Devaluation Versus Import Restriction as an Instrument for Improving Foreign Trade Balance

FOR ANY COUNTRY or trading area except a very small one, a devaluation can generally be expected to bring a decline in the foreign price of its exports. If, instead of devaluing, a country should achieve the same improvement in its foreign trade balance by a restriction of imports, whether through a tariff or by quantitative restrictions, the foreign price of its exports would not fall significantly.1 Consequently, the value of exports and imports that must be sacrificed in order to achieve a given improvement in the foreign balance through devaluation exceeds, for such a country or trading area, the value of the imports that must be sacrificed in order to achieve an equal improvement of the foreign balance through import restriction or a tariff. If, then, the welfare value of a dollar’s worth 2 of imports is equal to the welfare value of a dollar’s worth of exports, it costs the country less to use import restriction rather than devaluation as an instrument for improving the foreign balance, subject to the qualifications mentioned below.

Abstract

FOR ANY COUNTRY or trading area except a very small one, a devaluation can generally be expected to bring a decline in the foreign price of its exports. If, instead of devaluing, a country should achieve the same improvement in its foreign trade balance by a restriction of imports, whether through a tariff or by quantitative restrictions, the foreign price of its exports would not fall significantly.1 Consequently, the value of exports and imports that must be sacrificed in order to achieve a given improvement in the foreign balance through devaluation exceeds, for such a country or trading area, the value of the imports that must be sacrificed in order to achieve an equal improvement of the foreign balance through import restriction or a tariff. If, then, the welfare value of a dollar’s worth 2 of imports is equal to the welfare value of a dollar’s worth of exports, it costs the country less to use import restriction rather than devaluation as an instrument for improving the foreign balance, subject to the qualifications mentioned below.

FOR ANY COUNTRY or trading area except a very small one, a devaluation can generally be expected to bring a decline in the foreign price of its exports. If, instead of devaluing, a country should achieve the same improvement in its foreign trade balance by a restriction of imports, whether through a tariff or by quantitative restrictions, the foreign price of its exports would not fall significantly.1 Consequently, the value of exports and imports that must be sacrificed in order to achieve a given improvement in the foreign balance through devaluation exceeds, for such a country or trading area, the value of the imports that must be sacrificed in order to achieve an equal improvement of the foreign balance through import restriction or a tariff. If, then, the welfare value of a dollar’s worth 2 of imports is equal to the welfare value of a dollar’s worth of exports, it costs the country less to use import restriction rather than devaluation as an instrument for improving the foreign balance, subject to the qualifications mentioned below.

When import restrictions are already in force, the domestic value for welfare purposes of imports worth one dollar in the foreign market will be greater than the domestic value for welfare purposes of exports worth one dollar in the foreign market, say c times as great. It would then pay to restrict imports further only as long as c is less than the rate at which exports can be exchanged for imports on the margin in foreign markets. That rate is equal to 1e where e, the elasticity of trade, is defined as the ratio of the marginal to the average terms of trade. It accordingly pays to restrict imports rather than to devalue as long as c< 1e.

If, on the contrary, c>1e, that is, if the domestic value for welfare purposes of a dollar’s worth of imports is greater than the domestic value for welfare purposes of the exports that must be given up at the margin in order to obtain a dollar’s worth of imports, then it pays to devalue rather than to restrict imports further. Therefore, whether or not c is greater than 1e is the criterion of whether it is more desirable, from the viewpoint of the country concerned, to devalue or to restrict imports when an improvement in the foreign balance is required.

From the point of view of the rest of the world it is, however, better if the country concerned achieves any specified improvement in its foreign balance by devaluation rather than by import restriction. The assumption is made throughout this analysis that other countries will not retaliate, or at least that any retaliation to a given change in the foreign balance of the country concerned will be the same in form and extent whether devaluation or import restriction is used. When an improvement in the foreign balance is required, the final choice between import restriction and devaluation must, moreover, depend on many factors in addition to the trading advantage the country can gain. Such factors as international agreements to avoid restrictions, the desire to maintain stable exchange rates, the political power of groups with special interests, the greater difficulty of making necessary readjustments when one instrument is used rather than the other, the different impact on domestic prices and incomes, all affect the choice. The present paper attempts no contribution to the discussion of these problems, nor does it take into account the weight of these considerations on one side or the other. Its purpose is to assess the desirability, for the devaluing country, of devaluation, compared with restriction, as an instrument for improving the trade balance. It merely considers the relative welfare cost of the two instruments for improving the foreign balance3 from the point of view of the country’s getting the greatest trading advantage out of its foreign trade. Consequently, when in this paper a country is said to be better off by one method rather than another, the meaning is that the country gains more from foreign trade by one method than by another. Throughout the discussion full employment is assumed. In the absence of full employment, radically different considerations would govern the choice of instrument for improving the foreign balance.

