Equilibrium Stability in Open Economy Models

Martin Bodenstein∗∗ Federal Reserve Board September 2011

Abstract This paper derives analytical results for the relationship between the slope of the excess demand function and the dynamic properties around a deterministic steady state in a two country model. In models that admit multiple steady states, the sign of the slope of the excess demand function is positive for some steady states and negative for others. To obtain stationarity of the net foreign asset position under incomplete financial markets I introduce the stationary inducing devices analyzed in Schmitt-Groh´e and Uribe (2003) and Ghironi (2006). For portfolio costs, a debt-elastic interest rate, or an overlapping generations framework the equilibrium dynamics around a steady state are unbounded if the excess demand function for the foreign traded good is increasing in the good’s own price. Otherwise the dynamics are bounded and locally unique. By contrast, with Uzawatype preferences, the equilibrium dynamics around a steady state are shown to be bounded and locally unique irrespective of the sign of the slope of the excess demand function.

Keywords: equilibrium multiplicity, incomplete markets, open economy

JEL Classification: D51, F41

∗∗

Telephone (202) 452 3796. E-mail [email protected]. The views expressed in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. ∗

1

Introduction

As shown in Bodenstein (2011), models of the international business cycle admit multiple locally isolated deterministic steady states, if the elasticity of substitution between traded goods is sufficiently low.1 For the case of a model with two agents and two goods, standard theorems in general equilibrium theory imply that the slope of a good’s excess demand as a function of its relative price needs to change sign for multiple steady states to exist. This paper derives analytical results for the relationship between the slope of the excess demand function and the dynamic properties around a deterministic steady state in a two country model. Each country produces a distinct traded good that serves as a limited substitute for the other country’s good. Labor is the only factor of production and international financial markets are incomplete with a non-statecontingent bond being the only asset that trades internationally. In the standard model with incomplete markets the steady state is undetermined since the growth rate of marginal utility does not depend on the allocation of bond holdings. Absent arbitrage opportunities, the price of the non-state-contingent bond is equalized across countries implying that expected marginal utility growth is equalized across countries. In the deterministic steady state, this condition contains no information about the steady state values of the system and the system of equilibrium conditions becomes underdetermined. Any level of international bond holdings is a steady state. Furthermore, net foreign asset position follows a unit root process when the model is linearized around a deterministic steady state. 1

Consider an endowment economy with two countries and two traded goods that are imperfect substitutes as in Mas-Colell et al (1995). The countries are mirror images of each other with respect to preferences and endowments. One equilibrium always features a relative price of the traded goods equal to unity. However, there can be two more equilibria if there is home bias in consumption and the price elasticity of substitution between goods is low. If the price of the domestic good is high relative to the price of the foreign good, domestic agents are wealthy compared to the foreign agents. If the elasticity of substitution is low, foreigners are willing to give up most of their good in order to consume at least some of the domestic good, and domestic agents end up consuming most of the domestic and the foreign good. The reverse is true as well. Foreign agents consume most of the goods, if the foreign good is expensive in relative terms. Of course, these last two scenarios cannot be an equilibrium for high values of the elasticity of substitution. In the limiting case of perfect substitutability.

2

To obtain stationarity of the net foreign asset position in this setting, I introduce the stationary inducing devices analyzed in Schmitt-Groh´e and Uribe (2003) and Ghironi (2006): a portfolio costs, a debt elastic interest rate, endogenous discounting, and an overlapping generations structure. All four approaches analyzed here, resolve this indeterminacy by construction. Stationarity of international bond holdings follows as agents finance additional consumption out of a positive net foreign asset position and the economy moves towards its steady state. This paper studies the local dynamic properties under these stationarity inducing devices for different signs of the slope of the excess demand function of the foreign traded good. If agents face a portfolio cost for holding/issuing bonds, the equilibrium dynamics around a given steady state are unbounded (or unstable) if the excess demand function for the foreign traded good is increasing in the good’s own price in this steady state. However, if the excess demand function for the foreign traded good is decreasing in the foreign goods price, the equilibrium dynamics are bounded and locally unique (saddle-path stable). The same results apply for models with a debt-elastic interest rate or an overlapping generations framework of Ghironi (2006) and Weil (1989). By contrast, if preferences allow for an endogenous discount factor as in Uzawa (1968), the equilibrium dynamics around a steady state are shown to be bounded and locally unique irrespective of the sign of the slope of the excess demand function. Bodenstein (2011) also studies the relationship between equilibrium dynamics in the neighborhood of a steady state and the sign of the slope of the excess demand function. Whereas Bodenstein (2011) considers a richer class of models, all the results in that paper are only numerical. Here I derive with paper and pencil the exact nature of the relationship under investigation. The remainder of the paper is organized as follows. Section 2 presents the static model that underlies the analysis, proves the existence of the equilibrium, and relates the sign of the slope of the excess demand function to steady state multiplicity.

3

In section 3, the static model is extended to incorporate dynamics and I analyze the characteristics of the steady states and the local dynamics under the different stationarity inducing approaches. Section 4 offers concluding remarks.

2

Model with multiple equilibria

Each country produces one good that can be traded internationally without frictions. The two goods are assumed to be imperfect substitutes in the household’s utility function. Labor, which is supplied endogenously, is the sole factor of production and the population size is normalized to one. At this first stage, the model is static. Households maximize utility subject to the budget constraint max U (ci , li )

(1)

ci1 ,ci2 ci ,li

s.t. P¯1 ci1 + P¯2 ci2 ≤ P¯i wi li + P¯i Πi + dWi ,

(2)

where ci is given by a linear-homogeneous aggregator Hi (ci1 , ci2 ).2 Hi is assumed to satisfy ∂Hi ∂ 2 Hi ∂ 2 Hi Hij = > 0, Hiii = < 0, Hiji = > 0, ∂cij ∂c2ii ∂cij ∂cii and the Inada conditions lim Hi1 (ci1 , ci2 ) =

ci1→0

lim Hi1 (ci1 , ci2 ) =

ci1→∞

lim Hi2 (ci1 , ci2 ) = ∞,

ci2→0

lim Hi2 (ci1 , ci2 ) = 0.

ci2→∞

2

An aggregator that satisfies the restrictions imposed on Hi is given by the generalization of the CES aggregator as suggested by Dotsey and King (2005): [ ( ) ]ρ [ ( ) ]ρ (1 + η) ci1 αi2 (1 + η) ci2 1 αi1 −η + −η = . (1 + η) ρ αi1 ci (1 + η) ρ αi2 ci (1 + η) ρ This aggregator allows for the elasticity of substitution to be non-constant. For η = 0, one obtains the standard CES aggregator.

4

The strictly concave period utility function U (c, l) satisfies Uc > 0, Ul < 0 and Ucc < 0, Ull < 0, Ucl ≤ 0.

