (1 − s − β KK ) (18)

α1ΓL ≡ {[αKK +αKL

α0ΓL ≡ −1− +

(1 + αLL )β ΓL −θ(s − β LL )αΓK αKL (s − β LL )(1 + θαKK ) − θ(1 + αLL )(1 − s − β KK )

(19)

KK θ(1 + αLL ) αΓK (1 − s + β KL ) − αKL β ΓL 1−s−β − αKK β ΓL 1−s+β KL

(s − β LL )(1 + θαKK ) − θ(1 + αLL )(1 − s − β KK )

α∞ ΓL ≡ {θ(1 + αLL )[(1 + αLL )(1 − s − β KK ) + αLK (1 − s + β KL ) (1 − s − β KK )(s − β LL ) ] (20) +αKL (1 − s + β KL ) −(s − β LL ) (1 − θαKK αLL −θαΓK αKL ) −(1 + αLL )β ΓL }/(s − β LL )(1 + θαKK ) α−1 ΓL ≡ {θ(1 + αLL )[(2 + αLL )(1 − s − β KK )+ (αLK +αΓK ) (1 − s + β KL ) (1 − s − β KK )(s − β LL −β ΓL ) +αKK (s − β LL − β ΓL ) + αKL ] (21) 1 − s + β KL −2(s − β LL ) [1 − θαΓK αKL −θαKK αLL ] −2(1 + αLL )β ΓL } / {2(s − β LL ) (1 + θαKK ) −θ(1 + αLL )(1 − s − β KK )} αSΓLB is the lower root of equation S1 = SB , using (22) and (26). See Appendix 6.5. 29

6.4.2

Expressions for S1

S1 = 1 + θ (1 + αLL ) {(1 − s − β KK ) (αΓL −αLL ) + (1 − s + β KL )(αLK −αΓK ) 1 − s − β KK ](s − β LL +β ΓL )}/{θ (1 + αLL ) [αLK (1 − s + β KL ) −[αKK +αKL 1 − s + β KL 1 − s − β KK + (1 + αLL ) (1 − s − β KK ) + αKL (s − β LL )] (22) 1 − s + β KL −(s − β LL )(1 + αΓL (1 + θαKK ) − θαΓK αKL −θαKK αLL ) − (1 + αLL )β ΓL } (α0ΓL −αΓL ) [(s − β LL )(1 + θαKK ) − θ(1 + αLL )(1 − s − β KK )] (23) (α∞ ΓL −αΓL )(s − β LL )(1 + θαKK ) (αΓL −α1ΓL )θ(1 + αLL )(1 − s − β KK ) = 1+ (24) (α∞ ΓL −αΓL )(s − β LL )(1 + θαKK ) (α−1 −αΓL )[2(s − β LL )(1 + θαKK )−θ(1 + αLL )(1 − s − β KK )] = −1+ ΓL (25) (α∞ ΓL −αΓL )(s − β LL )(1 + θαKK )

S1 = S1 S1

1−D1 (+∞) 1−D1 (+∞) SB ≡ −2−T = −3−D is the value taken by S1 when ∆1 goes through 1 (+∞) 1 (+∞) point B, satisfying SB ∈ (−1, 0) under Assumptions 2.2 and 3, and is given by:

SB = 1+ 6.4.3

4(1 + αLL ) ∈ (−1, 0) −3(1 + αLL ) − (1 + θαKK )(1 + αΓL ) + θαΓK αKL

(26)

Expressions for critical values of σ 1 − s − β KK )] − θ[(1 − s − β KK )(1 + αΓL ) 1 − s + β KL +αΓK (1 − s + β KL )]}/[αLL − αΓL − θ[αKK (1 + αΓL ) − αΓK αKL ]] (27)

σ H1 = {s − β LL + β ΓL [1 + θ(αKK + αKL

1 − s − β KK ]+θ[(1 − s − β KK )(2 + αLL +αΓL ) 1 − s + β KL +(1 − s + β KL )(αLK +αΓK )]}/[2(2 + αLL +αΓL ) + 2θ(αKK (1 + αΓL ) − αΓK αKL )]

σ F = {(s − β LL −β ΓL )[2 + θ(αKK +αKL

σT ≡

(1 − s − β KK ) . αKK

30

(28)

