Profit and Efficiency Maximization
under single vs discriminating pricing


Single-pricing firm
A single-pricing firm sells every unit at a uniform price. Its total revenue (TR) reaches a maximum when the elasticity of demand = |1| before price falls to zero. Because it must lower the uniform price for every unit just to sell just one more unit, MR < P.

Discriminating-pricing firm
If the price searcher can sell each unit of output according the buyer's willingness to pay (i.e., the reservation price), the total revenue generated will be much larger. This total revenue generated through perfect price discrimination can be called total willingness to pay (TWP)*.

TWP reaches its maximum when price charged is equal to zero.

Marginal willingness to pay (MWP) is measured by the slope of tangent to TWP. Since each buyer is charged its reservation price, MWP coincides with the demand curve itself.

In addition, since the discriminating-pricing firm does not have to lower its price for all units just to sell one more unit, its marginal revenue (MRdp) is equal to MWP. And because MWP is the reservation price paid by each buyer (Pdp), MRdp = Pdp.

When the single-pricing firm and the discriminating-pricing firm are faced with the same demand curve, P along the same demand curve for the same output is identical. But the single-pricing price (Psp) applies to all units sold. The price-discriminating price (Pdp) applies to only the additional unit sold.

Maximum-profit output
Just like the single-pricing firm, the discriminating-pricing firm produces at an output level where MR = MC to maximize profit. But it is MRdp = MC. Since MRdp = Pdp, so Pdp = MC at the maximum-profit output.

Because the single-pricing firm must charge the same price for all buyers, MRsp < Psp. So when MRsp = MC, Psp < MC.

Therefore, the maximum-profit output under discriminating pricing is always larger than the maximum-profit output under single pricing. And under discriminating pricing, marginal benefit to the consumer is always equal to marginal cost to the producer (MB = MC). In other words, maximizing profit will also maximize efficiency under discriminating pricing.

Distribution of economic surplus
Economic surplus is the difference between what buyers are willing to pay (TWP) and what it costs sellers to produce (TC) under both single pricing and discriminating pricing. Economic surplus is maximized when MB (marginal benefit) = MC (marginal cost). This condition translates into P = MC where P is taken to mean the marginal benefit of the last unit bought. Economic efficiency is maximized when economic surplus is maximized. Under single pricing, economic surplus is divided between consumer surplus (TWP - TR) and economic profit (TR - TC). When the single-pricing maximum-profit output produces positive profit, both consumer surplus and economic profit are positive.

Under perfect price discrimination, all economic surplus becomes economic profit to the producer. Consumer surplus is equal to zero. Because every unit is sold at the buyer's reservation price, maximum profit for the discriminating-pricing firm is much higher and output is much larger. In fact, output exceeds the ATCmin level. So the higher output under price discrimination comes at the expense of consumer surplus. But more units can also be bought at different affordable prices by consumers.

When profit is zero/negative under single pricing vs discriminating pricing
Even when the single-pricing firm is just earning zero profit, the discriminating-pricing firm can still manage to earn positive profit by charging each unit according to the buyer's reservation price.

And when the single-pricing firm can no longer afford to stay in business, the discriminating-pricing firm may still be able to produce at a profit by capturing the entire consumer surplus.

*Haveman, R. H. "Common Property, Congestion, and Environmental Pollution," Quarterly Journal of Economics. May 1971: 278-287.
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