(3)

ci denotes final consumption, li labor, cij is the consumption of good j by a household located in country i. P¯i is the price at which good i is traded and wi is the wage in country i denoted in units of country i’s traded good. Real profits are Πi . dWi is an exogenous lump sum transfer to agents in country i with dW1 + dW2 = 0. This lump sum transfer is introduced to make the derivations easily applicable to the dynamic economy introduced in the section 3. Firms in country i produce the traded good i using a linear production technology F (A1 , li ) = Ai li . Defining the relative prices q¯ ≡

(4) P¯2 P¯1

and Φi (¯ q) ≡

P¯i , Pi

where Pi is the price of the

aggregate consumption good, a competitive equilibrium is given by Definition 1 (Competitive Equilibrium) : A competitive equilibrium is a collection of allocations ci1 , ci2 , ci , li , yi and prices q¯, Φi , wi , i = 1, 2, such that (i) for every household the allocations solve the household’s maximization problem for given prices, (ii) for every firm profits are maximized and (iii) the markets for labor and for the two traded goods clear.

2.1

Existence of the equilibrium

The analysis on the equilibrium existence in a production economy is based on Kehoe (1980, 1985). Let Li be the time endowment of agents in country i. The excess demand for a good is defined as the difference between the demand for a specific good and the aggregate endowment with this good. The economy’s endowment with goods 1 and 2 is zero, while the leisure endowments are L1 and L2 . I denote the excess demand for goods 1 and 2 by dc,i = c1i + c2i , i = 1, 2. The excess demand for leisure is given by dl,i = (Li − li ) − Li = −li and let d = (dc,1 , dc,2 , dl,1 , dl,2 ). 5

The production side of the economy is given by a 4 × 6 activity analysis matrix A. Each column of A represents an activity, which transforms inputs taken from the vector of aggregate initial endowments or from the outputs of other activities into outputs, which are either consumed or further used as inputs. Positive entries in an activity denote quantities of outputs produced by the activity; negative entries denote quantities of inputs consumed. Aggregate production is denoted by Ay ′ , where y is a 6 × 1 vector of nonnegative activity levels   −1 0 0 0 A1 0      0 −1 0 0 0 A2  , A=    0 0 −1 0 −1 0    0 0 0 −1 0 −1 with the first 4 columns of this matrix being free disposal activities. ( ) An equilibrium of this economy is a price vector pˆ = P¯1 , P¯2 , w1 , w2 that satisfies the following three properties: (i) pˆ′ A ≤ 0; (ii) there exists a nonnegative vector of activity levels yˆ such that Aˆ y ′ = d (ˆ p); and (iii) P¯1 = 1. The first condition requires that profits are nonpositive. The second one requires that supply equals demand. The third one is a price normalization. Existence of an equilibrium follows directly from applying standard fixed point theorems as shown in Kehoe (1985). Let the activity vector be y ′ = (0, 0, 0, 0, l1 , l2 ). Then

 c + c21  11   c12 + c22 d (p) − Ay ′ =    −l1  −l2





Al   11     A 2 l2 −     −l1   −l2

   .   



Using Walras’ Law, an equilibrium is a price vector such that z2 (p) = c12 (p) + c22 (p) − A2 l2 (p) = 0. For the model outlined above, the excess demand function for good 2 can be written solely as a function of the relative price q¯. Using the first order condi6

tions associated with a household’s optimization problem described in (1), linear homogeneity of Hi implies ( ) ci1 Hi1 ,1 = λi P¯1 , ci2 ( ) ci1 Hi2 ,1 = λi P¯2 ., ci2

(5) (6)

where λi is the Lagrange multiplier on the household’s budget constraint. Given the properties of Hi1 and Hi2 imposed above ( ) ci1 (st ) ˜i 1 . = H ci2 (st ) q¯

(7)

As the price of the final consumption good needs to satisfy Pi =

P¯1 ci1 + P¯2 ci2 , ci

(8)

the relative prices Φ1 and Φ2 are given by ( ( ) ) ˜ 1 q ¯ H ¯ 1 H1 q¯ , 1 P1 ∂Φ1 (¯ q) ] with Φ′1 (¯ q) = Φ1 (¯ q) ≡ = [ ( ) < 0, P1 ∂ q¯ ˜ 1 1 + q¯ q¯ H q¯ ( ( ) ) ˜2 1 , 1 q¯H2 H q¯ P¯2 ∂Φ2 (¯ q) ] with Φ′2 (¯ q) = Φ2 (¯ q) ≡ = [ ( ) > 0. P2 ∂ q¯ ˜ 2 1 + q¯ H

(9)

(10)

q¯

I normalize the price of the consumption basket in country 1 to unity, P1 = 1, and Φ1 (¯ q) denote P2 by q, the real exchange rate. q and q¯ are related as follows q = q¯Φ . q) 2 (¯

Finally, using (2) and (7) the demand functions for good 2 are ] [ 1 1 ) w1 l1 + dW1 , c12 = ( ( ) Φ1 (¯ q) ˜ 1 1 + q¯ H q¯ ] [ 1 1 ) w2 l2 + c22 = ( ( ) dW2 , q¯Φ1 (¯ q) ˜2 1 1 + 1 H q¯ q¯

7

(11)

(12)

and the excess demand function for good 2 can be written as z2 (¯ q , dW1 ) = c12 + c22 − y2 A1 l1 (¯ q , dW1 ) + Φ11(¯q) dW1 ( ) = ˜ H1 1q¯ + q¯ A2 l2 (¯ q , dW1 ) − q¯Φ11(¯q) dW1 ( ) + ˜2 1 1 + 1 H q¯ q¯ −A2 l2 (¯ q , dW1 ) .

(13)

Equation (13) already reflects that under perfect competition the zero profit condition implies Ai = wi . Furthermore, households’ optimal labor choices can be written as a function of q¯ and the exogenous lump sum transfer dW1 , by noting that c1 = Φ1 (¯ q ) A1 l1 + dW1 , Φ2 (¯ q) dW2 . c2 = Φ2 (¯ q ) A2 l2 + Φ1 (¯ q ) q¯

(14) (15)

and Ul (ci , li ) = −Φi (¯ q ) Ai . Uc (ci , li )

(16)

Thus, an equilibrium in the economy is fully summarized by the relative price q¯∗ , s.t. z2 (¯ q ∗ , dW1 ) = 0.

2.2

Multiplicity of equilibria

If all the equilibria of an economy are locally unique, the economy is referred to as regular. Kehoe (1980) provides general conditions for a production economy that ensure regularity. In addition, he shows that the number of equilibria in a production economy is odd. Let the index of an equilibrium pˆ be defined as    ¯ −J¯ B  . index (ˆ p) = sgn det  ′ ¯ −B 0 J¯ is formed by deleting the first row and the first column from Dd (ˆ p), the matrix of derivatives of the excess demand functions with respect to each price, if good 1 8

¯ is formed by deleting the first row from B (ˆ is the numeraire. B p), where B (ˆ p) is the submatrix of A whose columns are all those activities that earn zero profits at pˆ. Theorem 2 in Kehoe (1985) states that the sum of the indices across all equilibria ∑ equals +1, i.e., j index (ˆ pj ) = +1. Hence the number of equilibria in a regular economy is finite and odd. Despite substantial progress in the development of fixed point algorithms, it is in general impossible to find all the equilibria of an economy unless uniqueness of the equilibrium can be proven. What can be said about the number of equilibria in the model presented in this paper? Using Kehoe’s  ∂d /∂ q¯  c,2  J¯ =  ∂dl,1 /∂ q¯  ∂dl,2 /∂ q¯

approach,   0 0 0 A2    ¯  0 0  , B =  −1 0   0 0 0 −1

   , 

since wi = Ai . It turns out that   ¯ −J¯ B  = − ∂dc,2 + A2 ∂dl,1 = − ∂z2 . det  ∂ q¯ ∂ q¯ ∂ q¯ ¯′ 0 −B If the excess demand function as defined in (13) is downward sloping in each equilibrium, the equilibrium is unique. However, if an equilibrium with