6.4.4

Expressions for critical values of εγ

εγ H is such that D = 1, which is equivalent to: εγ H = 1+ {[αLL −αΓL −θ (αKK (1 + αΓL ) − αΓK αKL )] σ − (s − β LL ) (29) 1 − s − β KK −β ΓL [1 + θ(αKK +αKL ] + θ[(1 − s − β KK )(1 + αΓL ) 1 − s + β KL +αΓK (1 − s + β KL )]}/[σ(1 + θαKK ) − θ(1 − s − β KK )] εγ F is such that 1 + T + D = 0. After some computations, we obtain: εγ F = 1 + {−2[(2 + αLL +αΓL ) + θ(αKK (1 + αΓL ) − αΓK αKL )]σ (30) 1 − s − β KK ]+θ[(1 − s − β KK )(2 + αLL +αΓL ) +(s − β LL −β ΓL )[2 + θ(αKK +αKL 1 − s + β KL +(1 − s + β KL )(αLK +αΓK )]}/[σ(2 + θαKK ) − θ(1 − s − β KK )] εγ T is such that 1 − T + D = 0. After some computations, we obtain: (1 − s − β KK ) ] (s − β LL +β ΓL ) + (1 − s − β KK )(αLL − αΓL ) (1 − s + β KL ) + (αLK −αΓK ) (1 − s + β KL )}/(1 − s − β KK −σαKK ). (31)

εγ T = 1+{[αKK +αKL

6.5

Proof of Lemma

S1 can be written as in (22), but also, given the definitions of the critical values of αΓL in (18)-(21), in several different ways (see (23)-(25)), so that S1 = 0 for αΓL = α0ΓL , S1 = 1 for αΓL = α1ΓL , and S1 = −1 for αΓL = α−1 ΓL . Under Assumptions 1, 2.3 and (23), lim∞+ S1 = +∞ while lim∞− S1 = −∞, and αΓL →αΓL

αΓL →αΓL ∞ S1 < 0 iff αΓL < while S1 > 0 iff αΓL > αΓL . Under Assumptions 2.3 and ∂S1 1 4, using (24) we have ∂α < 0 and S1 > 1 for α∞ ΓL ≤ αΓL < αΓL , while using ΓL ∂S1 (23) we have that ∂α < 0 also for α0ΓL < αΓL < α∞ ΓL . Hence, S1 is decreasΓL 0 1 1 ∞ 0 ing in αΓL ∈ (αΓL , αΓL ) and we have αΓL > αΓL > α−1 ΓL > αΓL (see Figures 2-4). ∆1 moves counterclockwise when αΓL decreases from α1ΓL to α0, ΓL , with −1 1 0 |S1 | > 1 for αΓL < αΓL < αΓL and S1 ∈ (−1, 0) for αΓL < αΓL < α−1 ΓL .

α∞ ΓL

Accordingly, the half line ∆1 must go through point B at a unique critical SB B value αSΓL ∈ α0ΓL , α−1 ΓL such that S1 = SB ∈ (−1, 0) for αΓL = αΓL , with SB given in (26), and S1 ∈ (−1, SB ) for αΓL ∈ αSΓLB , α−1 ΓL while S1 ∈ (SB , 0) for SB 0 αΓL ∈ αΓL , αΓL . We now show that αSΓLB ∈ α0ΓL , α−1 ΓL is the lower root of equation S1 = SB . (1+θα )(αΓL −αL )−2(1+αLL ) Using Assumption 3.1 and (26), note that SB = (1+θαKK ) α −αΓL . L KK ( ΓL ΓL )+2(1+αLL ) 31

Using this expression and (25) and defining A ≡ 2 (1 + αLL ) [2(s − β LL ) − θ(1+αLL )(1−s−β KK ) ], S1 = SB can be written as z(αΓL ) = 0, where: (1+θα ) KK

z(αΓL ) ≡ θ(1 + αLL )(1 − s − β KK ) αΓL − αLΓL

α−1 ΓL − αΓL −

−1 L ∞ α−1 ΓL − αΓL A + 2 αΓL − αΓL (s − β LL )(1 + θαKK ) αΓL − αΓL

B Therefore, αSΓL must be a solution of z(αΓL ) = 0. Under Assumption 1, z is a concave polynomial of degree 2. Using also Assumption 3.1 and since as SB −1 −1 −1 0 shown above α∞ ΓL > αΓL , we have z(αΓL ) > 0. Hence αΓL ∈ αΓL , αΓL must be the lower root of z (αΓL ) = 0, or equivalently of S1 = SB .

