EconS 425 - Second-Degree Price Discrimination Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 46
Introduction Today, we ll cover our last type of price discrimination: second-degree price discrimination. Second-degree price discrimination can be implemented when a rm knows that subgroups exist within their market, but cannot identify which subgroup any individual belongs to. Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 46
Second-Degree Price Discrimination Second-degree price discrimination is much more subtle than the others we have studied thus far. Before, under rst-degree price discrimination, the rm had perfect information and could charge each person an individual price. Likewise, under third-degree price discrimination, the rm wasn t able to identify every individual consumer, but it could identify and sort which subgroups an individual belonged to. Eric Dunaway (WSU) EconS 425 Industrial Organization 3 / 46
Second-Degree Price Discrimination The di erences between consumers in second-degree price discrimination is not observable to the rm. These di erences are things like preferences and tastes. Employing techniques that we have learned thus far in rst and third-degree price discrimination won t be as e ective. We have no way of knowing if a consumer is lying about which group they belong to in order to get a better price. Eric Dunaway (WSU) EconS 425 Industrial Organization 4 / 46
Second-Degree Price Discrimination We can use some of the techniques from rst-degree price discrimination as a starting point to develop new pricing strategies. Namely, two-part and block pricing. Unfortunately, we will never reach the e ciency levels we saw under those pricing schemes. The monopolist will lose surplus to some of the consumers in order to "convince" them to reveal their true preferences. We may also have some quantity left out of the market, leading to deadweight loss. Eric Dunaway (WSU) EconS 425 Industrial Organization 5 / 46
Second-Degree Price Discrimination Let s build a model! Let θ i represent consumer i s inherent preferences for the rm s output, q. This information is private to the consumer. The total value that consumer i receives from consuming q units of the rm s output is given by v(θ i, q) where the following relationships hold, v(θ i, 0) = 0 v (θ i,q) q > 0 2 v (θ i,q) q 2 < 0 v (θ i,q) θ i > 0 and v(θ 2, q) > v(θ 1, q) if θ 2 > θ 1. Eric Dunaway (WSU) EconS 425 Industrial Organization 6 / 46
Second-Degree Price Discrimination The total surplus that consumer i receives is v(θ i, q) pq and if we calculate a rst-order condition with respect to quantity, we obtain v(θ i, q) q p = v(θ i, q) q p = 0 = v q (θ i, q) which is an expression for the inverse demand function for a consumer with preference θ i. p i (q) = v q (θ i, q) Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 46
Second-Degree Price Discrimination We can plot the total surplus that consumer i receives when purchasing q units of the good for the total expenditure of T. This allows us to develop indi erence curves which tell us all of the di erent combinations of q and T yield the same amount of surplus for consumer i. An important thing to remember is that these indi erence curves are not like the ones we are used to seeing them. Normally, we want more of everything. In this case, the consumer wants more quantity, q, but wants to pay less for it, T. Thus, utility increases as we move down and towards the right. Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 46
Second-Degree Price Discrimination T q Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 46
Second-Degree Price Discrimination T S 1 q Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 46
Second-Degree Price Discrimination T S 1 S 2 q Eric Dunaway (WSU) EconS 425 Industrial Organization 11 / 46
Second-Degree Price Discrimination If the monopolist could only charge one price to the market, we would follow the same steps as we have done before to derive a uniform price. Let N be the total number of consumers in the market with N 1 consumers having a preference level of θ 1 for the product while N 2 consumers have a preference level of θ 2. N 1 + N 2 = N and θ 2 > θ 1 > c, where c is the constant marginal cost of production for the rm. The rm knows the values of N 1, N 2, θ 1 and θ 2, but is unable to observe any individual i s value for θ i. Let the valuation for consumer i be expressed as follows, v(θ i, q) = θ i q q 2 2 for 0 q θ i Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 46
Second-Degree Price Discrimination v(θ i, q) = θ i q q 2 2 To gure out a uniform price, we need an aggregate demand function. Fortunately, we know the relationship between the valuation function and the inverse demand function, p(q) = v(θ i, q) q = θ i q and if we solve this expression for q, we have the demand function, q(p) = θ i p From here, we can derive an aggregate demand function. Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 46
Second-Degree Price Discrimination q(p) = θ i p Now, we simply add up all of the demand functions from all N consumers, Q(p) = N 1 i=1 organizing terms, (θ 1 p) + N 2 i=1 (θ 2 p) = N 1 (θ 1 p) + N 2 (θ 2 p) Q(p) = N 1 θ 1 + N 2 θ 2 Np Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 46
