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Self-Adjusting Free Trade:
A Generally Mathematically Impossible Outcome
October, 2009
Abstract
In this paper we show that even if: a) Marshall-Lerner conditions are universally satisfied, b) bi-lateral trade universally responds effectively and efficiently to exchange rate fluctuations, and c) freely floating exchange rates react quickly and in a normal direction to trade imbalances, free market forces cannot be expected to produce balanced and sustainable international trade. We do this by showing that: a) if there is a balanced trade solution to an exchange rate based international trading system this solution must be unique, and that b) the unique solution to a free trade system is generally mathematically unstable. The principled theoretical issue addressed by this paper is whether freely floating exchange rates and normal exchange rate response to trade deficits can, under the most ideal assumptions, produce a balanced international free-trade regime. If this is not possible, the free trade doctrine has no theoretical legitimacy.
JEL Classification: International Economics (F), Neoclassical Models of Trade (F11), Foreign Exchange (F31), Current Account Adjustment; Short-Term Capital Movements (F32)
Key Words: International Trade, Free Trade, Managed Trade, Exchange Rates, Fair Trade
1. Introduction
Can international trade be managed through a self-correcting free trade regime?
Conventional wisdom, going back at least to David Ricardo (1817), holds that, at least under properly highly idealized conditions, this is possible. However, in an earlier paper, we have shown that Ricardos model is in fact mathematically over-determined and thus generally has no feasible solution (citation omitted). Other analyses of international trade similarly refute the notion that freely floating exchange rates can produce balanced world trade see for example (Blecker, 1999) Eatwell and Taylor (2000), (Taylor, 2004), (Fletcher, 2004), (Shaikh, 1999, 2007).
The following paper attempts to add to the cumulative weight of these and other theoretical and empirical critiques of free-trade by addressing the purely logical issue of whether freely floating, or administered (but not coordinated across multiple countries), exchange rates can produce a stable, balanced, and sustainable international trade solution even if all of the standard assumptions are fully satisfied.
We do this by constructing a simple, but realistic and completely generalizable, short-term demand model, to show that free trade based on floating exchange rates will not generally produce sustainable balanced international trade. The basic problem is that though there is always a unique single internationally balanced trade solution this unique solution will not, except through extreme coincidence, be arrived out through free trade because it will always be (barring an almost impossible accidental configuration) mathematically unstable. Moreover, for the same reason, administrative efforts by individual nations to set their own exchange rates so as to achieve balanced trade, will almost always fail to produce world wide balanced trade, as exchange rate movements that may improve bilateral balance for individual nations are not generally consistent with the movements necessary to generate international balance. We conclude that sustainable benefits from international trade generally cannot be realized through purely market based free-trade, or through nationally administered exchange rate adjustment systems that are focused on bi-lateral trade balances, no matter how perfectly calibrated and responsive these might be.
Our argument proceeds as follows. In Section 2 we first show that any solution to a general multiple country and multiple good international trading system must be unique. This is demonstrated by analyzing the characteristics that any system of exchange rate based import demand equations would have to exhibit. In Section 3, we construct a three country international trade model and show that the unique international balance trade solution cannot consist of three bi-lateral trade balances. In Section 4 we show that the alternative possibility of a unique solution with three bi-lateral trade imbalances is generally unstable and thus also not a viable economic solution. In Section 5 we show that the same is true for a general N country model. The unique solution cannot be one of universal balanced bi-lateral trade as this is again over-determined and mathematically infeasible, so that the unique solution must include at least three off-setting bi-lateral trade imbalances. But this solution is inherently unstable and thus highly unlikely to be obtained, or maintained, in a free trade system. In Section 6 we conclude that market-led freely floating exchange-rate adjustments, even under the most ideal free trade assumptions, cannot be expected to lead to a balanced international trading system.
The principled theoretical issue addressed by this paper is whether freely floating exchange rates and normal exchange rate response to trade deficits can, in theory, produce a balanced international free-trade regime. If this is not possible, the free trade doctrine has no theoretical legitimacy.
