Tax policy and endogenous factor supply in a small open economy.

I. Introduction

Since a paper by Kemp and Jones [6], there have been few attempts at modeling the effects of variable factor supply in a simple trade model, with almost all papers confined to a two-good, two-factor, small-open-economy model. Papers were typically concerned with the effect of variable factor supply on the shape of the economy's production possibility set, and with the result derived in Kemp and Jones [6] that when one factor is endogenously supplied in a simple two-good, two-factor general equilibrium trade model, output supply functions may no longer be upward-sloping, as in Frenkel and Razin [5], and Martin [7; 8], for example. A more general treatment is given in Dixit and Norman [3] and Woodland [14], where in an m-good, n-factor general equilibrium trade model, it is assumed that a subset of the factors of production are endogenously supplied, but very few comparative statics experiments are explored. A recent paper by Mayer [9] synthesizes much of this earlier work in a two-good, two-factor general equilibrium model, and provides an initial attempt at modeling the employment effects of tariff changes.

Since the existence of variable factor supply in even a very simple general equilibrium trade model has been shown to complicate the simplest comparative statics results, the literature has paid very little attention to the exploration of the effects of various tax policy changes on factor supply. As a result, an important channel of the effects of trade tax changes on equilibrium has been largely ignored, that being the effect of trade tax changes on factor market equilibrium, and the concomitant effects on equilibrium in the small open economy. The question of the effects of variable factor supply is also important given the obvious attention that policy-makers and various interest groups pay to the employment effects and industry output effects of trade and factor tax policy changes.

Another important implication of the lack of attention paid to endogenous factor supply in international trade models is the inability of trade models to analyze the effects of taxes on factor supply. In the standard Heckscher-Ohlin trade model of a small open economy with an endowment of factors of production, factor taxes can have no real effects.

The objective of this paper is to use the general equilibrium trade model with endogenous factor supply developed in Dixit and Norman [3] and Woodland [14] to illustrate the implications of the presence of endogenous factor supply for many familiar tax policy experiments. We will also examine the effects of tax policy changes on factor market equilibrium. Results will be derived using duality theory in an m-good, n-factor trade model, so that the analysis will not be restricted to the case where there are only two goods and two factors. Assumptions of constant returns to scale, perfect competition, and the absence of externalities will be retained so as to concentrate on the effects of variable factor supply. The model is presented in section II.

In order to raise revenue, many governments collect taxes on factors of production whose level of supply ought most appropriately be modeled as being endogenous. For example, tax on labor income collected in Canada in 1986 implied an average tax rate of 17.8% of value added by labor. When some factors are endogenously supplied, these taxes will have real effects on equilibrium factor supply, so our first step must be to determine the employment effects of factor tax changes on endogenously supplied factors in a general equilibrium trade model. These effects are derived in section III, where we show that factor taxes will always reduce variable factor supply when those factors are normal. Since taxes on international trade will also affect equilibrium factor supply, it also becomes necessary to determine the way in which trade taxes distort the market for endogenously supplied factors. We determine the effects of trade taxes on the supply of endogenous factors in section III, and describe conditions under which trade tax changes will either raise or lower factor supply. Results in this section imply a generalization of results in Mayer [9], which uses a two-good, two-factor general equilibrium trade model with one endogenously supplied factor, and only considers the effects of tariffs on imports.

Once the effects of trade and factor taxes on general equilibrium have been determined, we can consider the welfare effects of trade and factor taxes when some factors are endogenously supplied. In section IV, we show that zero trade and factor taxes still constitute a Pareto-optimal equilibrium for a small open economy, so that increases in trade or factor taxes starting from an initial equilibrium with no taxes must reduce welfare. However, if there exist trade and factor taxes, then a reduction in trade taxes may lower welfare. We derive the characteristics of the vector of optimal second-best trade taxes when there are taxes on endogenously supplied factors, for a small open economy, and show how trade taxes can offset or augment the factor-market distortion imposed by a factor tax in a small open economy.