There are several different types of trade restrictions which have, or can have, identical results, not only on the trade balance, but on its components, the volumes and prices of imports and exports. In particular, a quantitative restriction of imports, a restriction of foreign exchange expenditure, a tax on imports, a tax on exports, or a set of differential exchange rates for imports and exports can all be used to achieve the same effects on foreign trade. In order to simplify the discussion, one type of restrictive instrument, namely a restriction of foreign exchange expenditure on imports, will be given primary attention. It should be borne in mind that, within the scope of this paper, any conclusion concerning restriction of imports can be applied to the other restrictive instruments.

The principal disadvantage of devaluation, compared with import restriction, is the fact that devaluation can usually be expected to have an adverse effect on the foreign price of exports. As a consequence, the devaluing country will lose by the devaluation import and export goods of greater value than the amount by which the foreign balance is improved. Before this relationship is examined in full generality, a simplified case will be considered.

A Simplified Case

Under certain simplifying assumptions, the improvement of the foreign balance brought about by a devaluation is given by the formula 4

dB=kM(ehmd+efxd1)(1)

where dB is the improvement in the foreign balance, measured in foreign currency; k is the extent of devaluation measured as the proportion by which the foreign currency value of home currency has fallen; M is the predevaluation total value of imports in foreign currency (here assumed to be equal to the predevaluation total value of exports); ehmd is the total elasticity of home import demand;5 and efxd is the total elasticity of foreign export demand.5

The formula is based on the following assumptions:

(i) k is small.

(ii) The domestic supply of export goods is perfectly elastic with respect to the domestic price of export goods.

(iii) The foreign supply of import goods is perfectly elastic with respect to the foreign price of import goods.

(iv) The prices of imports and exports in the devaluing country are equal to the corresponding foreign prices converted at the going rate of exchange.

(v) The initial value of imports is equal to the initial value of exports.6

Under these assumptions it may be shown that if the prices of imports and exports in the devaluing country can be taken as the measure of their value for welfare purposes, the country is better off if it restricts imports rather than devaluing unless either ehmd or efxd is infinite, in which case only an infinitesimal devaluation would ever be required.

In this connection the following criterion may be used to determine whether a country is better off by one method or another. Assume the value of a unit of exports for welfare purposes to be q and of a unit of imports to be p. Under simple conditions, q and p may be taken as the prices of export and import goods in the home market, though in more complicated situations other values of q and p may be used.

The welfare cost of any particular step taken to improve the foreign balance can be expressed as the welfare value of the additional exports taken away from domestic consumption, plus the welfare value of the reduction of imports, under the assumption that both exports and imports are changed so as to contribute to the improvement. Whichever way the trade moves, the welfare cost of a given change in the foreign balance involving small changes, dx and dm, in export and import volumes, respectively, with signs taken so as to be positive when the respective volume increases and negative when it is reduced, can be expressed as qdx—pdm. This formula applies to a small movement only, since a large change would probably alter the relative welfare value per unit of imports and exports.

If one method of achieving a certain amount of improvement in the foreign balance brings about changes of dx1 and dm1 in exports and imports, and a second method brings about changes of dx2 and dm2, the country may be said to be better off using the first method provided its welfare cost is less, i.e., provided qdx1 − pdm1 < qdx2 − pdm2. In the following examples the assumption will be made that the initial domestic prices of imports and exports represent their values per unit for welfare purposes. Suppose first that efxd = 1 and ehmd = 1. Then a 1 per cent devaluation will lead to a 1 per cent decline in the foreign price of exports and, since the foreign elasticity of demand is assumed to be unity, to a 1 per cent increase in the volume of exports. The devaluation will also lead to a 1 per cent rise in the domestic price of imports and therefore, ehmd being 1, to a 1 per cent decline in imports. The trade balance is accordingly improved by 1 per cent of the value of imports as indicated by formula (1). As the cost of this improvement the country has had to sacrifice the consumption of twice this value in import and export goods taken together. Had the country merely restricted imports by 1 per cent it would have improved its foreign balance as much as it did by the 1 per cent devaluation, and it would not have had to give up an additional 1 per cent of exports.

