In 1946, Coase rejected Hotelling-Lerner's solution for financing a nationalized monopoly on the grounds that any tax structure could distort relative prices. In situations where two-part tariffs are infeasible, Coase suggested average cost pricing as a noninferior solution to the above policy. This article shows that, in a general equilibrium model, it is possible to choose a distortionary Hotelling-Lerner's tax policy that is superior to average cost pricing.

The solution of Hotelling (1938) and Leiner (1944) to the problem of optimal policy for a nationalized monopoly consists of equalizing market price to marginal cost and financing the losses derived from economies to scale through nondistortionary (lump sum) taxation.[1] Coase (1946) has three objections to this solution. First, in the presence of increasing returns to scale, marginal cost pricing does not necessarily guarantee efficiency. Second, the Hotelling-Lerner solution could alter income distribution. Third, the nondistortionary character of taxes is actually a theoretical abstraction and taxes are, in general, distortionary. For this reason, Coase suggested average cost pricing as a better alternative for the case in which two-part tariffs are infeasible:[2] "[T]he claim which is made for the Hotelling-Lerner solution as inevitably superior to average cost pricing must therefore be rejected" (Coase, 1946, p. 181).

This article compares both solutions for the case of a single-output monopoly focusing on Coase's third objection. For that purpose, we consider the usual version of average cost pricing (the so-called Coase's solution) and a distortionary version of the Hotelling-Lerner solution, in which the government equates the price of the monopolized good to its marginal cost while losses arising from economies of scale are financed by proportional taxes or subsidies on the remaining private goods. As we will see, the formulation used will allow us to characterize average cost pricing as a particular outcome of the class of distortionary Hotelling-Lerner policy. This will allow us to compare both policies within the same framework, concluding that, in general, average cost pricing is not a second-best equilibrium. Hence, it is possible to choose a distortionary Hotelling-Lerner pricing scheme, through Ramsey's taxation, that improves welfare.

MODEL AND RESULTS

Let us consider an economy with three commodities:[3] leisure (the numeraire), denoted by x; the good offered by the nationalized monopoly, denoted by y; and a private good, denoted by z, produced under perfect competition. We make the following assumptions.

1. There are n consumers (indexed by i). For each consumer, w[sub i] represents the initial endowment of numeraire, (x[sub i], y[sub i], z[sub i]) is an element of R[sup 3] is his consumption vector, l[sub i] = w[sub i] - x[sub i] is his labor supply, and u[sub i]: R[sup 3]+ right arrow R represents his utility function, which is assumed to be strictly concave and increasing with demand correspondences for the private goods x[sub i] (*(This character cannot be represented in ASCII text)) and z[sub i] (*(This character cannot be represented in ASCII text)).[4]

2. There are r perfectly competitive firms (indexed by j) producing good z under constant returns to scale. We denote by z[sub j] the output of firm j, whose cost function is C[sub j] (z[sub j])=q z[sub j], q is an element of R+, where q is the constant marginal cost and market price. This means that the industry supply function is infinitely elastic, and consequently, in equilibrium, producers' surplus is zero. We also assume the usual market-clearing condition:

Multiple line equation(s) cannot be represented in ASCII text. (1)

3. The cost function of the nationalized monopoly is given by

Multiple line equation(s) cannot be represented in ASCII text.

where c and k are constants satisfying [Multiple line equation(s) cannot be represented in ASCII text]; and

[Multiple line equation(s) cannot be represented in ASCII text] Denoting by T the revenue yielded by taxes, the budgetary equilibrium for the government is given by

T + p y = C (y), (2)

where p is the price set by the government for the consumption of y. Note that for T = 0, equation (2) is just the price-average cost role. Finally, the usual feasibility condition in terms of labor is

Multiple line equation(s) cannot be represented in ASCII text. (3)

The following lemma states that the budgetary equilibrium for the government, equation (2), implies equilibrium in the labor market, equation (3), and vice versa.

Lemma 1. The budget of the government is balanced if and only if the feasibility condition for the numeraire good holds.

The proof, which is in the appendix, is an immediate consequence of Walras's law. This result allows us to use equation (2) instead of equation (3), which will be useful in subsequent calculations.

