## Types of Price Discrimination

• Price discrimination is the act of selling the same product at two or more prices
• Intuitively, this means charging two different prices for the same good
• Mathematically, price discrimination is present if $\frac{P_1}{MC_1} \neq \frac{P_2}{MC_2},$ where the subscripts denote two separate markets, consumers, etc.
• Economic definition vs legal definition

### First-degree price discrimination (perfect price discrimination)

• Assume that a monopolist has perfect information about every consumer
• A firm prices the good exactly at each consumer’s willingness to pay • Under perfect competition,
• Consumer surplus = A+B+C
• Producer surplus = 0
• Under monopoly,
• Consumer surplus = A
• Producer surplus = B • Under perfect price discrimination,
• Consumer surplus = 0
• Producer surplus = E

### Second-degree price discrimination

• Assume that a monopolist has limited information about consumers about their reservation prices
• Second-degree price discrimination is often associated with quantity discounts
• All consumers are offered the same price schedule, and they self-select into different price categories

Suppose that Ralph Lauren produces a pair of designer jeans and sell them with quantity discounts.

• MC=AC=$20 • A consumer pays$80, $60,$40, and $20 for the first, the second, the third and the fourth jeans that they buy. • Suppose 10 consumers buy in each of the four price categories • 10 consumers bought just one pair of jeans • 10 consumers bought two pairs of jeans • 10 consumers bought three pairs of jeans • 10 consumers bought four pairs of jeans • This means that 100 pairs of jeans were sold in total, 40 of them sold at$80, 30 of them sold at $60, 20 of them sold at$40, and 10 of them sold at \$20.
• In this example,
• Consumer surplus will be the summation of the red shaded regions
• Producer surplus will be 4000 (A+B+C)
• However, depending on the price schedule set by the producer, the welfare implication is not clear
• The second degree price discrimination could improve or reduce welfare compared with a uniform price policy ### Third-degree price discrimination

• Suppose a monopolist sells its output in two separate markets with distinct demand curves
• This causes the monopolist to make two decisions in order to maximize profits:
1. How much total output to produce
2. How much of that output to sell in each market
• Once these decisions have been made, prices will be set in the two markets according to their respective demands.

Under these conditions, the firm’s total profit is given by $\Pi = P_1Q_1+P_2Q_2-TC(Q),$ where $$Q=Q_1+Q_2$$ To maximize profit, we are solving for the following optimization problem of $\max_{Q_1, Q_2} \Pi$ This implies that \begin{aligned} \frac{\partial \Pi}{\partial Q_1} &=MR_1 - MC =0 \\ \frac{\partial \Pi}{\partial Q_2} &=MR_2 - MC =0 \end{aligned} With these conditions, we have $MR_1= MR_2 = MC$ • We have three separate graphs: market 1, market 2, and the combination of both markets
• The marginal revenue curves are summed horizontally to get a MR curve in terms of $$(q_1 + q_2)$$
• It’s important to note that increasing q in either market 1 or 2 will cause an increase in the total MC. Hence, we will optimize in the combined market $$(Q^* = q_1^* + q_2^*)$$ and determine $$q_1$$ and $$q_2$$ after. • Q is determined by the point where $$MR_1 = MR_2 = MC$$
• Markets 1 and 2 produce output their respective marginal revenue is equal to the marginal cost.
• The market with less elastic demand is able to charge a higher price → price discrimination
• Recall, the more elastic the demand curve, the more consumers are responsive to a change in price

Claim: if a firm price discriminates between two markets, the market with more inelastic demand will necessarily be charged a higher price

• Based on the optimal strategy for profit maximizing monopolist, they produce the quantity where $$MC=MR$$ \begin{aligned} MR&=P\left(1-\frac{1}{|\eta|}\right) \end{aligned}
• Generalizing our optimality condition to two separate markets, a price-discriminating monopolist sets $$MC=MR_1=MR_2$$ \begin{aligned} P_1\left(1-\frac{1}{|\eta_1|}\right) = P_2\left(1-\frac{1}{|\eta_2|}\right) \end{aligned}

### Solving for Optimal Production

• A firm with market power in multiple markets where the costs of production are the same is maximizing profits when the following condition holds: $MR_1 = MR_2 = MC$

Suppose a firm holds a monopoly in two separate markets. The distinct inverse-demand functions for both markets are given below: \begin{aligned} P_1 &= 96-3q_1 \\ P_2 &= 64-q_2 \end{aligned}

Production for the two separate markets is done from a central location so marginal costs are the same for producing in both markets: $MC = 0.5(q_1 + q_2) -16$

Assuming the firm is profit-maximizer, what will the output and pricing combinations for the two distinct markets $$(q_1^*, q_2^*, P_1^*, P_2^*)$$?

Step 1: Find $$MR_1$$ and $$MR_2$$ \begin{aligned} MR_1 &= 96-2\cdot 3q_1 = 96-6q_1 \\ MR_2 &= 64-2\cdot1q_2 = 64-2q_2 \end{aligned}

Step 2: Set $$MR_1=MR_2$$ and solve for $$q_2$$ as an equation for $$q_1$$ (or vice versa)

\begin{aligned} MR_1 &=MR_2 \\ 96-6q_1 &= 64-2q_2\\ 2q_2 &= 6q_1 - 32 \\ q_2 &= 3q_1 -16 \end{aligned}

Step 3: Plug your equation for $$q_2$$ into the marginal cost function and simplify so it’s in terms of $$q_1$$

\begin{aligned} MC &= 0.5(q_1+ q_2)-16 \\ &= 0.5(q_1+3q_1 -16) -16 \\ &= 2q_1 -24 \end{aligned}

Step 4: Set $$MR_1=MC$$ and solve for $$q_1^*$$

\begin{aligned} MR_1 &= MC\\ 96-6q_1&=2q_1-24\\ 8q_1&=120\\ q_1^*&=15 \end{aligned}

Step 5: Plug $$q_1^*$$ into $$q_2$$ equation we found in step 2 to find $$q_2^*$$

\begin{aligned} q_2^\ast&=3q_1^\ast -16\\ &= 3\cdot 15 - 16 \\ &=29 \end{aligned}

Step 6: Plug $$q_1^*$$ and $$q_2^*$$ into inverse-demand functions to find the prices charged in each market

\begin{aligned} P_1^\ast&=96-3q_1^* \\ &= \51 \\ P_2^\ast&=64-q_2^* \\ &= \35 \\ \end{aligned}

\begin{aligned} (q_1^\ast, P_1^\ast) &= (15, \51) \\ (q_2^\ast, P_2^\ast) &= (29, \35) \\ \end{aligned}