The volume of trade effects of trade and factor tax changes are given in section V. For a given trade tax change, volume of trade effects are derived when all factor supplies are fixed and when the supply of some factors are determined endogenously. The presence of variable factor supply is shown to magnify the volume of trade effects whenever the cross-price elasticities between an endogenously supplied factor and traded goods are sufficiently small. This result is particularly important when considering results from general equilibrium trade models describing the volume of trade effects of trade tax changes, such as those implied by the recently completed uruguay Round of the GATT or the North American Free Trade Area between Canada, the U.S., and Mexico, since these predicted volume of trade effects would generally be underestimated. Concluding comments are offered in section VI.

II. Dual Trade Model with Endogenous Factor Supply

In constructing the trade model, we must keep in mind the ultimate objective of carrying out some comparative statics experiments. To this end, we will not consider the case where there are more goods produced in equilibrium than there are factors of production, since then the equilibrium vector of output supplies would not be unique, and the local analysis using differentiation in subsequent sections could not be applied. In the interest of simplicity, we will also not consider the case where there are more factors than goods, since not much insight is added at the expense of a considerable degree of complication.(1) Instead we concentrate solely on the case where the number of factors equals the number of goods produced in equilibrium. In this "even" model, for a small open economy (SOE) unable to affect world output prices, the production sector will take output prices as given by world terms of trade, level of supply of the endogenous factors as given by the solution to the consumption sector's utility maximization problem, and the level of supply of the remaining factors as being exogenously given, and will determine input prices and level of output supply. The consumption sector will take output prices as given by world terms of trade, input prices as given by the solution to the production sector's cost minimization problem, and will determine the level of output demand and level of supply of the endogenous factors, subject to an income constraint given by the endowment of factors.

We describe an economy with m perfectly competitive industries producing outputs y[prime] = ([y.sub.1], [y.sub.2], ..., [y.sub.m]) with n factors of production. The consumption vector for the economy is represented by the vector [Mathematical Expression Omitted]. We let [Mathematical Expression Omitted] be the vector of domestic output prices corresponding to y and z. Suppose that a subset e [is less than or equal to] n of the factors of production are endogenously supplied, while the remaining n - e factors are exogenously supplied. The endowment vector can be written as:

[Mathematical Expression Omitted],

with v [is an element of] [R.sup.+], [Mathematical Expression Omitted]. We correspondingly decompose the factor price vector into [Mathematical Expression Omitted], [Mathematical Expression Omitted]. To incorporate the presence of endogenously supplied factors into the demand side of the economy, assume that there exists a single representative consumer in the economy who owns all of the productive factors, some of which he directly consumes. Since the consumer can substitute between consumption of some factors and consumption of goods, the supply of these factors will be variable. While there are other ways of modeling variable factor supply in a general equilibrium model [9], this method keeps the model relatively simple and facilitates the comparison of results to those derived in other representative agent models.

To enable us to analyze the effects of trade and factor taxes, we allow for the presence of ad valorem taxes or subsidies on international trade and the level of usage of the endogenously supplied factors.(2) All tax revenue is costlessly collected and redistributed to the representative consumer. The only trade policy instrument available to a nation will be an ad valorem tax or subsidy on exports or imports, summarized by the vector t[prime] = ([t.sub.1], [t.sub.2], ..., [t.sub.m]). Define the vector T[prime] [is equivalent to] (1 + [t.sub.1], 1 + [t.sub.2], ..., 1 + [t.sub.m]), and describe the vector of world output prices as p*[prime] = ([p*.sub.1], [p*.sub.2], ..., [p*.sub.m]). The vector of domestic prices can then be expressed as:

p[prime] = T[prime]D(p*),(2)

where D([center dot]) denotes a diagonal matrix with the elements of ([center dot]) on the main diagonal. If we define the vector of net exports for this economy as x = y - z [is an element of] [R.sup.m], then total trade tax revenue will be given by the expression:

TR = -t[prime]D(p*)x. (3)

If good i is imported, then an import tariff (subsidy) is represented by [t.sub.i] [is greater than] 0([t.sub.i] [is less than] 0), while if i is exported, then an export tax (subsidy) is reflected by [t.sub.i] [is less than] 0([t.sub.i] [is greater than] 0).