2. Exchange Rate Based International Trading Systems Have, at Most, One Balanced-Trade Solution
In the general, exchange rate based, N country import demand model, each country will produce, consume, export and import, Ni types of goods and services, of which Mi =< Ni are (at least partially) imported. Each country will therefore have mi import demand equations.
The free trade doctrine assumes that, under appropriate conditions, exchange rate adjustment will cause international trade to balance. The implicit assumption here is that if Marshall-Lerner conditions hold, and exchange rates are free to float and quickly respond to trade fundamentals in a normal way so that they depreciate in response to trade deficits and appreciate in response to surpluses, there will be price, or exchange rate, clearing of international trade. This implies that exchange rate adjustment will offset other factors influencing import demand and cause a free-trade system to move toward an equilibrium position that will be obtained under ceteris paribus conditions - if other non-price shift factors remain constant long enough.
The degrees of freedom calculus in this general case is as follows. Each country will have Mi (i = 1, ,N) import quantity variables and thus Mi import equations. In reduced form each equation will consist of an import quantity demanded that will be dependent on the N-1 exchange rates (one country will have a numeraire international medium of exchange currency), and on parameters or shift factors Qi that affect import demand for this good or service but that are independent of the exchange rates. Examples of these might include: income, income distribution, taste, and price elasticities of imports and exports. All told we will then have EMBED Equation.3 import quantities demanded as dependent variables, and EMBED Equation.3 independent parameters along with N-1 independent exchange rate variables, for a total of:
(2.1) EMBED Equation.3 EMBED Equation.3 +N-1
variables.
On the other hand, as each import quantity demanded in each country will have to satisfy an import demand equation, and each country (except for one see below) will have to satisfy a trade balance equation with all of its trading partners. Therefore for internationally balanced trade to occur we will have EMBED Equation.3 demand equations and N-1 trade balance equations for a total of:
(2.2) EMBED Equation.3 + N-1 equations.
Note that though trade must be balanced for each of the N countries, balanced trade in N-1 countries will cause trade to be balanced in the Nth country. Thus there only N-1 independent trade balance equations.
Due to the great diversity between countries there is no reason to believe that any commodity import demand equation for any country, including all of its long-term and other non-price related parameters, can, in a general case, be derived from the commodity import demand equations of other countries. We can therefore safely assume that in general the equations counted in (2.2) will all be independent equations.
As can be readily seen, subtracting (2.2) from (2.1) gives us exactly (but no more than) the EMBED Equation.3 degrees of freedom that we need for a general solution to this system that does not constrict the values of the parameters or shift-factors Qi in any way. Disregarding these longer term, non-price related, or at least short-term non-price related, parameters, we have a system of EMBED Equation.3 + N-1 independent equations and EMBED Equation.3 + N-1 unknowns.
This implies that if a solution exits, it will be unique, i.e. there is a single unique vector of exchange rates and import quantities demanded for any given set of non-exchange-rate dependent (at least in the short-term) parameters that will generate international balanced trade. As has been noted, given the diversity between nations dependent equations and redundant solutions where the imports of some countries can be derived from those of others, are not generally plausible.
This will be true for a linear demand system, as well as for localized solutions for non-linear demand systems as solutions to these systems will have to satisfy linear systems of partial differential equations to which the same degrees of freedom calculus will apply. It will also be true for any system of exchange rate based import equations regardless of whether these are based on a standard Neoclassical trade model with: preferences, technologies, and factor endowments given; Hecksher-Ohlin Neoclassical trade models which assume equal preferences and technologies but different factor endowments across countries; or a Romerian Neo-Marxist unequal exchange model which assumes highly unequal capital stock endowments but market-led price-based, or exchange rate, clearing of international trade balances in the short-term (Findlay, 1995) (citation omitted). Any international trade model which assumes price, or exchange rate, trade clearing, subject to other assumptions (for example regarding technology and preferences; or technology, and capital stock) and other possible long-term shift factors (like growth, real wage changes, or distributional changes in output or income), will satisfy (2.1) and (2.2).
3. A Three Country Free Trade Solution Cannot Include Any Bi-Lateral Trade Balances
The fact that a unique possible balanced trading solution may exist does not in itself mean that market forces will move international trading toward this solution. If the unique solution is not one that free market forces would lead to, there is no reason to believe that an equilibrium free trade solution can ever be obtained.
