Background

To see why increases in the minimum wage may not increase real wages consider what happens to the price a monopoly charges when consumers' incomes increase. As will be shown later, if consumers' income increases by x percent and a monopoly's marginal cost increases by the same x percent then the monopoly will raise its prices by x percent. This not because they have to increase their prices but because they can get the higher prices without any loss in sales.

Industries are not typified as monopolies (single sellers) per se, but they do involve what economists call monopolistic competition. Monopolistic competition includes the situation where the market is divided up into trade areas and within a trade area there is only a single seller. The single seller can function as a monopolist as long as the other competitors in the market also function as monopolists and the trade areas remain stable. But monopolistic competition is more general and includes the situation in which there is product differentiation.

Monopolistic competition characterizes most retail business, as in the case of supermarkets. In the case of services stations and fast food outlets there may be multiple sellers in nearby locations but they function as a cartel which accepts a division of the market and avoid price competition with each other. A cartel is just multiple firms functioning as a monopoly.

Provision of governmental sevices is typically a matter of monopoly per se although for smaller cities within a metropolitan area there is an element of monopolistic competition. Public sector monopolies do not function exactly in the same way as private sector monopolies but the respond increases in costs or increaces in what their clientel can pay.

A Simple Model of Pricing
by a Monopolist

Suppose the demand function of the monopoly is linear and is given as price p as a function of quantity sold q. A simple version of the analysis is given here and the general case is proven in the Appendix.

p = a − bq

where a and b are constants.

The marginal revenue MR is given as

MR = a − 2bq

Let marginal cost be a constant c.

The profit maximizing production for the monopolist is then given by

MR = MC
a − 2bq = c
and therefore
q = (a−c)/2b

The price established by the monopolist is then

p = a − b(a−c)/2b = a − ½(a−c)
and hence
p = ½(a+c)

The above can be applied to each of the N industries. Then column vectors P, A and C, for prices p_i, the demand parameters a_i, and marginal costs c_i satisfy

P = ½(A + C) = ½A + ½C

The above rule is that monopoly price is a simple average of marginal cost and maximum price that can be obtained for the good or service. For an oligopoly with effectively n competitors the rule would be

p = (1/(n+1))a + (n/(n+1))c

In other words price is a weighted average of a and c. The case of monopoly is just the case of n equal to 1. All that follows for a monopoly applies equally well for an oligopoly.

Marginal Costs

Let q_ij be the amount of the output of the j-th industry required per unit of output of the i-th industry and r_ik be the amount of the k-th labor service required per unit of output of the i-th industry, Then

C = QP + RW

where W is the column vector of wage rates.

The Levels of Consumer Demand

The levels of the inverse demand functions are given by the parameters a_i are assumed to be proportional to consumers income y; i.e.,

a_i = yg_i
and hence
A = yG

Consumers' income is assumed to be all wage income, thus

y = L^TW

where L is the column vector of the K employed labor amounts and L^T is the row vector which is the transpose of L. For the moment it is assumed that L is given exogenously.

Equlibrium Prices

The above expressions for A and C may be substituted into the equation for prices giving

P = ½G(L^TW) + ½( QP + RW)
which leads to
(I−½Q)P = ½[GL^T + R]W
P = ½(I−½Q)⁻¹[GL^T + R]W

where I is the identity matrix. Let the matrix premultiplying W in the above equation be denoted as M. Then P=MW.

Effect of Proportional
Increases in Wages

Suppose the government can set all wage rates and increases all wage rates by a proportional amount x; i.e.,

W' = (1+x)W

Then from the above the new prices are given by

P' = MW' = M(1+x)W = (1+x)MW = (1+x)P

Thus the ratios of prices to wages rates remain unchanged and likewise for the ratios of wages to prices, the real wage rates remain unaffected.

Here is the time series on the real wage rates

The previous analysis presumed that the utilization of labor services L remained constant when all wages were increased by a constant fraction x. Under this assumption prices all increase by the same constant fraction x. This means that for W'=(1+x)W

P' = (1+x)P
L' = L

is a solution.

Computation of the Effect of
a General Increase in Wages
on Employment

The vector of the utilization of labor services is given by

L = RX

The inverse demand functions have the form P=A−BX, were B is a diagonal matrix. In order for P' to equal (1+x)P at the same value of X when W→(1+x)W it must be that A'=(1+x)A and B'=(1+x)B. In particular this will give the same X* at which P=0 for W and (1+x)W. For more on the effect of a proportional income increase on A see Inverse Demand Functions.

Given that P=A−BX, outputs are given by

X = B⁻¹(A−P)

If B'=(1+x)B then

B'⁻¹ = (1/(1+x))B⁻¹

Therefore

X' = B'⁻¹(A'−P') = ½B'⁻¹(A'−C')
which reduces to
X' = ½(1/(1+x))B⁻¹((1+x)A−(1+x)C)
and further to
X' = = ½B⁻¹(A−C) = X

Since X'=X, L'=RX' is equal to L'=RX=L .

Thus as a result of W→(1+x)W

X → X
L → L
y →(1+x)y
A → (1+x)A
B → (1+x)B
C → (1+x)C

And real wage rates remain constant. Thus, to the extent that prices are set by monopolistic structures, an increase in nominal income through wage increases set by fiat is canceled out by price increases. Therefore it would not be surprising if real wage rates remained constant despite increases in the federal minimum wage. Here are the relevant.graphs.

Real increases in income in such an economy can only come through real changes in the economy such as increases in productivity, improved competition in terms of trade areas or reduced supplies of labor.

(To be continued.)

Conclusion

There may be market forces connected with general monopolistic competition throughout the economy that prevent measures like increases in the minimum wage rate from increasing the average real wage rate.

Any notion of price increases driving up wages is canceled out by wage increases driving up prices through increased costs and higher demand.

Appendix

Proof that in monopolistically structured markets an x proportional increase accompanied by an x proportional increase in marginal cost will lead to an x proportional increase in price. Let the inverse demand function be p(q) and the marginal cost function c(a). Then marginal revenue is equal to

r(q) = p(q) + q(dp/dq)
and profit maximization
requires q* be
such that
p(q*) + q*(dp/dq) = c(q*)

Now consider a case in which the new inverse demand function p'(q) is equal to (1+x)p(q) and the new marginal cost function is (1+x)c(q). The new marginal revenue function r'(q) is then

r'(q) = p'(q) + q(dp'/dq) = (1+x)p(q) + (1+x)q(dp/dq) = (1+x)r(q)

Therefore profit maximization requires a level of output q^ such that

r'(q^) = c'(q^)
which is equivalent to
(1+x)r(q^) = (1+x)c(q^)
and hence
r(q^) = c(q^)

This means that q^ is the same q*. Output does not change and hence

p'(q^) = (1+x)p(q*)

The price goes up by (1+x).

An inverse demand function p(q) is the price that consumers are willing and able to pay for the quantity q. This can be associated with their income. That is to say, if consumers income goes up by a proportional amount x then the inverse demand function rises by a proportional amount x then the inverse demand function rises by a proportional amount x. If the proportional increase in income comes from increases in wage rates then marginal cost will rise and roughly by a proportional amount x.