We will take the vector of ad valorem taxes on endogenously supplied factors as being given by f [is an element of] [R.sup.e]. To motivate this assumption, we could suppose that the government must raise some desired level of revenue, say for the provision of a public good.(3) The tax on the endogenously supplied factors will be paid by the factor owner. Since the number of goods equals the total number of factors, the gross wage w paid to the endogenously supplied factors is determined by the production sector's cost minimization problem. However, the net wage earned by the representative consumer for supply of the endogenous factor will vary according to the factor tax. We will define the net-of-tax return earned by the representative consumer per unit supplied of the endogenous factor as:

[Mathematical Expression Omitted].

Total factor tax revenue collected per unit of the ith endogenously supplied factor is [f.sub.i][w.sub.i], and total factor tax revenue is given by:

[Mathematical Expression Omitted].

Following the general equilibrium trade model with variable factor supplies in Dixit and Norman [3], we assume that there exists a twice continuously differentiable strictly quasi-concave utility function which represents preferences over consumption of goods and the endogenously supplied factors, U(z, v). The properties of the expenditure function corresponding to the consumer's expenditure minimization problem are noted in Dixit and Norman [3] and Woodland [14], so for our purposes it suffices to note that the output demand functions are given by:

[Mathematical Expression Omitted],

and the supply functions of the endogenous factors are given by:

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is the expenditure function. In equilibrium, the expenditure function is equal to total fixed factor income, so that [Mathematical Expression Omitted]. This is income from fixed factors in Woodland [15, 240], lump-sum income in Dixit and Norman [3, 63], and unearned income in Mayer [9, 107], modified to include income from factor and trade taxes.

To describe the supply side of the economy, we assume that each industry produces output [y.sub.i] using inputs [Mathematical Expression Omitted], i = 1, ..., m, and constant returns to scale technology summarized by the twice continuously differentiable, strictly quasi-concave production functions [f.sup.i]([a.sup.i]), i = 1, ..., m. Once the consumption sector has determined the equilibrium levels of supply of the endogenous factors v, the set of feasible production points can be summarized by an appropriately defined production possibility set. Given output prices, endowments and factor supplies, the activities of the production sector can be summarized by the revenue function [Mathematical Expression Omitted], whose properties are noted in Woodland [14] and Dixit and Norman [3]. The output supply and input price functions are given by:

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

In general equilibrium, total income must equal total expenditures. This condition is summarized by equation (9):

[Mathematical Expression Omitted].

The external sector of this small open economy can be summarized by the balance of trade function, defined as:

[Mathematical Expression Omitted].

Given our definition of net exports, the net export function is given by the derivative of the balance of trade function with respect to output prices:

[Mathematical Expression Omitted].

In equilibrium, trade is balanced, so that equation (10) is equal to zero, and all factors are fully employed, reflected by the condition [Mathematical Expression Omitted].

III. Employment Effects of Trade and Factor Taxes

We can solve for the change m endogenous factor supply caused by a given change in trade taxes or factor taxes by totally differentiating the representative consumer's factor supply function given in equation (7):

[Mathematical Expression Omitted].

A change in trade taxes dt will cause a change in domestic output prices dp according to the total differential of equation (2), and a change in factor taxes and trade taxes will both cause a change in the net-of-tax return to the endogenously supplied factors according to the total differential of equation (4). By substituting the total differential of the income-equals-expenditure constraint (equation (9)) for the change in utility dU, and simplifying using the envelope properties of the revenue and expenditure functions (equations (6)-(8)), we can rewrite the change in endogenous factor supply in equation (12) as:

[Mathematical Expression Omitted],

where [v.sub.[m.sub.f]] = -[E.sub.wU]/[E.sub.U] is the income effect on endogenous factor supply.(4) This is the general change in factor supply caused by any trade or factor tax change. We can isolate the effects of a factor tax change by considering the following example. Suppose we begin at an initial equilibrium where trade taxes are fixed and equal to zero, so that t = dt = 0.(5) If a factor tax change df results in the SOE producing the same m goods in equilibrium after the tax change as were produced before the factor tax change, then the factor price vector w does not change, so that dw = 0. Substituting these restrictions into equation (13) and premultiplying by df[prime]D(w) gives:

df[prime]D(w)dv = df[prime]D(w)[E.sub.ww]D(w)df + df[prime]D(w)[v.sub.[m.sub.f]][f[prime]D(w)dv]. (14)

