Question

A firm has production function f(z) = 2z
1/2 The price of the input is r and the price of
output is p.
a) Calculate the optimal output, input, and profit?

Answer 1

$f(z) = 2z^{\frac{1}{2}}$

Let:

price of input = r

price of output = p

Profit = Total revenue (TR) – Total cost (TC)

$Profit = p \times 2z^{\frac{1}{2}} – (rz) = 2pz^{\frac{1}{2}} – rz$

To find optimal profit we will differentiate profit function with respect to z.

$\frac{d(profit)}{dz} = 2p \times \frac{1}{2\sqrt{z}} – r = 0 \\ 2p = r2\sqrt{z} \\ p = r\sqrt{z} \\ Z^* = (\frac{p}{r})^2$

Optimal input:

$Z^* = \frac{p^2}{r^2} \\ Output = f(z) = 2(Z^*)^{\frac{1}{2}} \\ = 2(\frac{p}{r})^{2 \times \frac{1}{2}} \\ = 2\frac{p}{r} \; (optimal \; output) \\ Profit = 2p(\frac{p}{r})^{2 \times \frac{1}{2}} – r(\frac{p^2}{r^2}) \\ = 2p(\frac{p}{r}) – r \frac{p^2}{r^2} \\ = 2\frac{p^2}{r} - \frac{p^2}{r} \\ = \frac{p^2}{r} \; (optimal \; profit)$