Question

In: Economics

Consider the following production function: x = f(l,k) = lb kb where x is the output,...

Consider the following production function: x = f(l,k) = l^b k^b 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.

Expert Solution

(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 .

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.

Rahul Sunny answered 2 years ago

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