Question 11.8: A firm operates with the production function Q = 4K^0.6L^0.5......

The output constraint is 200 = 4K^{0.6}L^{0.5} and the objective function to be minimized is the total cost function TC = 15K + 8L. The corresponding Lagrangian function is therefore

G = 15K + 8L + λ(200 − 4 K^{0.6}L^{0.5} )

Partially differentiating G and setting equal to zero, first-order conditions require

\frac{\partial G}{\partial K } = 15 − λ2.4 K^{-0.4}L^{0.5} = 0      giving    \frac{15 K^{0.4}}{2.4L^{0.5}} = λ                       (1)
\frac{\partial G}{\partial L } = 8 − λ2K^{0.6}L^{-0.5} = 0      giving      \frac{4L^{0.5}}{K^{0.6}} = λ                            (2)
\frac{\partial G}{\partial \lambda } = 200 − 4K^{0.6}L^{0.5} = 0                           (3)

Setting (1) equal to (2) to eliminate λ

\frac{15 K^{0.4}}{2.4L^{0.5}} = \frac{4L^{0.5}}{K^{0.6}}
15K = 9.6L
1.5625K = L                     (4)

Substituting (4) into (3)

200 − 4 K^{0.6}(1.5625K)^{0.5} = 0
200 = 4 K^{0.6}(1.5625)^{0.5} K^{0.5}
 \frac{200}{4(1.5625)^{0.5}} =  K^{1.1}
40 =  K^{1.1}
K = \sqrt[1.1]{40} = 28.603434

Substituting this value into (4)

1.5625(28.603434) = L
               44.692866 = L

Thus the optimal input combination is 28.6 units of K plus 44.7 units of L (to 1 dp). We can check that these input values correspond to the given output level by substituting them back into the production function. Thus

Q = 4 K^{0.6}L^{0.5} = 4 (28.6)^{0.6}(44.7)^{0.5} = 200

which is correct, allowing for rounding error. The actual cost entailed will be

TC = 15K + 8L = 15(28.6) + 8(44.7) = £786.60