Price Discrimination

Price Discrimination

Uniform Pricing

• Occurs when a monopolist charges every consumer the same price.

• Some consumers pay less than they would have been willing to pay.

• Reservation price: the highest price a consumer would pay.

• The monopolist would like to charge each consumer their reservation price.

• Price Discrimination: Any scheme which attempts to sort consumers and then charge them differently.

Perfect Price Discrimination

• Occurs when a firm charges each consumer their reservation price (unless that price is below marginal cost (mc)).

• If a consumer purchases more than one item, then each item is sold for that item’s reservation price.

• Extracts all surplus from consumers.

Discrete Case Assumptions:

• Discrete goods (i.e. Cannot sell 2.5 items)

• Each consumer wants at most one item

• Reservation prices differ across consumers

• Inverse demand is found by ordering consumers by their reservation price

• $P(Q)$ is the Qth highest reservation price.

Discrete Example

• Consumer Value

  • Ann: 6

  • Bob: 5

  • Carol: 4

  • Dan: 3

  • Ellie: 2

  • Frank: 1

• MR = p because (e.g.) Ann and Bob are charged different prices.

• $mc = 3$

• Monopolist sells only to consumers with Value ≥ mc, and charges each consumer his/her value.

  • Ann pays 6, Bob pays 5, Carol pays 4, and Dan pays 3.

Continuous Demand

• Downward sloping demand

• Zero fixed cost

• Constant mc

• Per item profits are $P(q) − mc =⇒ π(q)$ is area between $P(q)$ and mc from 0 to q.

• Profits at q1 < q*$are$π(q1) = A

• Profits at q*$(where$P(q*) = mc$) are$π(q*) = A + B

• Profits at q2 > q*$are$ π(q2) = A + B − C =⇒$Profit is maximized at$q^$where$P(q*) = mc

Example - Perfect Price Discrimination

• $P(Q) = 100 − Q$

• $C(q) = \frac{1}{2} q^2$

• $mc(q) = q$

• $P(q^<em>) = mc(q^</em>) =⇒ 100 − q^* = q^<em>$ or $q^</em> = 50$

• $\bar{p} = P(q^*) = 100 − 50 = 50$

• Profits = A + B

• Area of a triangle is $\frac{1}{2} × 50 × 100 = 2500$

• Alternatively, R(q*) = A + (B + C)$and$π(q*) = R(q*) − C(q*)

• $= \frac{1}{2} (100 − \bar{p}) × q^* + \bar{p} × q^* − \frac{1}{2} (q^*)^2 = (100 − 50) × 50/2 + 50 × 50 − 50 × 50/2 = 2500$

Perfect Price Discrimination - Conclusions

• The monopolist extracts all surplus through pricing

• This leads the monopolist to maximize the total surplus

  • This is very bad for consumers

  • It is efficient

Pricing By Group

• Perfect Price Discrimination is generally impossible.

• However, it is often possible to charge different prices to different groups.

  • Pharmaceutical prices in NZ and USA

  • Senior discounts

  • Children’s prices

  • Group discounts

• In each of these cases, a firm:

  • can separate consumers into different markets

  • can (partially) sort market by demand conditions

  • can prevent (hamper) resale between markets

  • CANNOT differentiate between people in the same market.

• If we add (v) has market power, then we have the conditions for price discrimination (by group)

Two Market Model

• Our firm sells:

  • $q_1$ in market 1

  • $q_2$ in market 2

• It seeks to maximize $\pi(q<em>1, q</em>2) = R<em>1(q</em>1) + R<em>2(q</em>2) − C(q<em>1 + q</em>2)$

  • Where:

    • $R<em>1(q</em>1)$ is market 1 revenue

    • $R<em>2(q</em>2)$ is market 2 revenue

    • $C(q<em>1 + q</em>2)$ is joint production cost

Mathematical Reference

• If we wish to maximize $f (x<em>1, x</em>2)$ then we have two first order conditions:

  • $FOC(x<em>1): \frac{\partial f (x</em>1, x<em>2)}{\partial x</em>1} = 0$

  • $FOC(x<em>2): \frac{\partial f (x</em>1, x<em>2)}{\partial x</em>2} = 0$

• In ECON201 we ignore second order conditions for functions with two (or more) variables.

Two Market Maximization Problem

• The firm seeks to maximize $\pi(q<em>1, q</em>2) = R<em>1(q</em>1) + R<em>2(q</em>2) − C(q<em>1 + q</em>2)$

  • $FOC(q<em>1): R'(q</em>1) − C'(q<em>1 + q</em>2) = 0$

  • $FOC(q<em>2): R'(q</em>2) − C'(q<em>1 + q</em>2) = 0$

• Or $R'(q<em>1) = C'(q</em>1 + q<em>2) = R'(q</em>2)$

• Set MR in each market equal to common mc.

