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 Ricardo’s 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 import quantities demanded as dependent variables, and independent parameters along with N-1 independent exchange rate variables, for a total of:

(2.1) +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 demand equations and N-1 trade balance equations for a total of:

(2.2) + 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 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 + N-1 independent equations and + 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 K
P imports b from U and a from K
K 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 B’s currency from sales receipts than exporters from B to A will have of A’s currency. As exporters from A to B have to pay producers in country A in A’s 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 A’s 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 Say’s 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 eitherall 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) + 3

equations, where we are substituting the three “free market” trade balance equations (3.4) – (3.6), for N-1 or twoindependent 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) +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:

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 U’s only other partner K, so that (3.1) remains in balance:

(4.1) UP: y°b° > c°

(4.2) KU:d° > x°f°

Such that:

(4.3)U:y°b° – c° = d° – x°f°

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: y°a° < x°e°

Such that:

(4.5)P: x°e° – y°a° = y°b° - c°

(4.3) and (4.5) will then guarantee:

(4.6)K: x°e° - y°a° = d° –x°f°

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 “Say’s 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