Consider a firm with production function Q f K,L K 1 2 L 1 2 Q is the quantity produced, K is capital and L is labor Rental rate (or price) of capital is r 2 and wage of labor is w 6 Firm has a total budget of 100 This firm has the following restrictions Total cost of production should be LESS THAN OR EQUAL TO 100 Total spending on wages (i e wL ) should be LESS THAN OR EQUAL TO 30 The firm wants to produce as much as goods possible Write the problem of the firm ( 10 Points ) (Hint max s t ) Write the Lagrangian function of the problem ( 10 Points ) Write Kuhn Tucker Conditions Explicitly state Complementary Slackness Conditions ( 15 Points ) Find capital (K) and labor (L) which maximizes the firms production Also find maximum amount of goods this firm can produce ( 15 Points )

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Question: Consider a firm with production function Q=f K,L = K 1/2 L 1/2 Q is the quantity produced, K is capital and L is labor.

Consider a firm with production function Q=fK,L=K1/2L1/2

Q is the quantity produced, K is capital and L is labor. Rental rate (or price) of capital is r=2 and wage of labor is w=6. Firm has a total budget of 100. This firm has the following restrictions:

Total cost of production should be LESS THAN OR EQUAL TO 100
Total spending on wages (i.e. wL) should be LESS THAN OR EQUAL TO 30.

The firm wants to produce as much as goods possible.

Write the problem of the firm. (10 Points)

(Hint: max s.t. )

Write the Lagrangian function of the problem. (10 Points)
Write Kuhn-Tucker Conditions. Explicitly state Complementary Slackness Conditions. (15 Points)
Find capital (K) and labor (L) which maximizes the firms production. Also find maximum amount of goods this firm can produce. (15 Points)

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