A stochastic dynamic model of trade and growth: Convergence and diversification

Abstract

There is a growing literature that studies the properties of models that combine international trade and neoclassical growth theory, but mostly in a deterministic setting. In this paper we introduce uncertainty in a dynamic Heckscher–Ohlin model and characterize the equilibrium of a small open economy in such an environment. We show that, when trade is balanced period-by-period, the per capita output and consumption of a small open economy converge to an invariant distribution that is independent of the initial wealth. Further, at the invariant distribution, there are periods in which the small economy diversifies. Numerical simulations show that the speed of convergence increases with the size of the shocks. In the limit, when there is no uncertainty, there is no convergence and countries may specialize permanently. The paper highlights the role of market incompleteness, as a result of the period-by-period trade balance, in this setup. Through an analytical example we also illustrate the importance of country specific risk in delivering our results.

Introduction

The theory of neoclassical growth has traditionally been developed in the closed economy environment. Though, there has been some early attempts to integrate international trade and neoclassical growth (see for example, Bardhan, 1965, Oniki and Uzawa, 1965), it is only more recently that there has been a surge of interest in such dynamic trade models. However, most of the papers in this research area have restricted attention to the dynamics of an open economy in an environment without uncertainty. In this paper we analyze a stochastic dynamic model which integrates neoclassical growth and trade and characterize the equilibrium of a small economy in such an environment. We use this model to address two central issues—convergence of income, a recurring question in the growth literature and diversification of trade, a common concern in the trade literature. In doing so, we illustrate the interplay of trade, diversification of production and uncertainty in an incomplete markets setting.

Specifically we study the properties of a dynamic Heckscher–Ohlin model with uncertainty. Just as in the one-sector neoclassical growth model, we obtain income convergence across countries. We show that, when trade is balanced period-by-period, a standard assumption in deterministic Heckscher–Ohlin models, the per capita output and consumption of a small open economy converge to an invariant distribution that is independent of the initial wealth. Further, we find that when income of an economy is within the invariant distribution, there will surely be some periods in which the small economy diversifies.

The closed economy neoclassical growth theory, which has been a standard vehicle to answer the question of income convergence across countries, predicts that, as long as countries have the same preferences, technologies, and population dynamics, they will converge to the same level of per capita income from any (positive) initial wealth. The closed economy version of the neoclassical growth model has been extended to incorporate uncertainty by Brock and Mirman (1972), who show that countries will converge to the same invariant distribution of income irrespective of their positive initial wealth.

Bardhan (1965) and Oniki and Uzawa (1965), who study dynamic Heckscher–Ohlin models with fixed savings rates, were the first to study integrated models of growth and trade. More recently Chen (1992), Ventura (1997), Deardorff (2001), Cunat and Maffezzoli (2004b) and Bajona and Kehoe (2010) have used two-country models while Atkeson and Kehoe (2000) have used a small open economy model to study the dynamics of an open economy.¹ These recent papers have used deterministic dynamic Heckscher–Ohlin models—that is models with two or more tradable commodities produced with neoclassical production functions that differ in capital intensities—and have found that it is possible that income levels across countries do not converge (see Bajona and Kehoe, 2006 for more details about conditions under which countries do not converge in a two-country model). Although the models vary in details, trade-induced factor-price equalization, which leads to existence of multiple steady states and specialization, is a common feature across all of them.

In these models factor-price equalization ensures that the rate of return on capital is same in all countries that have their aggregate capital–labor ratios within the diversification cone. As a result, if the world economy, which is within the cone of diversification, is in a steady state, then all countries with different aggregate capital–labor ratios within the diversification cone must be in a steady state. Outside the diversification cone the rate of return depends on the country's own capital, and the diminishing returns to capital ensure that countries that start with a low capital–labor ratio outside the diversification cone, will grow until they reach the lower boundary of the diversification cone. Such countries will never enter the interior of the diversification cone and will permanently specialize in the production of tradable goods that are less capital intensive. The case for the countries which start with a capital–labor ratio above the diversification cone is symmetric. Thus, the initial conditions will determine the fate of a country in the long run.

What difference does uncertainty make? In our model, with country specific shocks and market incompleteness initial conditions do not matter. When the country's capital–labor ratio is outside the cone of diversification, the diminishing returns to its own capital push it towards the boundary of diversification cone. Inside the cone of diversification the rate of return is independent of the small country's capital, but the real rate of return still depends on the country's shocks. Now, since different countries face different shocks there would be mutual gains through risk-sharing if countries could borrow from and lend to each other. Borrowing constraint as a result of period-by-period trade balance prevents that.² Under these circumstances, with i.i.d. income shocks, the representative household in a small open economy self-insures by accumulating more capital when current income is high and de-accumulating capital when current income is low. Once the country reaches the diversification cone, a long sequence of negative (positive) shocks will push capital holdings of the representative household beyond the lower (upper) boundary of the diversification cone, where further declines (increases) in capital are resisted by changes in the returns to capital. As a result of these effects all small economies converge to the same invariant distribution of capital with its support including the entire diversification cone. Thus, the long run average income of the small economies will be the same and, further, those economies will diversify production at least in some periods.

We also simulate our model to determine how the speed of convergence depends on the size of the shocks that the economy faces. We find that the bigger the shocks are, the faster is the convergence. In the limit, when uncertainty vanishes, convergence disappears. This suggests that, if uncertainty is small, initial conditions play an important role in the development of a country: it takes a long time for initially poor countries to catch up with richer countries. The simulations, thus, provide a sense of continuity between the deterministic and the stochastic models. The simulations also illustrate the difference in the shape of the policy function between the deterministic and the stochastic models. Simulations are also useful to get a sense of what happens when the stochastic process is Markovian instead of i.i.d. Our theoretical results in this paper are proved using an i.i.d. process, but we find, in simulations, that even economies that face persistent shocks exhibit similar properties of convergence and diversification. Thus, it is likely that the results in this paper will carry over to an economy that faces persistent shocks.

In order to show the importance of the country-specific shocks we construct an example in which all small economies face the same world-wide productivity shock. In this example, we show analytically, that the small open economies, starting from different initial conditions never converge. This example makes it clear that it is not just the uncertainty that matters.

Our paper closely relate to the papers studying dynamic trade models mentioned earlier. There is another paper which requires mention. Datta (1999) also analyzes the income dynamics under uncertainty in a multi-sector small open economy. Datta considers a general environment with production sets and exploits its equivalence to the one-sector stochastic growth model to prove the convergence of income process. Datta's environment, however, does not allow for nontradable goods and for primary fixed factors, like labor. Further, she does not study, nor it is easy to extend her model to study, the dynamic interaction between capital accumulation and trade patterns. In contrast, we focus our attention on a neoclassical model with fixed factors and with nontradable goods, standard in growth and trade literature. This enables us to study patterns of growth, trade and specialization and relate our results to the existing literature.

Our paper, as explained earlier, also relates to the literature on income fluctuations problem, which studies savings decisions under uncertainty and market incompleteness. Clarida (1987), Aiyagari (1994), and Chamberlain and Wilson (2000) are just a few examples of papers that fall into this category.

This paper is organized as follows. In Section 2, we describe our model's environment. In Section 3, we report the equilibrium results for the model, including those on convergence and diversification. In Section 4, we simulate the model and discuss the speed of convergence. In Section 5, we present the analytical example with the world-wide shocks. Finally, we conclude in Section 6. All the proofs are collected in the Appendix.