Income Effects, Wealth Effects, and Multiple Equilibria. in Trade Models with Durable Goods

Transcription

1 Income Effects, Wealth Effects, and Multiple Equilibria in Trade Models with Durable Goods By Eric W. Bond and Robert A. Driskill Department of Economics Vanderbilt University December 1, 2007

2 I. Introduction One of Koji Shimomura s contributions to the theory of international trade was to extend the standard two good, two country static model of international trade to a dynamic model in which one of the goods was a durable consumption good. His first paper on the topic (Shimomura (1993)) emphasized the notion that if countries had identical and homothetic preferences, this model would generate predictions about the pattern of trade in the steady state that were similar to those obtained in static models. The primary role of durability in this case is to introduce new parameters (i..e. the rate of time preference and the rate of depreciation) as potential sources of comparative advantage that are absent from the case where goods are non-durable. In contrast, his second paper (Shimomura (2004)) suggested that the presence of a durable good could introduce a troubling lack of predictability into the standard trade model when tastes differed across countries. In that paper, he analyzed a case in which the durable good is inferior in one of the countries and showed that the model could have multiple steady states. Furthermore, one of the steady states could exhibit dynamic indeterminacy. Shimomura s result on the multiplicity of equilibria is reminiscent of the results from static trade models that multiple equilibria may arise when the exporting country has a higher propensity to consume its export good than does the importing country. In the static model, an increase in the relative price of a good results in substitution away from the good in all countries, but also transfers income from the importing country to the exporting country. When substitution effects are small and the difference in marginal propensities to consume is large, an increase in price can actually increase in the excess demand for the good as shown in Jones (1961). When the aggregate excess demand function has upward sloping regions, the result may be multiple equilibria. Shimomura s results on multiplicity of equilibria arise from a similar pattern of income effects, since in his model the exporter of the durable good has a marginal propensity to consume it of unity and the importer has a negative marginal propensity to consume. However, his result that there are a continuum of paths leading to the middle steady state equilibrium 1

3 contrasts sharply with the results from the static model, where such equilibria are always unstable. The latter result suggests that the dynamic model introduces additional issues of market stability that are not present in the static model. Our goal in this model is to explore the role of income effects in determining the number of and stability of steady state equilibria in a model with durable goods. We first examine the case in which trade must balance at each point in time, which is the case considered by Shimomura. We show that the steady states of this model can be illustrated using steady state offer curves, and that the effects of steady state prices on these offer curves can be decomposed into income and substitution effects as in the static model. We also show that the possibility of multiplicity of steady state equilibria arises when a country has a higher marginal propensity to consume its exportable than does its trading partner. As long as the consumption good is not inferior, the steady state pattern of trade between the countries can be predicted from the comparison of the relative prices in the autarkic steady state. These results suggest a strong link between the role of income effects in static models and those in dynamic models with a durable good in determining the uniqueness of the steady state equilibrium when trade is balanced at each point in time. We also discuss the relationship between the number of equilibria and the stability of the various equilibrium points, and show that there is no necessary relationship between the slope of the aggregate excess demand functions at a steady state equilibrium and the dynamic stability of the steady state. We show that examples can be constructed in which a dynamically unstable steady state is bounded on each side by a steady state that exhibits saddle path stability, a result similar to the outcome in the static model. However, Shimomura s example indicates that this is not the only possibility. One of the complicating factors that is introduced in the dynamic model is that an increase in the price of a good at a point in time leads to an incentive for intertemporal as well as intratemporal substitution. In the model with trade balance, the possibilities for intertemporal substitution are limited by the inability to smooth consumption over time through international capital markets. Therefore, we also 2

