10 points Suppose the production function is given by where

(10 points) Suppose the production function is given by where L is labor hours per week. 1n the short run, K is fixed at 100, K = 100, so the So, the conditional factor demand for labor is L(g) 1. f(L, K) LXK short-run production function is a. If capital rent is $10 per unit and wages are $5 per hour, find the total cost, variable cost, fixed cost, and short-run average variable cost functions (Hint: use the fact that the total cost is equal to wL+rK) Using the short-run total cost function from part (a) and the following short-run marginal cost: b. how much will the firm produce at a price of $102 How many labor hours will be hired per week? How much profit will be earned?

Solution

Ans a)

r=$10, w=$5 then Total Cost=wL+rK=5L+10(100)=1000+5L

C=1000+5(q^2/100)=1000+q^2/20=1000+0.05q^2

Variable Cost is cost varying with q hence we get

VC=0.05q^2

FC=1000 that is invariant with output

SRAVC=VC/q=0.05q

Ans b)

We have Variable Cost=0.05q^2 & FC=1000

Hence differentiating TC wrt q we get MC

dTC/dq=MC=d/dq(0.05q^2)=0.1q=1/10(q)

Ans c)

Now p=10 then Profit can be given as

Profit=p*10sqrt(L)-5L-1000

=100sqrt(L)-5L-1000

Differerntiating wrt L

=50/sqrt(L)-5=0

50/sqrt(L)=5

L=100

Hence Profit=100sqrt(L)-5L-1000

Hence Profit=100(sqrt(100))-5(100)-1000=-500 and q=10*sqrt(100)=100