A Tariff Game with Intermediate Imports

DIPANKAR PURKAYASTHA*

I. Introduct ion

Neoclassical economic theory views tariffs as a possible result of market distortions. A rationale for tariffs exists in the presence of externalities, political lobbying, and market power of large countries. Surprisingly, most studies ignore the fact that an overwhelming volume of international trade is in intermediate goods and this may have an effect on the choice of optimal tariffs. This paper constructs a model of two large countries, one of which is an exporter of intermediate goods. The relevant optimal tariffs are derived, and it is shown that in the presence of intermediate imports, the choice of tariffs is considerably different from the standard case involving final goods only. The negotiation locus is asymmetric, and a negotiated tariff reduction may never yield free trade.

H. A Model of Optimal Tariffs with Intermediate Goods

Assume ~ that in a two country world country 1 has a revealed export preference for good X~ (a final good), and country 2 has a revealed export preference for good K (an intermediate good). The non-traded goods are defined by the technologies:

X~ = X~(K J) i = j + 1, j = 1, 2, (1)

where a superscript denotes a country and a subscript denotes a commodity. Global endowment of K is confined to country 2 only. Home use of intermediate goods in country 2 is denoted by K 2 = K - K ~, where K is the total endowment of intermediate goods in country 2.

Let both countries impose ad valorem import tariffs at rates t~ (tariff imposed by countryj on good i). Let P~ be the domestic market price of good i in country j . The relationships among international prices are :

P~ = Pl ; (2)

P~ = Pk(1 + t~); (3)

P? -- PI (1 4- t•); and (4)

N = P,. (S)

Assume domestically in both countries that there is perfect competition with quasi-concave and continuously differentiable production functions and that there exists a unique price vector. It will be shown below that this set-up considerably modifies the choice of optimal tariffs.

*California State University at Fullerton. The author would like to thank Fred Inaba for helpful suggestions on this paper.

i The methodology of this section is based on MeMillan [1986].

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When K is a pure intermediate good, domestic perfect competition requires that the price of K equals the value of its marginal product

o r

i = j + 1, j = 1 , 2 (6)

[ ]1 OXi 2 (7)

P~ =/3J(1 + t { ) P k, /3J - [ 'b 'KJ ' j - - 1, 2; t ~ - - 0 .

Let e{ denote the excess demand function in countryj for commodity i (i = 1, k):

J = e~(P,, 1 t~). (8) e l g l ,

Excess demand is a function of world prices and tariff rates. Revenues from the tariffs are calculated as

R J = P,t~e~. (9)

The tariff revenue depends on the value of import demand and the rate of ad valorem tax. The indirect trade utility function (H J) for eountryj is assumed to be continuous, quasi-convex in (P~, P~+t), homogeneous of degree zero in P{, P~+t, and weakly increasing in R J:

H ~ = H i ( P / , P/+,, R J); j = 1, 2. (10)

Using equations (7), (9), and (10),

H t = H1(P1, 131Pk(1 + t~), tkPkek 1 (11)

2 l (12) H 2 = H2(P,(1 + t~), B 2 P~, l I el el)"

Note also the condition for the overall balance of world trade:

e/ = -e~ h j = 1, 2; h = 1, 2 ; j # h. (13)

In a noncollusive Cournot-Nash game, country 1 finds its best tariff by assuming that tt 2 does not change as a result of its action. Country 1 sets

= 0 . ot2

Equation (11), (13), and Roy's identity allows one to derive 2

2See MeMillan [1986]. For simplicity, 13 is assumed to be constant.

PURKAYASTHA: TARIFF GAME 53

1 (~I ~_ [t~ a P t ] - ' (14) ti = + 1 ) ' tkll

where 7f is country 2's elasticity of export supply. Note that this reduces to a familiar optimal tariff result when /3 ~ = 1. The optimal tariff for the capital exporter, however, does not change. Following an analogous procedure for country 2, one obtains its optimal tariff:

t? = --1 (15) ¢

where ~ ~ is country l ' s elasticity of export supply. Equations (14) and (15) show that, in general, the two countries would impose different optimal

tariffs even if their respective export supplies have the same elasticity. The intuition behind the asymmetry is simple enough: import restrictions affect the consumption of country 1 indirectly- through production, while such restrictions affect country 2's consumption directly. A comparison between equations (14) and (15) would show that while a large intermediate good exporter will always impose a tariff, a large intermediate good importer may opt for a subsidy (negative tariff) if the marginal productivity of home goods output with respect to the intermediate imports is small enough.

