9781118041581

where the binding constraint is Q 1 Q 2 25 and z again denotes the

Lagrange multiplier. Setting the appropriate partial derivatives equal to zero,

we find

Notice that the third condition is simply the original constraint. We now find

values that satisfy these three equations simultaneously: Q 1 10, Q 2 15, and

z 10. The values for Q 1 and Q 2 confirm our original solution. In addition,

note that the first two equations can be written as z  20 Q 1  40 2Q 2 , or

z M 1 M 2. In other words, the multiplier z represents the common value

of marginal profit (equalized across the two markets). The actual value of M

in each market is z 10. Thus, if the manager could increase total sales (above

25), he would increase profit by 10 per unit of additional capacity.

To sum up, the use of Lagrange multipliers is a powerful method. It effec-

tively allows us to treat constrained problems as though they were uncon-

strained.^3

Questions and Problems

1. The economist Arthur Laffer has long argued that lowertax rates, by
   stimulating employment and investment, can lead to increasedtax revenue
   to the government. If this prediction is correct, a tax rate reduction
   would be a win-win policy, good for both taxpayers and the government.
   Laffer went on to sketch a tax revenue curve in the shape of an upside-
   down U.
   In general, the government’s tax revenue can be expressed as
   R t B(t),where t denotes the tax rate ranging between 0 and 1
   (i.e., between 0 and 100 percent) and B denotes the tax base. Explain
   why the tax base is likely to shrink as tax rates become very high. How
   might this lead to a U-shaped tax revenue curve?


2. The economic staff of the U.S. Department of the Treasury has been
   asked to recommend a new tax policy concerning the treatment of the
   foreign earnings of U.S. firms. Currently the foreign earnings of U.S.
   multinational companies are taxed only when the income is returned to

L/ z 25 Q 1 Q 2 0.

L/ Q 2  40 2Q 2 z0;

L/ Q 1  20 Q 1 z0;

(^3) It is important to note that the method of Lagrange multipliers is relevant only in the case of
binding constraints. Typically, we begin by seeking an unconstrained optimum. If such an opti-
mum satisfies all of the constraints, we are done. If one or more constraints are violated, however,
we apply the method of Lagrange multipliers for the solution.
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