Optimal export taxes with an endogenous location.

Mai, Chao-cheng

1. Introduction

Export tax policy is one of the most debated issues in many developing countries. Those countries that have strong natural advantages in the production of primary commodities (such as agricultural and livestock products, coffee, jute, rubber, and others) have attained at particular times a position as dominant suppliers in international trade. They have often used export taxes on those commodities to obtain foreign exchange and/or government tax revenues (Renaud and Suphaphiphat 1971; Repetto 1972; Gomez-Sabaini 1990). For example, Gomez-Sabaini (1990) provided a detailed analysis of the reasons for the evolution of export taxes in the case of Argentina during the 1932-1987 period and pointed out that, in 1985, the share of export taxes in GDP rose to 1.85%, representing almost 64% of the revenue from foreign trade. Left (1969) followed an exportable surplus approach to develop the notion that export taxes have been used (such as in Brazil) for internal income distribution reasons. Renaud and Suphaphiphat (1971) used a simple econometric analysis to determine the magnitude of the increases in the domestic rice price and paddy price for separate types of rice and paddy as a consequence of the reduction or abolition of the rice export tax.(1) These analyses relied on a setting of either perfect competition or monopoly with constant returns to scale.

Recent papers on trade policy with increasing returns and imperfect competition have attracted attention because of their relevance to the new protectionism. Some articles going in this direction include Spencer and (1983), Brander and Spencer (1984a, b, 1985), Dixit (1984), Venables (1985), De Meza (1986), Eaton and Grossman (1986), Mai and Hwang (1987), Hwang and Mai (1988, 1991), Rodrik (1989), and others. Eaton and Grossman (1986), in particular, have argued that, whenever there is more than one domestic firm exporting to a foreign market, competition among them is detrimental to home-country social welfare. Hence, an export tax can be used to lower total domestic exports, shifting them closer to the output level with collusion. In this way, an export tax enables the home country to fully exploit its monopoly power in trade.

All of the above-mentioned studies analyze trade policies in a nonspatial world in which transportation costs between countries are assumed to be zero or insignificant. It is now widely recognized that geographic space is costly and that the world of economic space is underscored by imperfectly competitive markets (Losch 1954; Greenhut 1956; Churchill 1967) and that the level of trade restrictiveness can have important effects on the distribution of production both across countries and across regions within each country. Several recent studies have examined the implications of trade policies in the context of an open spatial economy. These include Benson and Hartigan (1983, 1984, 1987), Brander and Krugman (1983), Brander and Spencer (1987), Horstmann and Markusen (1990), Hatzipanayotou and Heffley (1991), and Anderson, Schmitt, and Thisse (1995). What is perhaps surprising is that none of the above-mentioned papers have their firms choose their locations between two different sites. In this respect, Krugman (1991a, b, 1993) and Krugman and Elizondo (1996) considered a model in which firms are located in a space and in which transportation costs play a critical role. Krugman and Elizondo (1996) formally demonstrated, in particular, that a fall in export taxes decreases the importance of the home relative to the foreign markets and shifts firm location outwards. Their result receives empirical support from Hanson (1997) in the case of Mexico. Nevertheless, all these works have a positive orientation, and it is believed that normative analyses are needed for exploring the welfare implications of trade policies.

The present paper goes a step further to conduct a normative analysis of an export tax policy and to make a comparison between optimal export taxes with endogenous location and those with exogenous location. More specifically, this paper presents a partial-equilibrium model in which N imperfectly competitive firms produce a homogeneous good for export to another country (i.e., there is no domestic consumption of the good) and in which there is no foreign production taking place. Each firm is assumed simultaneously to make both a Cournot output decision and a spatial location decision within national boundaries. The location decision balances the costs of transporting inputs to the production site against the costs of transporting output abroad. Within this framework, we investigate the impact of an export tax on both equilibrium output and plant location as well as solve the optimal export tax. These results will then be compared with those under an exogenous location. We find that the optimal tax is smaller and that the output effect of a tax increase is larger under an endogenous location as long as production is not constant returns to scale. We also investigate a free-entry version of the model and find that comparisons between exogenous and endogenous locations depend on the convexity/concavity of the demand function.

The paper is organized as follows. Section 2 of the paper sets forth a simple oligopoly model to derive the optimal export tax rule in the short run in which the number of domestic firms is exogenously given. Section 3 conducts a long-run analysis in which free entry is allowed and in which the number of domestic firms is endogenously determined. Concluding remarks are contained in the final section.

