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Critical Prices for
Supply and Demand
Decisions by Firms
Facing Fixed Prices

The point of this material is that the supply and demand schedules for a single firm facing fixed prices necessarily have
critical prices at which they collapse to zero. The collection of firms in a market likely have similar critical prices
so the supply and demand schedules for a market also have critical prices beyond which the supply and the demands are zero.

The Supply Function for a Firm Facing a Fixed Product Price

Let C(q) be the cost function for a firm which receives a price of p its product. Its profit is then

Π = pq − C(q)

It maximizes profit at an output q* where

C'(q) = p

The derivative of the cost function is usually denoted as the Marginal Cost function MC(q). The supply functions for the firm
is over some range of price the inverse of the marginal cost function. But the quantity supplied is zero if the maximum profit is
negative. If the protit is zero then

C(q) = pq
and thus
p = C(q)/q

Thus the lowest price at which any of the product is supplied is at the level of minimum average cost. Thus for a firm its supply as a function of price
is equal to the inverse of the marginal cost function down to a price equal to its minimum average price.

q(p) = (MC)^{−1}(p) for p≥min(C(q)/q)
q(p) = 0 for p≤min(C(q)/q)

At a price equal to min(C(q)/q) a firm is indifferent between supplying zero or the quantity
that makes average cost a minimum.

The Demand Function for Labor by
a Firm Facing a Fixed Wage Rate

Let Q(L) be the production for a firm which employs L workers. Suppose the firm pays a wage rate w and receives a price p
for its product.
Its profit is then

Π = pQ(L) − wL

It maximizes profit when it employs L workers such that

pQ'(L) = w

The quantity pQ'(L) is called the marginal revenue product of the firm. Let this function be denoted as MRL(L). Thus for
profit maximization MRL(L) = w.
The labor demanded by the is the inverse of the function MRL(w) up to some limit on the wage rate. That limit is the wage rate
that makes profit equal to zero. The profit is zero if the wage rate equals or exceeds the maximum revenue product of labor; i.e.,
the maximum value of (pQ(L)/L).

The demand for labor by a firm facing a fixed wage rate is then

L(w) = (MRL)^{−1}(w) for w≤max(pQ(L)/L)
L(w) = 0 for w≥max(pQ(L)/L)

Although the above was in terms of the demand of a firm for labor, it equally applies to the demand of the firm for
anything involved in its production.