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Real exchange rate volatility and international trade: a reexamination of the theory.

I. Introduction

One of the issues that have received considerable attention in the comparison of the properties of alternative exchange rate regimes is the effect of exchange rate risk on the volume of trade. It has been argued that the higher volatility of exchange rates witnessed since the adoption of the floating regime in 1973 has led to a decrease in international trade transactions. This is because most trade contracts are not for immediate delivery of goods; and since they are denominated in terms of the currency of either the importer or the exporter, unanticipated fluctuations in the exchange rate affect realized profits and hence the volume of trade. It is implicitly assumed that forward exchange markets that can help traders eliminate this type of variations in profits either are not available (as it is true for the majority of currencies because most are not fully convertible, thereby impairing forward markets) or for some reason they are not utilized to fully hedge exchange risk present in trade transactions.(1)

The empirical evidence, regarding the effect of exchange rate risk on trade, has at best been inconclusive. The large majority of the empirical studies are unable to establish a systematically significant link between exchange rate variability and the volume of international trade whether on an aggregate or on a bilateral basis. Abrams |1~, Akhtar and Hilton |2~, Cushman |4; 5; 6~ and Kenan and Rodrik |12~ find some significant negative effects of exchange volatility on exports. However, Bailey, Tavlas, and Uhlan |3~, Hooper and Kohlhagen |10~ and an International Monetary Fund Study |11~ do not find any supporting evidence for the depressing effect of exchange rate volatility on international trade. It is also interesting to note that, in many of these studies, a significant positive effect of exchange rate volatility on the volume of trade is found for some cases. However, the positive effect, believed to be at odds with the theory was either ignored or dismissed as a perverse result, since "as far as volumes are concerned, theoretical considerations are unambiguous in suggesting that increased uncertainty should reduce the level of trade" |11, 18~.

The purpose of this paper is to show that a positive effect of exchange rate variability on trade has a theoretical basis.(2) There can be no theoretical presumption that an increase in exchange risk has an adverse effect on trade. The key to this claim is the fact that nominal unhedged trade contracts are standard risky assets that can be analyzed in a conventional asset portfolio model. Consequently, whether an increase in the riskiness of the return on these assets--that is, an increase in the volatility of the exchange rate--increases or decreases investment (trade), will in general depend on the risk aversion parameter of the model. The existing work on the effects of exchange rate uncertainty on trade has employed, as recognized by Hooper and Kohlhagen |10~ a restrictive version of portfolio choice which leads to an unambiguously negative relation. Our theoretical analysis implies that the empirical evidence, rather than being a puzzle, can be reinterpreted as saying something about attitudes towards risk. Given the widely held presumption that exchange rate volatility is detrimental to the volume of trade, we feel that this paper makes a worthy contribution.

II. The Basic Model

In this paper we exploit the similarity of trade decisions to the portfolio-savings decisions under uncertainty. The model we employ is one with incomplete asset markets and ex ante trading decisions, in which the choice of exports(3) is made before the resolution of uncertainty in prices. To facilitate the exposition we analyze the behavior of a small open economy, first assuming that no forward exchange markets are available; later we introduce a forward market but we require the payment of a fee (commission) for participation in forward transactions. In both cases we show that increased riskiness affects the volume of trade, but the sign of this effect is ambiguous depending on the risk aversion parameter.

We employ the standard set up for a small country as in Cushman. Let the domestic country be endowed with some good Y, and the rest of the world be endowed with some different good Z. The representative domestic agent lives for two periods but consumes only in the second period. In the first period, t, she receives her share of output and decides how much of the domestic and foreign goods to consume in the future (t + 1). Because delivery of the domestic good takes time, she has to sign a trade contract now that specifies the quantity(4) of the exported good to be delivered in the next period. The terms of trade are uncertain(5) but their probability distribution is known. Imports in period t + 1 are financed by the contemporaneous sale of the domestic good; that is the trade balance is zero.

The domestic individual chooses an amount of exports X to maximize expected utility

q = Eu(Y - X, PX)

where P is the terms of trade (real exchange rate) and Y - X is consumption of the exportable good. The first order condition is:

E(-|u.sub.1~ + P|u.sub.2~) = 0

where |u.sub.1~ and |u.sub.2~ are the derivatives of the utility function with regard to the first and second arguments respectively.