2. Optimal Export Tax

The analysis in this paper is confined to a partial equilibrium setting and a Weberian triangle space.(2) Assume there are n domestic firms exporting an identical product to a foreign country. Each firm uses two transportable inputs L and K (which are located at A and B, respectively, in the domestic country) in the production of the output q which is sold, across the border C, at the foreign market F henceforth referred to as the foreign markets as pictured in Figure 1. Each firm chooses its optimal production location at E.(3) Define s and z as the distances of E from A and B, respectively, and h as the distance between E and F; 0 is the angle between FE and FA, [Beta] is the angle between FA and FB, and a and b are the lengths of FA and FB, respectively.

The production function of firm i is specified as

[q.sup.i] = f([L.sup.i], [K.sup.i]), i = 1, 2, ..., n, (1)

which is assumed to be homothetic. To simplify our analysis, we first derive the cost function by minimizing total costs subject to a given output level. That is,

[Mathematical Expression Omitted]

s.t. [q.sup.i] = f([L.sup.i], [K.sup.i]), (2)

where w and r are the base prices of L and K at A and B, respectively, and are assumed to be constant; k and rn are the constant transport rates of L and K, respectively; and s and z are the distances from A and B to the locus of production of the final product, respectively. By the law of cosines, the distances from the two input sites to E can be determined in terms of h and [Theta] as follows:

[s.sup.i] = [-square root of [a.sup.2] + [([h.sup.i]).sup.2] - 2a[h.sup.i]cos [[Theta].sup.i]]

[z.sup.i] = [-square root of [b.sup.2] + [([h.sup.i]).sup.2] - 2b[h.sup.i]cos([Beta] - [[Theta].sup.i]).] (3)

Shephard's lemma indicates that cost functions are separable in input prices and output if production functions are homothetic. In other words, the total cost function can be written as

[Mathematical Expression Omitted], (4)

where [W.sup.i] = w + k[s.sup.i] and [Mathematical Expression Omitted] are the delivered prices of L and K, respectively, and c is a function of [W.sup.i] and [Mathematical Expression Omitted], which are, in turn, functions of [Theta] and h. Note that each firm's location decision involves the two variables [Theta] and h. However, treating both [Theta] and h as decision variables would make the model mathematically tedious and obscure the basic focus of the paper. Hence, we assume throughout the rest of the paper that h is a variable while 0 is given at [Mathematical Expression Omitted]. This simplified assumption bears some empirical relevance. For instance, the FE line (and its extension to AB) in Figure 1 could represent an intercountry highway on which each firm in the domestic country chooses a location to set up its plant.

According to Equation 4, the average cost (AC) and marginal cost (MC) are derivable as

AC = C(q)/q = MH/q (5)

MC = [C.sub.q] = M[H.sub.q]. (6)

From Equations 5 and 6, we can define the following relation:

[Mathematical Expression Omitted]. (7)

In addition, we assume that there is no domestic consumption. The inverse demand function in the foreign market is given by

p = p(Q) = p([summation of] [q.sup.i] where i = 1 to n), [p.sub.Q] [less than] 0, (8)

where Q = [summation of] [q.sup.i] where i = 1 to n as the market is totally occupied by the n domestic firms.

We take a two-stage approach. In the first stage, the domestic government imposes an export tax in anticipation of domestic export. In the second stage, each firm in the domestic country chooses its location and produces and then exports its product to the foreign country. Given that the domestic country levies an export tax t, the second stage problem is expressed by the following profit function of a representative firm:

[[Pi].sup.i]([q.sup.i], [h.sup.i]) = [p(Q) - e[h.sup.i] - t][q.sup.i] - M([h.sup.i])H([q.sup.i] i = 1, 2, ..., n, (9)

where [e.sup.i] is the constant transport rate per unit of distance per unit of export.

