In: Economics
Consider the following production function: x = f(l,k) = lb kb where x is the output, l is the labour input, k is the capital input, and b is a positive constant.
Suppose b < 1/2.
(a) Set up the cost minimization problem and solve for the conditional labour and conditional capital demand functions. Let w and r be the wage rate and rental cost of capital respectively.
(b) Using your answer in (a), derive the cost function and simplify the function as much as you can.
(c) Use your answer in (b) to derive the marginal cost function.
(a) The production function is
. The cost of production would be
. The problem would be
Minimize
,
Subject to
.
The Lagrangian function to minimize the cost would be
, and the FOCs would be as below.
or
or
or
.
or
or
or
.
or
or
or
.
Comparing first two FOCs, we have
or
or
or
. Putting it in the third FOC, we have
or
or
or
, and since
or
, we have
or
, which are the required labor and capital demand.
(b) The cost function would be
or
or
or
is the required cost function.
(c) The marginal cost would be
or
or
or
. Note that since
, we have
, and hence
increases with x.