The term [Mathematical Expression Omitted] is the vector of changes in per unit factor tax revenue due to the given factor tax change df. The first term on the right-hand side is a quadratic form about a negative definite matrix (recall that the expenditure function is strictly concave in w), so that df[prime]D(w)[E.sub.ww]D(w)df [is less than] 0. This is the direct effect of the factor tax change on endogenous factor supply. Essentially, an increase in factor taxes lowers factor returns and therefore lowers endogenous factor supply. The second term represents the income effect of a factor tax change on the supply of the endogenous factors. When evaluated in the neighborhood of zero factor taxes, the vector f is a null vector, so that:

df[prime]D(w)dv = df[prime]D(w)[E.sub.ww]D(w)df [is less than] 0. (15)

In the neighborhood of zero factor taxes, changes in taxes on any endogenously supplied factors will be negatively correlated with changes in the supply of endogenous factors.(6)

A more general result can be obtained by rewriting equation (14) as follows:

df[prime]D(w)[I - [v.sub.[m.sub.f]]f[prime]D(w)]dv = df[prime]D(w)[E.sub.ww]D(w)df [is less than] 0, (16)

where I is the e x e identity matrix. If all endogenously supplied factors are normal, then an increase in income leads to a fall in factor supply (an increase in demand by the representative consumer for the endogenously supplied factor), so that [v.sub.[m.sub.f]] is a negative vector. If all factor taxes are positive, every element of the matrix in square brackets in equation (16) is positive, and we get the following result:

Result. Changes in factor taxes will be negatively correlated with changes in endogenous factor supplies.(7)

Only when factor supply is subsidized ([f.sub.i] [is less than] 0) might the income effect of a factor tax change on factor supply outweigh the substitution effect, in which case an increase in the factor tax would result in an increase in factor supply.

It is worth noting the seemingly counter-intuitive role played by the assumption that endogenously supplied factors are normal, since in partial equilibrium models, this assumption is a necessary condition to observe backward-bending factor supply curves. However, in our model the income effect derives from the change in income due to the endogenous general equilibrium changes in employment. In this case a factor tax increase which decreases the real return to the endogenously supplied factor will necessarily result in a decrease in factor supply.

To find the effect of a given trade tax change on supply of the endogenous factor, in the neighborhood of zero trade and factor taxes, substitute the restriction t = f = df = 0 into equation (13) and premultiply by dw[prime] to get:

dw[prime]dv = -dw[prime][E.sub.wp]D(p*)dt - dw[prime][E.sub.ww]dw. (17)

The second term in equation (17) is the negative of a quadratic form about a negative definite matrix (E is concave in w), and is therefore a positive scalar. If the first term is a positive scalar, then changes in the returns to the variable factor due to the trade tax change will be positively correlated to factor supply changes. This first term is dependant upon two elements. To illustrate, suppose there is only one endogenous factor, and consider the effect of applying a trade tax(8) on a single traded good.(9) If that good uses the endogenously supplied factor intensively, then the return to the variable factor will rise, dw [is greater than] 0.(10) Then if the taxed good and the variable factor are complements in consumption, an increase in the tax on the traded good will cause consumption of the variable factor to fall, so that supply of the variable factor to the production sector will rise, -[E.sub.wp] [is greater than] 0. That is, as sufficient condition for a trade tax to increase supply of an endogenous factor is to tax a good which uses the endogenous factor intensively and which is a complement in consumption for the endogenous factor. A sufficient condition for a trade tax to decrease supply of an endogenous factor is to tax a good which does not use the endogenous factor intensively and which is a substitute in consumption for the endogenous factor. These conditions imply a generalization of the result in Mayer [9, 111-12], from the two-good, two-factor model to the case where there can be any number of goods or factors, and the trade tax is not restricted to being a tariff on the imported good. Of course, a more general version of this result is summarized by equation (13) above, where the dimension from which tax policy instruments can be chosen is relaxed to include taxes on factors. In either case, the result that dw[prime]dv [is greater than] 0 will always hold as long as the cross-elasticity between demand for the endogenous factor and the taxed good is zero or sufficiently close to zero (since -[E.sub.[wp.sub.i]] = [Delta]v/[Delta][p.sub.i]). If this condition holds, then the result that dw[prime]dv [is greater than] 0 can be interpreted as follows:

Result. A change in the trade tax on a good will be positively correlated with changes in the supply of the endogenously supplied factors used intensively in production of that good, and will be negatively correlated with changes in the supply of other endogenously supplied factors of production.