We specify the functional form of T and the government's economic policy. We assume that taxes are distortionary, proportional, and anonymous, denoting t[sub 1] is an element of (-1,1) the tax (subsidy) rate on labor income and t[sub 2] is an element of (-1,1) the tax (subsidy) rate on good z. As the government chooses both the pair (t[sub 1],t[sub 2]) and the price p of the monopolized good, an economic policy will be given by the triple [Multiple line equation(s) cannot be represented in ASCII text] satisfying (2), which can be written as

Multiple line equation(s) cannot be represented in ASCII text. (4)

The above concept of economic policy allows us to characterize Coase's solution (price-average cost policy) as [Multiple line equation(s) cannot be represented in ASCII text], where C= C(y)/y, and the distortionary Hotelling-Lerner's solution as {c,t[sub 1],t[sub 2]}, where c is the marginal cost of good y and

Multiple line equation(s) cannot be represented in ASCII text. (5)

Note that both policies fulfill equation (4). Hence, by Lemma 1, under both policies the feasibility condition in terms of the numeraire holds. We are now ready to define the equilibria associated with the two different policies defined above.

Definition 1. Price-average cost equilibrium

The allocation [Multiple line equation(s) cannot be represented in ASCII text] is a price-average cost equilibrium if, given the price-average cost policy [Multiple line equation(s) cannot be represented in ASCII text], it satisfies (1) and (3), and [Multiple line equation(s) cannot be represented in ASCII text] solves the program

Multiple line equation(s) cannot be represented in ASCII text. (6)

where [Multiple line equation(s) cannot be represented in ASCII text]

Definition 2. Distortionary Hotelling-Lerner equilibrium

The allocation where [Multiple line equation(s) cannot be represented in ASCII text] is a distortionary Hotelling-Lerner equilibrium if, given the distortionary Hotelling-Lerner policy {c,t[sub 1],t[sub 2]}, it satisfies (1) and (3), and where [Multiple line equation(s) cannot be represented in ASCII text] solves the program

Multiple line equation(s) cannot be represented in ASCII text. (7)

with (t[sub 1],t[sub 2]) is an element of (-1,1)[sup 2].

The way in which the economic policy has been defined allows us to show, in the following lemma, that average cost pricing is a member of the distortionary Hotelling-Lerner pricing scheme.

Lemma 2. The distortionary Hotelling-Lerner policy {c, t, -t} with where [Multiple line equation(s) cannot be represented in ASCII text] generates a distortionary Hotelling-Lerner equilibrium that is equal to the price-average cost equilibrium.

Proof

Substituting {c, t, -t} into the consumers' budget constraint of program (7), the consumers' budget constraint of program (6) is obtained; therefore, under this policy, both equilibria are the same.

We will refer to the above distortionary Hotelling-Lerner policy as the price-average cost equivalent tax policy. With this result, we obtain average cost pricing as a particular outcome of the class of distortionary Hotelling-Lerner policies, allowing the comparison between the two equilibria. Note that in the price-average cost equivalent tax policy, t is proportional to the scale elasticity where [Multiple line equation(s) cannot be represented in ASCII text]. This means that t can be interpreted as a measure of the returns to scale in the production of the monopolized good.

We are now ready to define the second-best tax policy, adapting the well-known concept of optimal taxation stated by Ramsey (1927)[5] to our framework. First, we assume that the social welfare function is utilitarian. Then, given the set of feasible policies, the government chooses the tax values that maximize that social welfare function. Therefore, denoting by V[sub i](t[sub 1],t[sub 2]) the indirect utility function of the ith consumer, the second-best distortionary Hotelling-Lerner policy is defined.

Definition 3. Second-best distortionary Hotelling-Lerner policy

The distortionary Hotelling-Lerner policy where [Multiple line equation(s) cannot be represented in ASCII text] is a second-best distortionary Hotelling-Lerner policy if it solves the following program:

Multiple line equation(s) cannot be represented in ASCII text. (8)

As we can see, the social welfare function is created as the sum of the consumers' indirect utility functions. This is because, in equilibrium, private producers' surplus is zero and the budget of the government is balanced. Additionally, due to the interiority conditions assumed in assumption (1), the following lemma characterizes the first-order conditions of program (8).

Lemma 3. The second-best distortionary Hotelling-Lerner policy [Multiple line equation(s) cannot be represented in ASCII text] satisfies

Multiple line equation(s) cannot be represented in ASCII text. (9)

Multiple line equation(s) cannot be represented in ASCII text. (10)

The proof is in the appendix. Note also that according to Dierker (1991) and because of assumptions (1), (2), and (3), the second-order conditions are fulfilled.

Proposition 1. If at the second-best distortionary Hotelling-Lerner policy y > 0 for some consumer i, then price-average cost equilibrium is not a second-best distortionary Hotelling-Lerner policy.

