Suppose that a competitive company must produce beer using a Hops plant with a production function f:
Question:
Suppose that a competitive company must produce beer using a Hops plant with a production function f: R⟶R. (f ' >0 , f '' <0)
The beer has a price "P" in the market, and hops has a price of W. The company owns a x0 quantity of Hops from last year, which can spoil or grow.
The quantity of this plant that will be useful for production is x0 + θε, where ε is a random variable with zero mean and θ> 0. θε∈ (-x0, x0).
The firm can buy more x> 0 in the market and will maximize the expected profits.
a. Define the expected profits of this beer firm as a function of x and the other parameters. f(x0, p, w, θ).
Let x*(x0, p, w, θ)> 0 be the solution to the maximization problem.
b. Given the above, determine the FOC that solves x*.
c. Determine the sign of the following derivatives (make additional realistic assumptions to determine the signs if necessary) and conceptually explain the results.
- ∂x*/∂x0
-∂x*/∂P
- ∂x*/∂w
-∂x*/∂θ