Second-Degree Price Discrimination Q(p) = N 1 θ 1 + N 2 θ 2 Np Let s organize this a little bit. Factoring out N = N 1 + N 2 from every term on the right side, N1 Q(p) = N θ 1 + N 2 θ 2 p N 1 + N 2 N 1 + N 2 and let γ N 1 has preference θ 1, N 1 +N 2 represent the proportion of the population that Q(p) = N(γθ 1 + (1 γ)θ 2 p) Lastly, let θ m γθ 1 + (1 preference level, giving us γ)θ 2, which is the weighted mean Q(p) = N(θ m p) Eric Dunaway (WSU) EconS 425 Industrial Organization 15 / 46
Second-Degree Price Discrimination Setting up the monopolist s pro t maximization problem, max p pq(p) cq(p) = max p pn(θ m p) cn(θ m p) with rst-order conditon, Rearranging terms, N(θ m p) pn + cn = 0 2pN = θ m N + cn and solving for our equilibrium price, p = θ m + c 2 > c Thus, the rm is extracting some economic pro ts. Eric Dunaway (WSU) EconS 425 Industrial Organization 16 / 46
Second-Degree Price Discrimination p = θ m + c 2 We can obtain our market quantity by plugging price back into the aggregate demand function, Q = N(θ m p θm c ) = N 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 46
Second-Degree Price Discrimination p = θ m + c 2 Q = N θm 2 c Interestingly, our γ parameter lets us see how the price and market quantity change as a larger share of the market has the lower preference for the good. p γ Q γ = p θ m θ m γ = 1 2 (θ 1 θ 2 ) < 0 = Q θ m θ m γ = N 2 (θ 1 θ 2 ) < 0 so as γ increases, the uniform price must decrease as the rm caters to the segment with lower preferences, and the market quantity decreases as the overall preference for the good declines. Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 46
Second-Degree Price Discrimination Lastly, the surplus for consumer i is, S i (p) = v(θ i, q) pq = v(θ i, q(p)) pq(p) (θ i p) 2 = θ i (θ i p) p (θ i p) 2 = (θ i p) 2 (θ i p) 2 = (θ i p) 2 2 2 Notice that for any given price, the surplus for the consumer with higher valuation is always greater than the surplus for the consumer with lower valuation, i.e., (this is a general result) S 2 (p) > S 1 (p) Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 46
Two-Part Tari Let s take a look at what would happen if we tried to implement two-part pricing for our model. Recall that under two-part pricing, we charge a xed access fee, A, and then a price per each unit purchased, p. Under rst-degree price discrimination, we would set the access fee equal to the consumer surplus under perfect competition and set the unit price equal to marginal cost. Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 46
Two-Part Tari If we leave price set at marginal cost, we have two choices for the access fee. We could target the higher valuation consumer (θ 2 ) and set it equal to their surplus, S 2 (c). The price would be too high for the lower valuation consumer (θ 1 ), however, and they would not purchase. We could target the lower valuation consumer(θ 1 ) and set it equal to their surplus, S 1 (c). Everyone would enter the market, but the higher valuation consumer(θ 2 ) would retain signi cant consumer surplus. Can we do better? Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 46
Two-Part Tari Starting with our access fee equal to the surplus of the lower valuation consumer, S 1 (c) and the per unit cost equal to marginal cost, we have two new options. We could raise the access fee and lower the price per unit to try and obtain more surplus from the high type, while not losing any surplus from the low type. However, now we would be running at a loss for each unit sold (usually a bad thing). Or, we could lower the access fee and raise the price per unit, keeping it such that it claims all the surplus from the low type, while taking advantage of the fact that the high type purchases a higher quantity. The second option looks appealing. Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 46
Two-Part Tari If we set the unit price higher than marginal cost, c > 0, the quantity that the low type consumer purchases, q(p) will decrease. We can still calculate their remaining surplus using our surplus expression, S 1 (p) = (θ 1 p) 2 2 and that remaining surplus should be our access fee, i.e., A = S 1 (p). Eric Dunaway (WSU) EconS 425 Industrial Organization 23 / 46
Two-Part Tari With this information, we can calculate our new unit price using the pro t maximization problem, max p = max p (N 1 + N 2 )(A + pq(p) cq(p)) " # (θ 1 p) 2 N + (p c)(θ m p) 2 and taking a rst-order condition with respect to price, N [ (θ 1 p) + θ m p (p c)] = 0 and, solving for p, we have our unit price, p = θ m θ 1 + c > c Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 46
Two-Part Tari Lastly, to calculate our access fee, A = S 1 (p ) = (θ 1 (θ m θ 1 + c)) 2 and we have our complete two-part tari, 2 = (2θ 1 θ m c) 2 2 T = A + p q = (2θ 1 θ m c) 2 2 + (θ m θ 1 + c)q Eric Dunaway (WSU) EconS 425 Industrial Organization 25 / 46
Two-Part Tari T S 1 (c ) q 1 q 2 q Eric Dunaway (WSU) EconS 425 Industrial Organization 26 / 46