Proof

Assume to the contrary that the price-average cost equilibrium is generated by the second-best distortionary Hotelling-Lerner policy. Thus, by Lemma 2, the price-average cost equivalent tax policy [Multiple line equation(s) cannot be represented in ASCII text] (and its allocation [Multiple line equation(s) cannot be represented in ASCII text], with y[sup ac, sub i] > 0 for some i), would have to satisfy equations (9) and (10) in Lemma 3, substituting

Multiple line equation(s) cannot be represented in ASCII text. (11)

Multiple line equation(s) cannot be represented in ASCII text. (12)

where lambda[sub i] and mu are the Lagrange multipliers associated with programs (7) and (8), respectively. Subtracting (12) from (11),

Multiple line equation(s) cannot be represented in ASCII text. (13)

where

Multiple line equation(s) cannot be represented in ASCII text.

However, under the price-average cost equilibrium, [Multiple line equation(s) cannot be represented in ASCII text] hence (13) can be written as

Multiple line equation(s) cannot be represented in ASCII text. (14)

where beta[sub i] = lambda[sub i]/mu > 0; i = 1, 2, ..., n.

On the other hand, differentiating the budgetary equilibrium of the government [given by equation (5)] with respect to t[sub 1] and t[sub 2], we obtain

Multiple line equation(s) cannot be represented in ASCII text. (15)

and

Multiple line equation(s) cannot be represented in ASCII text. (16)

and substituting the price-average cost equivalent tax policy {c,t,-t} with [Multiple line equation(s) cannot be represented in ASCII text] (and its allocation) into (15) and (16),

Multiple line equation(s) cannot be represented in ASCII text. (17)

Multiple line equation(s) cannot be represented in ASCII text. (18)

therefore, subtracting (18) from (17) and simplifying,

Multiple line equation(s) cannot be represented in ASCII text. (19)

Finally, comparing (19) and (14), we obtain

Multiple line equation(s) cannot be represented in ASCII text.

which holds if and only if y[sup ac, sub i]= 0 for every i = 1, 2, ..., n, which is a contradiction.

Proposition 2. If at the second-best distortionary Hotelling-Lerner policy y[sub i] > 0 for some consumer i, there is at least one distortionary Hotelling-Lerner policy that is superior to average cost pricing.

Proof

As a consequence of Proposition 1, the price-average cost equilibrium is not a second best. Therefore, the second-best distortionary Hotelling-Lerner policy [Multiple line equation(s) cannot be represented in ASCII text] is superior to average cost pricing.

CONCLUSION

The purpose of this article was to analyze the issue of pricing the output of a single-product nationalized monopoly, comparing the well-known Hotelling-Lerner's solution to the average cost pricing, which had been supported by Coase as a noninferior alternative to the former due to the inevitably distortionary character of taxes. To address Coase's criticism, a distortionary version of the Hotelling-Lerner's solution was defined, consisting of setting the price of the nationalized monopolized good to its marginal cost and financing losses with distortionary taxes on the remaining private goods. This framework has allowed us to realize the significant role played by dimensionality in the model. Indeed, whereas with only two commodities both solutions would be the same, with three (or more) commodities, the degrees of freedom of the tax policy considered rises and average cost pricing is found as a member of the family of distortionary Hotelling-Lerner pricing schemes, allowing us to compare the two equilibria. This has made us conclude that, in general, average cost pricing is not a second-best equilibrium. Hence, it is possible to choose a distortionary Hotelling-Lerner pricing scheme, through Ramsey's taxes, that improves welfare.

APPENDIX

Proof of Lemma 1

Let us denote by v[sub ij] is an element of [0, 1] the participation of consumer i in the profits of firm j, where [Multiple line equation(s) cannot be represented in ASCII text] by T[sub i] the taxes paid by consumer i, with [Multiple line equation(s) cannot be represented in ASCII text]; and by s the price of good z. Then, the budget constraint of consumer i can be written as

Multiple line equation(s) cannot be represented in ASCII text.

Adding over consumers,

Multiple line equation(s) cannot be represented in ASCII text. (20)

On the other hand, firms' profits are given by

Multiple line equation(s) cannot be represented in ASCII text. (21)

Substituting (21) in (20), we have

Multiple line equation(s) cannot be represented in ASCII text. (22)

Therefore, if the market of the numeraire good is in equilibrium, equation (3), the left-hand side of (22) is equal to C(y), and equation (2) holds. On the other hand, if equation (2) holds, from the right-hand side of (22), we obtain the feasibility condition in terms of labor of equation (3).

Proof of Lemma 3

The Lagrangian associated with program (8) is

Multiple line equation(s) cannot be represented in ASCII text.

and the first-order conditions are

Multiple line equation(s) cannot be represented in ASCII text. (23)

Multiple line equation(s) cannot be represented in ASCII text. (24)

Now, let us consider the individual maximization problem under the distortionary Hotelling-Lerner policy, given by program (7). Denoting by lambda[sub i] (i = 1, ..., n) the Lagrange multipliers, the first-order conditions are given by

Multiple line equation(s) cannot be represented in ASCII text. (25)

Multiple line equation(s) cannot be represented in ASCII text. (26)

Multiple line equation(s) cannot be represented in ASCII text. (27)

and differentiating the individual budget constraint of program (7) with respect to t[sub 1] and t[sub 2], we have

Multiple line equation(s) cannot be represented in ASCII text. (28)

Multiple line equation(s) cannot be represented in ASCII text. (29)

On the other hand, let us define the indirect utility function as

Multiple line equation(s) cannot be represented in ASCII text.