In this stage, the representative firm maximizes profit by choosing its output and location. The Cournot-Nash equilibrium is characterized by the following first-order conditions:

[Mathematical Expression Omitted] (10)

[Mathematical Expression Omitted]. (11)

Assuming the stability conditions to be satisfied (Dixit 1986), we then have 2n equations in the first-order conditions to solve for [q.sup.i] and [h.sup.i], i = 1, 2, ..., n. Assuming further that the n firms are symmetrical, the equilibrium, which is assumed to be unique, can then be represented by the following two equations:

[[Pi].sub.q] = [p.sub.Q](nq)q + (p(nq) - eh - t) - M(h)[H.sub.q](q) = 0 (12)

[[Pi].sub.h] = -eq - [M.sub.h]H(q) = 0. (13)

Note that, in the short run, entry is prohibited and the number of the domestic firms is fixed. Now, we can solve for q and h by Equations 12 and 13 and evaluate the comparative static effects of an export tax on each firm's exports and location by totally differentiating Equations 12 and 13 such that

[Mathematical Expression Omitted] (14)

[h.sub.t] = dh/dt = -[[Pi].sub.hq]/D, (15)

where [q.sup.h] denotes the equilibrium output of each firm when its location is endogenously determined, and

D = [[Pi].sub.qq][[Pi].sub.hh] = [[Pi].sub.qh][[Pi].sub.hq]

[[Pi].sub.qq] = n[P.sub.QQ]q + ([n + 1)[P.sub.Q] - M[H.sub.qq]

[[Pi].sub.qh] = [[Pi].sub.hq] = -e - [M.sub.h][H.sub.q] = [M.sub.h](H/q - [H.sub.q])

[[Pi].sub.hh] = -[M.sub.hh]H.

Note that D [greater than] 0 and [[Pi].sub.hh] [less than] 0 by the second-order conditions. Because the effect of an export tax on exports is important in understanding the economic forces controlling the optimal location and export tax, we shall address this issue first. The following proposition clearly states the result.

PROPOSITION 1. In the short run with endogenous location, a rise in the export tax leads to a reduction in exports to the foreign country.

PROOF. From Equation 14, it immediately follows that [Mathematical Expression Omitted]. QED.

The intuition behind this result is as follows. As the level of the export tax increases, the effective marginal cost of exports becomes higher. Under the same demand conditions, this accordingly leads to a fall in exports. It is worth emphasizing that the effect of an export tax on exports with endogenous location is qualitatively similar to the effect on exports with exogenous location as derived by Eaton and Grossman (1986), but the difference with Eaton and Grossman is that they had foreign firm production. Nevertheless, there exists a quantitative difference between endogenous location and exogenous location. To probe deeper into the cause of the difference, let us apply the Le Chatelier Principle (Samuelson 1947; Silberberg 1990) to decompose Equation 14 into two parts,

[Mathematical Expression Omitted], (16)

where superscripts h and o denote that the associated variables are under endogenous location and under exogenous location, respectively. Hence, [q.sup.o] denotes each firm's equilibrium output when the location is given exogenously, [Mathematical Expression Omitted] represents the effect of an export tax on q for an exogenously given location, while [Mathematical Expression Omitted] represents the effect of a change in location h on q when the location is treated exogenously. These two effects are readily derivable by the total differentiation of Equation 12 with respect to q, t, and h,

[Mathematical Expression Omitted] (17)

[Mathematical Expression Omitted]. (18)

Manifestly, Equation 17 shows that, with exogenous location, an increase in the export tax results in a reduction in exports to the foreign county. Correspondingly, it follows from Equation 18 that moving the location farther away from the foreign market indicates that the cost of transporting the inputs relative to the cost of transporting the output decreases, causing the input/output ratio to rise. This in turn leads to an increase in the exports to the foreign country if the production function exhibits decreasing returns to scale. The reverse is true if the production function exhibits increasing returns to scale (Hwang and Mai 1990).

Next, comparing the exports under endogenous location with those under exogenous location, we obtain Proposition 2.

PROPOSITION 2. In the short run, the decline in output because of the export tax increase is greater with endogenous location than with exogenous location if the production function is not constant returns to scale. However, they are perfectly equivalent if the production function is constant returns to scale.

PROOF. Substituting Equations 15, 17, and 18 into Equation 16 and rearranging the terms yields

[Mathematical Expression Omitted] if the production function is (is not) constant returns to scale (henceforth CRS). QED. (19)

Proposition 2 can be explained by a diagrammatic analysis. Assume each firm produces output q and situates at the same location at the initial free trade equilibrium as shown in Figure 2a. Consider first the case of the CRS production function. Since [Mathematical Expression Omitted], we have [q.sup.h] = [q.sup.o] for any export tax t.(4) Nevertheless, when the production function is non-CRS, we have [Mathematical Expression Omitted], implying that the curve qh is steeper than the curve [Mathematical Expression Omitted] as indicated in Figure 2b. Thus, we can obtain [q.sup.o] [greater than] [q.sup.h] for any t [greater than] 0 (not shown in Figure 2b).