∂z2 ∂ q¯

> 0 is found

then there must be at least two more equilibria. In the model of this paper, totally differentiating equation (13) under the assumption dW1 = 0 delivers



˜′ H 1

( ) 1 q¯

1 q¯

− q¯



A l1 ∂l1 q¯  ∂z2 ( ) [ (1 ) ] = + ∂ q¯ ∂ q¯ l1 ˜ 1 1 + q¯ ˜ 1 1 + q¯ H q¯ H q¯ q¯   ( ) ˜′(1) H 2 q¯ ˜ 2 1 A2 l2  ˜ 1 + q¯ H H2 ( ) q¯ ∂l q¯  ] [ (q¯ ) ]− 2  . + [ ( )  ∂ q¯ l2  ˜ 2 1 + q¯ ˜ 2 1 + q¯ q¯ H H q¯ q¯

9

Let the (possibly variable) elasticity of substitution between traded goods in country i be denoted by εi (¯ q ) . Hence, equation (7) implies ( ) ( ) ˜ i′ 1 1 ∂ cci1 (¯ q ) H q¯ q¯ i2 ( ) . εi (¯ q) = =− ci1 ∂ (¯ q ) ci2 ˜i 1 H q¯

(17)

The slope of z˜2 in equilibrium (˜ z2 (¯ q ∗ ) = 0) can then be expressed as ( )   1 1 ˜ ε (¯ q ) H + 1 1 1 q¯ q¯ ε2 (¯ q) − 1 ∂l1 q¯ ∂l2 q¯  ∂z2 A 1 l1 ] ( ) ( ) + − + . |q¯=¯q∗ = − [ ( ) ∂ q¯ ∂ q¯ l1 ∂ q¯ l2 ˜ 1 1 + q¯ ˜1 1 1 + 1 ˜2 1 1 + 1 q¯ H H H q¯ q¯ q¯ q¯ q¯ (18) ∂li q¯ ∂ q¯ li

is the general equilibrium elasticity of labor with respect to the relative price

q¯. The next section relates the dynamic stability of the model to the sign of the slope of the excess demand function.

3

Dynamics model

In the dynamic extension of the model, time is discrete and each period the economy experiences one of finitely many events st . st = (s0 , ..., st ) denotes the history of events up through and including period t. The probability, as of period 0, of any particular history st is π (st ). The initial realization s0 is given. Intertemporal financial markets are exogenously incomplete in the sense that the only asset that is traded internationally is one non-state-contingent bond. The bond is in zero net-supply. The problem of the representative household in country i is given by ∞ ∑ ∑ ( ( ) ( )) ( ) β t U ci st , li st π st max ci (st ),li (st ), t=0 st ci1 (st ),ci2 (st ), Bi (st ) s.t. ( ) ( ) ( ) ( ) ( ) ( ) ( ) Pi st ci st ≤ P¯i st wi st li st + P¯i st Πi st ) ( ) ( ) ( +Bi st−1 − Q st Bi st . 10

(19)

(20)

Pi (st ) ci (st ) are the household’s total consumption expenditures which are equal to P¯1 (st ) ci1 (st ) + P¯2 (st ) ci2 (st ). Substituting Bi (st−1 ) − Q (st ) Bi (st ) for dWi (st ) in the budget constraint, Bi (st−1 ) denotes the (nominal) bond holdings that agent i has inherited from period t − 1. Q (st ) is the price of the bond. Given the assumptions on technology, preferences, and trade stated previously, the equilibrium dynamics are determined by: 1. the excess demand function for good 2 derived from equation (13) ( ( ) ( ) ( ) ( )) z2 q¯ st , B1 st−1 − Q st B1 st = 0,

(21)

where ∑ Uc (c1 (st+1 ) , l1 (st+1 )) ( ( ) ) Q st = β π st+1 |st , t t Uc (c1 (s ) , l1 (s )) t+1 t s

(22)

|s

2. the risk sharing condition ∑ [ Uc (c1 (st+1 ) , l1 (st+1 )) ] ( ) β π st+1 |st (23) t t Uc (c1 (s ) , l1 (s )) t+1 t s |s ] ∑ [ Uc (c2 (st+1 ) , l2 (st+1 )) q¯ (st ) Φ1 (¯ ( t+1 t ) q (st )) Φ2 (¯ q (st+1 )) π s |s (24). = β Uc (c2 (st ) , l2 (st )) q¯ (st+1 ) Φ1 (¯ q (st+1 )) Φ2 (¯ q (st )) t+1 t s

|s

Conditions (14)-(16) can be used to substitute out for consumption and labor in the risk sharing condition. In the standard model with incomplete asset markets, the deterministic steady state of the net foreign asset position is not determined. Furthermore, the net foreign asset position follows a unit root process when the model is linearized around a deterministic steady state. To pin down the net foreign asset position in steady state and remove the stationarity problem, I consider several of the approaches that have been put forward to induce stationarity in the approximate models, among others are: • portfolio costs as in Heathcote and Perri (2002), and Schmitt-Groh´e and Uribe (2003), 11

• debt elastic interest rates as in Boileau and Normandin (2008), Devereux and Smith (2007), and Schmitt-Groh´e and Uribe (2003), • endogenous discounting as in Mendoza (1991), and Schmitt-Groh´e and Uribe (2003), • overlapping generations as in Ghironi (2006). In the remainder of the section, I analyze the impact of the each of these stationarity devices on the equilibrium dynamics in the neighborhood of a deterministic state around which the model is linearized. The analysis does not assume uniqueness of the steady state and shows how the dynamic properties are affected by the presence of multiple steady states.

3.1

Bond economy with portfolio costs

Agents face a convex cost for holding/issuing bonds. The collected fees are reim( ) bursed to the agents by a lump-sum transfer Ti (st ). Γ Bi /P¯i denotes the portfolio costs in terms of country i’s traded good, where Γ′ (0) = 0, Γ′ > 0 for Bi > 0, Γ′ < 0 for Bi < 0 and Γ′′ (0) > 0, implying the new budget constraint ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Pi st ci st ≤ P¯i st wi st li st + P¯i st Πi st + Bi st−1 − Q st Bi st ) ( ( ) ( t) Bi (st ) ¯ − Pi s Γ ¯ t + Ti st . Pi (s ) The risk sharing condition thus changes to ∑ [ Uc (c1 (st+1 ) , l1 (st+1 )) Uc (c2 (st+1 ) , l2 (st+1 )) q (st ) ] ( ) β − π st+1 |st t t t t t+1 Uc (c1 (s ) , l1 (s )) Uc (c2 (s ) , l2 (s )) q (s ) st+1 |st ) ) ( ( B1 (st ) B2 (st ) ′ ′ = Γ −Γ . (25) P¯1 (st ) P¯2 (st ) The dynamics of the economy are again summarized by the (new) risk sharing equation (25) and the condition that the excess demand for good 2 is zero. As Γ′ = 0 for Bi = 0 and larger than zero otherwise, equation (25) implies that in any deterministic steady state B1 = B2 = 0. Consequently, if the excess demand 12