Differentiating it with respect to t[sub 1] and t[sub 2], we obtain

Multiple line equation(s) cannot be represented in ASCII text.

and taking (25), (26), and (27) into account, we have

Multiple line equation(s) cannot be represented in ASCII text.

from which it follows, using (28) and (29), that

Multiple line equation(s) cannot be represented in ASCII text. (30)

Finally, substituting (30) into (23) and (24) and simplifying, equations (9) and (10) in Lemma 3 are obtained.

An Example

To help the reader to understand better how the taxes are determined, let us consider the following example. Assume that

Multiple line equation(s) cannot be represented in ASCII text.

According to program (6), the price-average cost equilibrium is given by

Multiple line equation(s) cannot be represented in ASCII text.

On the other hand, according to program (7), the distortionary Hotelling-Lerner equilibrium is given by

Multiple line equation(s) cannot be represented in ASCII text.

where (dropping constants) the indirect utility function can be written as

Multiple line equation(s) cannot be represented in ASCII text. (A1)

Note also that, according to Lemma 2, the price-average cost equilibrium can be obtained by substituting in the above equilibrium the policy {c,t,-t} with

Multiple line equation(s) cannot be represented in ASCII text.

To characterize the second-best distortionary Hotelling-Lerner policy, and for the sake of simplicity, let us call [Multiple line equation(s) cannot be represented in ASCII text].

The Lagrangian associated with program (8) is given by

Multiple line equation(s) cannot be represented in ASCII text.

whose first-order conditions are

Multiple line equation(s) cannot be represented in ASCII text. (A2)

Multiple line equation(s) cannot be represented in ASCII text. (A3)

Multiple line equation(s) cannot be represented in ASCII text. (A4)

Clearing mu from (3), substituting it into (2) and (4), and simplifying, we obtain

Multiple line equation(s) cannot be represented in ASCII text.

Finally, using (1), we can see that the welfare gain from the second-best distortionary Hotelling-Lerner policy with respect to the price-average cost equilibrium is given by (B + G)1n(1 - k/(B + G))-B1n(1 -k/B) > 0.

AUTHOR'S NOTE: I am indebted to F. Beltran, L. Corchon, C. Pita, B. Valdes, A. Villar, and two anonymous referees for their helpful comments and suggestions. I am also grateful for financial assistance from the Spanish Ministry of Education CICYT Project PB93-0940 and the Instituto Valenciano de Investigaciones Economicas.

NOTES

1. A thorough foundation of this view can be found in Guesnerie (1975).

2. Two-part tariffs were claimed to be efficient by Coase; Vohra (1990) demonstrates that Coase's conjecture does not generally hold.

3. This is adopted for the sake of simplicity, and similar results can be reached with more than three goods. If there were only two goods, there would be one tax policy equivalent to average cost pricing (i.e., both solutions would be the same).

4. This is because, if one of the private goods were not demanded at some price, the model would become one with two commodities and both solutions would be the same (see Note 3).

5. More recent versions of this concept are provided by Atkinson and Stiglitz (1972) and by Sandmo (1976).

6. The proof uses nonincreasing returns to scale in the production of good z, and therefore pi[sub j] can be positive.

REFERENCES

Atkinson, Anthony B., and Joseph E. Stiglitz. 1972. The structure of indirect taxation and economic efficiency. Journal of Public Economics 1:97-119.

Coase, Ronald H. 1946. The marginal cost controversy. Economica 13:160-82.

Dierker, Egbert. 1991. The optimality of Boiteux-Ramsey pricing. Econometrica 59:99-121.

Guesnerie, Roger. 1975. Pareto optimality in non convex economies. Econometrica 43:1-29.

Hotelling, Harold. 1938. The general welfare in relation to problems of taxation and of railway and utility rates. Econometrica, July, 42-69.

Lerner, Abba P. 1944. The economics of control. New York: Macmillan.

Ramsey, Frank P. 1927. A contribution to the theory of taxation. Economics Journal 37:47-61.

Sandmo, Agnar. Optimal taxation. Journal of Public Economics 6:37-54.

Vohra, Rajiv. 1990. On the inefficiency of two-part tariffs. The Review of Economics Studies 57:415-38.