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The production function for widgets takes the following form: q = 4L + 6K a. What...

The production function for widgets takes the following form:

q = 4L + 6K

a. What is the least cost combination of L and K that the firm should employ to produce 48 widgets when w = 2 and r = 4.

b. Suppose the price of labor increases to w = 4 but the rental rate of capital is unchanged. If the firm still wants to produce 48 widgets at the lowest cost possible, should it alter its input mix? If so, what is the new combination of L and K that is optimal?

c. More generally, derive the firm’s factor demand functions and long run cost function when (i) w/r > 2/3; and when (ii) w/r < 2/3.  

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Answer #1

1- now here MRTSkl = MPL/MPK = 2/3

price ratio is 2/4 = 1/2

now as MRTS>PL/PK thus, L* = 48/2 = 24 AND K = 0

2-NOW PRICE RATIO IS 4/4 = 1

AS PRICE RATIO > MRTSkl

thus L* = 0 AND K * = 48/4 = 12

3- now as W/R> 2/3

L* = 0 AND K * =Q/r

as w/r<2/3

L* = Q/w and K* = 0

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