function for good 2 has n zeros in q¯, then the model with incomplete asset markets and portfolio costs has n deterministic steady states. Linearized Model To conduct local stability analysis, the model is log-linearized around a deterministic steady state with B1 = B2 = 0. Absent shocks to technology, equations (14) − (16) imply ′ q ) q¯ 1 ˆl1,t = ωq,1 Φ1 (¯ q¯t − ωb,1 [b1,t−1 − βb1,t ] , Φ1 (¯ q) c1 ′ q ) q¯ Φ2 (¯ q) 1 ˆl2,t = ωq,2 Φ2 (¯ q¯t + ωb,2 [b1,t−1 − βb1,t ] , Φ2 (¯ q) q¯Φ1 (¯ q ) c2 Φ′ (¯ q ) q¯ 1 cˆ1,t = [1 + ωq,1 ] 1 q¯t + [1 − ωb,1 ] [b1,t−1 − βb1,t ] , Φ1 (¯ q) c1 ′ q) 1 Φ (¯ q ) q¯ Φ2 (¯ cˆ2,t = [1 + ωq,2 ] 2 q¯t − [1 − ωb,2 ] [b1,t−1 − βb1,t ] , Φ2 (¯ q) q¯Φ1 (¯ q ) c2

(26) (27) (28) (29)

where q¯t denotes the percentage deviation of the relative price q¯ from its steady state value at time t. cˆi,t and ˆli,t are the deviations of consumption and labor from their respective steady state values. b1,t is the absolute deviation of country 1’s bond holdings. If b1,t > 0, country 1 is lending to country 2 in period t. ωb,i and wq,i are given by ωb,i

] Ul,i Ucc,i Ulc,i − Uc,i ] [ = [ Ul,i Ulc,i − Uc,i Ucc,i + Ucl,i −

Uc,i U Ul,i ll,i

−Uc,i l1i ] [ Ul,i Ulc,i − Uc,i Ucc,i + Ucl,i −

Uc,i U Ul,i ll,i

τi = [

[

], ],

ωq,i = −ωb,i + τi . With the assumptions on the utility function U (c, l) stated in (3), which are satisfied by almost all utility functions that are commonly used in macroeconomics, one obtains 0 < ωb,i < 1 and τi > 0. Using equations (26) − (29) the log-linear approximation of the excess demand function in equilibrium can be written as ∂z2 ∂z2 q¯q¯t + [b1,t−1 − βb1,t ] = 0 ∂ q¯ ∂dW1

(30) 13

with

 [

   ( ) ] ′ ′ ∂z2 q ) q¯  1 − ε2 (¯ q) Φ (¯ q ) q¯ ˜ 1 1 1 + 1 + ωq,1 Φ1 (¯ ( ) − ωq,2  2 q¯ = c12 ε1 (¯ q) H + ,  ∂ q¯ q¯ q¯ Φ1 (¯ q) Φ2 (¯ q)  ˜2 1 1 H q¯ q¯ ( ) 1 1 ˜ 2 1 1 A2 l2 H q¯ q¯ ∂z2 A 1 l1 ∂l1 1 ∂l2 1 Φ1 (¯ q) qΦ2 (¯ q) [ [ ( ) ( ) ( ) ( ) ] ] = + + − ∂dW1 ˜ 1 1 + q¯ ∂dW1 l1 ˜ 2 1 1 + 1 ∂dW2 l2 ˜ 1 1 + q¯ ˜2 1 1 + 1 H H H H q¯ q¯ q¯ q¯ q¯ q¯ { } ′ ′ Φ (¯ q ) q¯ Φ (¯ q ) q¯ 1 1 + [1 − ωb,1 ] 1 . − [1 − ωb,2 ] 2 = − q¯Φ1 (¯ q) Φ1 (¯ q) Φ2 (¯ q)

To arrive at the expression for

∂z2 q¯, ∂ q¯

the term in (18) is rewritten using equations

(26) and (27). Log-linearizing the risk sharing equation (25) and using the linearized excess demand function (30) delivers the system of linear difference equations x′t = (¯ qt , bt−1 ) and xt+1 = MP xt , where the coefficient matrix MP satisfies   ∂z2 ∂z2 q¯ ∂dW ∂ q¯ 1 ˇ ∂z ˇ Γ Γ ∂z2 ∂z   1 + ∂dW 2 d − ∂z2 q dq − ∂ q¯2 q¯db ¯db ∂dW1 q ∂ q¯ .  1 MP =  ∂z2  q¯ 1 β

and

∂ q¯ ∂z2 ∂dW1

1 β

[ ] Γ′′ (0) 1 ˇ Γ= 2 1+ , β Φ1 (¯ q) q¯

( )2 ∂z2 ∂z2 1 Φ′1 (¯ Φ′2 (¯ q ) q¯ q ) q¯ dq − q¯db = − 1 + [1 − ωb,1 ] − [1 − ωb,2 ] + Υc12 db < 0, ∂dW1 ∂ q¯ q¯Φ1 (¯ q) Φ1 (¯ q) Φ2 (¯ q)   ( )  ′ ′ ′ ′ Φ1 (¯ q ) q¯ Φ2 (¯ 1 1 Φ1 (¯ q ) q¯ ε2 (¯ q ) Φ2 (¯ q ) q¯ q ) q¯ ˜ ( ) Υ = −τ1 + τ2 − ε1 (¯ q ) H1 + > 0,  Φ1 (¯ q) Φ2 (¯ q) q¯ q¯ Φ1 (¯ q) q)  ˜ 1 1 Φ2 (¯ H 2

q¯

q¯

are composite parameters.3 The sign of the slope of the excess demand function plays the key role in determining the stability of the local dynamics around the steady state. Note that 3

For completeness, [ ( ) ] ′ Ucc,1 c1 Ucc,1 c1 Ucl,1 l1 Φ1 (¯ q ) q¯ 1 db = − − + ωb,1 Uc,1 Uc,1 Uc,1 Φ1 (¯ q ) q¯Φ1 (¯ q ) c12 [ ( ) ] ′ Ucc,2 c2 Ucc,2 c2 Ucl,2 l2 Φ2 (¯ q ) q¯ 1 + − + ωb,2 Uc,2 Uc,2 Uc,2 Φ2 (¯ q ) q¯Φ1 (¯ q ) c12

14

∂z2 d ∂dW1 q ∂z2 . ∂ q¯

−

ˇ= Γ

∂z2 q¯db ∂ q¯ ′′ Γ (0) β 2 Φ1 (¯ q)

is negative for all parameter values and independent of the sign of [ ] ˇ=0 1 + 1q¯ measures the importance of the portfolio cost. For Γ

one obtains the standard linearized international business cycle model with incomplete markets and non-stationary bond holdings. The following theorem summarizes the dynamic properties of the model. Theorem 1 Assume that agents face portfolio costs for holding/issuing bonds as described above. If the slope of the excess demand function is negative in a steady state, then this steady state is a saddle point. If the slope of the excess demand function is positive in a steady state, then such a steady state is unstable if Γ′′ (0) is sufficiently small, i.e., Γ′′ (0) < ∆P . Otherwise this steady state is a saddle point. I use the terms ”unbounded dynamics” and ”unstable steady state” as synonyms. Similarly, ”a saddle-path stable steady state” is also referred to as a steady state that is associated with ”bounded and locally unique dynamics.” Proof 1 The determinacy and the trace of the coefficient matrix MP are det (MP ) = ( ) ∂z2 q¯ ∂ q¯ 1 1 ˇ and tr (MP ) = ∂z2 Γ + 1 + β . Since β < 1, | det (MP ) | > 1. Remem∂z2 β ber that If

∂z2 ∂ q¯

∂z2 d ∂dW1 q

−

∂dW1 ∂z2

∂ q¯

dq −

∂ q¯

q¯db

q¯db < 0.