HI. GATT Games

In an important paper, Mayer [ 1981] has shown that the tariff negotiation locus in a final goods only model is given by:

T l + T IT 2 + T 2 = 0,

where T 1 and T 2 are the respective optimal tariffs of country 1 and country 2. Mayer's negotiation locus guarantees that if one country proposes free trade (say, T l = 0), the other should follow (T 2 = 0). The model in section II can be used to demonstrate that this symmetry does not hold in a model with intermediate inputs.

To show this, following Mayer [1981], optimal tariff indifference curves are constructed first. Rewriting the indirect utility function for country 1 where T~ is now the optimal tariff:

where y1 = p~ Xl the numeraire:

Z = ( P k / P , ) , P , - 1.

P~rfect competition in the domestic market ensures that

d X l + P ~ d X 2 = O.

v' = v' (e , , p: , r ' ) , (16)

1 1 1 + Pg X 2 + Tg Zek. The term of trade is defined as Z and good 1 is defined as

(17)

(18)

Noting that the nontraded good is consumed entirely at home, total differentiation of the indirect utility function yields:

d V = Vy Y ~. (19)

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Each country's excess demand functions depend on the terms of trade and trade taxes:

e{ = e{(Z, T{). (20)

Mayer's method can now be directly applied to equations (16) - (20) to derive the tariff indifference curve for country 1:

[dT~ ] = _ ~2T~ml + A (21)

dT~ 1 T~m2(¢ l - 1)

A m ~t + E2 _ 1; m J = Oe'dt{; [ j = 1, 2, at{

and that of country 2:

dT~ 2 [(1 + T2) - T2tel]rn 2' (22)

where ~ = elasticity of country l ' s imports with respect to Z and e2= elasticity of country 2's imports with respect to Z. Equating (21) and (22) yields the tariff negotiation locus:

2 T~A + A l = 0 . (23) (E 1 - 1)[ml(Z + e2)T~ + ~.2ml(z + 1)TkT1 +

The negotiation locus (23) shows the following: 1) If the Marshall Lerner condition holds (A > 0), and if E 1 ~ 1, free trade will not occur. 2) If the intermediate exporting country imposes no tariff, the intermediate importing country will impose a tariff. If the intermediate importing country imposes no tariff, the intermediate exporting country will impose a subsidy. In general, T~ > T~. 3) As country l ' s terms of trade improve (Z falls), country l ' s optimal tariff increases. These results demonstrate that in international tariff negotiations, a capital importer is unlikely to be an advocator of free trade.

IV. An Example

Nash solutions obtained from equations (14) and (15) will be unique as long as the relevant elasticities are single-valued throughout the relevant region. Notice that every time one country imposes a tariff, in general, it will alter the elasticity of its own demand for foreign exports. It is rather inconvenient to deal with such non-unique solutions) Gorman utility functions avoid this problem by generating iso-elastic offer curves. Tariffs based on a Gorman utility function displace the offer curve only by a constant and keep the elasticity unchanged.

Gorman utility functions do, however, impose severe restrictions on the pattern of utility functions and the endowments. The models developed in sections II and III do not generate iso-

3 Numerical examples of optimal tariff rates for a variety of production and utility functions are presented in Hamilton and WhaUey [1983]. The following example is a special ease of Johnson [1965].

PURKAYASTHA: TARIFF GAME 55

elastic offer curves and thus cannot be illustrated with the help of Gorman functions. Fortunately, however, the basic structure of intermediate imports can be incorporated in the Gorman form and the consequent asymmetry can be explored. One is still able to show that in a Cournot-Nash game, an intermediate importer's optimal tariff is likely to be more than the optimal tariff of an intermediate exporter.