The question is how a mean preserving increase in the riskiness of the real exchange rate affects the volume of trade (X). To answer this question we need to determine whether g = |u.sub.2~P - |u.sub.1~ is a convex or concave function in P |13~. If g is convex (concave) then an increase in riskiness increases (decreases) X. The condition for convexity is X|2|u.sub.22~ + XP|u.sub.222~ - X|u.sub.112~~ |is greater than~ 0 which in general depends on the form of the utility function employed (|u.sub.ii~ and |u.sub.iii~ are the second and third derivative of u with regard to the ith argument respectively). For instance(6), if the utility function is of the constant relative risk aversion family, (1/1 - |Alpha~)|C.sup.1-|Alpha~~, where C is consumption and |Alpha~ is the coefficient of risk aversion, an increase in riskiness decreases the volume of trade if and only if the coefficient of relative risk aversion is less than unity.

The intuition behind this result is straightforward. P here represents the return on the investment in X. As explained in |14~, an increase in risk makes the consumer less inclined to expose his/her resources to the possibility of a loss. This effect implies a decrease in X. On the other hand, higher riskiness makes it necessary to commit more resources to savings (sell more X) to protect oneself against very low consumption of the imported good in the next period. Which effect dominates depends on risk aversion.

Risk aversion also determines the relationship between the quantity of output committed to trade (X) and its expected return (the expected terms of trade, P). When the affect of uncertainty is positive (|Alpha~ |is greater than~ 1), the wealth effect from an expected improvement in the terms of trade dominates the substitution effect, leading to a decrease in exports (X).

III. A Forward Exchange Market

Transaction costs for relatively small volumes of forward exchange purchases are quite substantial. We will now show that the set of exporters that faces nonzero transaction costs will choose to only partly hedge exchange risk through participation in the forward market; this means increased riskiness will have an effect on the volume of trade.

The domestic agent chooses his/her optimal portfolio of trade contracts to maximize expected utility

Eu (|C.sub.1~, |C.sub.2~)

subject to the constraints

|C.sub.1~ = |Y.sub.1~ - |X.sub.1~ - |X.sub.2~

|C.sub.2~ = |P.sub.1~|X.sub.1~ + |P.sub.2~|X.sub.2~

where |X.sub.1~ is the number of units of the exportable good that is contracted to be delivered in period t + 1 at an unspecified (spot) price, |P.sub.1~, which is unknown as of period t; the remaining units of exports, |X.sub.2~, when delivered, will be sold at a prespecified forward price,(7) |P.sub.2~, which is set in period t. In other words, |X.sub.1~ units are subject to exchange risk (|X.sub.1~ is a risky asset), while |X.sub.2~ units are completely hedged against variations in the price of exportables (|X.sub.2~ is a safe asset).

Let |Mathematical Expression Omitted~. To simplify things we will assume that the rest of the world is risk neutral so there is no risk premium on forward contracts. Our results would remain intact if instead of a transactions cost we admitted the existence of a risk premium. The price specified in forward contracts is then |Mathematical Expression Omitted~ where R is a constant(8) transaction cost.

Substituting for |C.sub.1~ and |C.sub.2~ in the utility function and maximizing with respect to |X.sub.1~ and |X.sub.2~ we have

|Mathematical Expression Omitted~.

PROPOSITION I. If R = 0 and |Mathematical Expression Omitted~ then |X.sub.1~ = 0 (complete hedge).

Proof. If R = 0, (1) reduces to |Mathematical Expression Omitted~. This implies that |Mathematical Expression Omitted~ since |Mathematical Expression Omitted~. But |Mathematical Expression Omitted~. If hedging is completely costless, then risk averse individuals can completely eliminate all the risk that is due to real exchange rate changes by selling forward all their exports at the expected future price. Consequently, the volatility of the real exchange rate (terms of trade) does not affect the volume of trade.

PROPOSITION II. If R |is greater than~ 0 then |X.sub.1~ |is not equal to~ 0.

Proof. Now (1) becomes |Mathematical Expression Omitted~. Hence |X.sub.1~ |is greater than~ 0.

This is the standard case of portfolio choice with one risky (|X.sub.1~), and one risk free asset (|X.sub.2~). The analysis reduces to determining the sign of the second derivative of the g function with regard to the terms of trade. Again the results are ambiguous depending on the risk aversion parameter. For instance if the utility function is |Nabla~ = (1/1 - |Alpha~)|C.sup.1-|Alpha~~ and |Alpha~ |is less than~ 1 then increased riskiness decreases total exports (X) and also decreases the ratio of unhedged exports to total exports (|X.sub.1~/X). For |Alpha~ |is greater than~ 1 higher riskiness increases X but has an ambiguous effect on |X.sub.1~/X.