We now turn to the effects on the production location and state the following proposition.

PROPOSITION 3. In the short run, the production location of each firm is invariant with respect to a change in export tax if the production function exhibits CRS. However, the location moves farther away from (closer to) the foreign market as a result of an increase in the export tax if the production function exhibits increasing (decreasing) returns to scale.

PROOF. Since [M.sub.h] [less than] 0 by Equation 13, it follows from Equation 15 that

[Mathematical Expression Omitted]. (15[prime])

From Equations 7 and 15[prime], the proposition follows. QED.

The economic intuition underlying Proposition 3 is straightforward. As indicated by Proposition 1, each firm lowers its output (export) as export tax increases. When the production function is increasing (decreasing) returns to scale, the reduction in output implies that the input/output ratio rises (falls). Therefore, the burden of transporting the inputs increases (decreases) in relation to the burden of transporting the output, and hence the pulling powers of input sources increase (decrease) in relation to the pulling power of the output, thereby moving each firm's location away from (closer to) the foreign market. On the other hand, when the production function is CRS, the pulling powers of the input sources are balanced with the pulling power of the output so that the location is independent of a change in export tax.

Finally, let us consider the first stage problem of how the domestic government determines the optimal tax. First, we can establish the following.

PROPOSITION 4. In the short run, when there is only one firm in the domestic country, the optimal trade policy is laissez-faire. If there exists imperfect competition (i.e., n [greater than] 1), the domestic country has an incentive to impose an export tax on the domestic exporting firms. Moreover, the optimal export tax moves the industry equilibrium to what would, in the absence of an export tax, be the monopoly or collusion outcome.

PROOF. The welfare function can be specified as the sum of the total industry profit and the export tax revenue,

W = n[Pi] + tnq. (20)

The first-order condition for welfare maximization is given by

[W.sub.t] = n d[Pi]/dq [q.sub.t] + n [Delta][Pi]/[Delta]t + nq + tn[q.sub.t] = 0. (21)

Assuming the second-order condition to be satisfied, rearranging the terms in Equation 21 and noting [q.sub.t] [not equal to] 0, we obtain(5)

t = -(n - 1)[P.sub.Q]q. QED. (22)

Clearly, the magnitude of the optimal t is determined not only by the number of domestic firms but also by the size of q. If n = 1, then t = 0. Hence, if there is only one exporter in the domestic country, the rent earned from the foreign market is already at its maximum. Under such a circumstance, the optimal policy is laissez-faire. On the other hand, if n [greater than] 1, the total export is too much from the exporting government's point of view and the optimal export tax is positive. In the limiting case, when n gets very large (i.e., when the market is approaching perfect competition), then t approaches -nq[P.sub.Q]. This simply indicates that the optimal export tax under perfect competition is larger than that under nonperfect competition. Moreover, substituting Equation 22 into the first-order condition for optimal output and rearranging the terms, we have

[P.sub.Q]Q + P - eh - M[H.sub.q] = 0, (23)

which is exactly the first-order condition had the market been a monopoly.

Apparently this result holds true regardless of whether the production location is determined endogenously or exogenously.

We are now in a position to compare the magnitudes of t when h is exogenous or endogenous. The following proposition clearly states the result.

PROPOSITION 5. In the short run, the optimal export taxes under both endogenous and exogenous locations are perfectly equivalent if the production function is of CRS. However, it is lower under endogenous location than under exogenous location if the production function is not CRS.

PROOF. This can be accomplished by using Figure 2a and b. Assume that the endogenous location is the same as the exogenous one at the initial free trade situation. Hence, at t = 0, we get [Mathematical Expression Omitted], as indicated in Figure 2. If the production function is CRS, [q.sup.h](t) coincides with [q.sup.o](t) as shown in Figure 2a, and the optimal export tax is the same under both endogenous and exogenous location. In contrast, if the production function is not CRS, the optimal tax with endogenous location is lower than that with exogenous location, as indicated in Figure 2b. The intuition is of itself clear. As [Mathematical Expression Omitted], the domestic government requires less export tax to reduce output from [Mathematical Expression Omitted] to [q.sup.*] when the location is endogenously determined. Note that [q.sup.*] is the output of each individual film when all the exporting firms collude and act jointly as a monopolist. This is also the output of each exporter under optimal intervention. QED.

