# Isoelastic demands and constant markup

Why if the monopolists face isoelastic demands, profit maximization(for monopolists) implies that each of them sets a constant markup over the marginal cost?

This is a common situation when you deal with an endogenous R&D growth model. You have a unique homogeneous final good produced competitively. The production of this final good is described by a CES production function or simply a Cobb-Douglas aggregator. This homogenous final good is produced by a unit continuum of intermediates (or a range of intermediates evolving endogenously) produced in monopolistic competition. Then, the demands faced by the monopolists (producing each of them an intermediate) are isoelastic. Why does it imply constant mark up over the MC ?

• Please provide some more details Aug 31, 2022 at 8:16

Let's go for steps:

Profit function : $$\Pi(Q) = P(Q)Q -C(Q)$$

Where $$\Pi(Q)$$ are profits and $$P(Q)$$ is the inverse demand function, i.e., the price at which $$Q$$ can be sold given the existing demand, and $$C(Q)$$ is the cost function.

Then, profit max implies: $$\Pi'(Q) = P(Q) + \frac{dP}{dQ}Q - C'(Q)=0$$

Note that $$P(Q) + \frac{dP}{dQ}Q$$ is marginal revenue(MR), and

$$C'(Q)=0$$ is marginal cost.

Profit maximization implies MR=MC

Then, MR can also be written as $$P(1+\frac{dP}{dQ} \frac{Q}{P}$$)

but the second term in parenthesis is just the reciprocal of the price elasticity of demand ( call it $$\epsilon_p$$). You can rewrite MR as:

$$P(1+ \frac{1}{\epsilon_p})=P(\frac{1+\epsilon_p}{\epsilon_p})$$

Then, equating MR to MC and solving for P:

$$P=(\frac{\epsilon_p}{1+\epsilon_p})MC$$

Then, letting $$\eta$$ be the reciprocal of the price elasticity of demand:

$$P=(\frac{1}{1+\eta})MC$$

Thus a firm with market power chooses the output quantity at which the corresponding price satisfies this rule (markup rule).

Now, if a monopolist faces isoelastic demand, this means that the price elasticity of demand is constant and also $$\eta$$ is constant (because it is the reciprocal of the price elasticity of demand). Indeed, the term in parenthesis,$$\frac{1}{1+\eta}$$, is simply the markup (and it is constant over the MC).