< 0, tr(MP ) > 1 + β1 > 0 and the modulus of one eigenvalue is larger than

1, while the other one is smaller than 1. Given that bond holdings are the only state variable, the system is saddle-path stable. If

∂z2 ∂ q¯

> 0, the modulus of each eigenvalue is larger than 1 for |tr (MP ) | <

1 + det (MP ), requiring that ∆P ≡ −2β

(1 + β) Φ1 (¯ q) ] [ q¯ 1 + 1q¯

∂z2 2 d − ∂z q¯db ∂dW1 q ∂ q¯ ∂z2 ∂ q¯

> Γ′′ (0) > 0.

and Φ′1 (¯ q ) q¯ Φ′ (¯ q ) q¯ − [1 − ωb,2 ] 2 Φ1 (¯ q) Φ2 (¯ q) ) 2 Ull,i Ucc,i −Ulc,i U li ] [ ωb,i = − [ + Ucl,i Ul,i c,i

dq = −¯ q Φ1 (¯ q ) c12 db + 1 + [1 − ωb,1 ] with db < 0 as −

Ucc,i ci Uc,i

( +

Ucc,i ci Uc,i

Ulc,i − U

c,i

15

] li U Ucc,i + Ucl,i − Uc,i Ull,i Uc,i l,i

> 0.

Otherwise, the modulus of exactly one of the eigenvalues is larger than 1, while the other one is smaller than 1. Hence for Γ′′ (0) < ∆P the system is unstable whenever ∂z2 ∂ q¯

> 0.4 Γ′′ (0) measures the sensitivity of the portfolio costs in the neighborhood of the

steady state. In most applications, this sensitivity is low. If Γ′′ (0) is assumed to be very large, the economy behaves similarly to an economy without international financial markets. In the latter, any steady state is saddle-path stable. Hence, any steady state can be turned into a saddle point in the model with portfolio costs if the marginal costs of portfolio holdings increase strongly enough as the economy deviates from the steady state. However, given that the model with portfolio costs is supposed to behave closely to the original (non-stationary) model, it is common practice to specify portfolio costs that are small and that do not change dramatically in the neighborhood of the steady state. Such specifications are also in line with actual portfolio costs.

3.2

Bond economy with debt elastic interest rate

Under the assumption of a debt elastic interest rate consumers in the two countries face different prices for the bond. The spread between the prices is a function of international bond holdings. Following Devereux and Smith (2007), the interest rate differential is of the form ( ) ( ) ( ( ) ) ¯1 , R1 st = R2 st Ψ B1 st+1 − B

(31)

The roots of the characteristic equation that is associated with M satisfy P (λ) = λ2 −λtr (M )+det (M ) . For convenience, I summarize the necessary and sufficient conditions such that none, exactly one or both eigenvalues λ lie in the unit circle: 4

i. if | det (M ) | < 1 and |tr(M ) | < 1 + det (M ), the modulus of all eigenvalue is smaller than 1, ii. if | det (M ) | > 1 and |tr(M ) | < 1 + det (M ), the modulus of all eigenvalues is larger than 1, iii. if | det (M ) | < 1 and |tr(M ) | > 1 + det (M ) or | det (M ) | > 1 and |tr(M ) | > 1 + det (M ), the modulus of one eigenvalue is larger than 1, while the other one is smaller than 1.

16

¯1 is a reference where the function Ψ (B1 (st+1 )) satisfies Ψ (0) = 1 and Ψ′ < 0. B level of debt for country 1, which is set to zero. When country 1 is a net borrower, it faces an interest rate that is higher than the interest rate in country 2. When country 1 is a lender, it receives an interest rate that is lower. Using the first order condition with respect to bond holdings and the relationship between bond prices and interest rates, R1 (st ) = R2 (st )

∑

1 Ri (st )

= Qi (st ), I obtain

Uc (c2 (st+1 ),l2 (st+1 )) q (st ) π (st+1 |st ) Uc (c2 (st ),l2 (st )) q(st+1 ) ∑ Uc (c1 (st+1 ),l1 (st+1 )) t+1 |st ) st+1 |st β Uc (c1 (st ),l1 (st )) π (s

st+1 |st

β

( ( ) ) ¯1 . (32) = Ψ B1 st+1 − B

The dynamics of the economy are described by (32) and the condition that the excess demand for good 2 is zero, i.e., z2 (¯ q , B1 (st−1 ) − Q1 (st ) B1 (st )) = 0. In a steady state, equation (32) implies B1 = B2 = 0 given the assumption Ψ (0) = 1. Hence, values of q¯ that are solutions to z2 (¯ q , 0) = 0 in the static model, are steady states in the model with a debt elastic interest rate just like in the model with portfolio costs. The linear dynamics of the model with a debt elastic interest rate are given by xt+1 = MB xt , where  Ψ′ (0)  1− β MB =  

∂z2 q¯ ∂ q¯ ∂z2 ∂z2 d − q¯db ∂dW1 q ∂ q¯ ∂z2 1 ∂ q¯ q¯ β ∂z2 ∂dW1

′ − Ψ β(0)

∂z2 ∂dW1 ∂z ∂z2 d − ∂ q¯2 q¯db ∂dW1 q

1 β

  . 

As in the model with portfolio costs the stability of a steady state is linked to the slope of the excess demand function. Theorem 2 Assume that the interest rate differential between the two countries is debt elastic as described above. If the slope of the excess demand function is negative in a steady state, then this steady state is a saddle point. If the slope of the excess demand function is positive in a steady state, then this steady state is unstable if Ψ′ (0) is sufficiently large, i.e., 0 > Ψ′ (0) > ∆D . Otherwise this steady state is a saddle point. 17

Proof 2 The proof follows the same steps as for Theorem 1 with the difference that ∆P is replaced by ∆D where ∆D ≡ 2 (1 + β)

∂z2 2 d − ∂z q¯db ∂dW1 q ∂ q¯ ∂z2 q¯ ∂ q¯

< Ψ′ (0) < 0.

The condition Ψ′ (0) > ∆D implies that the interest rate does not react too strongly to changes in the bond holdings. Hence, to the extent that the model with a debt elastic interest rate is supposed to behave similar to the original model, any steady state for which the excess demand function is upward sloping is unstable.