Assume three countries: Portugal (P) which uses Euros (E), United Kingdom (K) which uses pounds ( ) , and US (U) which uses Dollars ($). Assume that the Dollar ($) is the international currency,and that exchange rates are: y=$/E and x=$/ .
Assume the following trade flows (all in measured in importers currencies) shown in Figure 1 below:
Uimports c from P and d from KP imports b from U and a from KK imports e from P and f from U
Trade balance equations (converted to $ with importers paying in domestic currency) are then: exports imports (3.1) U: yb + xf = d + c (3.2) P: c + xe = ya + yb (3.3) K: d + ya = xf + xe
This confirms the dependence of the trade balance equations as if the balances for U and Pare added together and yb + c is subtracted from both sides, the balance for K falls out.
(3.1) to (3.3) are purely static mathematical constraints that any exchange rate or price- based solution to world trade must satisfy. Under ideal free trade conditions (3.1) (3.3) are supposed to come into balance based on exchange rate fluctuations. The exchange rate adjustments in-turn are supposed to be driven by the supply and demand for each currency as determined by the needs of trading partners, so that exchange rates gravitate toward values that equalize bi-lateral trade between countries.1
More specifically we assume that under free trade there will be normal exchange rate effects, i.e. that trade deficits will cause currency depreciation and trade surpluses currency appreciation. These exchange rate effects are supposed to be a direct result of trading needs. If country A has a surplus with country B, exporters from A to B will have more of Bs currency from sales receipts than exporters from B to A will have of As currency. As exporters from A to B have to pay producers in country A in As currency for these goods, and vice versa for exporters from B to A, both sets of exporters need to exchange most of their sales receipts back to the currency of the country in which the goods were produced. Country As trade surplus with country B will thus result in an excess supply of currency B and demand for currency A, leading to a depreciation of currency B and an appreciation of currency A.2
Normal effects also imply that currency depreciation will reduce a trade deficit, and currency appreciation will shrink a trade surplus, and that these free trade induced exchange rate effects will be strong enough and occur quickly enough to induce free market self-adjustment toward balanced trade. Standard trade texts invariably claim that this normal response will occur when the Marshall-Lerner conditions are satisfied.
We note in passing that as these conditions do not take into account trade induced changes in overall output (they implicitly assume Says law), this is not generally true, especially for smaller developing countries see for example Taylor (2004, p. 253-257). However, for the purpose of this paper we will ignore these Keynesian effects of trade imbalance and show that even if the M-L conditions are satisfied and produce normal, timely and effective, trade responses, a free trade equilibrium will not generally obtain.
Under these conditions there are two possible types of equilibrium solutions to (3.2) (3.3). Either every bi-lateral trade relationship will be balanced:
(3.4) yb = c
(3.5) d = xf
(3.6) ya = xe
or all three of these bi-lateral trading relationships will be out of balance. Below we show that either all three of (3.4), (3.5) and (3.6) are satisfied or none are. It is not possible to have only one or two equalities, or inequalities.
The special case of (3.4) (3.5) is not generally feasible as based on (2.2) we would then have for the three country case:
(3.7) EMBED Equation.3 + 3
equations, where we are substituting the three free market trade balance equations (3.4) (3.6), for N-1 or two independent generic trade balance conditions, for example: (3.1) and (3.2), as stipulated by (2.2).3
The number of unknowns, however, remains as calculated in (2.1). For the three country case the number of variables will be:
(3.8) EMBED Equation.3 EMBED Equation.3 +2
including import quantities demanded, the long-term parameters, and the two exchange rates.
This leaves us with one missing degree of freedom as (3.8) minus (3.7) gives:
EMBED Equation.3
So that in this case international free trade can only be balanced if one of the long-term parameters (for example an import price elasticity for a particular country and commodity) is a function of the values of the other parameters.
Moreover, note that as the number of free trade bi-lateral trade balancing constraints like (3.4) (3.6) increases, degrees of freedom are reduced as equations like (4.3) (4.5) below are set to zero rather than to an arbitrary constant, causing an increasing over-determination problem for the system.
