Question

4. You own an 8,000 ft^2 (K = 8,000) factory that produces N95 masks and you produce them by combining labor and capital (your factory) as follows: Q = 4K^(1/3)L^(2/3), where Q is the number of masks produced per week. Assume there are many competing factories and all factories are identical. Until two months ago, the price of masks in this perfectly 1 competitive market was $1.50 each. Today the price is $3.00 ea. The rental value of your factory is $.50 per ft^2/week. Answer the following 2 questions: a)How much profit do you expect to make each week if the price remains at $3.00 ea.? b)If a new identical factory costs $1 million and it is operational within one month, should you purchase it today?

Answer 1

Given production problem :-

where Q is the no. of masks produced per week, K = Capital, L = Labor
(if we compare given production function with generic form, , we get A = 4 and = 1/3)

K = 8000, Rental Cost per week = $ 0.5*8000 = $4000, Rental Rate of Capital = 0.5

(a)
Using formula
Rental Rate of Capital = MPK(Marginal Product of Capital) = = 0.5
i.e. (1/(3*8000))Q = 0.5
So, Q (no. of masks produced) = 12000 per week

Labor Cost = = (2/3)*12000 = $8000

Total Sales/Revenue = $3 * 12000 = $36000
Total Cost = Labor Cost + Rental Cost = $4000 + $8000 = $12000

So, Profit per week = (Revenue - Total Cost) per week = $36000 - $12000 = $24000

(b)
If an identical new factory costs $1million and operational within one month, we will recover our expense in approximately (1+(1000000/(24000*4))) = 11.42 months
So, we should purchase it.

If you have any doubt, feel free to ask in comment box. Please do upvote.