I need help with QUESTION #5 ONLY Consider a firm with the production function,

Question

I need help with QUESTION #5 ONLY

Consider a firm with the production function, q = (K^1/2 + L^1/2)^2 In the short-run, the level of capital is fixed. Determine the equations for MPL and AP_L. Solve for the short-run cost function (i.e. total costs as a function of output) Determine the equations for MC, ATC, A VC, and AFC. Using the same production function as in question 4, suppose that the firm is now operating in long-run. Solve for the long-run cost function (i.e. total costs as a function of input prices and output). Consider your answer from questions 5a and 4b. How does short-run total cost compare to long-run total cost when the fixed level of capital in the short-run equals the optimal level from the long-run cost minimization problem? Prove your statement mathematically.

Explanation / Answer

a)

The long run cost minimization problem can be solved using Lagrangian method.

The cost structure of the firm is given by

C = rK + wL

Setting Lagrangian would imply:

Minimize C = rK + wL - (K0.5L0.5)2

Finding the partial derivatives and setting them equal to zero gives

w - MPL = 0

w = 2(K0.5L0.5)/L0.5

r - MPK = 0

r = 2(K0.5L0.5)/K0.5

The first two equations give

K = (w/r)2L

Substitute this value in the production function to get

L = (r/w+r)2q

K = (w/w+r)2q

These are the optimum values of L and K that minimizes the cost.

Total cost function now becomes

C = rK + wL

C = rw2q/(w+r)2 + wr2q/(w+r)2

b) The short run cost has both the fixed cost and the variable cost. The long-run cost has no fixed cost.

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.