6.6

Existence of σH2

s−β LL σ H2 is a value of σ ∈ 1+α , +∞ such that the half line ∆(σ) goes through LL point C, i.e. εγ H = εγ T (see (29) and (31)). We can see geometrically that, if σ H2 exists it must be such that σ H2 ∈ (σ H1 , +∞) and, for σ slightly higher than σ H1 , the half line ∆(σ) crosses (BC) on the left of C. Also, the half line ∆(σ) can only cross C if it has a slope S ∈ (0, 1). Consider first αKK = 0. s−β LL Then S ∈ (0, 1) for σ ∈ 1+α , +∞ , but lim S = 1. Hence, it is possible LL σ→+∞

that the half line ∆(σ), for any (σ H1 , +∞), always cross the line (BC) on the left of C, so that σ H2 does not exist. Using (18), (19), (29), (31) and Assump[(s−β LL )−θ(1+αLL )(1−s−β KK )](α1ΓL −α0ΓL ) s−β LL tion 3.2, we can write σ H2 = 1+α + . 1 LL (1+αLL )(αH ΓL −αΓL ) Under Assumptions 1, 2.3 and 4 the numerator is always positive. Hence s−β LL 1 if αH , +∞ , i.e., the half line ∆(σ) ΓL ≤ αΓL there is no σ H2 ∈ 1+αLL always cross (AC) above C, and εγ H < εγ T for σ > σ H1 . Otherwise, if 1 αH ΓL > αΓL , there is a unique value σ H2 ∈ (σ H1 , +∞) and we have εγ H < εγ T for σ H1 < σ < σ H2 , and εγ H > εγ T for σ > σ H2 . Consider now αK,K = 0. Note that εγ H = εγ T can be written as g(σ) = 0, where: g(σ) ≡ αKK (1 + θαKK )(αH ΓL − αΓL )(σ − σ T )(σ − σ H1 )

+ [σ(1 + θαKK ) − θ(1 − s − β KK )](1 − s − β KK ) α1ΓL − αΓL .

(32)

Therefore σ H2 is a solution of g(σ) = 0. When αKK < 0, S ∈ (0, 1) for s−β LL σ ∈ 1+α , +∞ (see (15)). Since lim S ∈ (0, 1), we can see geometrically LL σ→+∞

that a solution σ H2 ∈ (σ H1 , +∞) must exist and the number of these solutions is odd. As, under Assumptions 1, 3.2 and 4, g(σ) is a concave polynomial of degree 2, i.e. has at most two solutions, and g (σ H1 ) > 0, we deduce the 32

uniqueness of σ H2 and that σ H2 is the higher root of g(σ) = 0. Accordingly, εγ H < εγ T for σ H1 < σ < σ H2 , and εγ H > εγ T for σ > σ H2 . Finally, consider that αKK > 0. Under Assumption 5, σ T > σ H1 , so that, by (15) and (28), S ∈ (0, 1) when σ ∈ [σ H1 , σ T ) and, if σ H2 exists, then σ H2 ∈ (σ H1 , σ T ). For σ ≥ σ T , since S ≥ 1, the half line ∆(σ), starting within triangle (ABC), crosses (BC) on the left of C. Therefore, the existence of σ H2 ∈ (σ H1 , σ T ) is not ensured, and the number of solutions σ H2 ∈ (σ H1 , σ T ) that satisfy g(σ) = 0 must be even. Under Assumptions 1, 3.2 and 4, g(σ) describes a convex parabola with g(σ H1 ) > 0, g(σ T ) > 0 and g(+∞) = +∞. Hence, either g(σ) = 0 has two solutions (requiring g ′ (σ H1 ) < 0) or none (as it happens for instance when g ′ (σ H1 ) ≥ 0)). As g ′ (σ H1 ) ≥ 0 is equivalent to 1 H σ T (αH ΓL − αΓL ) ≤ σ H1 (αΓL − αΓL )

(33)

when this inequality is satisfied, there is no solution to g(σ) = 0. Under Assumption 3.2, this inequality is always satisfied when α1ΓL ≥ αH ΓL . In this case for all σ > σ H1 the half line ∆(σ) always goes above point C and Assumption 6 is satisfied.

6.7

Existence of σH3

Using (29) and (30), we have that εγ H ≥ εγ F ⇔ h(σ) ≥ 0, where: h(σ) ≡ [σ(2 + θαKK ) − θ(1 − s − β KK )](αH Γ,L − αΓL )(σ − σ H1 ) + 2[σ(1 + θαKK ) − θ(1 − s − β KK )](αΓ,L − αLΓL )(σ − σ F )

(34)

