San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

A General Theorem that Price Changes
in an Economy of Any Market Structure
are a Weighted Average of Those Due to
Shifts in the Demand Function and Those
Due to Shifts in the Cost Functions

## Background

The purpose of this material is to prove a theorem concerning price change in any market of profit maximizing enterprises no matter what the market structure is. The theorem is that the change in prices is a weighted average of the price changes resulting from shifts in the demand functions and shifts in the cost functions. The weights depend upon the competitive structure of the market; i.e., whether the market is monopolistic or oligopolistic and whether the products of the market are differentiated or undifferentiated.

The mathematics for the general case is a bit complex so the analysis for the simple case of a monopoly is given first.

## 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) = ½a + ½c

Thus for a monopolist the price set gives an equal weight to the the price it can get as it does to the cost of producing an additional unit. When a monopolist raises its price it may simply be because it can get a higher price and has nothing to do with higher costs.

Although there are few products marketed by a strict monopoly the results of the analysis apply more broadly because retail businesses operate as monopolies in the areas around their locations. That situation is known as monopolistic competition.

The above result implies that

#### Δp = ½Δa + ½Δc and further Δp/p = ½Δa/p + ½Δc/p

Likewise the above analysis can be applied to each of N monopolistic industries. Then column vectors P, A and C, for prices pi, the demand parameters ai, and marginal costs ci satisfy

#### P = ½(A + C) = ½A + ½C and likewiseof course ΔP = ½ΔA + ½ΔC

The above rule is that monopoly price is a simple average of marginal cost and the 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 again 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.

## The General Case

The general case involves multiple products whose prices and output are given by the column vectors P and Q. The degrees of substitutability are represented by the demand functions

#### Q = D(P)

For the analysis it is more convenient to work with the inverse demand functions. These will be assumed to be of the form

#### P = P0 − F(Q)

Changes in the demand function parameters P0 result in shifts the relationships between the price of a product and the quantity produced and consumed.

The profit function for the firm producing the i-th product is

#### πi = piqi − (C0i + C1iqi)

It is presumed here and now that cost functions are simple linear functions of outputs.

The first order condition for a maximum profit for the i-th firm is

#### (∂πi/∂qi) = pi +(∂pi/∂qi) qi − C1i = 0

In matrix notation for the market this is

#### P + (∂P/∂Q)Q − C1 = 0

where 0 is the zero vector.

The term

#### (∂P/∂Q) reduces to −((∂F/∂Q)J

where J=(∂qj/∂qi) represents the expectations of each firm about the reactions of the other firms to its increase in production. Of course (∂qi/∂qi)=1 for all i but (∂qj/∂qi) may or may not be zero for j≠i. The Cournot assumption is that for j≠i all such terms are zero. In the von Stackelberg Leader-Follower Model of a duopoly the follower firm makes the Cournot assumption but the leader presumes that when it increases production by one unit the follower firm decreases production by a half unit.

For typographic convenience let the matrix (∂F/∂Q) be denoted as G.

Thus the conditions for profit maximizations are, in matrix form,

#### P −GJQ −C1 = 0or, equivalently P0 − F(Q) −GJQ −C1 = 0

If the parameters P0 and C1 change there will have to be corresponding changes in Q and P. Those changes have to satisfy these conditions

#### dP0 − G(Q)dQ −H(Q)dQ − dC1 = 0

where H is the matrix of the derivatives of the elements of GJ with respect to the components of Q.

The above equation can be rewritten as

#### (G + H)dQ = dP0 − dC1

Let K denote the inverse of the matrix (G+H). Thus

The condition

requires that

#### dP = dP0 − G(Q)dQ

The substitution of the previous expression for dQ into this last equation gives

#### dP = dP0 − G(KdP0 − KdC) or, equivalently dP = (I − GK)dP0 + GKdC1

where I is the identity matrix. This says dP is a weighted average of the prices changes dP0 due to the shifts in the demand functions and shifts in marginal costs dC1; i.e.;

#### dP = R0dP0 + R1dC1where R0 + R1 = I

This is a relationship that prevails for all market structures. The structure of a market affects only the weights. This surprisingly general relationship extends to the whole economy with its multitude of products and market structures.

## A Special Case

There is a special implication of the above result. Suppose the price changes dP0 due to shifts in the demand functions is equal to the price changes dC1 due to shifts in the cost function. Say their common value is dF; i.e.,

#### dP0 = dC1 = dF

Suppose the minimum wage increased by x percent. If that increase prevailed throughout the goods and services required to produce a commodity. Then dC1/p would equal x.

The parameter dp0 may be proportional to average disposable income in the market. Then if all incomes increase by the same proportion as the minimum wage then dp0/p would equal x. Thus dp/p would equal x for all commodities and there would be no change in the real wage rate and no change in employment.