This is an extreme example because of the elasticities chosen, but the principle applies as long as efxd ≠ ∞. Thus, if ehmd = 2 and efxd = 3, then a 1 per cent devaluation will, according to (1), improve the foreign trade balance by 4 per cent of the predevaluation value of imports. This involves a 2 per cent cut in imports and a 3 per cent increase in exports. An import restriction of 4 per cent would have an equally favorable influence on the trade balance, and exports would not have to be increased. The net gain by use of import restrictions is accordingly 1 per cent of the value of imports or exports. That is, using the devaluation instrument, the cost of improvement in the trade balance is 2 per cent of the value of imports plus 3 per cent of the value of exports; using the import restriction mechanism the cost is 4 per cent of the value of imports, and the net gain by use of the latter method is the difference, which, by the assumption that the initial value of imports equals the initial value of exports, is 1 per cent of either.

The conclusion that an import restriction which brings about an improvement of the foreign balance equal to that brought about by a devaluation of proportion k costs less than the devaluation by the amount kM is in fact independent of the values of the demand elasticities as long as assumptions hold for which (1) is valid.7 For under those assumptions a devaluation in the proportion k will bring about an improvement of the foreign balance of kM(ehmd + efxd − 1) through a decline of the value of imports, at predevaluation prices, by kMehmd, and an increase of the value of exports, at predevaluation prices, by kMefxd. These changes will be partially counterbalanced by a fall of the foreign price of exports in the proportion k, which explains the presence of the −1 in formula (1), so that the import proceeds of predevaluation exports are reduced by kM through the terms of trade deterioration.

A restriction of expenditure on imports by kM(ehmd + efxd − 1) will achieve the same improvement in the foreign balance. The loss to the country through devaluation will be kM(ehmd + efxd) in the total value of imports and exports, while the loss through restriction of imports will be kM(ehmd + efxd − 1) in the value of imports. The difference, kM, represents the loss through the decline of the foreign price of exports as a consequence of the devaluation. The import restriction does not have this result. Therefore, irrespective of the values of ehmd and efxd, as long as the conditions required for the validity of (1) are satisfied, a devaluation in the proportion k will cost the country kM more than would an import restriction that would yield an equal improvement in the foreign balance.

This reasoning is appropriate to a welfare judgment only if the costs concerned measure the welfare values involved. The assumption has been made that if given amounts of imports and exports are of equal value at predevaluation prices, they are of equal welfare value to the country. Suppose this is not so, but that a dollar’s worth of imports is deemed by those guiding the country’s affairs to be worth c times as much as a dollar’s worth of exports, the dollar prices being the foreign prices. Then a restriction of imports by kM(ehmd + efxd − 1) is to be valued as costing, in welfare terms, ckM(ehmd + efxd − 1), while the cost of the devaluation is to be rated as ckMehmd + efxd Then the restriction of imports is to be preferred only if

ckM(ehmd + efxd − 1) < ckM ehmd + kM efxd

or cefxdc < efxd

or c<111efxd, provided efxd > 1

Of course, if efxd < 1, restriction of imports is always to be preferred, for then the more that is exported, the less is obtained for the exports. Therefore, no matter how much more valuable imports are than exports, a restriction of imports is better than a devaluation which would stimulate exports, if a greater volume of exports will bring in smaller foreign proceeds.

The result of the analysis so far may be interpreted as follows: With a given deficit8 a restriction of imports by m units would, under the assumption of perfectly elastic foreign supply of imports, permit a reduction of foreign expenditure by mpm, where pm is the foreign price of a unit of imports. Let us assume that m is chosen so that mpm=$1. This would permit a reduction of exports by some quantity, x, since there are fewer imports for which payment is to be made out of export proceeds. A decrease of x units of exports will reduce export receipts in a foreign currency by approximately xpx(11efxd), an approximation that is valid as long as x is small.9 Then, since x is the amount by which exports can be reduced so that the reduction of export proceeds will just equal the 1 dollar reduction of import expenditures:

xpx(11efxd)=$1orxpx=111efxd

Therefore, the right-hand expression in the last equation indicates the value of export goods that could be retained if the value of imports were reduced by 1 dollar. Thus, if efxd = 2, the expression on the right equals 2, which means that for every dollar’s worth of imports restricted, 2 dollars’ worth of exports can be retained. It is then worth while to restrict imports provided a dollar’s worth of imports is worth less than 2 dollars’ worth of exports. Therefore, if c denotes the ratio of the welfare value of a dollar’s worth of imports to the welfare value of a dollar’s worth of exports, it is worth restricting imports as long as c<111efxd but it is worth expanding imports and paying for them with increased exports as long as c>111efxd.

