Question

Consider a firm in a perfectly competitive market which chooses an input 𝑥 to maximize its profit 𝜋 = 100𝑓(𝑥) − 𝑤𝑥. 

i) Specify the first order condition for the profit maximization problem. (4 marks) ii) Suppose there is a differentiable function 𝑥(𝑤)that satisfies the implicit function in part (i) above. Using the chain rule, find 𝑥′(𝑤). (4 marks) iii) Suggest conditions on the function 𝑓(𝑥) so that 𝑥 ′(𝑤) < 0

Answer 1

i)

$$\frac{\delta\pi}{\delta x}=100 f'(x)-w$$

ii)

$$\frac {\delta \pi}{\delta x}=100f'(x)-xw'(x)-w$$

iii)

$$w'(x)<0$$

$$w'(x)=\frac{100f'(x)-w(x)}{x}$$