IV. Optimal Trade Taxes in the Presence of Factor Taxes

Now consider the effect of factor taxes on domestic welfare in the presence of trade taxes. We continue to assume that world output prices p* are fixed, that the number of goods produced in the initial equilibrium is equal to the total number of factors of production, and that all goods continue to be produced after the tax change. If the initial vector of trade taxes is t, and the initial vector of taxes on the endogenously supplied factors is f, then the effect of a change in trade taxes dt and factor taxes df on welfare is found by totally differentiating the income-equals-expenditure constraint in equation (9), noting that a trade tax change will cause domestic output prices to change, which implies that domestic input prices will also change, and using the envelope properties of the revenue and expenditure functions in equations (6)-(8):

[E.sub.U]dU = f[prime]D(w)dv - t[prime]D(p*)dx. (18)

The final term in equation (18) is the familiar volume of trade effect due to the trade tax change. The first term is the effect of the tax change on variable factor supply, where f[prime]D(w) is the vector of per unit factor taxes which are redistributed lump-sum to the representative consumer.

Of course, it must still be optimal for the SOE to set all trade and factor taxes equal to zero. This can be seen by noting that equation (18) is set to zero when we substitute t = f = 0. In the neighborhood of zero trade and factor taxes, any increase (decrease) in the tax on an endogenously supplied factor causes factor supply to fall (rise), and any increase (decrease) in the trade tax on any good causes net exports of that good to fall (rise). As a result, any movement away from zero trade and factor taxes causes a fall in welfare. If we consider an initial equilibrium where trade taxes are all zero, trade taxes are not changing, but there exists a tax on factor supply, so that f [is greater than] 0, then the reduction in welfare due to the factor tax is a direct result of the fact that the tax causes the factor to be under-supplied in equilibrium due to the factor market distortion.

If there does exist a factor tax, the resulting factor-market distortion can be offset by a tax on trade. For any given factor tax vector f, we can solve for the optimal second-best vector of trade taxes, denoted to, by setting equation (18) equal to zero and solving:(11)

[t[prime].sub.o] = f

Given world output prices p* and domestic input prices w, the vector [t.sub.o] is dependent upon the level of factor taxes, and upon the responsiveness of variable factor supply and net exports to trade taxes, represented by the matrices M([Delta]v/[Delta]t) and M[([Delta]x/[Delta]t).sup.-1], respectively. All other things being equal, the larger is the tax on any variable factor, the larger will be the factor market distortion which the trade tax vector [t.sub.o] must offset, implying a larger value for the optimal second-best vector of trade taxes. The matrix M([Delta]x/[Delta]t) is composed of elements which are a function of trade elasticities, so that ceteris paribus, optimal trade taxes are larger (smaller) on those goods which have smaller (larger) trade elasticities. Then whether a trade tax or subsidy is required to offset the factor market distortion is dependent upon the matrix M([Delta]v/[Delta]t), which describes the effect of a trade tax change on supply of the variable factor, and was discussed earlier in section III.

To illustrate, consider the two-good, two-factor version of this model, where one factor is endogenously supplied, and suppose without loss of generality that the first good is imported. Assume that the endogenously supplied factor is normal. If there exists a factor market distortion due to a tax f on the endogenously supplied factor, what is the second-best trade tax required to offset the factor-market distortion? In this two-good model, only relative prices are important, so it is sufficient to determine the optimal second-best trade tax on the imported first good, [Mathematical Expression Omitted]. The diagonal elements of the matrix M[([Delta]x/[Delta]t).sup.-1] are positive since the balance of trade function is convex in output prices. If the imported good uses the endogenously supplied factor intensively, then it would be optimal to have an import tariff on good 1, [Mathematical Expression Omitted], since the resultant increase in the return to the variable factor would increase its supply ([Delta]v/[Delta][t.sup.1] [is greater than] 0). On the other hand, if the imported good does not use the variable factor intensively, then [Delta]v/[Delta][t.sup.1] [is less than] 0, so it would be optimal to subsidize imports, increasing production of the exported good, and increasing variable factor supply. In this way, equation (19) indicates whether it is necessary to tax or subsidize trade in a particular good in order to offset the factor-market distortion introduced by the tax on the endogenously supplied factor.