So far, a criterion has been established of how large imports and exports should be with a given deficit. In order to use this as a criterion of choice between a devaluation and a restriction of imports when an improvement of the foreign balance is required, the problem need merely to be turned around. Assume that the desired improvement in the foreign balance were achieved entirely by an import restriction; would that lead to a value of c so large that at a given deficit it would pay to expand imports and exports by devaluation, i.e., is c>111efxd?10 If that is so, then import restriction has been carried too far and some devaluation might have been used. Conversely, if initially the desired improvement were achieved by a devaluation, would that lead to a value of c so small that c<111efxd, in which case some import restriction plus upward revaluation would be in order? If a satisfactory adjustment has been made and c is approximately equal to 111efxd, then any further improvement of the foreign trade balance might best be achieved by a combination of devaluation and import restriction to preserve the equality.

Of course, in actual practice, in addition to the difficulties of applying the formula because of ignorance of the elasticity and welfare values involved, other considerations are important in governing the choice at issue.

The General Case

The foregoing considerations are all special examples of a more general proposition. In order to formulate that proposition, the elasticity of trade, e, may be defined as the ratio of the marginal terms of trade to the average terms of trade. The elasticity of trade may be expressed in terms of the conventional elasticities of foreign demand for exports and supply of imports as:

e=11efxd1+1efms(2)

where efxd is the elasticity of the foreign demand for exports and efms is the elasticity of the foreign supply of imports.11 Let us, for simplicity of expression, refer to prices in terms of foreign currency as dollar prices. Then the following general proposition may be advanced:

Proposition. If a country wishes to improve its foreign trade balance, it will be better off, in welfare gained from international trade, if it achieves this end by a restriction of imports rather than by a devaluation, until the restriction is carried to the point where the domestic value, for welfare purposes, of a dollar’s worth of import goods in the foreign market is equal to 1e times the domestic value, for welfare purposes, of a dollar’s worth of export goods in the foreign market.

This conclusion applies with special force to a very large country or area, such as the sterling area or the dollar area taken as a whole. For such an area, the foreign supply of imports and the foreign demand for exports can be expected to be of low elasticity, so that import restriction might be carried quite far before it would pay to devalue. For a small country or area, however, it is more likely that foreign supply and demand are elastic; if this is so, there is very little terms of trade advantage in import restriction. Furthermore, if the foreign elasticities are highly elastic, no more than a very small devaluation would ever be required.

Corollary I. If domestic price ratios reflect the relative values of import and export goods for welfare purposes, then restriction of imports is preferable to devaluation as an instrument for improving the foreign balance, until restriction is carried to the point where the price ratio of export goods to import goods in the home market is e times the price ratio of export goods to import goods in the foreign market.

Formal proof of this proposition and its corollary are given in the Appendix. The following explanation is advanced, not as a rigorous proof, but as an attempt to indicate the general relationships underlying the proposition.

The term 1e can be very simply conceived as the number of dollars’ worth of export goods that could be retained if, with a given foreign currency deficit, imports were reduced by 1 dollar’s worth, with both exports and imports valued at the foreign prices before the restriction. Thus, if 1e is 2, at current prices, a reduction of imports by 1 dollar’s worth would permit a reduction of exports by 2 dollars’ worth, with an unchanged deficit. In other words, 1e measures the terms of trade in foreign trade between a dollar’s worth of imports and the number of dollars’ worth of exports that must be given for the imports, account being taken of the tendency for prices to change when more or less exports are offered or imports demanded.

If c > 1e, a dollar’s worth of imports, even though it requires the sacrifice of 1e dollars’ worth of export goods, is more than worth the sacrifice because the domestic welfare value of the dollar’s worth of imports is greater than the domestic welfare value of 1e dollars’ worth of exports. An expansion of imports is accordingly desirable, and can usually be achieved through a devaluation. Therefore, c > 1e implies that a devaluation is desirable from welfare considerations. Conversely, c < 1e implies that, at the margin, too great a value of exports is being given for each dollar’s worth of imports, and import restrictions are in order. Whether c is greater or less than 1e therefore is the criterion of whether devaluation or import restriction is in order.

