Public goods production, nontraded goods and trade restrictions.

Hatzipanayotou, Panos

I. Introduction

When lump-sum taxes are available, the first-best rule for public good production requires that the sum of the marginal rates of substitution be equal to the marginal rate of transformation (e.g., Samuelson [13, 387-89]). In this case, the social marginal cost of the public good equals its private marginal cost of production. When distortionary taxation is used to finance the production of public goods, Pigou [12] argues that the social marginal cost exceeds the marginal cost of production because of the induced indirect cost due to raising revenue through distortionary taxation. Stiglitz and Dasgupta [14, 151-74], Atkinson and Stern [3, 119-28], and Wildasin [15, 227-43] among others, demonstrate that in certain cases (e.g., when the taxed and public goods are complements in consumption), Pigou's argument fails and the social marginal cost may fall short of the marginal cost. A number of authors (e.g., King [10, 273-91], and Batina [5, 125-33]) examine these and related issues in a many-persons economy. In a trade theoretic context, Feehan [7, 155-64] derives the efficiency rule for the production of a pure public good when it is financed through tariff generated revenue. Abe [1, 209-22] examines the welfare effects of tariff reform programs when tariff revenue is used to finance the production of a pure public consumption good.

We build a small open economy trade model where two traded goods, one exported and one imported, one nontraded private good, and one public consumption good are produced. The production of the public good is financed through lump-sum taxes. Imports are restricted through voluntary export restraints (VERs), quotas, or tariffs.(1) Within this framework, and relative to previous studies, the present paper makes the following contributions. First, it shows that the Samuelson's rule is valid under an import quota and violated under a tariff or a VER. Second, it offers a different interpretation as to why the social and private marginal cost of the public good may not be the same. Third, it shows that the difference between the social and private marginal cost of the public good not only depends on the complementarity/substitutability between the public and restricted (i.e., imported) good, but also on the complementarity/substitutability between the imported and nontraded and between the nontraded and public good. That is, it depends on the "general equilibrium" substitutability/complementarity between the imported and the public good. Specifically, when imports are restricted by VERs (respectively, tariffs) the marginal cost understates (overstates) the social marginal cost when the public and imported goods are general equilibrium complements, and overstates (understates) it when they are general equilibrium substitutes.

II. The Public Good Economy with Trade Restrictions

Consider a small open economy with a representative consumer, producing three private goods - one exported, one imported and one nontraded good - and one public consumption good that is provided by the government to the representative consumer free of charge. Two internationally immobile factors - capital and labor - are used in the production of the four goods. The production functions are homogeneous of degree one and concave in the two factors. The full employment condition requires that

[v.sup.p] + [v.sup.g] = v, (1)

where [v.sup.p] and [v.sup.g] are the amounts of factors used in the production of the private and public goods, and v is the vector of fixed factor endowments (i.e., dv = 0).

Good and factor markets are perfectly competitive and trade is subject to alternative import restrictions (VERs, quotas, or tariffs). As usually assumed, tariff revenue is lump-sum distributed to the consumer, VER rents are captured by foreign exporters, and quota rents are captured by domestic importers.(2)

Let [p.sup.*] and p denote the world and domestic relative price of the imported good. The difference between the two prices, denoted by t(= p - [p.sup.*]), may be due to a tariff, an import quota, or a VER. Since the country is small in world goods markets, changing the import constraints or the level of public good production does not affect the world prices of the traded goods.

Individual utility depends on the consumption of the four goods, which are assumed normal. The private expenditure function, E(p, q, g, u) represents the minimum expenditure needed to achieve a level of utility u at a relative price of the imported good p, relative price of the nontraded good q and level of public good consumption g. The derivatives of the expenditure function with respect to p and q (i.e., [E.sub.p], and [E.sub.q]) are the compensated demands for the imported and nontraded goods. An increase in the consumption of the public good reduces the expenditure on the private goods needed to achieve a level of utility u. Thus, [E.sub.g] is negative. In the public economics literature, -[E.sub.g] is called the consumer's marginal willingness to pay for the public good [10, 273-91]. Throughout the analysis, subscripts denote partial derivatives.