4 examine the uniqueness and stability of steady states in the case where countries have access to international capital markets. The opening of international capital markets allows an efficient allocation of the durable good across countries at each point in time, which implies that the shadow value of an increment of capital must grow at the same rate in each country at each point in time. There will be a continuum of steady state consumption allocations that are consistent with a competitive equilibrium with international capital markets, and which of these allocations is realized will depend on the initial distribution of the durable good across countries. Thus, a country that initially receives a relatively large fraction of the durable good (relative to its share of world output) will run a trade surplus in the steady state. Interestingly, the introduction of international capital markets ensures that any steady state equilibrium will exhibit saddle path stability. The remaining issue to consider is whether the can be more than one path consistent with equilibrium in the presence of perfect capital markets. We first show that if preferences are homothetic and exhibit a constant elasticity of intertemporal substitution, there can be only one path consistent with an initial distribution of the durable good. The uniqueness of the equilibrium path follows from the fact that prices are independent of the distribution of income between countries, because each country has the same marginal propensity to consume out of wealth for each good at every point in time. In contrast, multiple steady states may be consistent with an initial distribution of the durable good if preferences differ across countries sufficiently that there are multiple equilibria with balanced trade. II. A Two Country Dynamic Model of Trade with a Durable Good We analyze a continuous time two country dynamic model in which there are two sectors, one that produces a non-durable good, denoted by c, and the other that produces a durable consumption good, denoted A (autos). The flow of utility obtained at a point in time in country i is represented by the concave i i i function u (c, A ). Country i is assumed to have a fixed output of good j (j = c, A) at each point in time. 3

5 Letting denote imports of good j at time t, the level of consumption of the non-durable will be and the evolution of the stock of the durable good will be It will be assumed in this section that there is no international lending and borrowing, so that trade must balance in each period. Choosing the durable good as the numeraire and letting p(t) be the relative price of c, the trade balance condition requires that (1) Given a path p(t) for the world price of the non-durable, the excess demands for country i will be the solution to the optimization problem (2) Letting ë be the costate variable associated with the durable good, the necessary conditions require that and. These conditions can be used to show that along the opti marginal rate of substitution between the non-durable and the durable will be equated to the relative price of the respective service flows, (3) The denominator on the right hand side of (3) is the relative service price of the durable good in country i, which consists of the interest and depreciation costs less the rate of capital. The rate of increase in the 4

6 costate variable, which is the shadow value of an increment of the durable good, can be interpreted as the capital gain on holding the durable good in country i. The relative price on the right hand side will be equated across countries only if the costate variable is growing at the same rate in each country. The price path p(t) will be an equilibrium for the world economy if the solutions to (2) satisfy for all t. A steady state equilibrium for the world economy will be a price and stocks such that for i = 1,2. Letting denote the excess dema country by country i in the steady state, the steady state stock of capital in country i will be. Evaluating (3) at the steady state values yields of c and (4) In the steady state, the service price of the flow of services is equated across countries under the assumption of identical discount factors and depreciation rates. Equations (1) and (4) can be solved to yield steady state excess demand functions, which are the levels of net imports of good j by country i that are consistent with optimizing behavior at a constant relative price for the non-durable and per period trade balance. In deriving these import demand functions, it will be useful to write the trade balance condition as T = pm + m. This balance condition allows us to capture i i i c A an income effect by allowing for an income transfer of T to country i. Totally differentiating this trade balance condition and (3), we obtain the effect of a change in the steady state price and the level of transfer on steady state imports of good c,. (5) 5

7 where must be negative by the quasi-concavity of preferences. The second term in parentheses will be positive if, which will be referred to as the case in which good c is a normal good. This terminology is chosen because good c will be a normal good in the standard two good static model iff i. It can be seen from (5) that this condition will ensure that Ä > 0, and also that ain increase in the steady state transfer of income to country i will result in an increase in imports of good i. Equation (5) provides a decomposition of the effect of a price change on excess demand into substitution effects and income effects that is similar to that obtained in the static case. Assuming that the consumption good is non-inferior, which guarantees Ä < 0, the first term in the effect of a price change in (5) must be negative. This term is the substitution effect, which captures the effect of a price change at a fixed level of purchasing power. The second term in parentheses is the effect of a change in income on steady state imports of c. Since c is assumed to be a normal good, then the income effect will tend to reduce (raise) the demand for good i in response to a price increase if good c is imported (exported) by country i. A similar decomposition can be derived for the effect of a change in p on steady state excess demand for the durable good. These excess demand functions can be used to derive a steady state offer curve diagram for country 1, which is illustrated in Figure 1. The dotted line 0A denotes the trade balance constraint for 0 i country i when the steady state price is p and T = 0. Point A represents the chosen trade bundle { } under the assumption that country 1 is an exporter of good c in the steady sta Figure 1 also illustrates a representative trade indifference curve over the steady state bundles, derived from. It should be noted that in contrast to the standard static offer curve deriva 6