Alternatively, we could rewrite equation (19), noting that each element of equation (19) is a scalar, since we are considering an example where there is only one endogenous factor and a trade tax on the single imported good:

[Mathematical Expression Omitted],

where [[Eta].sub.vt] is the elasticity of supply of the variable factor with respect to the trade tax, and [[Eta].sub.i] is the import demand elasticity.(12) The term {w [center dot] v}/{p* [center dot] x} is the ratio of the value of variable factor supply (gross of taxes) over the value of trade in the imported good (at world prices). In this simple illustrative example, this equation gives us an idea of the value of the optimal import tax for given values of these specific parameters. For example, if we use a value of [[Eta].sub.i] = -1.19 for the import demand elasticity, if the share of value added by labor to the value of trade is (w [center dot] v)/(p* [center dot] x) = -2.08, and if there exists a tax on the variable factor of f = 0.178, then the optimal import tax as a function of the elasticity of supply of the variable factor with respect to the trade tax is given in Table I.(13) Only positive values are given for the factor supply elasticity [TABULAR DATA OMITTED] with respect to the price of the imported good, since according to the data set in Wigle [13], imports into Canada are relatively labor-intensive.

V. Volume of Trade Effects of Trade and Factor Taxes

We can solve for the volume of trade effects of a tax policy change by differentiating the net export equation (11) with respect to trade and factor taxes to get:

dx=[G.sub.pv]dv + [E.sub.pw]D(w)df + [[G.sub.pp] - [E.sub.pp]]D(p*)dt + [v.sub.[m.sub.f]][f[prime]D(w)dv - t[prime]D(p*)dx] - [E.sub.pw][I - D(f)]dw. (21)

We can describe the volume of trade effects of a factor tax change by evaluating equation (21) in the neighborhood of zero trade and factor taxes as follows:

dx = [G.sub.pv]dv + [E.sub.pw]D(w)df. (22)

We can decompose the change in net exports into a change from the production and consumption side of the economy. If a factor tax is introduced (df [is greater than] 0), supply of the endogenous factor will fall (dv [is less than] 0). Output of those goods which do not use the taxed factor intensively will rise ([G.sub.[p.sub.i][v.sub.j]] [is greater than] 0). A sufficient condition for a factor tax to increase exports (decrease imports) is that the good be a complement in consumption with the taxed factor ([E.sub.[p.sub.i][w.sub.j]] [is less than] 0). Alternatively, if cross-price elasticity between the taxed factor and any good is sufficiently small, then the effect of a factor tax change on net exports will be given only by the pattern of relative factor intensity.

A potentially more interesting and important result, given attempts to model the effects of trade tax changes arising from negotiations such as those under the Uruguay Round of the GATT or the North American Free Trade Area between Canada, the U.S., and Mexico, is the effect of trade tax changes. If we evaluate equation (21) in the neighborhood of zero trade and factor taxes, then if df = 0, dt [is not equal to] 0:

dx = [G.sub.pv]dv - [E.sub.pw]dw + [[G.sub.pp] - [E.sub.pp]]D(p*)dt. (23)

In a model where the supply of all factors is fixed, dv = 0 and [E.sub.pw] is a null matrix, so we get the standard result that dt[prime]D(p*)dx [is greater than] 0. (Recall that G is strictly convex in p and E is strictly concave in p, so that a quadratic form about [G.sub.pp] - [E.sub.pp] will be strictly positive.) That is, the imposition of an import tariff (dt [is greater than] 0) will cause imports to fall (dx [is greater than] 0), and so on. With variable factor supply, this result is complicated by the presence of the first two terms in equation (23). To illustrate, suppose only one factor is endogenously supplied, and consider the effect of the imposition of a tariff on an imported good, so that d[t.sub.i] [is greater than] 0, [x.sub.i] [is less than] 0. The effect of the term [G.sub.pv]dv will always magnify the volume of trade effect, since if the imported good uses the endogenous factor intensively, [G.sub.[p.sub.i]v] [is greater than] 0, and the return to the endogenous factor will rise, so that dv [is greater than] 0. If not, then [G.sub.[p.sub.i]v] [is less than] 0 and the return to the endogenous factor will fall, so that dv [is less than] 0. This effect will be magnified further when the imported good and the endogenous factor are complements in consumption, since then dw [is greater than] 0 and -[E.sub.wp] = -[E.sub.pw] [is greater than] 0.(14) This allows us to state the following result:

Result. The volume of trade effect of a trade tax change will always be magnified when a single factor is endogenously supplied:

* when the trade tax is imposed on a good which uses the variable factor intensively and is a complement in consumption for the variable factor,

* or when the trade tax is imposed on a good which does not use the variable factor intensively and is a substitute in consumption for the variable factor.

