Formulation of the Problem

Taxes or subsidies may affect market prices. The manner in which they do so may depend upon the structure of the market; i.e., the degree of competition. The standard categories of market structure are monopoly, oligopoly, monopolistic competition and competitive. The case of the impact of taxes and subsidies in a competitive market is covered elsewhere. Here the focus is on monopolized markets. The word monopoly means one seller and this is not precise enough. It is possible that there is only one seller in an industry because that seller keeps the price so low that no other firms want to enter the market at that price. An example of that situation is the aluminum market in which Alcoa (the Aluminum Company of America) was the only producer of aluminum until after World War II when the Federal Government artificially created the entry of Kaiser and Reynolds Aluminum into the market. There are numerous examples of there being only one store of a particular type in towns too small to support two stores of that type. Monopoly is better than Zero-opoly. The detrimental element of monopoly occurs when it is a protected monopoly and hence does not have to worry about potential competiton. The theory of protected monopoly is covered more thoroughly elsewhere.

In the simplified case of constant unit cost (average and marginal) and a straight line demand function the monopoly output is one half of what the competitve output would be. The price rises to a level that is half way between the competitive price and the maximum price for the market; the maximum price is the price that would reduce the quantity demanded to zero. This is shown in the following diagram.

In the diagram the upper red line is the demand fuction. The demand line extends from the quantity demanded if the price were zero, q_max, to the price at which the quantity demanded is reduced to zero, p_max. The quanities p_c and q_c are the price and output that would prevail under competition. In the absence of externalities in the production and consumption of the product, q_c and p_c, would be the socially optimum levels of production and price.

The lower red line is the marginal revenue line which represents the benefit to the monopolist of a unit increase in production. The monopolist chooses a level of output where marginal revenue is equal to margina cost. That ouput is labeled q_mon in the diagram and the price charged is p_mon. The monopoly price is only partially tied to cost. The other consideration is what the traffic will bear. The monopolist does not charge all the traffic will bear because that would reduce the quantity demanded to one unit. Instead the monopolist sets price and output to maximize profit. In the simple model being considered above that price is half way between p_c and p_max. Since p_max is the maximum price that anyone will pay for one unit of the product the monopoly price may well be far above the marginal cost.

The General Case

Let p be the price paid by the consumers of a product provided by a protected monopoly. Let q be the quantity produced and consumed and let C(q) be the cost function. Let t be the excise tax paid by the producer. The profit π of the monopolist-producer is Let f(q) be the inverse demand function. The price received by the producer is the price paid by the consumer less the excise tax; i.e., p−t.

π = q(f(q)−t) − C(q)
 

Thus the maximum profit is achieved at an output q* such that

dπ/dq = (f(q*)−t) + q*f'(q*) − C'(q*) = 0 (first order condition)
and
d²π/dq² = 2f'(q*) + q*f"(q*) − C"(q*) < 0 (second order condition)
 

Now consider an increase in the tax and its impact on quantity and the price paid by consumers. The differentiation of the profit-maximizing condition with respect to t yields:

f'(q*)(dq*/dt) −1 + f'(q*)(dq*/dt) +q*f"(q*)(dq*/dt) − C"(q*)(dq*/dt) = 0
which may be solved for (dq*/dt) to yield
dq*/dt = 1/[2f'(q*) + q*f"(q*) − C"(q*)]
 

Note that the expression in the square brackets in the denominator is guaranteed by the second order condition to be negative so (dq*/dt) is also negative.

The impact of an increase in the tax rate on the price paid by consumers is also determined

dp*/dt = f'(q*)(dq*/dt) = f'(q*)/[2f'(q*) + q*f"(q*) − C"(q*)]
and since f'(q) is negative, (dp*/dt) is positive.
 

Upon dividing the numerator and denominator in the expression for (dp*/dt) by f'(q*) one obtains

dp*/dt = 1/[2 + q*(f"(q*)/f'(q*)) − C"(q*)/f'(q*)]
 

With a little algebraic manipulation it can be shown that d(p*-t)/dt is negative; i.e.,

dp*/dt − 1 = 1/[2 + q*(f"(q*)/f'(q*)) − C"(q*)/f'(q*)] − 1
dp*/dt − 1 = −[1 + q*(f"(q*)/f'(q*)) − C"(q*)/f'(q*)]/[2 + q*(f"(q*)/f'(q*)) − C"(q*)/f'(q*)]
 

Since the denominator of the above fraction is negative and the numerator is more negative the ratio is positive. Hence d(p*-t)/dt is the negative of a positive ratio and is therefore negative. In other words, the tax drives down the price received by the producer.

If the demand function is a straight line then f"(q*)=0. If marginal costs are constant then C"(q*)=0. Under those two condition then dp*/dt=1/2 and d(p*-t)/dt=−1/2.

For a subsidy s the effects are that dp*/ds is negative and d(p*+s)/ds is positive. For the special case of a straight line demand function and constant marginal cost, dp*/ds = −½ and d(p*+s)/ds = +½. In other words, the benefit of the subsidy is divided equally between the monopolist and the consumers.

The following diagrams show the effect of a subsidy for the simplified case considered above.

If the government pays a subsidy s in a protected monopoly market where the price without the subsidy is p_mon, the price paid by consumers does not fall to (p_mon-s). Instead the monopoly price rises to p_mon+½s so the price to consumers falls to p_mon−½s. Thus the higher the subsidy the higher the price set by the monopolist and the more consumers feel they must have a subsidy. The consumers do benefit from the subsidy but only to an extent equal to half of the subsidy. The other half of the subsidy goes to the monopoly. Of course the consumers ultimately pay for the government subsidy in terms of taxes. Thus the consumer/taxpayers have a net increase in costs equal to the one half of the subsidy that goes to the monopoly.

(To be continued.)