• Two stage problem

• Opportunity cost

Sample Problem

• $P<em>1(q</em>1) = 6 − \frac{q_1}{2}$

• $P<em>2(q</em>2) = 9 − q_2$

• $C(q<em>1 + q</em>2) = q<em>1 + q</em>2$

• Maximize profit if firm:

  • can price discriminate

  • cannot price discriminate

Part (a)

• Recall if $P = a − b ∗ q$, then $MR = a − 2b ∗ q$

• $MR<em>1 = R'</em>1(q<em>1) = 6 − q</em>1$

• $MR<em>2 = R'</em>2(q<em>2) = 9 − 2q</em>2$

• $mc = C'(q<em>1 + q</em>2) = 1$

• Our first oder condition was $R'<em>1(q</em>1) = C'(q<em>1 + q</em>2) = R'<em>2(q</em>2) =⇒ 6 − q<em>1 = 1 = 9 − 2q</em>2$

• $=⇒ q<em>1 = 5$ and $q</em>2 = 4$

• We see that $p<em>1 = 6 − 5/2 = \frac{7}{2} = 3.5$ and $p</em>2 = 9 − 4 = 5$

Part (b)

• Now firm must charge the same price in both markets.

• We model this situation by constructing a composite demand curve and requiring the firm to choose a single quantity $q = q<em>1 + q</em>2$.

• Even though we model the firm as choosing output, the composite demand is constructed by adding quantities purchased at each given price. That is $Q(p) = Q<em>1(p) + Q</em>2(p)$.

• This amounts to adding the curves together horizontally

Creating Composite Demand

• We start with Markets 1 and 2

• P > 6 =⇒ sales only in Market 2

• P < 6 sales in both markets

• Note flatness of composite demand

Composite Demand - math

• $P<em>1(q</em>1) = 6 − \frac{q<em>1}{2} =⇒ Q</em>1(p<em>1) = 12 − 2p</em>1$

• $P<em>2(q</em>2) = 9 − q<em>2 =⇒ Q</em>2(p<em>2) = 9 − p</em>2$

• $p ≥ 6 =⇒ Q(p) = 9 − p$, and $p ≤ 6 =⇒ Q(p) = 21 − 3p$

• But it is easier to work with inverse demand

• $q ≤ 3 =⇒ P(q) = 9 − q$

• $q ≥ 3 =⇒ P(q) = 7 − \frac{q}{3}$

Composite Demand - MR and mc

• $q ≤ 3 =⇒ P(q) = 9 − q$

• $q ≥ 3 =⇒ P(q) = 7 − \frac{q}{3}$

• $=⇒ MR = 9 − 2q$

• $=⇒ MR = 7 − \frac{2}{3} q$

• Note: slopes changes, and MR jumps at 3.

• If Marginal cost is ’High,’ then we use the q < 3 part of MR.

• If Marginal cost is ’Low,’ then we use the q > 3 part of MR.

• We have two possible solutions

• When we start, we don’t know which picture is relevant, so we must start as if this is the correct picture.

• First step is solve for the two values of q at which MR = mc.

• Sometimes, one of these values will not make sense, and we can throw it out.

• If that does not happen, then we must compare profits at both values

Solving the Problem

• $q ≤ 3 =⇒ P(q) = 9 − q$

• $q ≥ 3 =⇒ P(q) = 7 − \frac{q}{3}$

• $=⇒ MR = 9 − 2q$

• $=⇒ MR = 7 − \frac{2}{3} q$

• We need to solve for MR = mc = 1.

• q < 3 then we have $9 − 2q = 1$

• $=⇒ q = 4$ which violates q < 3 =⇒ this cannot be the answer.

• q > 3 then we have $7 − \frac{2}{3} q = 1$

• $=⇒ q = 9$ which is valid.

• $=⇒ q = 9$ is the solution.

• This implies that $p = 7 − \frac{9}{3} = 4$

Confirming the Solution

• We compare here the choices of $q = 9$ and $q = 4$

• The profits at $q = 9$ are $P(9) ∗ 9 − C(9) = 4 ∗ 9 − 9 ∗ 1 = 27$.

• What we mean by the profits at $q = 4$ is messy.

• We found $q = 4$ as the optimal choice from making sales only in Market 2, so that is how we will evaluate the profits.

• Under this approach, the profits from $q = 4$ are $P_2(4) ∗ 4 − C(4) = (9 − 4) ∗ 4 − 4 = 16$. This confirms that $q = 9$ is the solution.

Price Discrimination by Group - Conclusions

• Price discrimination and profits

• Price discrimination and cross market subsidization.

• Price discrimination and parallel trade

• Pharmaceuticals