How far a devaluation should be carried can best be seen by first considering how far the import restriction should be carried. As imports are restricted, the value for welfare purposes of a dollar’s worth of import goods can be expected to rise in the restricting country. Of course the elasticity of trade will also change as imports are restricted, but probably not to a major degree. In any case, import restrictions will have been carried far enough when the value for welfare purposes of a dollar’s worth of imports is 1e times the value for welfare purposes of a dollar’s worth of exports, that is, when c = 1e. From this point on, any further improvement of the trade balance can best be achieved by a combination of devaluation and import restriction. For then the welfare cost of devaluation is the same as that of trade restriction.

If the trade restriction is imposed as an ad valorem tariff, and welfare values are assumed equal to domestic prices, no adjustment of the restriction will be required as devaluation is carried on as long as the value of e does not change. If the trade restriction takes the form of a quantitative restriction of imports, then it would be appropriate to adjust these restrictions as the devaluation proceeds so as to preserve the ratio of welfare values of a dollar’s worth of imports to a dollar’s worth of export goods at the figure 1e. That is, if import restrictions are unchanged and the currency devalued, exports will normally be increased and the domestic price of export goods can be expected to rise, as can their value for welfare purposes. If e has not changed, import restriction will no longer be at the optimum, since after devaluation the welfare value of a dollar’s worth of import goods will be less than 1e times the welfare value of a dollar’s worth of export goods. Consequently, whenever it is necessary to devalue from a previous optimum position, it is also appropriate to reduce the amount of imports permitted. This occurs automatically if the trade restriction is imposed in the form of an ad valorem tariff.

The fact that a devaluation unaccompanied by restriction of imports may have adverse terms of trade effects merely illustrates the more general principle that it is always to a country’s advantage to restrict imports to the point where the domestic value of a dollar’s worth of imports is worth, in welfare terms, 1e times the domestic welfare value of a dollar’s worth of exports.12 This holds whether or not the country is trying to improve its foreign balance. If the country is not attempting such improvement, it can reduce exports while it is restricting imports and so free the domestic resources for the production of goods for home consumption. If the country has not imposed the optimum amount of import restriction, and it is faced by the necessity of improving its foreign balance, it is better off to impose, first, the optimum restriction of imports. Only if the optimum restriction brings an insufficient improvement in the foreign balance is the country well advised to devalue. In that case the deterioration of the foreign terms of trade as a result of the devaluation does not carry over into the domestic sphere because a dollar’s worth of imports is, in the home market, worth more than a dollar’s worth of exports. However far the devaluation goes, it does not pay to impose further trade restrictions if, at the margin, the domestic welfare value of a dollar’s worth of imports is at least 1e times the domestic welfare value of a dollar’s worth of exports.

These considerations lead to:

Corollary II. If the value for welfare purposes of a dollar’s worth of imports is greater than 1e times the value for welfare purposes of a dollar’s worth of exports, the country will gain from devaluation as long as devaluation permits a greater volume of imports and exports with a given foreign balance.

The foregoing considerations concerning the relative cost of restriction and devaluation look entirely to the foreign market; that is, e depends on only the foreign elasticities of demand and supply. But surely the desirability of restriction or devaluation must in some way depend on the domestic, as well as on the foreign, elasticities. This is seen to be the case once consideration is given to the question of how far an import restriction or devaluation should be carried rather than to the question of which method is to be preferred at a particular point.

It has been shown that an import restriction is to be preferred as long as c < 1e. But as import restriction is applied and import goods become scarce relative to export goods, the value of c will increase. The value of e may also change since the elasticity of foreign supply and demand can be expected to vary as the volume of trade changes; for simplicity, however, let the assumption be made that the value of e remains unchanged over the ranges of trade which will be considered here.

The rate at which c increases as imports are reduced depends on the domestic elasticities, particularly on the elasticity of domestic demand for imports. If that demand is highly inelastic, then only a small import restriction will suffice to increase greatly the price or welfare value of a unit of import goods relative to the price or welfare value of a unit of export goods. Under such circumstances there is little practical advantage in import restrictions even if c < 1e, for a small import restriction will cause so great a rise of c as to lead to c > 1e. Therefore, only a very small restriction is in order, and devaluation appears to be the appropriate method of improving the foreign balance.

