I was helping some students study for an exam–the thing I was best at in my life was taking tests, so I always find ways to go back to that activity–and I encountered what may be a subtle inconsistency in the way economists typically solve these types of problems. But, before I decide whether I’ve actually stumbled on something worth pursuing, I first want to make sure other professional economists went down the same path I did.
So, in the comments, if you want to participate please tell me whether you have a degree (and what level) in economics, and then what you think the answers are to the questions.
There are 1,000 firms. Each firm must decide whether to produce corn or wheat. For firm i, the production function for corn is given by:
C_i = 2L_i^(2/3)
where _ denotes subscript and ^ denotes exponentiation. L_i is the number of units of labor hired by firm i.
For wheat the production function is:
W_i = ( 160L_i^(1/2) ) / n
where n is the number of firms who have chosen to go into wheat production.
The market prices for corn and wheat output are $3 and $5 per unit, the wage rate for labor is $4, and the interest rate is 10% per year. Each firm is a price taker in whatever line it goes into.
==> Calculate the profit-maximizing levels of output for a corn- and a wheat-producing firm, respectively.
==> In equilibrium, how many of the 1,000 firms will go into corn and how many into wheat production?