8 slope of the trade indifference at the chosen bundle for country 1, pä/(ñ+ä), is flatter than the slope of the price line. This lack of tangency reflects the fact that the optimization problem does not maximize the value of steady state consumption. Since the slope of country 2's trade indifference curve will also be pä/(ñ+ä) at the consumption point, the steady state equilibrium consumption bundle will be on the contract curve for the steady state level of the durable good as indicated by (3). Thus, the lack of tangency does not represent an inefficiency in the allocation of goods between countries. Country 1's offer curve is shown by the curve 0AB, which is the locus of trade bundles obtained as i the relative price p is varied. The autarkic steady state price, denoted q, is the solution to. The excess demand function must be decreasing in p in the neighborhood of, since the substitution effects dominate the income effects when trade volumes are sufficiently small. This also ensures that the autarkic steady state equilbrium price is unique. Assuming that good A is also a normal good, must be increasing in p because the substitution and income effects will work in the same direction for the importable. Combining these results establishes that the steady state offer curve for country 1 must be upward sloping in the neighborhood of. However, could be decreasing in p if the income effects are sufficiently large, leading to a backward bending offer curve as illustrated in Figure 1 at point A. A. Existence and Uniqueness of Steady State Equilibria A price will be a steady state equilibrium if it satisfies. The of the excess demand function and the uniqueness of the autarky price that > (< 0) for p < (>). Assuming WLOG that the autarky price of c is lower than in country 1, there will exist a p satisfying E(p) = 0. Furthermore, it must be the case that country 1 exports the consumption good at the steady state. This establishes the existence of a steady state equilibrium with trade, and that the steady state trade pattern can be predicted from the comparison of the autarkic steady state prices when the non-durable is not inferior. 7

9 This equilibrium value will be unique if E (p) < 0 for all p, where (6) The first term in parentheses is the sum of substitution effects across countries, which must be negative. The second term in parentheses is proportional to the difference in marginal propensities to consume good c between the exporting country and the importing country. A sufficient condition for E (p) < 0 is that the marginal propensity to consume good c be higher in the country that is importing good 2. Shimomura (1993) examined the case in which the marginal propensities to consume were identical in the two countries, which guarantees the existence of a unique solution. In Shimomura (2004), the marginal propensity to consume good c is zero in the importing country and exceeds 1 in the exporting country. This creates the possibility of an equilibrium with E (p) > 0 if substitution effects are sufficiently small. Asimple example can be used to illustrate the possibility of multiple equilibria. Suppose that home 2 * 2 country preferences are given by U = c + á0a - á1a and foreign preferences by U* = A + â0c* - â1c*. If endowments are chosen such the home country has comparative advantage in good c, then each country has a marginal propensity of 1 for consuming its exportable. Figure 2 illustrates the offer curves for a parameterization for which there are 3 steady state equilibria. Home exports (foreign imports) of good c are measured along the horizontal axis and home imports (foreign exports) of good A are measured along the vertical axis. The autarky steady state relative prices are.043 in the home country and.506 for the foreign country. The steady state equilibrium prices in this example are p = {.081,.139,.286}. Since the home country is the exporter of good c, its steady state consumption of both goods is higher the higher is the steady state price of the good c. 8

10 B. Local Dynamics around Steady States The possibility of an upward-sloping aggregate excess demand function for good c results from the fact that an increase in the relative price of c will transfer income to the country with a higher marginal propensity to consume c. With a Walrasian price adjustment process that raises the price of good c when it is in excess demand, stabilility of the equilibrium requires E (p) < 0. In the case where income effects are such that there are three equilibria, the middle equilibrium will be unstable and hence would not be observed. However, the equilibria with the highest and lowest prices would both be stable, and theory does not provide guidance as to which of these extreme equilibria would be observed. Uniqueness and stability of equilibrium of market equilibrium is ensured by the assumption of identical and homothetic preferences, which eliminates the income effects on aggregate demand resulting from a price change in (6). In contrast to the static model, where the adjustment process is exogenously imposed, the dynamic model provides a characterization of the adjustment process as part of the solution to the country s 1 2 optimization problem. If we assume that the initial distribution of the durable good satisfies A + A = ù /ä, the evolution of the state variables ë, ë and A are given by W A (7) i Utilizing the necessary conditions u - ë p in the respective countries and the world market clearing condition i c 1 for the consumption good, we can solve for the values of c and p that satisfy the necessary conditons as a function of the state variables. Inserting these expression, c (ë, ë, A ) and p(ë, ë, A ), into (7) yields a 3 9