4. Any Feasible Three Country Free Trade Solution Is Generally Unstable
We can therefore assume that the unique equilibrium trade solution includes imbalanced bi-lateral trade balances for at least one of the three, possible, country pairs.
Suppose, without loss of generality, that the unique international trade solution is one where U has a trade surplus with P.4 For (3.1) to remain in balance, this imbalance has to be perfectly offset by trade with Us only other partner K, so that (3.1) remains in balance:
(4.1) UP: yb > c
(4.2) KU: d > xf
Such that:
(4.3) U: yb c = d xf
Where () designates the unique balanced trade equilibrium values generated by exchange rates: y and x, leading to equilibrium import (or export) quantities demanded: b, c, d and f.
Moreover, this solution will also determine equilibrium values for ao and eo since for the international system to be in balance, P will than have to have just the right surplus with K, so that (3.2) will hold:
(4.4) KP: ya < xe
Such that:
(4.5) P: xe ya = yb - c
(4.3) and (4.5) will then guarantee:
(4.6) K: xe - ya = d xf
so that (3.3) will hold as well. This shows that one bi-lateral trade imbalance inevitably leads to three bi-lateral trade imbalances.
But can such a solution be a stable equilibrium?
In static terms the excess Euro demand for Dollars generated by (4.1) will equal the excess Dollar demand for Pounds generated by (4.2), and both of these (in Dollar terms) will equal the excess Pound demand for Euros (notice that in this case P has a surplus with K) from (4.4), so that if the excess Dollars from (4.2) were exchanged for the excess Euros from (4.1), and these were then exchanged for the excess Pounds from (4.4), all parties would be satisfied and exchange rates would remain at levels satisfying (4.1), (4.2), and (4.4).
Problems appear however, when one looks at the detailed behavioral dynamics that are supposed to make this work. First note the holders of the excess Euros from (4.1) want Dollars, the holders of the excess Dollars from (4.2) want Pounds, and the holders of the excess Pounds from (4.4) want Euros so that none of these currency quantities demanded match-up. They are equal (in dollar values) but not in the barter, or elementary Says Law sense, of an exchange of shoes for bread which produces an inseparable demand for bread and supply of shoes, and demand for shoes and supply of bread.
This implies that for the exchange rates that lead to the system (4.1) (4.6) to remain constant, or for supply and demand for each currency to match, an international medium of exchange, say gold, must be introduced so that Euros can be exchanged for gold and then gold for Dollars, Dollars can be exchanged for gold and gold for Pounds, and Pounds for gold and gold for Euros, cancelling out all the intermediate gold transactions and balancing the exchange rates.5
The ultimate source of demand for, or supply of, any currency is thus irrelevant.6 Currency affects that originate from any one of (4.1), (4.2), or (4.4) will be transmitted through gold to any other of these equations. As will be seen below, this is a critical point that is generally overlooked in simple real barter based or in two-country trade models. A surplus supply of Euros and excess demand for Dollars that may originate in the hands of importers of US products into Portugal, from a US trading surplus with Portugal, does not just depreciate the value of the Euros that these importers receive or appreciate the dollars used to pay the US producers of these products, rather these currency affects are transmitted to Dollars and Euros in general including those that are used in other international trading transactions. As the direction and rate of change of the currency movements originating from one trade imbalance are more or less independent of other trade imbalances - particularly between countries that have no, or marginal, trading relations with the countries in the original imbalance, these currency adjustments will not, except accidentally, be those that would be required to bring the other trading relations that they impact into balance.
Suppose for example that for some short-term reason that does not affect the long-term parameters of the system, Us trade surplus with P, that was initially in the state of equilibrium specified in (4.1), increases, so that (4.1) becomes:
(4.7) UP: yb > c
Where:
(4.8) yb - c > yb - c
This will increase the excess supply of Euros and demand for Dollars because of the surplus of Euros that importers of goods from the US into Portugal will receive but need to convert to Dollars to pay US producers, relative to the Dollars that the importers of goods from Portugal into the US will receive and want to convert to Euros to pay Portuguese producers. As Pounds are not used in this trade the demand and supply of Pounds will not be affected. Thus relative to some constant standard like gold: a) the Dollar will appreciate, b) the Euro will depreciate, and c) the value of the Pound will remain unchanged. Though the decline of y=$/E will be greater, this will cause both y=$/E and x=$/ to decline.
