A manufacturing firm\'s production function is Q = 10K0.50.5 ere Q is the quanti

Question

A manufacturing firm's production function is Q = 10K0.50.5 ere Q is the quantity of output (units), K is the amount of capital used (machine-hours), The firm rents capital at a rate of R = $4 per hour and hires labor at W = $16 per hour. The firm plans to produce product of labor is P5 a is the mount of labor hired (worker-hours). For this production function, the margin 0.5 5L r 0.5 abor is MPand the marginal product of capital is MPK 720 units of output. How much labor and how much capital should the firm use to minimize its cost, of production 0. Suppose that a technological improvement increases the firm's total factor productivity, so the pro- duction function is now 0.5 for which the marginal product of labor is MP. = 6P , and the marginal product of capital is MPs. How much labor and how much capital should the firm use now to minimize the cost of producing 720 units of output? c. How does the technological improvement change the firm's costs of producing 720 units of output? d. Use calculus to show that the marginal products of labor and of capital of the new technology are as given.

Explanation / Answer

Q= 10K0.5L0.5 ----------- (1)

MPL= 5K0.5/L0.5 , MPk = 5L0.5/K0.5

  R= $4 per unit

W= $16 per hour.

Firm plans to produce 720 units of output.

(a) Cost- minimization condition : MPL/MPK = W/R

5K0.5/L0.5 / 5L0.5/K0.5 = 16/4

K/L = 16/4

K= 4L --------- (2)

Now put equation (2) in (1)

WE get , Q = 720 = 10(4L)0.5 (L)0.5

720= 10(2) L

L= 36

Now put this in equation (2) we get, K=4 L

K= 4(36)

K= 144

So. the firm use 36 worked hours of labours and 144 machine hours of capital to minimize its cost of production.

(b) With technological improvement , MPL= 6K0.5/L0.5 and MPK = 6L0.5/K0.5

  Now MPL/MPK = W/R [Cost minimisation condition]

6K0.5/L0.5/ 6L0.5/K0.5 = 16/4

K/L = 16/4

K=4L

But now the production function is Q = 12K0.5L0.5

So put K=4L in this production function.

Q = 720 = 12(4L)0.5(L)0.5

720= 24L

L= 30

Now, put L=30 in K=4L , we get,

K= 4(30)

K= 120

Now, the firm use 30 worked hours of labours and 120 machine hours of capital to minimize its cost of production.

(c) Technological improvement change the firm's cost of producing 720 units of ouput positively.

Firms cost is given by WL + RK

Earlier without technological improvement, Firms cost = 16(36) + 4(144) Because[L=36 and K=144]

= 576 + 576 = 1152

Now, with technological improvement , Firms cost = 16(30) + 4(120) Because[L=30 and K =120]

= 480 + 480 = 960.

So, after technological improvement there will be less firms cost for production.

(d) Q= 12K0.5L0.5

So to find the marginal products of labour and capital , do the partial differentiation of production function with respect to labour and capital respectively.

MPL = 0.5(12 K 0.5 ) L-0.5 [by partial differentiate of Q with respect to L]

  = 6 K0.5/ L0.5

MPK = 0.5(12L0.5) K-0.5 [by partial differentiate of Q with respect to K]

= 6L0.5/K0.5

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