Suppose that the production function of a representative firm in perfect competi

Question

Suppose that the production function of a representative firm in perfect competition is given as X=K0.25L0.75 and the budget of this firm is defined as PL+PK=C for arbitrary C

a. Derive the Long-run expansion path

b. Suppose that PL (wage) is 1 and PK (rent for capital) is 2. Current the available budget limit is 12 and plant size (fixed level of capital) is Ko=3. Find the demand for labor. Is this cost minimizing level of labor employment?

c. Now suppose that the plant size (level of capital) can be adjustable under same input price & budget condition in question (b). Find the demand for labor.

d. Now drop previous assumption about price of labor, price of capital and budget limit, and just focus on the general case. Find the formula of total cost function, marginal cost function, and average cost function in the long-run.

(Hint: TC(X)= (PL)L + (PK)K. Plug input demand functions into L and K of TC function. Input demand functions are derived by plugging rearranged production function with respect to L or K into the formula of expansion path.)

Explanation / Answer

a) Expansion path :

MUL/MUK = 0.75K/0.25L

PL/PK = 1/2

equating MUL/MUK and PL/PK

0.75K/0.25L = 1/2

K = 0.25L/1.5

THe above is the expansion path.

b) Budget Constraint : 12 = (PL)L + (PK)K or 12 = (1)L + (2)3

or 12 = L + 6

L = 12-6 = 6 labors.

Hence demand for labor will be 6.

At cost minimizing level MPL/MPK = PL/PK

0.75K/0.25L = 1/2

Putting L = 6 and K = 6

0.75/0.25 is not equal to 1/2 hence it is not cost minimizing demand of labor.

c) If plant size is adjustable,

MUL/MUK = 0.75K/0.25L

PL/PK = 1/2

equating MUL/MUK and PL/PK

0.75K/0.25L = 1/2

K = 0.25L/1.5

Putting this value in budget function, we get the following

12 = L + 2(0.25L/1.5)

12 = L + 0.5L/1.5

L =18/2 = 9 labors

and K = 0.25(9)/1.5 = 1.5 units.

Demand for labor is 9 now.

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