IV. Extensions

The analysis so far has been conducted using a single individual who does the importing and exporting and the consuming of both products. In addition, there is no production in the model. In this subsection we investigate whether the main result on the ambiguity of the relationship between uncertainty and the volume of trade still obtains when one looks more explicitly at the exporting decisions of the producers.

The simplest case to consider is that of a domestic producer who has a fixed supply of a good which she can sell at either a known and fixed price on the domestic market or an uncertain price on the foreign market (due to exchange rate uncertainty and the absence of a forward market). Suppose there is a mean preserving spread in the distribution of the exchange rate. Will this induce her to shift her sales more toward the domestic market?

Nominal profits are given by the following equation

|Pi~ = p(1 - s)x + ep*sx

where |Pi~ is profits, x is the (fixed) supply of the good, p its domestic nominal price (measured in units of the domestic currency) and p* its foreign nominal price (in units of the foreign currency); e is the exchange rate; s is the share of the good that is sold on the foreign and 1 - s the share sold on the domestic market. We assume that p and p* are known and fixed,(9) that e is uncertain and that the exporting decision must be made before the resolution of uncertainty (that is the exported is not paid immediately when the order is placed).

Let real profits be defined as |Pi~ = |Pi~/p and also let q = ep*/p. The producer chooses s to maximize the expected value of a concave function G of real profits, i.e |max.sub.s~ G(|Pi~). The first order condition is

EG|prime~|(1 - s)x + sx~(q - 1) = 0

where G|prime~ is the derivative of G with regard to s. Diamond and Stiglitz |8~ and Levhari |13~ present methods for studying mean preserving spreads can be used to examine the effects of an increase in exchange rate volatility on the volume of trade (sx). Again the effect depends on whether the function G|prime~( )(q - 1) is convex or concave in s which in turn depends on the properties of the risk aversion function.

For example, suppose that the profit function is given by the constant relative risk aversion function G(|Pi~) = |1/(1 - a)~||Pi~.sup.1-a~. It can be easily calculated that the sign of the second derivative of the first order condition, that is sign {|d.sup.2~(G|prime~(|Pi~)(q - 1)/d|q.sup.2~)} = sign {s(q - 1)(a - 1) - 2}. A sufficient condition that a mean preserving spread in the distribution of the exchange rate reduces sales abroad is that a |is less than~ 1 (low risk aversion). For large values of a one cannot rule out the possibility that higher uncertainty will be associated with an increase in exports.

In the example we have just gone through, there is still no production. We now briefly describe the effects of greater uncertainty in a model with two goods and an endogenous production decision.

Let the domestic producer engage in the production of two goods, y which is a domestic import competing good selling at a fixed (domestic currency) nominal price of p, and x which is the exportable selling abroad at the price of ep* (where the exchange rate, e, is again assumed to be uncertain). Let y = f(x) be the production transformation curve. Nominal profits are then given by the equation

|Pi~ = |pf(x) + ep*x~.

The producer chooses a production level x (and hence y) to maximize the expected value of a concave function of real profits, EG(|Pi~), where |Pi~ = |f(x) + qx~, q = |ep*.sub.y~/p. The first order conditions are given by the following equation

EG|prime~||Pi~~|f|prime~(x) + q~ = 0.

Again the same method can be used to establish the ambiguity in the direction of change in the volume of trade. There is absolutely no qualitative difference between this case and the earlier one of the exchange economy.

The same technic can be used to deal with the effects of uncertainty in more involved environments. For instance, let the domestic producer engage in the production of two goods, y which is a domestic nontradeable selling at a fixed (domestic currency) nominal price of |p.sub.y~, and x which can be sold at home at the fixed nominal price of |p.sub.x~ or abroad at the price of |ep*.sub.x~ (where the exchange rate, e, is again assumed to be uncertain). Nominal profits are then given by the equation

|Pi~ = ||p.sub.x~(1 - s) + |ep*.sub.x~s~x + |p.sub.y~f(x).

The producer is now choosing x and s. Again, with the use of specialized profit and production functions one can examine the effects of a spread in the distribution of e.

1. The bid ask spread in forward markets is in general larger than in spot markers. Moreover, if the hedge decision in endogenous and trade is only partly hedged due to a risk premium, it is not clear that exchange rate volatility is detrimental.

2. De Vries and Viaene |7~ construct a mean-variance model which generates both positive and negative effects. Their model, however, emphasizes the role of the net trade position rather than risk aversion issues.

