- An Antiquarian Note on Optimal Tariffs
The notion that a single country with power to influence world commodity prices might improve its well-being by imposing positive or negative tariffs on some of its imports and exports, thus abandoning free trade, goes back to Robert Torrens (1821, 1844) and John Stuart Mill (1844); and we owe the more precise notion of a nonnull optimal tariff vector to Charles Bickerdike (1906, 1907) and F. Y. Edgeworth (1908). It was recognized that the existence of such a vector is jeopardized if that country’s trading partners choose to retaliate by imposing tariffs of their own or if world markets are noncompetitive or otherwise distorted. Subject to those qualifications, the existence of a nonnull optimal tariff vector was accepted with remarkably little dissents. 1
The calm was disturbed when, fifty years ago, Jan de V. Graaff (1949, 53) suggested that free trade might be the most advantageous commercial policy for a country, even when the country is in no sense small, when there are no distorting elements (commodity and income taxes, incompleteness of markets, externalities, nonconvex production sets, and public goods), and when there is no threat of retaliation by its trading partners. The suggestion was later repeated in Graaff’s much-admired book on welfare economics (Graaff 1957, 130). The free-trade outcome [End Page 415] was described as exceptional but held to be perfectly possible under standard assumptions; and with that assessment trade theorists appear to have agreed, without delay and without exception.
That the conventional wisdom of more than a century was so readily abandoned is surprising, especially since Graaff failed to supply a clear argument in defense of his suggestion. He seems to have simply assumed that the cross effects of price changes could be such as to justify his conclusion. In the present note it will be shown that, under standard assumptions, the optimal tariff vector of a single country can be null only in the uninteresting case in which, under free trade, the country does not trade. Thus, effectively, we will restore the conventional wisdom circa 1949 and, with it, the common sense that an optimizing country with market power will exercise that power.
Let there be n traded commodities and m trading countries of which the first is called the home country and the rest, collectively, the foreign country. 2 The aggregate foreign excess demand functions are denoted by Z* i = Z* i (p*1, . . . , p* n ), i = 1, . . . , n, where p* j denotes the world price of commodity j. It is assumed that the home country is never small, even locally; that is, it is assumedfor each (p*1, . . . , p* n ) > 0 and for some i and j, |∂Z* i /∂p* i | ≠ ∞. The requirement that the international payments of the foreign country be in balance is then expressed by the equation
Since (1) is satisfied for any (p*1, . . . , p* n ) > 0, we may differentiate, obtaining
The upper boundary of the convex set of home production possibilities is defined by the equation
where Xj denotes the home output of commodity j. The task of the home country is that of choosing outputs and prices such that its utility U[X 1−Z*1(p*1, . . . , p* n ), . . . , Xn −Z* n (p*1, . . . , p* n )] is a maximum subject to (3). (The world prices are manipulated by means of the home country’s import and export taxes.) Introducing the Lagrange multiplier λ, the task is to maximize U[X 1−Z*1(p*1, . . . , p* n ), . . . , Xn −Z* n (p*1, . . . , p* n )] + λT(X 1, . . . , Xn ) with respect to p*1, . . . , p* n , X 1, . . . , Xn , and λ. Among the first-order conditions for an internal maximum, we find
where Ui ≡ ∂U/∂(Xi − Z*i). However, in a competitive equilibrium Ui = µpi and pi = πip*i , where pi denotes the home price of commodity i and πi is 1 plus the rate of import duty if commodity i is imported and the inverse of 1 plus the rate of export duty if commodity i is exported. Thus equation...