11 equation system of differential equations. 1 The usual approach to stability of a steady state equilibrium in the dynamic model is to linearize the dynamic system (7) and evaluate its properties at the steady state equilibrium. If the steady state equilibrium is a saddle point, then there will be unique path along which the system evolves on its path to the steady state. This is the analog of stability in the static model, and will arise if the Jacobean of the linearized sysem has one negative (real) root and two roots with positive real parts. A second possibility is that the steady state is unstable, in which case any values starting in the neighborhood of the steady state would move away from that equilibrium. This corresponds to the unstable equilibrium of the static model. In Bond and Driskill (2007) we examine the issue of dynamic stability in the model with quasi-linear preferences discussed in relation to Figure 2. We show that in the case where there are 3 steady equilibria, the high and low price steady states are saddle points but the middle price equilibrium point is an unstable point. For initial endowments for the durable that are sufficiently high (low), the only stable path in this case is to the highest (lowest) equilibrium price. For some intermediate values, on the other hand, there will be stable paths leading to both the lowest and highest price steady states. This outcome is close in spirit to that obtained in the static model, where an unstable equilibrium is bounded above and below by stable equilibria. We also show that if the home utility function is transformed by taking log(u), then the high and low price equilibria are saddle points but the middle equilibrium is a sink. The middle price equilibrium will thus exhibit dynamic indeterminacy, in that there will be a continuum of paths leading to the steady state as in Shimomura (2004). This represents an additional form of multiplicity beyond that involving the 1 W W In the more general case where A ù A /ä, the evolutions of the world stock of durables is given by. This equation can be combined with (7) to obtain a four equation system. Since the W differential equation governing the evolution of A is independent of the other state variables, the eigenvalues of this 4 equation system will consist of the three eigenvalues of (7) plus ä. Thus, the dynamics are effectively those of the 3 equation system. 10

12 steady states, since the path that approaches a particular steady state is not unique. These examples are special in that they use the fact that the preferences of one of the countries is linear in the durable good, which means that the value of the costate variable will be constant at all points in time. This simplifies the dynamics to allow treatment of a two equation system. The fact that even with these simplifications we cannot obtain general conclusions about the relationship between the slope of the aggregate excess demand function at the steady state and the dynamic stability of the equilibria suggests that general results are unlikely to be obtained. However, the examples do point to the importance of the intertemporal substitution in dynamic behavior, since the log preferences that generate indeterminacy have a lower degree intertemporal substitution than do the linear preferences. This contrasts with the static model, where the intertemporal elasticity of substitution plays no role. The importance of intertemporal substitution also suggests that it is useful to extend the analysis to the case in which countries can borrow internationally. III. Perfect Capital Markets Case In this section we examine the case in which there is an international capital markets in which households can lend or borrow at a world interest rate r(t). Country i will then face an intertemporal budget constraint requiring that the present value of its trade deficit, evaluated using the discount factor from world capital markets, equal 0. (8) Assuming that the existing stock of durable goods can be moved costlessly between countries at a point in time, the market clearing condition for durables at a point in time will require that, whe A W(t) is the world stock of durables at time t and. 11