3. We could have alternatively assumed ex ante import decisions without affecting any of the main points. Notice, however, that in the incomplete asset market set up of our model, we cannot allow the writing of contracts on both importables and exportables because this may violate the zero balance of trade assumption.

4. Specifying both quantity and price would amount to the implicit assumption of the availability of a forward market. It would be feasible to allow for more realistic contracts which specify both quantity and price. This would, however, require modifying our framework to make riskiness operate thorough the currency denomination of the trade contract (as they do in practice). While both models are very similar and have identical implications for trade, the latter is harder to analyze because it includes nominal risk so one needs to justify the presence of money in the economy (i.e., through a cash-in-advance constraint or other acceptable mechanisms).

5. Generally speaking price uncertainty has to be induced by more basic stochastic elements, like technological or monetary disturbances, since prices are endogenous to the world economy. However, since our model is essentially a partial equilibrium (the small open economy assumption), the source of uncertainty need not be specified.

6. If the utility function is of the form

U = |(Y - X).sup.a~|(PX).sup.b~,

then the condition for convexity is given by

|X.sup.2~|(b/X) - a/(Y - X)~b(b - 1)|(Y - X).sup.a~|(PX).sup.b-2~ |is greater than~ 0.

The response of trade volume to exchange rate volatility depends, in addition to the parameters of the utility function, on the volume of trade.

7. In reality, forward exchange markets determine the nominal, rather than the real exchange rate. Ours is a simplifying assumption whose realism depends on the significance of other sources of real exchange rate variability. With relatively stable (sticky) or predictable goods nominal prices, a nominal exchange contract will approximate a real exchange contract.

8. We can make R a function of the size of forward transactions, |X.sub.2~, without affecting any of the main results.

9. We assume that p and p* are fixed with regard to changes in s in the general equilibrium of the model so that we can study mean preserving spreads. Otherwise, one would also get changes in the mean (the relative price ep*/p).

References

1. Abrams, Richard K., "International Trade Flows under Flexible Exchange Rates." Federal Reserve Bank of Kansas Economic Review, March 1980, 3-10.

2. Akhtar, Akbar M. and R. Spence Hilton, "Effects of Exchange Rate Uncertainty on German and U.S. Trade." Federal Reserve Bank of New York Quarterly Review, Spring 1984, 7-16.

3. Bailey, Martin J., George S. Tavlas, and Michael Ulan, "Exchange Rate Variability and Trade Performance: Evidence for the Big Seven Industrial Countries." Weltwirtschaftliches Archiv, 1986, 466-76.

4. Cushman, David O., "The Effects of Real Exchange Rate Risk on International Trade." Journal of International Economics, August 1983, 45-63.

5. -----, "Has Exchange Risk Depressed International Trade? The Impact of Third Country Exchange Risk." Journal of International Money and Finance, September 1986, 361-79.

6. -----, "US Bilateral Trade Flows and Exchange Risk during the Floating Period." Journal of International Economics, May 1988, 317-30.

7. De Vries, Casper and Jean Marie Viaene, "International Trade and Exchange Rate Volatility." European Economic Review, 1992.

8. Diamond, Peter and Joseph Stiglitz, "Increases in Risk and Risk Aversion." Journal of Economic Theory, June 1974, 337-60.

9. Gotur, Padma, "Effects of Exchange Rate Volatility on Trade: Some Further Evidence." International Monetary Fund Staff Papers, September 1985, 475-512.

10. Hooper, Peter and Steven W. Kohlhagen, "The Effect of Exchange Rate Uncertainty on the Prices and Volume of International Trade." Journal of International Economics, November 1978, 483-511.

11. International Monetary Fund. "Exchange Rate Volatility and World Trade." International Monetary Fund Occasional Paper 28, July 1984.

12. Kenen, Peter B. and Dani Rodrik. "Measuring and Analyzing the Effects of Short-Term Volatility in Real Exchange Rate." Working Papers in International Economics, G-84-01, Dept. of Economics, Princeton University, March 1984. An abridged version in the Review of Economics and Statistics, May 1986, 311-15.

13. Levhari, David. "Optimal Savings and Portfolio Choice under Certainty," in Mathematical Methods in Investment and Finance, edited by Giorgio Szego and Karl Shell. Amsterdam: North-Holland, 1972, pp. 34-48.

14. Sandmo, Agnar, "The Effect of Uncertainty on Saving Decisions." The Review of Economic Studies, June 1970, 353-60.
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Author:Zilberfarb, Ben-Zion
Publication:Southern Economic Journal
Date:Apr 1, 1993
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