The geometry of protectionism in the imperfect substitutes model: a reminder.

I. Introduction

This note aims to clarify the estimation of the cost of restricting imports of a good which substitutes imperfectly for goods produced and consumed at home. It reviews the estimation technique pioneered and used by the U.S. Federal and International Trade Commissions[6; 8]. This ingenious technique is valid, provided the relevant demand and supply functions are linear in prices and provided the relevant goods display zero income elasticities of demand. The base case assumes a small country. We review this method because several recent texts[1; 4] and studies[2; 3] use invalid techniques(1) to estimate the costs of protectionism in the imperfect substitutes model.

II. ITC Method

Figure 1 illustrates the ITC method. A small country restricts imports of good 0, raising the internal price(2) from [P.sub.0b] to [P.sub.0a] and reducing consumption from [Q.sub.0b] to [Q.sub.0a]. The demand curve [D.sub.0b] takes as given the before-protection, equilibrium prices of the other goods consumed domestically. The demand curve [D.sub.0a] takes as given the after-protection, equilibrium prices of these goods. The ITC technique estimates the consumer loss as [P.sub.0b][P.sub.0a]cd and the social loss of a tariff or equivalent quota as cde.

This technique appears peculiar and incomplete. The estimated consumer loss is an area beside neither [D.sub.0b] nor [D.sub.0a]; instead, the consumer loss is estimated as an average of the losses derived from these curves (provided the demand function is linear). Also, the technique appears to disregard the consumer and producer gains or losses in the markets for non-tradables(3) which substitute for or complement the import. However, these potential criticisms are invalid: precisely because we measure the consumer loss as an average, the disregard of gains or losses in other markets is legitimate.

III. A Basic Result

Here we demonstrate the above assertion. Let PS and CV denote the increase in the national producer surplus and the compensating income variation for national consumers associated with the change in prices induced by the import restriction. Then, the assertion is confirmed by the result:

Result: PS = CV - ([P.sub.0b][P.sub.0a]cf + [P.sub.0b][P.sub.0a]gd)/2.

That is, the consumer loss which remains after accounting for the ITC average ([P.sub.0b][P.sub.0a]cd) equals the producer gain in the non-tradables markets.

Our proof uses the following notation. Suppose there are N non-tradables, whose prices and quantities are indexed ([P.sub.i], [Q.sub.i]), i = 1, ..., N. ([P.sub.ib], [Q.sub.ib]) and ([Pi.sub.ia], [Q.sub.ia]) are the before- and after-protectionism equilibrium prices and quantities, with ([delta][Pi.sub.i], [delta][Q.sub.i]) [equivalent] ([P.sub.ia], [Q.sub.ia]) - ([P.sub.ib], [Q.sub.ib]). Assume linear demand and supply functions:

[Mathematical Expression Omitted]

in which [a.sub.ji] = [a.sub.ij] and [b.sub.ji] = [b.sub.ij] for all i,j.

Finally, in assigning producer gains ([ps.sub.i]) and consumer losses ([cv.sub.i]) to individual markets, we must assume an order for the shifts in the demand and supply curves used to measure the income gains or losses. Without loss of generality, we measure the consumer loss in market i as the area beside the demand curve for good i given the after-protection prices for goods (0, 1, ..., i - 1) and given the before-protection prices of goods (i + 1, ..., N). We order the shifts in the nontradable supply curves in the same fashion. Thus, in a typical market i, the consumer loss (for i [greater than or equal to] 0) and producer gain (for i [greater than or equal to] 1) are given by expressions (1),

[Mathematical Expression Omitted]

in which [cv.sub.i] is the consumer loss in market i measured as an area beside the good i demand curve which takes as given the before-protection prices of all other goods.

