A firm produces and sells a single product

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A firm produces and sells a single product. Let

q

denote the number of units produced and sold,

p(q)

the price (in dollars per unit) required to sell

q

units,

R(q)=q\,p(q)

the total revenue,

C(q)

the total cost, and

P(q)=R(q)-C(q)

the total profit. Answer the following conceptual questions about these functions. (a) Explain why the condition

P'(q)=0

at an interior critical point corresponds to the statement that marginal revenue equals marginal cost. (b) Describe how the signs of

P'(q)

to the left and right of a critical point determine whether that point yields a local maximum of profit. (c) Interpret the economic meaning of

P''(q)<0

at a production level

q=q^*

. What does this second–derivative condition tell the firm about profit sensitivity in the neighborhood of

q^*

? (d) Suppose instead that the firm wishes to minimize the average cost

AC(q)=\dfrac{C(q)}{q}

for

q>0

. Explain why the condition

\dfrac{d}{dq}AC(q)=0

is not algebraically equivalent to setting

C'(q)=AC(q)

, yet the latter equality still emerges when evaluating the derivative. Provide an economic interpretation of the equality

C'(q)=AC(q)

at the production level that minimizes average cost.