The maximum value of private goods production, for given p and [v.sup.p], is represented by the gross domestic product function [R.sup.*](p, q, [v.sup.p]). The derivatives of this function with respect to p and q (i.e., [Mathematical Expression Omitted] and [Mathematical Expression Omitted]) are the supply functions of the imported good and the nontraded good, and with respect to [v.sup.p] (i.e., [Mathematical Expression Omitted]) is the marginal revenue product of factors. Factor markets are in equilibrium when

[Mathematical Expression Omitted], (2)

where w is the vector of factor rewards.

The unit cost function of the public good, denoted by [C.sup.g](w), is homogeneous of degree one and concave in w (e.g., [Mathematical Expression Omitted] and [Mathematical Expression Omitted]). Using the properties of the [C.sup.2] function, and equations (1) and (2), the private gross domestic product function for a given level of public good provision is given by

R(p, q, g) = [R.sup.*](p, q, [v.sup.p](p, q, g)). (3)

The R(p, q, g) function has the following properties: (i) [Mathematical Expression Omitted], (ii) [Mathematical Expression Omitted], (iii) [R.sub.g] = -[C.sup.g], (iv) [Mathematical Expression Omitted], [Mathematical Expression Omitted], and (vi) [Mathematical Expression Omitted].(3) In the two-traded-good, two-factor Heckscher-Ohlin model, given world good prices, as long as the economy is incompletely specialized in all goods, the prices of factors are determined independently of changes in factor endowments and the price of the nontraded good, e.g., [Mathematical Expression Omitted], and [Mathematical Expression Omitted].

We assume that the government selects the level of the public good to be produced and raises the appropriate amount of revenue for its financing through lump-sum taxation. The government budget constraint requires that the lump-sum tax revenue T, equals the cost of the public good. That is,

T - g[C.sup.g](w) = 0. (4)

The country's budget constraint requires that private expenditure (i.e., E(p, q, g, u)) equals income from the production of private goods (i.e., R(p, q, g)) plus income from the production of the public good (i.e., g[C.sup.g]), plus revenue from trade restrictions (1 - [Rho])t[Z.sub.p], minus the lump-sum taxes (i.e., T). That is,

E(p, q, g, u) = R(p, q, g) + g[C.sup.g](w) + (1 - p)t[Z.sub.p] - T, (5)

where [Rho] is the fraction of the revenue from trade restrictions that accrue to foreigners. It is assumed that [Rho] = 0, in the case of a tariff where it is assumed that tariff revenue is lump-sum distributed to domestic residents, and in the case of a quota where it is assumed that quota rents are captured by domestic residents (e.g., importers). In the case of a VER, it is assumed that VER rents accrue to foreigners, and thus [Rho] = 1.

The condition that imports are restricted by VERs, or quotas is expressed as

[Z.sub.p](p, q, g, u) = [E.sub.p](p, q, g, u) - [R.sub.p](p, q, g) = [Z.sub.p]. (6)

Finally, the equilibrium in the nontraded good market requires that the excess demand equals zero. That is,

[Z.sub.q](p, q, g, u) = [E.sub.q](p, q, g, u) - [R.sub.q](p, q, g) = 0. (7)

III. Public Good and Welfare

Using equations (4)-(7), gives the welfare effects of an increase in the public good under the three trade regimes as follows:

[Mathematical Expression Omitted], (8)

(du/dg) = - ([E.sub.g] + [C.sup.g]), and (9)

[Mathematical Expression Omitted], (10)

where [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [E.sub.u] = 1 by choice of units. Local Walrasian stability requires that At and [[Delta].sub.v] are positive. When [Mathematical Expression Omitted] is positive (negative) we say that the imported and the public good are "general equilibrium" complements (substitutes).(4)

Equation (8) shows the welfare effect of an increase in the level of public good production in the presence of VERs. Equation (9) shows the welfare effect of an increase in the level of public good production in the presence of import quotas. In both the VER and quota cases, the endogenous variables are u, p, q, and T while the exogenous variable is g. Equation (10) shows the welfare effect of an increase in the public good production in the presence of a tariff. In this case, the endogenous variables are u, q, and T. Note that under a tariff, holding its rate constant and given the small country assumption, the domestic prices of traded goods are unaffected by changes in the level of the public good.