The decline in y will, under our free trade assumptions reduce Us surplus with P, moving (4.7) back toward the equilibrium values of (4.1).
But, at the same time the decline in x, or appreciation of the Dollar vis a vis the Pound, will under free trade assumptions, cause Ks surplus with U to increase so that (4.2) will change to:
(4.9) KU: d > xf
Where:
(4.10) d - xf > d - xf
This will in-turn increase the value of the Pound and reduce the value of the dollar, raising x=$/ and moving (4.7) away from (4.1). This increase in x will be greater or smaller than the decline in x from (4.7). Depending on the speed and (Dollar) magnitudes of the export and import responses in (4.7) and (4.9) to exchange rate fluctuations, (4.7) may or may not move closer to (4.1), and (4.9) may or may not move closer to (4.2).
Finally, unless they net out to zero, the changes in x and y will cause the /E=y/x ratio to change, leading to an indeterminate change in (4.4).
As there is absolutely no reason why these changes should move these disequilibrium import quantities back to the unique balanced trade position of (4.1), (4.2) and (4.4), we conclude that the unique free trade solution (4.1), (4.2) and (4.4) is unstable. Therefore there generally will be no free market forces moving international trade toward a (unique) balanced trade position, or maintaining the system in this balanced position if it is accidentally generated. A three country balanced free trade equilibrium will thus be a generally impossible model outcome.
5. A Balanced N Country Free Trade System is also not Mathematically Viable
A balanced trade solution to a general N country free trade system that is made up entirely of bi-lateral balanced trade is also not feasible as in such a system each distinct bi-lateral pair of trading partners and currencies will have to balance so that the number of these equations will equal the number of distinct groups of two that can be drawn from N countries or: EMBED Equation.3 . A free trade system of N countries will thus have:
(3.9) EMBED Equation.3
equations,
but only:
(3.10) EMBED Equation.3 EMBED Equation.3 unknowns.
As EMBED Equation.3 for all N>2, this system will be over-determined, and increasingly so, for all multiple country trading systems with three or more countries.
Also, and again as in the three country case, a balanced N country free trade solution with bi-lateral trade imbalances is generally unstable and thus a generally impossible outcome.7
In Section 4 we demonstrated that if one pair of countries has a trade imbalance at least three countries must have precisely off-setting imbalanced bi-lateral trade if international trade is to be balanced. We have also shown that if any one of these bi-lateral trade inequalities changes, the currency value of the two currencies involved will change (relative to a third currency), and this will necessarily impact the other two bi-lateral trade relations. Finally, we have pointed out that there is no reason, other than pure happenstance, that these currency appreciations and depreciations will lead to the kind of perfectly offsetting trade expansions or contractions between these three countries that would be necessary to reestablish the old balanced international trading system. To the contrary, as individual country satisfaction of the Marshall Lerner conditions simply ensures a qualitatively normal trade response to exchange rate fluctuation, and does not guarantee that exchange rate elasticities of imports between three (or more) countries will have the precise values necessary to re-establish international balance if such a balance were perturbed, there is no reason to assume that a market-led free trading system will ever come into balance, and if it does, that such a balanced solution will be stable.
It follows that any N country balanced trade solution that has one bi-lateral trade imbalance must have at least three bi-lateral trade imbalances. If, as is likely, the unique balanced trading solution to an N country trading system includes more than three bi-lateral imbalances, the above logic will apply to more than three countries. Due to the inseparability of the currency impacts of any bi-lateral trade imbalance, a perturbation (due to short-term causes) of any bi-lateral trade relationship from an initial international balanced trade position will cause exchange rate changes that will affect other bi-lateral trading relationships in indeterminate ways that will not, except coincidentally, restore the unique international trade balance consistent with the non exchange rate dependent (in the short-term) parameters of the system.