3.3

Bond economy with endogenous discounting

The concept of preferences with intertemporal dependencies was introduced by Uzawa (1968). Uzawa preferences fall into the broader class of recursive preferences. The subjective discount factor is assumed to be a decreasing function of the period utility level, i.e., agents become more impatient as current utility rises. I consider two specifications. In the first case agents do not take into account the effects of their choices on the discount factor, in the second case they do. No internalization The problem of the representative household is given by ∞ ∑ ∑ ( ) ( ( ) ( )) ( ) θi st U ci st , li st π st max ci (st ),li (st ), t=0 st ci1 (st ),ci2 (st ), Bi (st ) s.t. ( ) [ ( ( ) ( ))] ( t ) θi st+1 = βi U ci st , li st θi s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Pi st ci st ≤ P¯i st wi st li st + P¯i st Πi st + Bi st−1 − Q st Bi st .

with βi′ (Ui ) < 0. The equilibrium dynamics are fully summarized by z2 (¯ q (st ) , B1 (st−1 ) − Q (st ) B1 (st )) =

18

0, where ∑ ( ) [ ( ( ) ( ))] Uc,1 (c1 (st+1 ) , l1 (st+1 )) ( t+1 t ) Q st = β1 U c1 st , l1 st π s |s (33) Uc,1 (c1 (st ) , l1 (st )) t+1 t s

|s

and the risk sharing condition ∑ [ st+1 |st

] [ ( t )] Uc (c1 (st+1 ) , l1 (st+1 )) [ ( t )] Uc (c2 (st+1 ) , l2 (st+1 )) q (st ) ( t+1 t ) β1 U s − β U s π s |s = 0. 2 Uc (c1 (st ) , l1 (st )) Uc (c2 (st ) , l2 (st )) q (st+1 ) (34)

Equation (34) implies that in a steady state the discount factors are equalized across countries [ ( ( ) ( ))] [ ( ( ) ( ))] β1 U c1 st , l1 st = β2 U c2 st , l2 st .

(35)

As βi is strictly decreasing in Ui , the utility function is strictly concave, and the technology is concave, there is a unique allocation and a unique price q¯ that solves (35). The initial allocation of bond holdings is determined from the zero excess demand condition for good 2. In contrast to the two models discussed previously the steady state of the model with endogenous discounting is unique irrespective of the sign

∂z2 . ∂ q¯

Furthermore, this steady state does not necessarily feature zero bond

holdings. However, the functional forms of β1 and β2 can always be calibrated such that the unique steady state features B1 = B2 = 0. The local dynamics around the unique steady state are approximated by xt+1 = ME xt with



 1+ ME =  

where

∂z2 g ∂dW1 q

−

∂z2 ∂z g − ∂ q¯2 q¯gb ∂dW1 q ∂z2 ∂z d − ∂ q¯2 q¯db ∂dW1 q ∂z2 1 ∂ q¯ q¯ β ∂z2 ∂dW1

∂z2 q¯gb ∂ q¯

 0  ,  1 β

= Υc12 gb > 0. Υ and

∂z2 d ∂dW1 q

−

∂z2 q¯db ∂ q¯

are as defined in (31).

Theorem 3 Assume that the agents’ discount factors are endogenous and strictly decreasing in the current utility level. Furthermore, agents do not internalize the 19

effects of their choices on their discount factors. Then the unique steady state is a saddle point irrespective of the sign of the slope of the excess demand function. [ ] ∂z2 ∂z g − ∂ q¯2 q¯gb ∂dW1 q 1 Proof 3 The determinacy and the trace are det (ME ) = β 1 + ∂z2 and ∂z2 tr(ME ) = − 1−β β

∂z2 ∂z g − ∂ q¯2 q¯gb ∂dW1 q ∂z2 ∂z2 d − ∂ q¯ q¯db ∂dW1 q

+ 1 + det (ME ), respectively. Since

d − ∂ q¯ q¯db ∂dW1 q ∂z2 ∂z g − ∂ q¯2 q¯gb ∂dW1 q ∂z2 ∂z d − ∂ q¯2 q¯db ∂dW1 q

<0

irrespective of the sign of the slope of the excess demand function, the modulus of exactly one eigenvalue is smaller than 1. With bond holdings being the only state variable, the dynamic system is saddle-path stable. Notice that it is crucial to assume that the endogenous discount factor is decreasing in the utility level. Otherwise it is

∂z2 g ∂dW1 q

−

∂z2 q¯gb ∂ q¯

< 0 and |tr(ME ) | <

1 + det (ME ) irrespective of the slope of the excess demand function. In this case, both eigenvalues would be larger than 1.

With internalization If agents internalize the effects of their consumption and labor decisions on the discount factor, the risk sharing condition is given by ∑ [ ( ) Uc,1 (st+1 ) − η1 (st+1 ) βc,1 (st+1 ) ] ( ) t+1 t π s |s β1 s t Uc,1 (st ) − η1 (st ) βc,1 (st ) st+1 |st ] ∑ [ ( ) Uc,2 (st+1 ) − η2 (st+1 ) βc,2 (st+1 ) q¯ (st ) Φ1 (¯ ( ) q (st )) Φ2 (¯ q (st+1 )) t = π st+1 |st . β2 s t t t t+1 t+1 t Uc,2 (s ) − η2 (s ) βc,2 (s ) q¯ (s ) Φ1 (¯ q (s )) Φ2 (¯ q (s )) t+1 t s

|s

(36) ηi is the Lagrangian multiplier on the law of motion for the discount factor in country i and it evolves according to ∑ [ ( ) ( ) ( ) ( )] ( ) ηi st = −Ui st+1 + βi st+1 η st+1 π st+1 |st .

(37)

st+1 |st

Again, a steady state requires that the discount factors are equalized across countries, i.e., β1 (U (st )) = β2 (U (st )). Therefore, the model with internalization always has a unique steady state. 20

As equations (36) and (37) reveal this model has two additional state variables, i.e., ηˆ1,t and ηˆ2,t . The linearized dynamic system for the model with endogenous discounting and internalization is hence given by xt+1 = MI xt and x′t = (¯ qt , ηˆ1,t , ηˆ2,t , bt−1 )′ . The 4 × 4 coefficient matrix MI is defined as   1 −zm1 zm2 0   [ ]  aϕ1  m2 1  m1 − zaϕ1 zaϕ1 m1 0  β 1   [ ] MI =  aϕ2 , m1 1  m  −zaϕ + zaϕ 0 2 2 m2 β2 2    ∂z2 q¯  1 ∂ q¯ 1 0 0 ∂z 2 β β ∂dW1

where z =

∂z2 ∂dW1

(

1−

1 β

)

2 (dq + gq + hq ) − ∂z q¯ (db + gb + hb ) ∂ q¯ ] [ ∂z2 q¯ 1 ∂ q¯ , a = 1 + ∂z2 q¯Φ1 (¯ q ) c12 ∂dW