Let the term “import goods” be applied to all goods of the sort that are imported, whether actually imported or produced domestically. Then it may be said that the domestic demand for imports depends not only on the domestic demand for import goods but also on the domestic supply of import goods. That is, the domestic demand for imports at any particular price is equal to the domestic demand for import goods at that price minus the domestic production of import goods that is forthcoming at that price. If either domestic demand for import goods or domestic supply of import goods is highly elastic, the domestic demand for imports will be highly elastic and import restrictions may be carried very far before the price or welfare value of import goods rises so high that c > 1e.

Similar reasoning serves to establish the conclusion that even if c > 1e only a small improvement of welfare cost is to be expected from a devaluation if the domestic supply of exports is inelastic. Accordingly, even though it is technically correct to say that whether c < 1e or c > 1e is the criterion as between import restriction and devaluation from the point of view of minimizing the welfare cost of imports or exports foregone, nevertheless, for practical purposes qualifications may be made depending on the domestic elasticities. In particular, even if c < 1e import restriction will not bring any significant benefit if the domestic demand for imports is highly inelastic; and even if c > 1e, not much welfare gain can be expected from devaluation if the supply of exports is inelastic.

Qualifications

Some further general qualifications should be borne in mind in assessing the significance of this analysis.

Less than full employment

If, in the country concerned, there is unemployment of resources that for some reason cannot be dealt with by domestic policies, then devaluation may be more attractive relative to import restriction than appears from the foregoing discussion. The relative disadvantage of devaluation, when it is at a disadvantage, is that it leads to too cheap a price for exports. If additional exports can be produced with resources that would otherwise be unemployed, or if, in any case, an increase in exports will not in either the short or the long run be at the expense of domestic consumption or investment, then the low price of exports to the foreigner need not be a disadvantage of devaluation. This situation can be fitted into the above analysis by assigning a low welfare value to a unit of exports so that c will be large and, presumably, in general c > 1e, indicating the preferability of devaluation over restriction.

Foreign reaction and retaliation

From an international point of view, it is preferable that countries trying to improve their foreign balances do so by means of devaluation rather than by trade restriction. A devaluation by one country generally offers more advantageous terms of trade to the rest of the world; precisely because of this, import restriction may be preferable to devaluation, from the viewpoint of the country’s own self-interest. But if all countries restrict imports so that in the end all net foreign payment balances are unchanged, all countries will be worse off, while if all devalue so that foreign balances are unchanged, all can be expected to be at least as well off as before.

In any situation involving import restrictions (or tariffs), there is some pattern of relaxation of the restrictions (possibly involving some compensation from one country to another) by which another situation can be reached such that every country involved will be better off. For this and other reasons, exchange rate adjustment is to be preferred to import restriction as a general policy even though under certain circumstances outlined above a country can gain more from import restriction than from devaluation.

Appendix: Formal Proof of Proposition

The conventional formula for the effect of a devaluation runs in terms of total elasticities. Consider a devaluation in the proportion k so that 1 unit of home currency, initially worth r units of foreign currency, is worth r(1—k) units of foreign currency after the devaluation. Further, assume k to be small, and let dm, dx, dp, and dq represent the changes brought about in m, x, p, and q by the devaluation, where m, x, p, and q denote respectively the predevaluation quantity of imports, quantity of exports, foreign price of imports, and foreign price of exports. Let ph, qh, dph, and dqh be respectively the home prices of imports and exports and the changes therein associated with the devaluation.

Then the total elasticities are defined as follows:

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It is assumed, while considering the devaluation, that home prices of imports and exports equal the corresponding foreign prices converted at the going rate of exchange.1 Accordingly:

p = phr and p + dp = (ph + dph)r(1−k)

or,ifkissmallsothatkdphphcanbeneglected,dphphdpp=k(A5)

Similarly, from the corresponding relationships involving q,

dqhqhdqq=k(A6)

The change in the foreign balance in foreign currency is given approximately by

dB=(qdx+xdq)(pdm+mdp)(A7)

The terms at the right of the equality sign in (A-7) can, by use of the definitions of the elasticities and of relations (A-5) and (A-6), be expressed in terms of the elasticities and k.