Of course, these are sufficient conditions. Even if these conditions do not hold, the volume of trade effect would still be magnified as long as the cross-price elasticity of demand for the variable factor with respect to the taxes good were zero or sufficiently close to zero. An important implication of this result is that standard general equilibrium trade models (computable general equilibrium models, for example) will generally underestimate the volume of trade effects of a given trade tax change if factor supplies are all modeled as being fixed.

VI. Conclusion

By modifying the standard general equilibrium trade model of a small open economy to incorporate the presence of endogenously supplied factors, we have been able to add insight to some oft-considered comparative statics problems. Modeling factor supply decisions as the outcome of the consumption sector's utility maximization problem has enabled us to show how changes in domestic trade taxes affect the supply of factors of production. This has enabled us to supplement the well-known direct effects of trade tax changes on trade and welfare with the indirect effects of trade tax changes through changes in factor supply. Sufficient conditions were derived under which the volume of trade effect of trade tax changes were always magnified due to the presence of variable factor supply.

The model also allows for the consideration of a problem which trade models were hitherto unable to consider. With fixed factor supplies, taxes on factors of production in a Heckscher-Ohlin model of a small open economy cannot have any real effects, since factor supplies cannot change in response to factor taxes. When factor supplies are endogenously determined, the imposition of taxes on factors of production leads to a factor market distortion which causes factors to be under-supplied. This distortion leads to a reduction in welfare, which implies a role for trade taxes to offset the factor market distortion and raise domestic welfare, even in a small open economy. We derive an expression for this vector of second-best trade taxes, and provide a simple "back-of-the-envelope" calculation of optimal import taxes for Canada, given import demand elasticities and non-zero factor taxes, as a function of the factor supply elasticity with respect to the import tax. We also show that zero trade and factor taxes must still be welfare-maximizing for a small open economy facing given world output prices.

1. For a complete treatment of the case where there are more factors than goods in this model, see Waschik [12].

2. We will not consider taxes on the exogenously supplied factors, since it is well known that such taxes will have no real effects in a model with a representative consumer where revenue from such a tax is costlessly collected and redistributed to the representative consumer.

3. For example, a simplified version of the model in Atkinson and Stiglitz [1, 398-99], has the government choosing a wage tax to maximize utility of the consumption sector subject to the requirement that the government raise some given amount of revenue.

4. An issue is made in Mayer [9] as to whether or not fixed factor income is endogenous or exogenous in the literature dealing with variable factor supply. A completely specified general equilibrium model should incorporate the effects of endogenous changes in fixed factor income, and these are reflected by the income effects in equation (13).

5. Trade taxes are chosen to be equal to zero at the initial equilibrium so as to avoid complicating the following results with the generally indeterminate income effects [v.sub.[m.sub.f]], and to isolate and emphasize the effects of factor taxes.

6. If there were only one endogenously supplied factor, df and dv would be scalars. We could then say that a tax (subsidy) on the endogenously supplied factor would cause its supply to fall (rise). However, as is typical for results derived in higher dimensional trade models, since df and dv are vectors in general, the sign on the scalar df[prime]D(w)dv in equation (15) enables us to determine only the correlation between elements of df and elements of dv in general.

7. In the aforementioned literature on trade with variable factor supply, the ability to observe perverse output effects (downward-sloping output supply curves) depends upon the existence of backward-bending factor supply curves [8, 550]. The result we have just derived implies that factor supply curves must be upward-sloping, since factor tax changes are inversely related to factor price changes. Our more definite result obtains because of our earlier assumption that the number of factors equals the number of goods, so that wages are determined by the production sector. If this were not the case, our result would be complicated by the presence of terms describing the substitutability and complementarity between factors and goods [9, 108; 12, 27].