∂z2 ∂dW1

,

1

Φ′i (¯ q ) q¯ , βi Φi (¯ q) ηi βi′ = , 1 − ηi βi′

ϕi = Uc,i ci mi

βi′

for i = 1, 2 and { } η1 β1′ β1′′ Φ′1 (¯ 1 η2 β2′ β2′′ Φ′2 (¯ q ) q¯ q ) q¯ hb = Uc,1 c1 − Uc,2 c2 , ′ ′ ′ ′ 1 − η1 β1 β1 Φ1 (¯ q) 1 − η2 β2 β2 Φ2 (¯ q ) q¯Φ1 (¯ q ) c12 hq = −¯ q Φ1 (¯ q ) c12 hb . To analyze the stability properties of the model with internalization I first define the concept of the direct impact of a wealth increase on the intertemporal marginal rate of substitution. Consider the intertemporal rate of substitution in country 1 given by ( ) ( ( )) 1 − η1 (st+1 ) β ′ (U1 (st+1 )) Uc,1 (st+1 ) IM RS st+1 = β U1 st 1 − η1 (st ) β ′ (U1 (st )) Uc,1 (st )

(38)

The direct impact of a wealth increase on the intertemporal marginal rate of substitution is defined as the change in the IMRS to a change in wealth keeping current 21

and future prices and future allocations constant. Using equations (14) and (16), I obtain ∂IM RS (st+1 ) |direct ∂dW1 (st ) [ ( ) ] ( t) ( t+1 ) Ucc,1 (st ) c1 (st ) Ucc,1 (st ) c1 (st ) Ucl,1 (st ) l1 (st ) 1 = − + + ω s IM RS s 1b Uc,1 (st ) Uc,1 (st ) Uc,1 (st ) c1 (st ) ( ′ ) ( ) ( ) β (U1 (st )) η1 (st ) β ′ (U1 (st )) β ′′ (U1 (st )) + + Uc,1 st IM RS st+1 . (39) t t ′ t ′ t β (U1 (s )) 1 − η1 (s ) β (U1 (s )) β (U1 (s )) The first term in equation (39) measures the direct impact of the wealth transfer on the marginal utility of consumption under the assumption that the marginal rate of substitution between leisure and consumption is held constant. Under the assumptions on the utility function the term is positive. The increase in the wealth of the agents of country 1 lowers the labor supply and increases consumption. Consequently, the marginal utility of consumption, Uc,1 (st ), rises. This effect operates towards a rise of IM RS. The second term measures the effect of the wealth increase on IM RS through the endogeneity of the discount factor. There are two effects. First, as consumption and leisure rise in the current period, so does utility U1 (st ). As the discount factor is decreasing in the utility level this effect operates towards a decline of the IM RS. Furthermore, the change in the discount factor effects the IM RS also through its impact on the discounted future utility summarized in η1 (st ). Absent assumptions on β ′′ this expression cannot be signed. If the discount factor is constant,

∂IM RS1 (st+1 ) |direct ∂dW1 (st )

> 0. Hence, if the discount

factor βi does not react too strongly to changes in Ui (st ), the effect will still be positive leading to theorem 4. Theorem 4 Assume that the agents’ discount factors are endogenous and that agents internalize the effects of their choices on their discount factors. Irrespective of the sign of the slope of the excess demand function, any steady state is a saddle point if the discount factor does not react too strongly to changes in bond holdings. 22

Proof 4 The characteristic equations that is associated with MI simplifies to ( )2 ( [ ] ) 1 1 1 2 − λ− λ − + 1 + za [ϕ2 − ϕ1 ] λ + = 0. β β β With three state variables, ηˆ1,t , ηˆ2,t and bt−1 , the dynamic system is saddle-path stable if the modulus of exactly three eigenvalues is larger than 1. Since two of the four eigenvalues are equal to β1 , stability of the system requires za [ϕ2 − ϕ1 ] > 0 or ) ( 1 . za [ϕ2 − ϕ1 ] < −2 1 + β A sufficient condition for stability is Φ1 (¯ q ) c12 (db + hb + gb ) < 0 as it implies that za [ϕ2 − ϕ1 ] > 0 (db + hb + gb ) q¯Φ1 (¯ q ) c12 ( ) ( ′ ) [ ] ′ Ucc,1 c1 Ucl,1 l1 β1 η1 β1′ β1′′ Φ1 (¯ Ucc,1 c1 q ) q¯ + + ωb,1 + + Uc,1 c1 = − ′ ′ Uc,1 Uc,1 Uc,1 β1 1 − η 1 β1 β1 Φ1 (¯ q) [ ( ) ( ′ ) ] ′ ′ ′′ Ucc,2 c2 Ucl,2 l2 β2 Ucc,2 c2 η 2 β2 β2 Φ2 (¯ q ) q¯ − − + + ωb,2 + + Uc,2 c2 ′ ′ Uc,2 Uc,2 Uc,2 β2 1 − η 2 β2 β2 Φ2 (¯ q) Under the assumption that

∂IM RSi (st+1 ) |direct ∂dWi (st )

> 0, i = 1, 2, each of two the expres-

sions in brackets is positive and therefore Φ1 (¯ q ) c12 (db + hb + gb ) < 0.

3.4

Overlapping generations

Following Weil (1989), Ghironi (2006) shows that a model with an overlapping generations structure can also overcome the non-stationarity problem. Following his work, I assume that each country is populated by a continuum of infinitely lived households of measure Ni (st ) which grows at the exogenous and constant rate n. The key departure from the standard representative agent framework lies in the assumption that newly born households come into being without financial assets. 23

More specifically, at time t0 , the representative consumer in country i born in period v ∈ (−∞, t0 ) maximizes the intertemporal utility function Utv0

=

∞ ∑

[ ( ) ( ( ))] β t−t0 ζ log cvi st + (1 − ζ) log 1 − liv st

t=t0

subject to the intertemporal budget constraint ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Pi st cvi st ≤ P¯i st wi st liv st + P¯i st Πvi st + Biv st−1 − Q st Biv st . (40) The consumption index of a household of generation ν is given by cvi (st ) = Hi (cvi1 (st ) , cvi2 (st )). cvij (st ) is the amount of good j that the representative household of country i born in period v consumes in time t. All other variables are defined analogously. In order to be able to aggregate consumption across generations, the period utility function for households is the log of the Cobb-Douglas function. Following the aggregation steps presented in Ghironi (2006), the equilibrium conditions of the model are given by the market clearing condition for good 2 and a condition that relates cross county consumption dynamics, which is the analogue of the risk sharing condition in the previous models ( )−1 ( )−1 n t+1 n t+1 t+1 t+1 t+1 c (s ) c (s ) c1 (st+1 ) c (s ) q (st ) 1 2 2 1+n 1+n − = − . (41) c1 (st ) c1 (st ) c2 (st ) c2 (st ) q (st+1 ) ci denotes average consumption of households in country i and cti denotes consumption of newly born households. Consumption of newly born households is a (constant) fraction of their human wealth, i.e., cti (st ) = ζ (1 − β) hi (st ) where ( t) ( ( t )) ( t ) ( t ) Pi (st+1 ) ( t+1 ) hi s = Φi q¯ s Ai s + Q s hi s , Pi (st ) ( )−1 n t+1 ( t) c1 (st+1 ) − 1+n c1 (st+1 ) . Q s = β 1 c (st ) 1+n 1

(42) (43)

Intuitively, as newly born households have no financial assets a deterministic steady state requires that international bond holdings are zero. As the model with portfolio 24

costs and a debt elastic interest rate, the overlapping generations model preserves the steady states of the static model. The equilibrium system can be summarized as xt+1 = MO xt and and



