Please explain how you got your answers! :) A competitive firm uses two inputs a

Question

Please explain how you got your answers! :)

A competitive firm uses two inputs and has a production function f(x1,x2)=4L^1/2M^1/2, where L is the number of units of Labor, and M is the number of machines used. The cost of labor is $16 per unit and the cost of machines is $4 per unit.

a) Does this production function exhibit increasing, decreasing, or constant returns to scale? Explain why.

b) Calculate the marginal product of inputs x1, and x2.

c) What is the ratio fo x1 to x2 required to produce the output in the cheapest way possible?

d) How many units of x1 and x2 will the firm use to produce y units of the output?

e) What is the cost of poducing y units of the output for this firm?

Explanation / Answer

a)if we increase input by factor of n then production function will be n^1/n^1/2 =n^1/2 hence prodction increase is n^1/2 which is less than n .. hence production function exhibit decreasing rate of return

here X1 is L and X2 =M

b)marginal input for X1 (labour) =4/2M^1/2

marginal input of X2 =-2L/3M^3/2

c)for cheapest ,production marginal input for X1 +marginal input for X2 =0

4/2M^1/2 =2L/3M^3/2

L/M=3

.......

d) y =4L^1/2M^1/2

y =4*3M/2M^1/2

M=y^2/36

hence L =3M =y^2/12

e)hence cost =16*y^2/12 +4*y^2/36 =13y^2/9

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