San José State University

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A General Theorem on Price Changes
in an Economy of Any Market Structure

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 produccing 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.

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

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)

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.

In any case the conditions for profit maximization are, in matrix form,

P −GJQ −C1 = 0
or, 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 = P0 − C1

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

dQ = KP0 − KC1

The condition

P = P0 − F(Q)

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 + R1dC1
where
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.

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

Then

dP = dF

That is to say, if dP0 is equal to dC1 then the market price change is exactly to their common value; not any multiple of it or any fraction of it but exactly it.


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