9781118041581

PRICE AND ADVERTISING Suppose the firm’s profit function is

The partial derivative of profit with respect to P is

Notice that when we take the partial derivative with respect to P, we are treat-

ing A as a constant. Thus, 4A and A^2 disappear (Rule 1) and 2PA becomes 2A

(Rule 2). The partial derivative of profit with respect to A is

Setting each of these expressions equal to zero produces two equations in two

unknowns. Solving these simultaneously, we find that P 3 and A 5. Thus,

profit is maximized at these values of P and A.

Constrained Optimization

In the previous examples, the decision variables were unconstrained, that is,

free to take on any values. Frequently, however, decision variables can be

changed only within certain constraints. Consider the following example.

A SUPPLY COMMITMENT A firm is trying to identify its profit-maximizing

level of output. By contract, it already is committed to supplying at least seven

units to its customer. Suppose that its predicted profit function is given by 

40Q 4Q^2. The firm seeks to maximize subject to Q 7. Setting marginal

profit equal to zero, we have d/dQ  40 8Q 0 so that Q 5. But this value

is infeasible; it violates the contract constraint. The constrained maximum occurs

at Q 7, where d/dQ 6. Note that, since marginal profit is negative, profit

would decline if Q were increased. Thus, the firm would like to raise profit by

decreasing Q, but this is impossible due to the binding contract constraint.

A different kind of constrained optimization problem occurs when there

are multiple decision variables.

PROFITS FROM MULTIPLE MARKETS A firm has a limited amount of output

and must decide what quantities (Q 1 and Q 2 ) to sell to two different market seg-

ments. For example, suppose it seeks to maximize total profit given by

subject to Q 1 Q 2 25. Setting marginal profit equal to zero for each quantity,

we find that Q 1 20 and Q 2 20. But these desired quantities are infeasible; the

(20Q 1 .5Q 12 )(40Q 2 Q 22 ),

/ A 4 2A2P.

/ P 2 4P2A

 20 2P2P^2 4AA^2 2PA.

Appendix to Chapter 2 Calculus and Optimization Techniques 69

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