3. The Long-Run Analysis

In the long run, free entry and exit are allowed and each domestic firm can earn only normal profit. In this section, we intend to demonstrate that policy conclusions depend critically on industry structure assumptions (oligopoly with no entry vs. oligopoly with free entry). It is recognized that the free entry Cournot-Nash equilibrium requires that both the zero-profit condition (i.e., P = AC) and the profit-maximizing condition (i.e., MR = MC) be satisfied. In particular, the AC curve needs to be downward sloping, That is, the production technology is increasing returns to scale. Otherwise, the zero-profit condition, which requires the demand curve to be tangent to the AC curve at equilibrium, is not satisfied and, as a result, the number of firms would not be well defined. Therefore, all equilibria must fall within the IRS region in the long-run equilibrium. It is worth mentioning that the existence of increasing returns along with free entry oligopoly is now well recognized in the literature (Horstmann and Markusen 1986; Markusen and Venables 1988; Konishi 1990).

Under such a situation, the number of domestic firms is determined endogenously by the following zero profit condition:

[Pi] = [p(nq) - eh - t]q - M(h)H(q) = 0. (24)

The long-run equilibrium can be determined by the system of Equations 12, 13, and 24. By totally differentiating this system with respect to q, h, n, and t and applying Cramer's rule, we get the following comparative static results:(6)

[q.sub.t] = 1/[D.sub.3] [[[Pi].sub.hh][q.sup.3][p.sub.QQ]] (25)

[Mathematical Expression Omitted] (26)

[n.sub.t] = 1/[D.sub.3]{q[[D.sub.2] - (n - 1)[P.sub.Q][[Pi].sub.hh]]}, (27)

where [D.sub.3] is the relevant Hessian determinant and [Mathematical Expression Omitted]. Note that [D.sub.3] [less than] 0, [D.sub.2] [greater than] 0, and [[Pi].sub.hh] [less than] 0 by the stability conditions.

The output effect of an export tax under free entry oligopoly is now stated as follows.

PROPOSITION 6. In the long run with endogenous location, a rise in the export tax will decrease (not change, increase) output per firm if the demand function is convex (linear, concave).

PROOF. It immediately follows from Equation 25 that [Mathematical Expression Omitted] if [Mathematical Expression Omitted]. QED.

Hence, in the long run, a rise in the export tax will not necessarily reduce output per firm, as it depends on the shape of the demand function in question. This is contrary to the situation in the short run. The economic interpretation behind this proposition is given as follows. A specific export tax does not change the slope of the demand curve at any Q but will necessitate a higher p in equilibrium for firms to break even. In the case of linear demand (i.e., [p.sub.QQ] = 0), raising p will not alter the slope of the demand curve and so the required tangency between demand and average cost must occur at the same output level for each firm. When demand is concave (i.e., [p.sub.QQ] [less than] 0), raising p lowers the absolute value of the slope of the demand curve, implying that the point of tangency occurs at a larger output level for each firm (a flatter point on AC). For a convex demand curve (i.e., [p.sub.QQ] [greater than] 0), the opposite applies.

We now turn to Equation 26. The result of the impact of an export tax under free entry oligopoly can be summarized in the following proposition.

PROPOSITION 7. In the long run with endogenous location, the optimum production location is invariant with respect to a change in export tax if the demand function is linear. When the production function exhibits increasing returns to scale, the optimum location moves toward the foreign market as a result of a rise in the export tax if the demand function is convex.

PROOF. The first part of the proposition follows from Equation 26 that [h.sub.t] = 0 if [p.sub.QQ] = 0, while the second part of the proposition follows from Equation 26 that [h.sub.t] [less than] 0 if [(H/q) - [H.sub.q]] [greater than] 0 and [p.sub.QQ] [greater than] 0. QED.

The economic reasoning for this result is quite clear and will not be repeated here.

We can also compare the effect of the tax on industry output under endogenous location with that under exogenous location and establish the following proposition.

PROPOSITION 8. In the long run, the industry output effect of a rise in export tax with endogenous location is larger than (equal to, smaller than) that with exogenous location if the demand function is concave (linear, convex).