Under all three trade regimes, there is a direct welfare effect of an increase in the level of public good that depends on the magnitude of the consumer marginal willingness to pay for the public good relative to its unit cost of production. If the former exceeds the latter, fiscal expansion has a direct positive effect on welfare in all three regimes. In the case of a tariff or a VER, however, an indirect welfare effect of an increase in the public good occurs, due to changes in tariff revenue or VER rents. This indirect effect depends on the general equilibrium complementarity/substitutability between the imported and the public good. If -[E.sub.g] [successor] [C.sup.g] and [Mathematical Expression Omitted], then an increase in the level of the public good causes an overall increase in welfare under a tariff, but may decrease welfare under a VER.(5) Intuitively, when [Mathematical Expression Omitted], an increase in the public good production raises excess demand for the imported good, thus revenue from a tariff or a VER. Higher tariff revenue, due to increased imports, accruing to domestic residents induces an additional positive effect on welfare. Higher VER rents, due to higher demand for imports and thus a higher domestic price for the imported good, accruing to foreigners induces a negative effect on welfare.(6) On the other hand, if [Mathematical Expression Omitted], increased public good production improves welfare under a VER, but may reduce welfare under a tariff.

Equation (9) shows that under an import quota, the welfare effect of an increase in the level of the public good is independent of the effect of fiscal expansion on quota rents. In the case of quotas, if [Mathematical Expression Omitted], an increase in the level of the public good increases the demand for imports, the price of the imported good, and quota rents accruing to domestic residents, thus having a positive effect on welfare. But, the increase in the domestic price causes a loss equal to the difference between the loss in consumer surplus minus the gain in producer surplus. These two effects exactly cancel.(7)

IV. Optimal Rule for Public Good Production

Setting (du/dg) = 0 in equations (8)-(10) gives the optimal rule for public good production in the presence of a VER, an import quota, and a tariff, when lump-sum taxes are used to finance its production. That is,

[Mathematical Expression Omitted] (11)

-[E.sub.g] = [C.sup.g], and (12)

[Mathematical Expression Omitted]. (13)

The right-hand-side of equations (11), (12) and (13) shows the social marginal cost of the public good under the alternative import restrictions.(8) Equations (11)-(13) show that the first-best (i.e., Samuelson's) rule for the optimal production of the public good is always valid under an import quota. In the presence of a VER or a tariff, Samuelson's rule does not hold and the marginal cost of the public good production overstates or understates its social marginal cost, even if the production of the public good is financed through lump-sum taxes.(9) For example, if [Mathematical Expression Omitted], then under a tariff (VER) the unit cost of production overstates (understates) its social marginal cost.

PROPOSITION. Assume that imports are restricted by a tariff, a quota, or a VER, and that the production of the public good is financed through lump-sum taxes. Then, Samuelson's rule for the optimal production of the public good always holds under an import quota. Under a VER (tariff), the marginal cost of the public good understates (overstates) its social marginal cost if the imported and the public good are general equilibrium complements, and overstates (understates) it if they are general equilibrium substitutes.

Intuitively, the social marginal cost of the public good equals the private marginal cost plus an indirect positive or negative effect from the policy induced changes in revenue that are created from the trade restrictions. Equations (11)-(13) show that the reason why the modified Samuelson rule for the optimal provision of public goods is not the same under the various import restrictions is the asymmetric welfare effect of the policy induced changes in revenue from such trade restrictions. For example, when [Mathematical Expression Omitted] is positive, as discussed earlier, an increase in the public good production increases tariff revenue or VER rents. Since the VER rents (tariff revenue) accrue to foreigners (domestic residents), the indirect effect on domestic welfare is negative (positive) under a VER (tariff).(10) Thus, the marginal cost of the public good understates (overstates) its social marginal cost under a VER (tariff). In the case of an import quota, Samuelson's rule is valid, since, as shown earlier, welfare is not affected by changes in quota rents due to fiscal expansion.

