Question - A price-taking firm has the production function Q = f(z1, z2). The output price is P and the input price is w1 and w2. There are two unusual things about this firm. First, rather than maximizing profit, this firm maximizes revenue. Second, the firm is cash constrained. In particular it has only C amount of cash on hand before producing and as a result, its total expenditures on inputs cannot exceed C.
(1) Write down the firm's maximization problem.
(2) Suppose an econometrician observed the firm's revenue under various output prices, input prices, and level of the financial constraint and has estimated the firm's revenue as a function of these parameters (assume that the firm behave optimally in terms of its objective of maximizing revenue):
R(P, w1,w2, C) P(γ + ln C - αlnw1 - (1 - α)ln w2) (γ and α are scalars whose values are known).
What is the firm's optimal use of input z1(P, w1, w2, C)? Hint: think about the related properties discussed in consumer theory.