By definition, σ H3 is a value of σ such that εγ H = εγ F , therefore it must be a solution of h(σ) = 0. Since h(σ) is a polynomial of degree 2, the equation h(σ) = 0 has at most two solutions. We limit our analysis to configuration (iii) since σ H3 is only relevant under this configuration. Since ∆(σ) is positively sloped pointing upwards, it can only go through point B, if its initial point in ∆1 is on the left of the line (AB), i.e. σ H3 < σ F . Also, the polynomial h(σ) is a convex function of σ since, under Assumptions 1 and 3, the coefficient of the quadratic term σ 2 , (2 + θαKK ) (αH ΓL −αΓL )+2 (1 + θαKK ) (αΓL − s−β LL L αΓL ), is positive. We can see geometrically that if there is a σ H3 > 1+α LL s−β LL then it must satisfy 1+α < σ < σ < σ . Using (34), and Assumption H1 H3 F LL 3, we see that in this configuration h(σ F ) > 0 and h (σ H1 ) < 0. Therefore, there is a unique σ H3 ∈ (σ H1 , σ F ) such that h (σ H3 ) = 0, given by the higher root of h (σ) = 0. By continuity, we have that εγ H > εγ F for σ F > σ > σ H3 , and εγ H < εγ F for σ H1 < σ < σ H3 .

33

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[25] Schmitt-Grohé, S. and M. Uribe (1997), ”Balanced- Budget Rules, Distortionary Taxes, and Aggregate Instability,” Journal of Political Economy, 105, 976-1000. [26] Seegmuller, T. (2008), “Taste for Variety and Endogenous Fluctuations in a Monopolistic Competition Model,” Macroeconomic Dynamics, 12, 561-577. [27] Seegmuller, T. (2009), “Capital-labour Substitution and Endogenous Fluctuations: a Monopolistic Competition Approach with Variable Markup,“ Japanese Economic Review, 60, 301-319. [28] Volkerink, B., J-E Sturm and J. de Haan (2002), "Tax Ratios in Macroeconomics: Do Taxes Really Matter?", Empirica, 29, 209-224. [29] Weder, M. (2004), “A Note on Conspicuous Leisure, Animal Spirits and Endogenous Cycles,“ Portuguese Economic Journal, 3, 1-13. [30] Woodford, M., (1986), “Stationary Sunspot Equilibria in a Finance Constrained Economy,” Journal of Economic Theory, 40, 128-137.

36

∆(σ = +∞)

6 D

 @

@

@ @ B@

∆(σ ∈ (s, +∞)) 

1 @ @

@

@ @ -1@

@ @

C

0 @ @ A @ -1@ @

 

T

1

@

@ @

Fig 1. Perfect competition

@

@ @

@

* 

6 D

@

@

@ @ B@

∆1

1 @ @

 @ -1@ @

0 @

 

∆(σH1 ) *  3 ∆(σH2 )         C    

@ @

 



T

1

@ @ A @ -1@ @

@

@ @

@

@ @

@

Fig 2-a. Configuration (i ), with αK,K 6 0

6 D

@

@

∆1  

@ @ B@

1 @ @

@

@ @ -1@ @

 0 @

@ @ A @ -1@ @





 

      C

T

1

@

@ @

∆(σT )  ∆(σH1 ) * 

@

@ @

Fig 2-b. Configuration (i ), with αK,K > 0

@

6 D ∆(σF ) 1  iP P   @P P  ∆(σH1 ) ∆1@PP PP *  @ PP  3∆(σH2 )  P @  PP   P 1 C P B@ PP  @ PP  PP @ PP @ P @ @ -1@ 1 0 T @ @ @ @ A @ -1@ @ @ @ @ @ @ @ @

Fig 3-a. Configuration (ii ), with αK,K 6 0

6 D ∆(σF ) 1  

iP P  P  @P ∆(σH1 ) ∆1@PP PP *  @ PP  PP  @ PP  1 P B@ PP @ PP PP @ PP @ P @ @ -1@ 1 0 @ @ @ @ A @ -1@ @ @ @ @ @

∆(σT ) 

C

T

@ @

@

Fig 3-b. Configuration (ii ), with αK,K > 0

6 D

@

@

∆(σH3 ) @ 1 ∆(σF )   : @ *∆(σH2 )    1    @  yXXB X   C  X  @  X ∆1 XX   @ XXX  XXX @  XXX  @ X XXX @ -1@ 1 0 T @ @ @ @ A @ -1@ @ @ @ @ @ @ @ @

Fig 4-a. Configuration (iii ), with αK,K 6 0

6 D ∆(σT ) @



@

∆(σH3 ) @ 1 ∆(σF )   : @   1    @ yXXB X  XX@  ∆1 XX  @ XXX XXX @ XXX @ XXX X @ -1@ 1 0 @ @ @ @ A @ -1@ @ @ @ @

C

T

@

@ @

@

Fig 4-b. Configuration (iii ), with αK,K > 0