In a closed economy model where the government imposes commodity taxes to finance the provision of a public good, Atkinson and Stiglitz [4] show that the optimal rule is MRS = ([P.sub.g] - t[X.sub.g])/(1 + e), where (1 + e) is the elasticity of tax revenue to the tax rate, 1/(1 + e) is the social marginal cost of the public funds (on a per dollar basis) and t[X.sub.g] is the effect on tax revenue resulting from the complementarity and substitutability between public and private goods. In the present model, MRS = -[E.sub.g], [P.sub.g] = [C.sup.g] and the marginal social cost of a dollar of tax revenue is a dollar since there is lump-sum taxation. One could then interpret the term [Mathematical Expression Omitted], in equation (13), as the effect of changing g on tariff revenue. In equation (15), however, an equivalent interpretation cannot be offered for the term [Mathematical Expression Omitted]. Therefore, for the second right-hand-side terms of equations (11)-(13), zero for the case of equation (12), we propose a broader terminology of which the "effect of changes in g on revenue from trade taxes" is a special case. Specifically, in the case of import quotas and VERs, it is the prices among other things, that adjust to an exogenous shock (e.g., changes in g), while in the case of tariffs it is the volume of imports. In the case of a quota, however, changes in the price of the imported good do not affect directly welfare, while they do in the case of VERs. Similarly, in the case of a tariff, changes in imports affect welfare. Thus, a broader interpretation of the second right-hand-side term of equations (11)-(13) is that it captures, through the general equilibrium complementarity and substitutability between private and public goods, the effect of changing g on the welfare cost created by the trade restrictions.

V. Concluding Remarks

The modified Samuelson rule, as far as we know, has been examined only in models where all goods prices are fixed. In these models the relationship between the social marginal cost and the marginal cost depends only on the complementarity/substitutability between the public and the taxed goods. In a model such as the one developed in this paper, where prices for some goods are fixed and for some (i.e., nontraded good, imported good under quantitative trade restrictions) are variable, however, complementarity/substitutability between the public and the taxed goods is not enough. The relationship between the social marginal cost and the marginal cost depends also on (i) the complementarity/substitutability between the imported and nontraded good (i.e., fixed and variable price goods), between the public and nontraded goods (i.e., variable price good and public good), and (ii) on the prevailing trade regime.

Michael S. Michael University of Cyprus Nicosia, Cyprus

Panos Hatzipanayotou Aristotelian University of Thessaloniki Thessaloniki, Greece

Appendix: Derivations of Equations (8)-(10).

Differentiating equation (4) we obtain:

[Mathematical Expression Omitted]. (14)

Differentiating equation (5), holding world prices constant (i.e., d[p.sup.*] = 0), and using the properties of the R(p, q, g) function, we obtain:

[Mathematical Expression Omitted]. (15)

Under a VER constraint, dt = dp, [Rho] = 1, and assuming unchanged VER level [Mathematical Expression Omitted], equation (15) becomes:

[Mathematical Expression Omitted]. (16)

Under an import quota, dt = dp, [Rho] = 0, and assuming unchanged quota level [Mathematical Expression Omitted], equation (15) becomes:

[Mathematical Expression Omitted]. (17)

Finally, under an import tariff, [Rho] = 0, and dt = dp = 0 assuming a constant tariff rate. Then, equation (15) becomes:

du + dT - td[Z.sub.p] = -[E.sub.g]dg. (18)

Differentiating equation (6) we obtain:

[Z.sub.pp]dp + [Z.sub.qq]dq + [Z.sub.pg]dg + [Z.sub.pu]du = d[Z.sub.p]. (19)

Differentiating equation (7) we obtain:

[Z.sub.qp]dp + [Z.sub.qq]dq + [Z.sub.qg]dg + [Z.sub.qu]du = 0. (20)

Then, under a VER, equations (14), (16), (19) and (20) constitute a system of four equations in four unknowns (i.e., u, p, q, T) in terms of the policy parameter g. Using Cramer's rule to solve this system, we obtain, among other results, equation (8) of the paper.

