microeconomics, homeworkI have seen the explanation that if the monopolist produced at the inelastic part of the demand curve, then (s)he could raise the price. This is the rationale:
So, total cost would necessarily go down, total revenue would necessarily go up, and as a result profit would necessarily go up.
I do not understand why optimal production occurs when demand elasticity is equal to -1 (unit elasticity) and there is no MC (marginal cost is 0), yet here it is just the elastic part of the demand curve.
Is this the reason: If we were operating at any point on the elastic part of the demand curve, then decreasing price would increase quantity demanded (by a higher proportion in relation to price), but would also drive up costs, and since we don’t know how the two balance out, we can’t tell which is preferable (unless we are given more data)?
Like everyone else, a monopolist maximizes profit by producing where marginal revenue equals marginal cost.
Unlike the case for most other people, the monopolists marginal revenue curve is NOT the same as the demand curve.
Instead, as the following question shows, the marginal curve slopes downward FASTER than the demand curve.
http://economics.stackexchange.com/questions/17/why-does-the-marginal-revenue-curve-slope-downward-at-twice-the-rate-of-demand-i
Most non-monopolists wouldn’t want to produce where the demand is inelastic. But that relation is stronger for a monopolist for reasons just described.
$MR = MC$ comes from profit maximization, although it also has a really nice intuition. In micro’s question with $MC = 0$, the profit maximization is equivalent to revenue maximization.
\begin{gather} R = P \cdot Q \\ \frac{dR}{dQ} = \frac{dP}{dQ} \cdot Q + 1 \cdot P = 0 \\ \frac{dP}{dQ} \cdot \frac{Q}{P} = -1 \\ \therefore \eta = -1 \end{gather}
So the monopolist with $MC = 0$ prefers to sell at unit elasticity.
By definition of elasticity, on the inelastic portion of the demand curve, charging a higher price increases revenue (units sold decline, but the percentage decline in units sold is lower than the percentage increase in the price).
Selling fewer units reduces total cost.
Profit is total revenue minus total cost.
We just showed that, for all (output, price) points on the inelastic portion of the demand curve, the monopolist can choose to charge a higher price (sell fewer units) and earn higher revenue while incurring lower cost. That is, cannot maximize profits on the inelastic portion of the demand curve, because, as we showed, it is possible to increase profits by charging a higher price (i.e. selling fewer units).
Therefore, the profit maximizing price-output combination can only be on the elastic portion of the demand curve where earning more revenue does imply incurring higher costs.
Of course, this argument is valid for any type of producer. By definition, competitive firms face a perfectly elastic demand curve (regardless of the shape of the market demand curve, so the point is moot. For, say, Cournot dupololists, you have to look at the demand curve faced by the firm, rather than the market demand curve.
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