13 With this formulation of the market for the durable good, it is convenient to use the fact that to rewrite (8) as (9) i where A 0 is country i s initial holding of the durable good. The left hand side of this expression is country i s wealth, which is the sum of the value of the initial stock of durables and the present value of its endowment income. The right hand side is the present value of expenditure on consumption of non-durables and the services from durables, where the cost of capital services is r(t) + ä. Note that with perfect international capital markets, we can treat households as renting the service of durable goods in competitive 2 i rental markets whose rental prices are equalized across countries. Country i will choose its time path of c i and A to maximize the discounted value of utility subject to its budget constraint (9), where prices r(t) and p(t) are treated as exogenously determined on world markets. A competitive equilibrium in this case will be a solution to the following social planner s problem: (10) for the appropriate choice of. Letting ø be the multiplier associated with the constraint on country 2's 2 The ability to move durables across countries can be thought of as arising from the operation of international markets for used durable goods. A similar interpretation of the durable good market can be applied to the analysis of the previous section, although in that case the rental price will differ across countries because the capital markets are not integrated and interest rates may differ across countries. Similarly, the inability to borrow i in the previous section prevented jumps in A (t) because countries must finance all purchases of durables out of current income. 12

14 utility, the necessary conditions for c and A yield The marginal rate of substitution between non-durables and durables will be equalized at each point in time with perfect capital markets, so that the allocation at each point in time a point on the contract curve for allocation of the given world supplies of the non-durable, ù + ù, and the durable, A (t), between the two 1 2 W c c countries. This contrasts with the case in which trade must balance in each period, where the equalization of marginal rates of substitution occurs only in the steady state. By inverting the necessary conditions for choice of c and A in the planner s problem, we can W 1 express the efficient allocations to country 1 as a function of ø and A in a competitive equilibrium, c (ø, W 1 W A ) and A (ø, A ). It is straightforward to show that the utility of the allocation to country 1 (country 2) is decreasing (increasing) in ø, which reflects the fact that variations in ø sweep out the contract curve 1 W 1 W between 1 and 2. In addition, it can be shown that c (ø, A ) [A (ø, A )] will be decreasing in ø if the non- durable [durable] good is a normal good in both countries. Letting ì be the costate variable associated with W A in the planner s problem, the solution to the planner s problem yields the dynamic system (11) describing the evolution of the state and costate variables. With international lending, the dynamic system is simplified because the dynamics can be described by the evolution of the world capital stock and its costate variable. The distribution of capital across countries at a point in time does not matter to the properties of the dynamic system because of the ability to reallocate the existing stock of capital across countries at a point in time. The characteristic roots of the dynamic system in (11) will be ñ + ä and - ä, so the steady state will have a saddle path for any value of ø. Under the assumption of fixed outputs of the durable good, the steady state level of the world capital stock must be (ù + ù )/ä for all ø. The planner s solution will require equating the marginal rates 1 2 A A 13

15 of substitution across countries, which can be used to solve for the price of the non-durable that will decentralize the planner s solution in a competitive equilibrium, (12) Differentiation of (12) establishes that p (ø) > 0 iff country 2 has a higher marginal propensity to consume good c. This can also be used to define the steady state trade deficit for country 1, (13) Note that if T(ø) = 0, then that corresponding price p(ø) is a steady state equilibrium price for the case in which trade balance must hold in every period. Thus, the steady state equilibria derived in the previous section will also be competitive equilibria for the model with international lending and borrowing for some sets of initial conditions. However, (13) indicates that there will also be a continuum of solutions to the planner s problem in which there is a steady state trade imbalance. Using (11), the above solution to the planner s problem yields time paths of consumption levels in i W i W each country, {c (ø, A (t)), A (ø, A (t))}, that are parameterized by the weight placed on country 2 welfare and the initial value of the world capital stock, A W 0. The country optimization problem can then be used to W W derive the time path of prices, {p(ø, A (t)), r(ø, A (t))}, for which these consumption choices will be optimal. Substituting these solutions into the wealth constraint (9), this time path of consumption will be a i competitive equilibrium for initial stocks of the durable good A 0 that satisfy 14 (14)

