Question

Let y$\mapsto$p(y) be the C⁽²⁾ inverse demand function facing a monopoly, where y$\in$$\mathbb{R}$₊₊ is its rate of output, and let y$\mapsto$C(y) be the C⁽²⁾ total cost function of the monopoly. Assume that p(y)>0, p'(y)<0, and C'(y)>0 for all y$\in$$\mathbb{R}$₊₊, and that a profit maximizing rate of output exists. Total revenue is therefore given by R(y)=p(y)y. Given that question uses an inverse demand function, the elasticity of demand, namely $\epsilon$(y), is defined as $\epsilon$(y)= 1/p'y p(y)/y.

Why is $\epsilon$(y)<0?

Prove that price is greater than marginal cost at the profit maximizing rate of output. Use the product rule when differentiating R(y)=p(y)y

Prove that the monopoly sets price in the elastic portion of its demand curve at the profit maximizing rate of output.

Answer 1

in monopoly equilibrium, MR = MC

& MC can't be negative