6. Conclusion
We conclude that there is no reason why purely market led exchange rate price adjustments will produce balanced free trade. To the contrary, it is reasonable to assume that trade imbalances will induce an arbitrary subset of consistent, or inconsistent, currency adjustments at varying rates with varying impact on import demands, and that these adjustments will not, except coincidentally, lead to internationally balanced trade. And, if by chance, a unique internationally balanced trade position is obtained, it is not likely to last as, except by accident, it will be mathematically unstable for almost any perturbation.
For example Europe (in late 2007) had a trade surplus with the U.S. but a deficit with China. The Euros appreciation vis a vis the Dollar was expected to (eventually) reduce Europes trade surplus with U.S., but the Yuans relatively slower appreciation vis a vis the Dollar led to a Euro appreciation relative to the Yuan that increased Europes deficit with China leading to an overall European trade deficit. A larger and more rapid appreciation of the Yuan could have conceivably rebalanced this system, but it appeared unlikely to come about from free market forces (Palley, 2007a, 2007b).
This problem and the prisoners dilemma issues that are raised by problems like this, have been identified as a theoretical issue for particular cases of linked exchange rates by mainstream economists see for example (Ogawa and Takatoshi, 2002), but the general conclusion, that movement toward a world wide free trade equilibrium solution, even under the most ideal Neo-Classical assumptions, is theoretically impossible except through extreme coincidence, has been ignored.
International trade cannot be balanced through exchange rate fluctuations. Active trade management policies that restrict imports or exports are necessary to achieve balanced and sustainable international trade. Automatic self adjustment of international trade through market-led Free Trade is not generally possible. Similarly administered exchange rates tied to individual country trade deficits or surpluses will not, except by accident, generate a sustainable world trading system.
Endnotes
1) Needless to say, actual short-term, and even long-term, exchange rate adjustments do not generally follow this simple rule. Our concern in this paper is with simple free trade ideology how the doctrine is justified in introductory economics and public policy courses, and with the underlying theory of market fundamentals that may have some relationship with exchange rate movements, at least in traders minds, over the long-term. Actual current exchange rates are as much, or more influenced, by capital movements, and currency speculation. See (Blecker, 1999) for a critical review of these more sophisticated trade models.
2) I apologize to readers for going over these well known causal links in such great detail. As the papers fundamental argument is based on the details of these mechanisms it appears important to explicitly delineate these basic points even at the risk of pedantic redundancy.
3) As has been noted (3.3) is a dependent equation as it can be derived from (3.1) and (3.2). It thus does not reduce the number of degrees of freedom. In contrast, (3.6) cannot be derived from (3.4) and (3.5) and therefore reduces the number of degrees of freedom by one.
4) The instability demonstration described in the text below would also clearly apply to (3.4) (3.6) were such a solution mathematically feasible. As it is not, for expositional simplification we restrict ourselves to solutions composed of strict inequalities in this section - see discussion below.
5) We could specify that one of the currencies, for example Dollars, are the international currency but this would lead to an asymmetry between the currencies that would complicate the story without adding to it in any substantive way.
6) Again, I ask the readers indulgence for belaboring the obvious here. This is again part of an effort to highlight points that are so simple that their import appears to have been hitherto overlooked or ignored.
7) As has been noted in Section 4 (see footnote 5), the instability argument below clearly also applies to solutions that include on or more bi-lateral trade balances.
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he0Jjhe0JUh3Wjh3WUhsh$ho 6B*phhEkho 6B*phhr_ho 6B*phhr_ho B*phhr_ho B*]phhr_ho 5B*\phhr_ho B*\phUhr_ho ho ho B*ph$Real Exchange Rates and the International Mobility of Capital, Working Paper No. 265, The Jerome Levy Economics Institute of Bard College.
___________ . (ed.) 2007. Globalization and the Myths of Free Trade: History, theory, and empirical evidence. Routledge Press, London.
Takatoshi, Ito and Ogawa, Eiji. 2002. On the Desirability of a Regional Basket Currency Arrangement, Journal of Japanese and International Economics 16(3) p. 317-334, Sept.
Taylor, Lance. 2004. Reconstructing Macroeconomics. Harvard University Press, Cambridge, MA.
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