16 If the right hand side of this expression is monotonic in ø, then there will be a unique competitive equilibrium price path from a given initial condition. A. Steady State Paths with Identical and Homothetic Preferences We first consider the case in which the utility function in each country is, where ö is homogeneous of degree one in c and A and f is a concave function. The marginal rate of substitution can then be expressed as, where h < 0. Using this property in the necessar W W conditions for the social planner s problem yields the result that for all values of ø, c /A = c /A = ù c /A at each point in time. The implied price of services of the non-durable relative to the durable in a market equilibrium, p/(r+ä) = h(ù W c W /A ), will thus be independent of the value of ø. The index ö can be i W i W W interpreted as a composite consumption good, with â (ø, A ) = c (ø, A )/ù c denoting country i s share of world consumption in the solution to the planner s problem. 1 2 With homothetic preferences, the necessary condition for choice of c requires that f (ö ) = øf (ö ). In the case where these preferences exhibit a constant intertemporal elasticity of substitution, for ã 0, this condition simplifies to ö /ö = ø 1 2-1/ã. In this case country 1's share of world consumption will be constant along any optimal path, so that country 1 s share of world consumption can be expressed as a i decreasing function of ø, â (ø). It then follows that the path of prices and interest rates can be expressed as W W p(a (t)) and r(a (t)), since prices and interest rates will be independent of the world distribution of income. This follows from the fact that with a constant intertemporal elasticity of substitution, the marginal propensity to consume out of wealth will be the same in all countries. Therefore, redistribution of wealth between countries will have no impact on aggregate demands. Since country 1's share of world consumption is decreasing in ø and prices are independent of ø, the 1 right hand side of (14) must be decreasing in ø. It then follows that for a given initial endowment pair {A 0, 15

17 2 A 0 }, there will be a unique value of ø at which (14) is satisfied. This establishes that there is a one to one relationship between initial endowments of the durable goods and competitive equilibrium paths when preferences are identical and homothetic and exhibit a constant elasticity of intertemporal substitution. B. Heterogeneous Preferences We conclude by showing that when marginal propensities to consume differ across countries, there may be multiple steady state equilibria that are consistent with an initial distribution of the capital stock. We illustrate this using the example with quasi-linear preferences illustrated in Figure 2, U = c + á0a - á1a 2 * 2 and U* = A + â0c* - â1c*.. Solving the necessary conditions for choice of c and A in the social planner s W -1 problem yields c = ù c + (â1ø) - â 0/â 1 and A = (á 0 - ø)/ á 1, where ø is the weight placed on foreign welfare in the planner s problem. The consumption of the home country is constant for all t in a solution to the planner s problem, and consumption of both the durable and non-durable is decreasing in ø. Since the W world supply of c is a constant, c* = ù c - c will also be a constant in any solution. Foreign country consumption of the durable will be increasing (decreasing) over time in an optimal solution if the world * supply is increasing (decreasing). Since foreign country welfare is linear in A, its consumption will vary over time to allow the home country to smooth its consumption of services of the durable good. The fact that the home country has an interior solution for c along the optimal path means that r(t) = ñ in any decentralized solution. Since U c/u A = 1/(á 0 - á1a) = p/(r+ä) in a decentralized solution, we have p = (ñ+ä)/ø in a competitive equilibrium that yields the home consumption levels associated with the solution to the planner s problem. This result reflects the fact that a higher welfare level for the foreign country must be associated with a lower price for the home country s exportable in a competitive equilibrium. Substituting these prices into (14), we can solve for the value of the home country s initial endowment that are consistent with competitive equilibrium for a particular value of ø. The potential non-monotonicity of this expression in this case follows from the fact that higher value of ø is associated with a lower 16

18 expenditure level for the home country, but also with a lower relative price for its export good. Figure 3 illustrates the relationship between ø and [(p(ø)(c(ø) - ù ) + (ñ + ä)a(ø) - ù ]/ñ, which is c A the present value of the difference between home country expenditure and consumption services and endowment income on the right hand side of (14), evaluated at the parameter values used in Figure 2. For min max A (A, A ), there will be three values of ø that are consistent with he home country budget constraint. 0 These three values of ø correspond to competitive equilibria, with p = (ñ+ä)/ø along the equilibrium path. The multiplicity arises because the value of home country wealth depends on p, which affects world demand for the consumption good. An increase in the expected future value of p will raise the wealth of the home country, which is an exporter and p, and reduce the wealth of the foreign country. This wealth effect will have a positive effect on the world demand for the durable, since the home country has a higher marginal propensity to consume c out of wealth than does the foreign country. This creates the possibility of multiplicity of equilibria if substitution effects are low, an effect which is similar to that obtained in the case 3 where trade must balance at each point in time. Note however that this multiplicity will not arise for all values of A. For A > A 0 0 max, the wealth of the home country has a sufficiently large component of the durable good that the home country lends is a net lender. Exports are sufficiently low in this region that the terms of trade effect of changes in c are small. In fact, for sufficiently large A 0 the home country will become a sufficiently large lender that it becomes an importer of c on the path to the steady state. For A < A 0 min, the initial wealth of the home country contains very small component of the durable good. In this region the consumption for the non-durable is low and demand is more elastic in this region. As a result, there is a monotone relationship between ø and next expenditure in this region. This example illustrates how heterogeneity of tastes can lead to multiple equilibrium paths in the 3 Since foreign welfare is linear in A*, the initial foreign endowment, A 0*, will play no role in the issue of whether there are multiple equilibria. 17