Thus

dqq=kehxsefxd+ehxs(A8)
dxx=kefxdehxsefxd+ehxs(A9)
dpp=kehmdehmd+efms(A10)
dmm=kefmsehmdefms+ehmd(A11)

Substituting these values in (A-7) and setting X=xq, M=mp,

dB=k{Xehxs(efxd1)efxd+ehxs+Mehmd(efms+1)ehmd+efms}(A12)2

Note that if X=M, and if ehxs=efms= ∞,

dB=kM(efxd+ehmd1),theLerner-Robinsonformula.(A13)

Suppose an import restriction is imposed, of proportion such that after the restriction the expenditure of foreign currency on imports is M(1−j). Let a value be chosen for j such that the reduction of foreign currency expenditure on imports equals the improvement of the foreign balance brought about by the devaluation previously considered, i.e.,

jM=dB(A14)

Let the changes in quantities and prices which result from the import restriction be denoted by subscript 2, and the changes resulting from devaluation be denoted by subscript 1. The new, restricted, value of imports in foreign currency will equal the new amount of imports multiplied by the new foreign price of imports:

M(1j)=(m+dm2)(p+dp2)(A15)

If mp is substituted for M in (A-15), and if dm2·dp2 may be ignored

j=dp2p+dm2m(A16)

Now assume that the same elasticities that governed the price and quantity relationships in the case of devaluation also apply to changes in prices and quantities brought about by a restriction of imports. Of course, when imports are restricted the domestic demand for imports will no longer be satisfied at a price equal to the foreign price converted to domestic currency at the going rate of exchange. Definition (A-4) may accordingly be ignored, and equation (A-10) may be dropped and for it may be substituted the equation expressing the import restriction (A-14).

By use of definition (A-3) in (A-16)

dm2=jmefms1+efms(A17)

Since j has been chosen so that the improvement of the foreign balance is the same as that resulting from a specified devaluation in the proportion k, (A-14) and (A-17) can be combined to get

pdm2=dBefms1+efms(A18)

This can be considered as the cost of the import restriction to the restricting country. In order to obtain an improvement of dB dollars in the foreign currency value of the foreign balance, the country must give up dBefms1+efms dollars’ worth of imports; if devaluation is used as the instrument for improving the foreign balance, the country must give up −pdm1 dollars’ worth of imports and qdx1 dollars’ worth of exports. From (A-9) and (A-11)

pdm1=kMefmsehmdefms+ehmd(A19)

and

qdx1=kXefxdehxsefxd+ehxs(A20)

Assume that, for welfare purposes, a dollar’s worth of imports is worth c times as much as a dollar’s worth of exports. Then if the cost, in welfare terms, of the devaluation is denoted as C1, and of the import restriction as C2,

C1=qdx1cpdm1=kXefxdehxsefxd+ehxs+ckMefmsehmdefms+ehmd(A21)
C2=cpdm2=cdBefms1+efms(A22)

Then if in (A-22) the value of dB from (A-12) is substituted,

C2=ck{Xehxs(efxd1)efxd+ehxs+Mehmd(efms+1)ehmd+efms}efms1+efms

Therefore,

C1C2=kX(ehxsehxs+efxd)[efxdcefms(efxd1)1+efms](A23)

Since all the elasticities are defined so as to be normally positive, C1 will normally be greater than C2 provided the term within the brackets is positive. That term will be positive as long as

c>1+1efms11efxd(A24)

By definition of the elasticity of trade

e=dmdx/qp(A25)

where dm and dx are subject to the constraint of a fixed foreign balance so that

qdx+xdq=pdm+mdp(A26)

and the variation of p and m is restricted by the foreign supply function for imports, and the variation of q and x is restricted by the foreign demand function for exports.

From (A-26),

dx(q+xdqdx)=dm(p+mdpdm)(A27)

Therefore,

e=1+xdqqdx1+mdppdm=11efxd1+1efms(A28)

Accordingly, (A-24) and (A-28) indicate that the condition that a restriction of imports would cost less, in welfare terms, than a devaluation bringing an equal improvement in the foreign balance is c < 1e.

*

Mr. Alexander, Assistant Chief of the Statistics Division, is a graduate of Harvard College and the Harvard Graduate School. He was formerly Assistant Professor of Economics at Harvard College, and during the war was connected with the Office of Strategic Services. He has been consultant to the U.S. Treasury Department, U.S. State Department, Economic Cooperation Administration, and U.S. Department of Defense.

1

If the country’s imports significantly influenced other countries’ demand for the country’s exports, there would be a tendency for import restrictions to lower the price of exports. Even in this case, however, it can be shown that the conclusion of this paper, as expressed in its Proposition (p. 387), is valid.

