In the November 2015 debates for the campaign for the nomination of the Democratic Party candidate for the Presidency each candidate advocated a substantial increase in the national minimum hourly wage rate; two to $15 and one to $12. They argued that this was justified on the basis that that real wage had not increased in something like 30 years. It somehow did not occur to any of them since the national minimum wage had increased several times over the period that increases in the national minimum wage is not an effective way to increase real wages.

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 is 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.

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.

And it is not just monopolistic firm which raise prices when the perceive that the public has higher nominal income and therefore can pay more. Tuition at educational institutes is a prime example but also local taxes experience such rises.

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)

Here is the graphical version of the above derivation

If consumers' income goes up by a proportional fraction x the amount that consumers can pay for each quantity q goes up to q(1+x). This is equivalent to the constant a going up to a(1+x). For more on this see Inverse Demand Functions

There are of course other costs besides labor costs, but in a closed economy with fixed stocks of capital equipment the only short-run influences on marginal costs are wage rates. Thus if wages go up by a fraction x then marginal cost goes up by a like fraction; i.e., to c(1+x). Therefore the new monopoly price p' is

p' = ½(a(1+x)+c(1+x)) = (1+x)[½(a+c)] = (1+x)p

The graphical version of the above is

Thus, to the extent that prices are set by monopolistic structures, the increase in nominal income is canceled out by price increases. Therefore it would not be surprising if real wage rates remained constant,

The above analysis suggest that increases in wage rates lead to price increases. On the other hand increases in prices could lead to demands for higher wages by organized. So wages increases could lead prices or prices increases could lead wage increases, or the combination of the two effects could eliminate any lead-lag relationship.

Here are the rates of change of an index of wage rates in general

And this one is for wages and salaries computed seasonably over a period of a year.

Here are the corresponding rates of increase of the Consumer Price Index

And finally here is the time series on the real wage rates

When the annual rates of changes of the Wage Index and the Consumer Price Index are plotted together, as shown below, there is no definite lead-lag relationship between them. This may be because the lead or lag period is less than one year or that there multiple relationships between wage rates and inflation that eliminate any clear-cut leading of one rate over the other.

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.