19 case where there are perfect capital markets and countries have a higher marginal propensity to consume their export good. It is useful to compare the dynamic results for this case with those obtained in Bond and Driskill (2007) using the same set of home and foreign preferences, but without international capital markets. In that case the possibility also arose that there would be multiple paths from an initial endowment to a steady state. However, the convergence could only be to the extreme equilibria (i.e. highest price and lowest price) when there were three steady state equilibria. Convergence to the middle equilibrium was not possible because it was dynamically unstable. IV. Conclusions Our results illustrate the role played by income and wealth effects in generating multiple steady state equilibria in international trade models with durable goods. In the absence of international capital markets, the trade balance requirement will be imposed. Our results showed that in the case where trade must balance at each point in time, there will in general be only a finite number of steady state allocations of the durable good between countries that are consistent with a steady state. We have used steady state offer curves to show that as long as the consumption good is not inferior, the conditions on income effects required for the equilibrium to be unique are essentially the same as those in the static model. The major difference that arises when durable goods are introduced is that a richer set of dynamic outcomes is possible for describing the behavior of the economy in the neighborhood of the steady state. In particular, knowing the slope of the aggregate steady state excess demand function is not sufficient to characterize the local properties of the steady state equilibrium, in contrast to the static model with a Walrasian price adjustment process. One question for future work would be to derive a sufficient condition for saddle path stability when the steady state equilibrium is unique. In the presence of international capital markets, in contrast, we find a continuum of possible allocations of the stock of the durable good that are consistent with a steady state. Each of these outcomes 18

20 exhibits saddle path behavior, with the particular steady state outcome depending on the initial allocation of the durable good between countries. When preferences are identical, homothetic and exhibit a constant intertemporal elasticity of substitution, the path of prices will be independent of the initial allocation of the durable between countries and there will be a unique steady state associated with each initial value for the stock. However, multiple steady states may be consistent with a given initial point when preferences are heterogeneous. The explanation for this multiplicity is similar to that obtained in the static model. The potential for multiplicity requires countries to have a higher marginal propensity to consume their export goods. The evidence on home bias in consumption, suggesting that suggests that this is an 19

21 References Bond, Eric W and Robert Driskill (2007), Multiplicity and Stability of Equilibrium in Trade Models with Durable Goods, Vanderbilt University, manuscript. Jones, Ronald W (1961), Stability Conditions in International Trade: A General Equilibrium Analysis, International Economic Review, 2, pp Shimomura, Koji (1993), Durable consumption goods and the pattern of international trade, in Trade, Welfare, and Economic Policies: Essays in Honor of Murray C. Kemp, H. Herberg and N.V.Long eds., Michigan University Press Shimomura, Koji (2004), Indeterminacy in a dynamic general equilibrium model of international trade, in The Development Process of Rapidly Growing Economies: From Theory to Empirics, M. Boldrin, B-L Chen and P. Wang eds., Cheltenham, UK, Edward Elgar Publishing Inc, Chapter7,

22 Figure 1 Steady State Offer Curve for the Exporter of Good c 21

23 Figure 2: An Example of Multiple Steady State Equilibria with Quadratic Preferences 2 * 2 Home Preferences: U = c + 6A - 3A ; Foreign preferences: U* = A + 2.4c* - c* * * ñ =.02, ä =.2, ù c = 5, ù c =.1, ù A = 1/16, ù A = 1; 22

24 min max Figure 3 Multiple Steady State Paths with International Capital Markets for A (A, A ) 0 Parameters values as in Figure 2 23