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This question is bothering me a lot... I think the answer must be really simple, but I haven't been able to answer it.

In a monopoly, $p=\frac{c}{1-1/|\epsilon|}$, where $p$ is price of the good, and $c$ is the constant marginal cost, and $|\epsilon|$ is the absolute value of price elasticity of demand.

Why is this equality not valid for inelastic demand $|\epsilon|<1$ ? If $|\epsilon|<1$, then we get a negative price... Where in the deduction of this formula, from the profit maximization firm's problem, do we have to assume that $|\epsilon|>1$?

Edit: So for the FOC we need $|\epsilon|>1$, but where in the description of the profit maximization firm's problem do we need implicitely the condition?

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The expression $p=\frac{c}{1-1/\epsilon}$ is derived on the assumption that an interior solution is feasible, i.e. FOC is set to equal zero. But inelastic demand curve may not yield an interior solution at all, i.e. it may be impossible to set FOC equal to zero.

Perhaps this example will help. Consider a demand curve that features constant price elasticity: $q=p^{-\epsilon}$, or its inverse $p=q^{-1/\epsilon}$, where $\epsilon$ captures the price elasticity of demand. We set $\epsilon<1$, so that the demand is inelastic everywhere.

Assuming constant MC at $c$, let's maximize the profit of a monopolist facing this demand: \begin{equation} \max_q\; pq-cq=q^{1-1/\epsilon}-cq. \end{equation} The FOC is \begin{equation} \left(1-\frac1\epsilon\right)q^{-1/\epsilon}-c<0. \end{equation} Note that since $\epsilon<1$, the entire LHS is negative, and therefore we cannot have an interior solution.

On a more intuitive level, when a monopolist is operating at a point on the demand curve that is inelastic, it could always increase profit by raising price (or lowering quantity):

  • since demand is inelastic, 1% increase in price will cause a less than 1% reduction in quantity, so total revenue will go up;
  • with less quantity, total cost will also go down.

With more revenue and less cost, the monopolist will simply not be content at operating on the inelastic segment of the demand.

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