1 β

−

− 1−β uβ

1−β uβ

1−β uβ

 ( )  u MO =  e1 1−β − 1 + f1 −e1 + n1 1−β e1 β  ) ( u − 1 − f2 e2 −e2 + n1 1−β −e2 1−β β

x′t

(

ˆ 1,t , h ˆ 2,t , bt−1 = q¯t , h

   , 

with Φ′1 (¯ q ) q¯ Φ′2 (¯ q ) q¯ − , Φ1 (¯ q) Φ2 (¯ q) [ ] ∂z2 q¯ ′ q ) q¯ ζ ∂ q¯ 1 + n 1 − β 1 Φ1 (¯ − , ∂z2 n β u Φ1 (¯ q) c1 ∂dW 1 [ ( ∂z2 q¯ )] a ′ Φ (¯ q ) 1+n1−β 1 Φ (¯ q ) q¯ ζ 1−a 2 ∂ q¯ − 2 − , ∂z 2 n β u Φ2 (¯ q) c2 Φ1 (¯ q ) q¯ ∂dW1 [ ] ∂z2 q¯ ′ ζ ∂ q¯ 1 Φ1 (¯ q ) q¯ −β − , ∂z2 βn Φ1 (¯ q) c1 ∂dW 1 [ ] ∂z2 q¯ a ′ Φ (¯ q ) Φ (¯ ζ 1 q ) q¯ 2 ∂ q¯ − 2 − β 1−a . − ∂z βn Φ2 (¯ q) c2 Φ1 (¯ q ) q¯ ∂dW2

u = 1+ e1 = e2 = f1 = f2 =

1

With some algebra effort, it can be shown that in any steady state [ ′ ] a q) ζ ζ 1−a Φ2 (¯ Φ1 (¯ q ) q¯ Φ′2 (¯ q ) q¯ 1 + =− ζ −ζ , c1 c2 Φ1 (¯ q ) q¯ Φ1 (¯ q) Φ2 (¯ q ) Φ1 (¯ q ) q¯c12 and (

a q) ζ ζ 1−a Φ2 (¯ + c1 c2 Φ1 (¯ q ) q¯

)

∂z2 q¯ ∂ q¯ ∂z2 ∂dW1

> −1.

The dynamic properties of the overlapping generations framework are summarized by: Theorem 5 Assume an overlapping generations structure in which newly born agents have no financial assets. If the slope of the excess demand function is negative in a steady state, then this steady state is a saddle point. If the slope of the 25

)′

excess demand function is positive in a steady state, then this steady state is unstable provided that agents are sufficiently patient. Otherwise such a steady state is a saddle point. Proof 5 The eigenvalues of MO are found as the solution to the characteristic equations 0 = −λ3 { } ) ( 21−β 1 1−β + − (e1 + e2 ) λ2 + − n β β uβ {[ ] ( )2 ( )} 1−β 1 1 1−β 1 2 1−β u λ + (e1 + e2 ) − (f1 + f2 ) − − −1 β n u β n nu β 1−β ( )2 [ ] ( )2 11−β 1 1−β 1 1−β + − + (f1 + f2 ) . (44) n β β uβ nu β 1−β 1 β n

is one solution to the characteristic equation. Thus, ( ) 1 right hand side of (44) can be written as the product of λ − 1−β and β n It is easy to show that λ =

( { ( ) } { [ ] }) 1−β 1 1−β 1−β 1 1−β 1−β 2 λ − + − − (e1 + e2 ) λ + − + (f1 + f2 ) . nβ β uβ nβ β uβ uβ ˜ O that goes along with the second term are Determinacy and trace of the matrix M [ ( ) ∂z2 q¯ ] a ( ) Φ (¯ q ) ζ ζ 1 − β 2 ∂ q¯ ˜O = 1+ + 1−a det M , ∂z2 nuβ c1 c2 Φ1 (¯ q ) q¯ ∂dW1 ( ) ∂z2 q¯ a ( ) ( ) Φ (¯ q) ∂ q¯ ˜ O = 1 − β ζ + ζ 1−a 2 ˜O . tr M + 1 + det M ∂z2 uβ c1 c2 Φ1 (¯ q ) q¯ ∂dW 1

( As

ζ c1

+

ζ c2

a Φ 1−a 2

) (¯ q)

Φ1 (¯ q )¯ q

∂z2 q¯ ∂ q¯ ∂z2 ∂dW1

( ) ˜ > −1 in any steady state det MO > 0

If the slope of the excess demand function is negative in the steady state, i.e., ( ) ( ) ( ) ˜ O > 0 and tr M ˜ O > 1+det M ˜ O . If the excess demand function > 0, tr M

∂z2 q¯ ∂ q¯ ∂z2 ∂dW1

is downward sloping in the steady state, exactly one of the remaining eigenvalues is larger than one in absolute value. If the excess demand function is upward sloping in the steady state, i.e.,

∂z2 q¯ ∂ q¯ ∂z2 ∂dW1

< 0, both eigenvalues are outside the unit circle unless the

26

( following holds (implying that x = { −

ζ c1

+

ζ c2

a Φ (¯ q) 1−a 2

Φ1 (¯ q )¯ q

)

∂z2 q¯ ∂ q¯ ∂z2 ∂dW1

is sufficiently negative)

} 1−β 1−β 1−β x+1+ [1 + x] > 1+ [1 + x] , uβ nuβ nuβ nuβ + (1 − β) x < −2 . (2 + n) (1 − β)

As x > −1, a sufficient condition for both eigenvalues to be outside the unit circle is given by 1 +

Φ′1 (¯ q )¯ q Φ1 (¯ q)

−

Φ′2 (¯ q )¯ q Φ2 (¯ q)

>

1 1−β . 2 β

Hence, unless agents are very impatient steady

states for which the excess demand function is upward sloping are unstable. Thus, the overlapping generations model displays similar properties as the model with portfolio costs or a debt elastic interest rate.

4

Conclusions

Steady state multiplicity implies that the sign of the slope of the excess demand function can be either positive or negative in steady state. The models with portfolio costs, a debt elastic interest rate, or an overlapping generations structure preserve the multiplicity of stead states, whereas the steady state is unique under endogenous discounting. Avoiding functional forms as much as possible, I show with paper and pencil in a model that a steady states for which the slope of the excess demand function is negative, the local equilibrium dynamics are always saddle-path stable. However, if the slope is positive, the equilibrium dynamics are unstable for models that impose a portfolio cost, a debt elastic interest rate, or an overlapping generations structure. By contrast, the local dynamics are saddle path stable for models with endogenous discounting. Table 1 summarizes the results of the analysis:

27

Model portfolio cost # steady states equal to static ( ) 2 sign ∂z <0 ( ∂ q¯ ) 2 sign ∂z >0 ∂ q¯

debt elastic

endog. dcf.

endog. dcf.

overlapping

interest rate

(no int.)

(int.)

generations

equal to static

unique

unique

equal to static

model

model

model

saddle stable

saddle stable

saddle stable

saddle stable

saddle stable

unstable

unstable

saddle stable

saddle stable

unstable

Table 1: Summary of results

28

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