Assume that the Gorman trade utility functions for country 1 and country 2 are given by equations (24) and (25), respectively. (A and B are constants of the model.) Let Country 1 export XI and let country 2 export K:

U' = gX~ - AX"I

U 2 = pX~ - B X ; .

Each country must use K, the intermediate good, to produce X2 or X 3 under the technology

X i = g 0 .

The balance of trade condition is given by

P X 1 = K .

(24)

(25)

(26)

(27)

Using equations (24) - (27), first derive the competitive offer curves for country 1 and country 2, respectively:

(28) /c --

X1 = O B K °~. (29)

The intersection of equations (28) and (29) would correspond to a free trade equilibrium. The social planner in each country realizes that if the partner country does not retaliate, she

could trade at any point on her partner's offer curve. In a Cournot-Nash game, country 1 must then maximize (24) subject to (29). Similarly, the social planner in country 2 must maximize (25) subject to (28). The offer curves of the social planners in country 1 and in country 2 are thus derived as (30) and (31), respectively:

| K

r = (30)

x , = (B )K "°. (31)

Realizing that these are optimal offer curves, the social planners would now take measures to displace competitively determined (28) and (29) to socially determined (30) and (31), respectively. If one assumes that this is done by means of an ad valorem import tariff, the respective tariff- distorted offer curves now become [_:]1

K = -/X~°(1 + t~) ~ (32)

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X 1 = OBK°'(I + t~). (33)

A comparison of (30) with (32) and (31) with (33) would reveal that the necessary ad valorem Cournot-Nash tariffs imposed by the social planners are asymmetric. They are derived as:

t (34) tk = Or, - 1

2 g - 1. (35) tl = ~

Consider the following propositions.

Proposition 1. The Nash equilibrium volume of trade is less than the perfect competition volume of trade. (This can be shown by direct substitution of the equilibrium values into the utility function.)

Proposition 2. The Nash equilibrium utilities and terms of trade can be either more or less than the perfect comPetition utilities and terms of trade. (This can be shown by direct substitution into (27).)

Proposition 3. A higher output dasticity of intermediate goods in the home good sector will tend to reduce the optimal tariff imposed by the capital exporting country, but will increase the optimal tariff imposed by the importing country. If the countries possess equal market power, a capital exporter's tariff will be less than a capital importer's tariff. If the output elasticity of intermediate goods is high enough (0 > ~¢), the capital exporter will impose a subsidy on its imports. On the other hand, if the output elasticity of intermediate goods is low enough (0 < l /v ) , the capital importer will subsidize its imported capital.

The results confirm the conventional wisdom that a bilateral monopoly will reduce outputs traded but may or may not produce a level of welfare higher than the competitive equilibrium for a single country. Moreover, if the intermediate goods exhibit higher output elasticities, countries exporting such goods will be less inclined to impose a tariff.

V. Concluding Comments

The model demonstrates the importance of the home good sector in the choice of a tariff. Notice that two countries facing the same export supply elasticity may impose different optimal tariffs depending on the technologies in the home good sectors. The results also shed some light onto the often misunderstood behavior of tariff preferences in North-South trade. Generally speaking, given the current international economic order, the capital-rich Northern countries supply capital goods to the capital scarce South, and the South exports primary goods to the North. Assuming a I00 percent rate of depreciation, the model demonstrates why the capital exporting North is less likely to limit trade by means of a tariff. The structure of trade and the implied external economies of scale may be the underlying reason why the North is relatively more enthusiastic about tariff reductions.

REFERENCES Hamilton B.; WhaUey, J. "Optimal Tariff Calculations in Alternative Trade Models and Some Possible

Implications for Current World Trading Arrangements," Journal of International Economics, 15, 1983, pp. 323-48.

Johnson, H. "Optimal Intervention in the Presence of Domestic Distortions," Trade, Growth, and Balance of Payments, Chicago: Rand MeNally, 1965.

Mayer, W. "Theoretical Considerations on Negotiated Tariff Adjustments," Oxford Economic Papers, 33, 1981, pp. 135-53.

McMillan, J. Game Theory and International Economics, Switzerland: Harwood Academic Publishers, 1986.