Question

Isoquant Analysis. A firm with production function q = K^1/4L^1/4 operates with variable labour and variable capital. The firm sells output at a competitive price p = 80 and hires labour at w = 2 and capital at r = 0.5. This firm

1. will produce q =
2. will hire L =
3. will lease capital K =
4. will incur total cost C =
5. will earn profits π =

Now the price rises to p = 120. This firm

1. will produce q =
2. will hire L =
3. will lease capital K =
4. will incur total cost C =
5. will earn profits π =

Answer 1

A firm will hire labor and capital such that:

MPL/MPK = W/r

Where MPL is the marginal product of labor

MPK is marginal product of capital

W is wage rate

r is cost of hiring capital

q= K1/4 L1/4

MPL= differentiation of q with respect to L= 1/4 K1/4 L-3/4

MPK= differentiation of q with respect to K= 1/4 L1/4 K-3/4

MPL/MPK= K/L

Condition:

K/L = 2/0.5

0.5K= 2L

K= 4L Equation 1

Use it in the production function:

q= (4L)1/4 L1/4

q= 41/4 L1/4+1/4

q/41/4 = L1/2

Squaring both sides

q2/42/4 = L

L= q2/2 Optimal quantity of labor

Use this in equation 1

K= 4L

K= 2q2 Optimal quantity of K

Total cost= LW + Kr= 2(q2/2 )+0.5(2q2)= q2 + q2 = 2q2

Marginal cost= Differentiation of total cost with respect to q= 4q

In perfectly competitive market, optimal quantity arises where:

P=MC

80= 4q

q= 20 Optimal quantity that firm produce

L= q2/2 = 400/2= 200 Optimal quantity of labor

K= 2q2 = 2*400= 800 Optimal quantity of K

Total cost= 2(400)= 800

Optimal Profit= P*q - TC= 80*20-800= 800

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If price rises to 120:

MC= 4q

P=120

Condition:

P=MC

120=4q

q= 30   Optimal quantity that firm produce

L= q2/2 = 900/2= 450 Optimal quantity of labor

K= 2q2 = 2*900= 1800 Optimal quantity of K

Total cost= 2(900)= 1800

Optimal Profit= P*q - TC= 120*30-1800= 1800