Academic journal article Atlantic Economic Journal

# A Comparison of Ad Valorem and per Unit Taxes in Regulated Monopolies

Academic journal article Atlantic Economic Journal

# A Comparison of Ad Valorem and per Unit Taxes in Regulated Monopolies

## Article excerpt

I. Introduction

The comparison of ad valorem and per unit taxes in an unregulated monopoly market was begun nearly a century ago by Wicksell [1896]. In a well known paper, Suits and Musgrave [1953] extended Wicksell's analysis using a more general cost function. Most recently, Yang [1993] extended their result to the model of third-degree price discrimination. Unfortunately, for the policymaker, these studies apply only to an unregulated monopoly which is quite rare. This paper compares the effects of these taxes on the price and output of a rate-of-return regulated monopolist(1) and compares the firm's decisions when the revenue or the output from both taxes is held constant. In general, either tax will reduce factor-employment, output and hence raise the price. This type of monopoly is quite common for regulated utilities such as electric, gas, telephone, and water.

In the era of budgetary exigency, the Suits-Musgrave result plays an important role in the choice among taxes. The analytical results developed in this paper are intended to fill a void in the existing monopoly taxation theory.

II. The Unit Tax Model

Based on the formulation by Averch and Johnson [1962], it is assumed that a unit tax (u) is imposed on a rate-of-return regulated monopolist whose goal is to maximize after tax profits ([Pi]) subject to a rate-of-return constraint or:

Maximize [Pi] = (p - u) Q - wL - rcK, (1)

s.t. (p - u) Q - wL - scK [is less than or equal to] 0, (2)

p [is greater than or equal to] 0, Q [is greater than or equal to] 0, K [is greater than or equal to] 0, L [is greater than or equal to] 0, (3)

where: Q = f(L, K)(2) = output, p = output price;(3) L = labor input; w = wage rate; r = cost of financial capital; c = cost of physical capital; K = capital input; and s = allowed rate of return.

From equations (1) and (2), the Lagrangian function is:

Z = (p - u) Q - wL - rcK + [Alpha][scK + wL - (p - u)Q].

The first-order conditions of this interior maximization problem are:

[Z.sub.L] = ([R.sub.L] - u[Q.sub.L] - w) (1 - [Alpha]) = 0, (4)

[Z.sub.K] = ([R.sub.K] - u[Q.sub.K]) (1 - [Alpha]) + [Alpha]sc - rc = 0, and (5)

[Z.sub.[Alpha]] = scK + wL - (p - u) Q = 0, (6)

where R = pQ, and subscripts denote partial derivatives.

From equation (5), it is determined that:

[Alpha] = (rc - [R.sub.K] + u[Q.sub.K]) / (sc - [R.sub.K] + u [Q.sub.K]) [is not equal to] 1, (7)

since s [is greater than] r. Hence, from equation (4), [R.sub.L] - u[Q.sub.L] - w = 0.

Following the approach of Baumol and Klevorick [1970] and Yang and Fox [1994], one can derive the comparative statics of this model by forming the bordered Hessian:

[Mathematical Expression Omitted],

where the subscripts denote the partial derivatives.

The second-order conditions require that the value of the determinant H must be positive or:

[absolute value of H] = -[(u[Q.sub.K] - [R.sub.K] + sc).sup.2] (1 - [Alpha]) ([R.sub.LL] - u[Q.sub.LL]) [is greater than] 0. (9)

Unfortunately, equation (9) is not sufficient to guarantee [Alpha] [is less than] 1, as would be the case of the original A-J model without the tax. For this reason, the conventional assumption is that the monopolist has a strictly concave after-tax profit function or:

[R.sub.LL] - u[Q.sub.LL] [is less than] 0 and

[Mathematical Expression Omitted].

With [R.sub.LL] - u[Q.sub.LL] [is less than] 0, it follows immediately that [Alpha] [is less than] 1. From equation (7), it is found that:

sc - [R.sub.K] + u[Q.sub.K] [is greater than] 0. (11)

Using Cramer's rule, one obtains the following comparative statics:

[Mathematical Expression Omitted];

dK/du = [-Q ([R.sub.LL] - u[Q.sub.LL])]/[(u[Q.sub.K] - [R.sub.K] + sc) ([R.sub.LL] - u[Q.sub.LL])] [is less than] 0; and (13)

[Mathematical Expression Omitted]. …

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