Question

Q1. A monopolist firm produces a special soft drink. There are 20 H-type and 40 L-type consumers with the following demand functions: PH = 16 – QH PL = 12 – QL (Note that these are individual-level demand functions.) The marginal cost of producing 1 unit of product is $4. The firm knows that there are 2 types of consumers but cannot identify who is who. The firm can sell its products either in single units, or can pack them (4-unit pack, 6-unit pack, etc.).

a) Write the profit function for the monopolist firm (with restrictions, if any).

b) What is the best pricing and packing policy for the monopolist firm?

Answer 1

A monopolist wants to set price such that at that price quantity demanded is max; conversely that for a given quantity the price charged is max.

Because the two demand functions (and thus the marginal revenue functions also) are essentially parallel to each other, at any quantity there are two prices - let's say Lower and Higher. This divides the (positive) number line into three parts: p <= lower, lower <=high, p > higher. At the first price, both markets are captured. In the intermediate range, only H-market is captured. In the third region, sales are zero.

Below is the profit function. But the additional constraint is that because the monopolist can't differentiate, they can't pursue price distribution. So although marginal revenues are different across the markets, a single price is set so as to capture the greatest surplus in both markets.

A Lagrangian is set up for constrained optimization. It could be said that in the abstract the Lagrangian is really the profit function being maximized. Better just call it the firm's objective function.

Interesting that so little output (3 and 7) when the market sizes are more than twice (20 and 40) in magnitudes. The excess demand persists because (in the absence of price discrimination) supplying additional units would bring down prices for ALL units sold in that market and thus the firms profit.

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If only even-quantity packs are permitted (e.g. NOT packs of 3, 5 etc) then it's best to only sell in packs of 4.