The result follows readily. First, note that equilibrium in each non-tradable market requires [delta][Q.sub.i.sup.s], which implies (2) for each i [greater than or equal to] 1:

[Mathematical Expression Omitted]

Thus, (1b) and (2) permit the market i producer surplus to be re-written as (3):

[Mathematical Expression Omitted]

Summing the surpluses over the individual markets, we see that the latter terms evaporate (i.e., each sum contains one of each distinct cross produce [b.sub.ij][delta][P.sub.i][delta][P.sub.j], because of the symmetry of cross price derivatives). The aggregate producer surplus reduces to expression (4):

[Mathematical Expression Omitted]

Second, we sum the consumer losses over all markets to obtain the aggregate consumer loss of (5):

[Mathematical Expression Omitted]

Comparing (4) and (5) and using the symmetry of cross price derivatives, we see that PS and CV are related as:

Finally, we see from Figure 1 that:

[cv.sub.0] = [P.sub.0b][P.sub.0a] (7a)

[Mathematical Expression Omitted]

Observations (6) and (7) establish the basic result.

IV. Summary

The ITC geometry is convenient, for it depicts the deadweight loss solely with prices and quantities in the market which is taxed directly. The technique is applicable to a variety of problems in public finance: it is a special case of a result of Just, Hueth, and Schmitz[5] who show that the loss from a tax can be computed from areas beside the general equilibrium(4) supply and demand curves of the taxed good. Extensions of the ITC method to the large county case are straightforward. For example, when the supply curves of some imports slope upward, the Figure 1 loss due to the restriction on good 0 imports must be supplemented by the producer surplus gains of the foreign suppliers whose prices are affected. Thus, in estimating the U.S. loss from the VER on Japanese cars, Dinopoulos and Kreinen[2] correctly include the increased profits of European suppliers of cars to the U.S. with the direct losses of U.S. consumers of Japanese cars.(5) However, these authors incorrectly attribute an additional loss to repercussions in the U.S. market for U.S. cars.

References

[1.] Appleyard, Dennis and Alfred Field. International Economics. Boston: Richard D. Irwin, 1992, pp. 352-53. [2.] Dinopoulos, Elias and Mordechai E. Kreinen, "The Effects of the US-Japan Auto VER on European Prices and US Welfare." Review of Economics and Statistics, August, 1988, 484-91. [3.] Hufbauer, Gary, Diane Berliner, and Kimberly Ann Elliot. Trade Protection in the United States: 31 Case Studies. Washington: Institute for International Economics, 1986. [4.] Husted, Steven and Michael Melvin. International Economics. New York: Harper and Row, 1990, 183-85. [5.] Just, Richard, Darrell Hueth, and Andrew Schmitz. Applied Welfare Economics and Public Policy. Englewood Cliffs, New Jersey: Prentice Hall, 1982, pp. 192-96, 451-57. [6.] Morkre, Morris and David Tarr. Staff Report on the Effects of Restrictions on United States Imports: Five Case Studies and Theory. Washington: Federal Trade Commission, 1980. [7.] Pugel, Thomas, "Review of Trade Protection in the United States." Journal of Economic Literature, March, 1988, 120-22. [8.] U.S. International Trade Commission. The Economic Effects of Significant U.S. Import Restraints, Phase I: Manufacturing. Publication No. 2222. Washington: 1989, Ch. 2, Appendix D.

(1.) Pugel [7] points out of helpful discussions with Dennis Appleyard, Alfred Field, and A. Myrick Freeman. (2.) The numeraire is the good which displays a constant marginal utility. (3.) In this small country case, the internal prices of other tradables are insensitive to the restriction on good 0. (4.) That is, the equilibrium conditions for markets 1, ..., N give the equilibrium prices [P.sub.i], i [greater than or equal to] 1 as functions of [P.sub.0]. The general equilibrium supply and demand curves for good 0 relate [Q.sub.0] to [P.sub.0] alone by replacing the prices of other goods with their general equilibrium dependence on [P.sub.0]. In our example, the line connecting d and c is the general equilibrium demand curve. (5.) If goods 0 and 1 are Japanese and European cars and if European cars are supplied competitively with no tariff, the U.S. loss is [delta][P.sub.0]([Q.sub.0b] + [Q.sub.0a])/2 plus [delta][P.sub.1]([Q.sub.1b] + [Q.sub.1a]/2. Dinopolous and Kreinen [21 use [Q.sub.0a] and [Q.sub.1a] rather than the average quantities.