9781118041581

total (40) exceeds the available supply (25). The manager must cut back one or

both outputs. But how should she do this while maintaining as high a level of

profit as possible? To answer this question, observe that if output is cut back in

each market, the marginal profit in each market will be positive. What if the man-

ager chose outputs such that marginal profit differed across the two markets, say,

M 1 M 2 0? If this were the case, the manager could increase her total profit

by selling one more unit in market 1 and one less unit in market 2. She would

continue to switch units as long as the marginal profits differed across the mar-

kets. At the optimal solution, marginal profits must be equal. Thus, 

/ Q 1 



/ Q 2 must hold as well as Q 1 Q 2 25. Taking derivatives, we find the first

condition to be 20 Q 1  40 2Q 2. Solving this equation and the quantity con-

straint simultaneously, we find that Q 1 10 and Q 2 15. This is the firm’s opti-

mal solution.

THE METHOD OF LAGRANGE MULTIPLIERS The last two problems can be

solved by an alternative means known as the method of Lagrange multipliers.To

use the method, we create a new variable, the Lagrange multiplier, for each

constraint. In the subsequent solution, we determine optimal values for the

relevant decision variables and the Lagrange multipliers. For instance, in the

supply commitment example, there is one constraint, Q 7. (We know the

constraint is binding from our discussion.) To apply the method, we rewrite

this constraint as 7 Q 0, create a new variable, call it z, and write

In short, we have formed L (denoted the Lagrangian) by taking the original

objective function and adding to it the binding constraint (multiplied by z). We

then find the partial derivatives with respect to the two variables, Q and z, and

set them equal to zero:

Solving these equations simultaneously, we find that Q 7 and z 16. The

value of Q is hardly surprising; we already know this is the best the manager can

do. The interpretation of the Lagrange multiplier, z, is of some interest. The

value of the multiplier measures the marginal profit (M16) at the con-

strained optimum.

To apply the method in the multiple-market example, we write

L(20Q 1 .5Q 12 )(40Q 2 Q 22 )z(25Q 1 Q 2 ),

L/ z 7 Q0.

L/ Q 40 8Qz0;

40Q4Q^2 z(7Q).

Lz(7Q)

70 Appendix to Chapter 2 Calculus and Optimization Techniques

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