8. In this case, a trade tax will be a tax on a traded good whose effect is to raise the domestic price of that good relative to its world price, so that we needn't specify whether the good is imported or exported.

9. This result is illustrated for the case of one endogenous factor to simplify the example. The tax is applied to only a single traded good because otherwise the effect of the tax on factor prices may be indeterminate, since more than one factor intensity term would be needed to determine the factor price change resulting from the trade tax change.

10. Since we may have more than two goods and two factors, our usual notion of factor intensity becomes ill-defined. We use the notion of generalized factor intensities [4], where good i is defined as using factor j intensively if [Delta][y.sub.i]/[Delta][v.sub.j] [is greater than] 0.

11. A possible interpretation of this problem could be that the government solves a two-stage problem, first solving for the vector of optimal second-best trade taxes [t[prime].sub.o] by solving equation (19) for any given factor tax, and then solving for the vector of trade and factor taxes which yield a desired level of revenue FTR + TR. At least in developed countries, it is arguably the case that trade taxes are motivated more by the desire to affect (protect) domestic employment than to raise revenue.

12. Note that the last two terms in this expression must have the same sign, since if the good is imported (exported), the value of trade will be negative (positive) and the import demand (export supply) elasticity will be negative (positive). This implies that the sign on the optimal trade tax is determined only by the sign of the factor supply elasticity with respect to the trade tax.

13. The value of [[Eta].sub.i] = -1.19 for the import demand elasticity is taken from Wigle [14, 557]. The factor tax of f = 0.178 is calculated by taking the ratio of direct taxes on persons to labour's share of Gross Domestic Product in Canada in 1986, as reported in Statistics Canada Taxation Statistics [11] and the Bank of Canada Review [2], respectively. The value of (w [center dot] v)/(p* [center dot] x) = -2.08 for the ratio of the value of labour supply to the value of trade in the imported good is calculated using labour's share of Gross Domestic Product in Canada in 1986, as reported in the Bank of Canada Review [2], and the value of imports in Canada in 1986, as reported in Statistics Canada National Income and Expenditure Accounts [10].

14. In this case, consumption of the endogenous factor falls along with consumption of the imported good whose price is rising due to the trade tax, so that supply of the endogenous factor to the production sector will rise.

References

1. Atkinson, Anthony B. and Joseph E. Stiglitz. Lectures on Public Economics. Maidenhead: McGraw-Hill, 1980.

2. Bank of Canada Review. Ottawa: Bank of Canada, August 1992.

3. Dixit, Avinash and Victor Norman. Theory of International Trade. Cambridge: Cambridge University Press, 1980.

4. ----- and Alan D. Woodland, "The Relationship Between Factor Endowments and Commodity Trade." Journal of International Economics, November 1982, 201-14.

5. Frenkel, Jacob A. and Assaf Razin, "Variable Factor Supplies and the Production Possibility Frontier." Southern Economic Journal, January 1975, 410-19.

6. Kemp, Murray C. and Ronald W. Jones, "Variable Labour Supply and the Theory of International Trade." Journal of Political Economy, February 1962, 30-36.

7. Martin, John P., "Variable Factor Supplies and the Heckscher-Ohlin-Samuelson Model." Economic Journal, December 1976, 820-31.

8. -----, and J. Peter Neary, "Variable Labour Supply and the Pure Theory of International Trade: An Empirical Note." Journal of International Economics, May 1980, 549-59.

9. Mayer, Wolfgang, "Endogenous Labour Supply in International Trade Theory: Two Alternative Models." Journal of International Economics, February 1991, 105-20.

10. Statistics Canada. National Income and Expenditure Accounts, Cat. No. 13-531, Ottawa: Supply and Services Canada, 1987.

11. -----. Taxation Statistics, Cat. No. 61-208, Ottawa: Supply and Services Canada, 1987.

12. Waschik, Robert G. "International Trade Theory with Endogenous Factor Supply." unpublished Ph.D. Thesis, University of Western Ontario, 1990.

13. Wigle, Randall, "General Equilibrium Evaluation of Canada-U.S. Trade Liberalization in a Global Context." Canadian Journal of Economics, August 1988, 539-64.

14. Woodland, Alan D. International Trade and Resource Allocation. Amsterdam: North-Holland, 1982.