PROOF. The effect of the export tax increase on industry output when the location is given exogenously is derivable as follows:

d[Q.sup.o]/dt = [q.sup.2]/[D.sub.2] (2[p.sub.Q] - M[H.sub.qq]) [less than] 0. (28)

since the sign of the term (2[p.sub.Q] - M[H.sub.qq]) is normally negative (Seade 1980).

Moreover, the effect of the tax on industry output under endogenous location can be calculated by Equations 25 and 27 as follows:

[Mathematical Expression Omitted]. (29)

Furthermore, subtracting (28) from (29) yields

[Mathematical Expression Omitted]. (30)

Finally, comparing the magnitudes of t when h is exogenous or endogenous, we are ready to state the following proposition.

PROPOSITION 9. In the long run, the optimal export tax is higher (lower) under endogenous location than under exogenous location if the demand function is convex (concave). They are perfectly equivalent if the demand is linear.

PROOF. Since each firm earns zero profit in the long run, the welfare function consists only of total tax revenue,

W = nqt = Qt. QED. (31)

Clearly, the objective of maximizing social welfare by choosing t is reduced to that of maximizing total tax revenue, which is represented by the rectangle in Figure 3. As can be seen from the figures, when the demand is convex (concave), then the schedule of d[Q.sup.h]/dt is flatter (steeper) than that of d[Q.sup.[convolution]/dt, implying that [t.sup.h] is greater (smaller) than [t.sup.[convolution].(7) Nevertheless, [t.sup.h] is equal to [t.sup.[convolution] if the demand is linear.

4. Concluding Remarks

Export tax is one of the most important trade policies in developing countries with strong natural advantages in the production of primary commodities. However, little attention has been paid to the economic effects of export taxes outside the polar cases of monopoly and perfect competition. This paper presents a simple Cournot competition model to examine how the inclusion of economic space affects an export tax policy and to compare optimal export taxes under endogenous plant location with those under exogenous location in the short run and in the long run. This paper also brings in the neglected role of transport costs in plant location, and the reader emerges with a more complex but yet more insightful picture of the implications of an export tax. The framework adopted in this paper would also seem to open up possibilities for further research along similar lines in the future involving other trade policy instruments. Moreover, since the output effect of export tax does not differ from exogenous-location models with constant returns to scale but does differ with nonconstant returns, our analysis deepens one's understanding of the implications of new trade theory's departure from the constant returns assumption of traditional trade theory.

In any event, this paper presents some new results concerning export tax policy with oligopoly and the role of transportation costs in the case of free entry and spatial markets. As Krugman (1993) noted, traditional trade theory and policy make little use of the tools provided by location theory and economic geography. Thus, opportunities for research in this field abound, and it is hoped that this paper will have some effect in stimulating them.

We would like to thank two anonymous referees and the editor for very useful suggestions, which led to a substantial improvement in the paper.

1 As a matter of fact, export tax policy has been adopted by some developed countries. For example, Begley et al. (1998) considered the economic consequences of a series of events culminating in a 15% export tax being imposed on softwood lumber shipped from Canadian to U.S. markets as provided for in the 1986 Memorandum of Understanding. The paper suggested that the welfare implications of export tax may be important for both countries.

2 As is widely recognized in location literature, all intermediate locations are excluded in a linear space (see Sakashita 1967). To ensure the existence of an interior solution, we start with the Weberian (1929) triangle space (e.g., see Moses 1958; Khalili, Mathur, and Bodenhorn 1974; Mai and Hwang 1992) and also abstract from another interesting space where consumers spread along a line or a plane.

3 It is assumed that each domestic firm should select its production location within the territory of the domestic country, including its choice of location at the border C. As pointed out by a referee, the home versus the foreign location decision is an interesting question and would constitute another paper.

4 For graphical convenience, the curves are drawn as linear. Of course, this simplification does not change the qualitative results of the paper.

5 Also see Helpman and Krugman (1989) for the treatment with exogenous firm location.

6 The derivations of Equations 25, 26, and 27 are available from the authors upon request.

7 Note that only the normal case of (d[Q.sup.h]/dt) [less than] 0 is depicted in Figure 3. In the extreme case of (d[Q.sup.h]/dt) [greater than] 0, W, derived from Equation 31 is necessarily positive, implying a corner solution for t. If this is the case, then [t.sup.h] [greater than] [t.sup.[convolution] regardless of the concavity of the demand curve in question.

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