Under an import quota, equations (14), (17), (19) and (20) constitute the system of equations in the four unknowns (i.e., u, p, q, g) in terms of the policy parameter g. Solving this system as previously noted, we obtain, among other results, equation (9) of the paper.

Finally, under a tariff, substituting equation (19) into (18), and using (14) and (20), the resulting system of three equations in the unknowns u, q, T is solved in terms of the policy parameter g to obtain equation (10).

1. It is well established in the trade literature that, in the context of a small open economy, despite their negative welfare implications trade restrictions may exist for a variety of (valid/invalid) economic or non-economic reasons, e.g., "fair" play by domestic industries against "unfair" trade practices by foreigners, government revenue, infant industry protection, etc. Recently, Hillman and Ursprung [9, 729-45] using a model of political competition between candidates contesting elective office show that no candidate has an interest in formulating trade policy position using a tariff if a VER is a policy option.

2. When a country imposes an import quota, quota rents accrue to holders of the import licenses. That is, in the importing country, quota rents are captured either by domestic importers or by the government when it auctions the import licenses. Often, the rights to sell foreign goods in the domestic market are assigned to governments or individuals in the exporting countries. In this case, where quota rents accrue to foreigners, the analysis is similar to the case of VERs. Generally, however, quota rents accrue to domestic importers.

3. This method of deriving the R(p, q, g) function is due to Abe [1, 209-22], who calls it "restricted GNP" function. For a proof of the properties of this function in the case where only traded goods exist see Abe [1, 209-22]. See also Abe [2, 875-85].

4. Sufficient condition for the imported and public goods to be general equilibrium complements is that the imported and the public good are net complements (i.e., [Z.sub.pg] is positive), the imported and the nontraded good are net complements (i.e., [Z.sub.pg] is positive), and the public and the nontraded good are net complements (i.e., [Z.sub.qg] is positive).

5. Bradford and Hildebrant [6, 111-31] argue that ". . . some public goods are so complementary to certain private goods that the value of the former would be zero without the latter . . ." [p. 116]. Examples include, public highways and transportation vehicles, air safety and air travel, public television and television sets, and so on.

6. When VERs are imposed, the domestic price of the imported good increases. Consequently, domestic residents suffer a loss equal to the difference between the loss in the consumer surplus minus the gain in producer surplus. This loss equals the VER rents that accrue to foreigners. When the increase in the level of the public good raises the imported good price, it also raises this loss (i.e., the difference between the loss in consumer surplus minus the gain in producer surplus).

7. Hatzipanayotou and Michael [8, 727-45] demonstrate, in a different framework, a similar asymmetry regarding the welfare effect of policy induced changes in revenue from import restrictions.

8. Batina [5, 125-33], in a model of a closed economy with heterogeneity and distorted by an arbitrary system of commodity taxes, derives a modified Samuelson rule when the production of the public good is financed through a poll tax. His rule [p. 130, equation (5)] resembles the optimal rule for public good production, financed through lump-sum taxes, under a tariff (i.e., equation (17)). The focus of his analysis is on the interpretation of the modified Samuelson rule. The focus of our analysis, however, is to extend these results beyond the case of direct price distortions (e.g., to the case of an import quota, or a VER), and in the presence of goods with variable prices (i.e., nontraded).

9. Note that under constant returns to scale, the unit cost and the marginal cost of production are equal.

10. See footnote 6 as to why an increase in VER rents has a negative effect on domestic welfare.

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