2

Throughout this paper it is assumed that the dollar is the foreign currency involved so that the word “dollar” may be substituted for the more awkward expression “unit of foreign currency” which would otherwise be appropriate.

3

The term, improvement of foreign trade balance, is taken here to denote an increase in the quantity X-M, where X denotes the value of exports and M the value of imports. These values will usually be measured in terms of foreign currencies. For simplicity, changes in the balance of payments arising from other than trade are ignored.

4

Based on a formula attributed to A. P. Lerner, The Economics oj Control (New York, 1946), p. 378. See also Joan Robinson, “The Foreign Exchanges,” Essays in the Theory of Employment (2nd ed., Oxford, 1947), p. 143, fn., reprinted in Readings in the Theory of International Trade, H. S. Ellis and L. Metzler, eds. (Philadelphia, 1949), p. 93, fn. 10. The above formula applies equally well to a change in the foreign balance expressed in foreign currency or home currency provided M is measured in the appropriate currency. A more general formula also presented by Mrs. Robinson applies only to the change in the home currency value of the foreign balance. The more general formula used in this paper—formula A-12, given in the Appendix—applies to the change in the foreign balance measured in foreign currency.

5

So defined as to be positive if a price increase is associated with a reduction in the amount that will be bought; see Appendix for definition of total elasticity.

6
More generally, without making assumption (v), formula (1) would have to read:
dB=kMehmd+kX(efxd1)(1)

In the analysis from page 385 on, the difference between the costs of the two measures depends only on the effect on the foreign export prices, so that (v) can be dropped without affecting the generality of the argument.

7

Assumption (v) need not hold; see footnote 6.

8

See footnote 6.

9

This is the familiar relationship: if n is the elasticity of demand, marginal revenue equals average revenue multiplied by (11n).

10

It is assumed here that efxd > 1.

11

These must be defined as total elasticities (see Appendix). They are so defined that the elasticities will normally be positive.

12

In terms of a tariff this means that the optimum tariff is, at the ad valorem rate t=1e1=1efms+1efmd11efmd.This formula was first derived, in a slightly different form, by Bickerdike as the basis of a short article in the Economic Journal, XVI, 1906, pp. 529 ff., but since the presentation in that article was verbal and graphic, the formula was not published there, but was first published in Bickerdike’s review of Pigou’s Protective and Preferential Import Duties, in the Economic Journal, XVII, 1907, p. 101. The formula’s application was further extended and qualified by Edgeworth, Economic Journal, XVIII, 1908, pp. 393 ff., and pp. 541 ff., reprinted in Papers Relating to Political Economy, II (1925), pp. 340 ff. The general idea involved, although not the formula, was developed by Edgeworth in 1894; cf. Papers, II, p. 39. Further discussion of the problem appeared in A. P. Lerner, “The Diagrammatical Representation of Demand Conditions in International Trade,” Economica, II (n.s.), August 1934, pp. 319 ff. and The Economics of Control (New York, 1944), pp. 382-85; N. Kaldor, “A Note on Tariffs and the Terms of Trade,” Economica, VII (n.s.), November 1940, pp. 377 ff.; T. de Scitovsky, “A Reconsideration of the Theory of Tariffs,” Review of Economic Studies, IX (1941-42), No. 2, pp. 81 ff. (reprinted in Readings in the Theory of International Trade, Philadelphia, 1949); Joan Robinson, “The Pure Theory of International Trade,” Review of Economic Studies, XIV (1946-47), No. 2, pp. 98 ff.; R. F. Kahn, “Tariffs and the Terms of Trade,” Reviews of Economic Studies, XV (1947-48), No. 1, pp. 14ff.; I.M.D. Little, “Welfare and Tariffs,” Review of Economic Studies, XVI (1948-49), No. 2, pp. 65 ff.; J. de V. Graaff, “On Optimum Tariff Structures,” Review of Economic Studies, XVII (1949-50), No. 1, pp. 47 ff.

1

For the purposes of the proof which follows, it would be sufficient for home prices to be proportional to foreign prices.

2

The derivation of (A-12) follows that presented by Mrs. Robinson, op. cit., p. 142, fn. (Blakiston, p. 91, fn.), except that (A-12) applies to the foreign balance measured in foreign currencies while Mrs. Robinson’s formula applies to the foreign balance measured in domestic currency.