9781118041581

FIGURE 5.3

Producing Output

at Minimum Cost

The firm produces 636

units at minimum cost

at point B, where the

isoquant is tangent to

the lowest possible

isocost line. Point B

corresponds to10 units

of labor and 8 units

of capital.

The ratio of marginal products exactly matches the ratio of input prices.^7 (If

one input is twice as expensive as another, optimal usage requires that it have

twice the marginal product.) This relationship can be rearranged to read

MPL/PLMPK/PK.

206 Chapter 5 Production

(^7) The same condition is derived readily using the method of Lagrange multipliers introduced in the
appendix to Chapter 2. The problem is to minimize TC PLL PKK subject to F(L, K) Q 0 ,
where Q 0 denotes a given level of output. The Lagrangian is £ PLL PKK z(Q 0 F(L, K)). The
optimality conditions are £/ L PLz( F/ L) 0, £/ K PKz( F/ K) 0, and £/ z 
Q 0 f(L, K) 0. Dividing the first condition by the second yields PL/PK( F/ L)/( F/ K) 0.
It follows that PL/PKMPL/MPK, after recognizing that MPL F/ L and MPK F/ K.
16
6
8
10
12
14
Amount of Capital
06810121416
Amount of Labor
Isoquant Q = 636
Other
isocost
lines
Input cost = $220
Optimal input mix
B
c05Production.qxd 9/5/11 5:49 PM Page 206