Two-Part Tari T S 1 (c ) q 1 q 2 q Eric Dunaway (WSU) EconS 425 Industrial Organization 27 / 46
Two-Part Tari T S 1 (c ) S 1 (p * ) q 1 q 2 q Eric Dunaway (WSU) EconS 425 Industrial Organization 28 / 46
Two-Part Tari As we can see, moving from two-part pricing to a two-part tari, the low type consumer still has all of their surplus captured by the monopolist. For the high type consumer, they move to a lower surplus level, as more of their surplus becomes captured. In fact, the high type consumer is actually at a higher surplus level than they would be at if they pretended to be a low type consumer. Eric Dunaway (WSU) EconS 425 Industrial Organization 29 / 46
Two-Part Tari T S 1 (p * ) q 1 q 2 q Eric Dunaway (WSU) EconS 425 Industrial Organization 30 / 46
Two-Part Tari T T 3 T 2 S 1 (p * ) q 1 q 2 q Eric Dunaway (WSU) EconS 425 Industrial Organization 31 / 46
Two-Part Tari Since the high type consumer purchases q 2 units of the good for a total cost of T 2, the fact that they would be willing to purchase the same q 2 units at a total cost of T 3 tells us that our two-part tari is not pro t maximizing. At a total cost of T 3, the high type consumer is as well of as if they pretended to be a low type consumer (which implies that they still have positive surplus). Unfortunately, this is the best we can do under a two-part tari. Using an approach based on block pricing, we may be able to do better, though. Eric Dunaway (WSU) EconS 425 Industrial Organization 32 / 46
Menu Pricing Remember from block pricing, we set a quantity, q and a corresponding cost for that quantity, T for each individual consumer. We can do something similar when we can t observe the types of consumers we have, but we might run into the problem that one type of consumer will pretend to be the other. If we design the "menu" of quantities and total costs well, we can set it such that each consumer will self select into the bundle designed for them. Let s use the same setting with our two types of consumer to see what happens. Eric Dunaway (WSU) EconS 425 Industrial Organization 33 / 46
Menu Pricing Using menu pricing, the rm s pro t maximization problem is, max q 1,q 2,T 1,T 2 N 1 (T 1 cq 1 ) + N 2 (T 2 cq 2 ) = max q 1,q 2,T 1,T 2 N [γ(t 1 cq 1 ) + (1 γ)(t 2 cq 2 )] which is subject to four separate constraints. Eric Dunaway (WSU) EconS 425 Industrial Organization 34 / 46
Menu Pricing First, we have two participation constraints, v(θ 1, q 1 ) T 1 {z } Consumer 1 s surplus v(θ 2, q 2 ) T 2 {z } Consumer 2 s surplus 0 (PC 1 ) 0 (PC 2 ) These two constraints guarantee that both types of consumer actually want to buy the packages that we design for them. If either of these constraints don t hold, it would be better for that type of consumer to simply not buy anything. Eric Dunaway (WSU) EconS 425 Industrial Organization 35 / 46
Menu Pricing Second, we have two incentive compatibility constraints, v(θ 1, q 1 ) T 1 {z } Consumer 1 s surplus if they consume their bundle v(θ 2, q 2 ) T 2 {z } Consumer 2 s surplus if they consume their bundle v(θ 1, q 2 ) T 2 {z } Consumer 1 s surplus if they pretended to be consumer 2 v(θ 2, q 1 ) T 1 {z } Consumer 2 s surplus if they pretended to be consumer 1 (IC 1 ) (IC 2 ) These two constraints guarantee that the bundles we design are the best choice for both consumers. If either of these constraints don t hold, it would be better for one consumer to pretend that they are the other and consume their bundle instead. Eric Dunaway (WSU) EconS 425 Industrial Organization 36 / 46
Menu Pricing v(θ 1, q 1 ) T 1 0 (PC 1 ) v(θ 2, q 2 ) T 2 0 (PC 2 ) v(θ 1, q 1 ) T 1 v(θ 1, q 2 ) T 2 (IC 1 ) v(θ 2, q 2 ) T 2 v(θ 2, q 1 ) T 1 (IC 2 ) The nice thing is that at most, two of these constraints will bind, i.e., are equal. If a constraint doesn t bind, we don t need to worry about it at all in our problem. Almost every time, it s one participation constraint, and the opposite incentive compatibility constraint. Our challenge is guring out which ones do bind, which requires some logic. Eric Dunaway (WSU) EconS 425 Industrial Organization 37 / 46
Menu Pricing I can prove that PC 2 doesn t bind by contradiction. First, let s assume it does, v(θ 2, q 2 ) T 2 = 0 If I substitute this into IC 2, v(θ 2, q 2 ) T 2 v(θ 2, q 1 ) T 1 0 v(θ 2, q 1 ) T 1 and remember that v(θ 2, q 1 ) > v(θ 1, q 1 ) from our original assumptions. I can write which is a contradiction of PC 1. 0 v(θ 2, q 1 ) T 1 > v(θ 1, q 1 ) T 1 Eric Dunaway (WSU) EconS 425 Industrial Organization 38 / 46
Menu Pricing Intuitively, if I design my menu such that we just barely get the high type consumer into the market, and if we re also going to make sure that our menu is incentive compatible (each consumer wants the package designed for them), then no matter what bundle I design for the low type consumer, they would be better o not buying at all. Therefore, we can t target the high type consumer and we need to focus on the low type. We can say that PC 1 will bind in equilibrium. Interestingly, if you try this proof with PC 1, you ll nd that PC 2 is always satis ed. Eric Dunaway (WSU) EconS 425 Industrial Organization 39 / 46
Menu Pricing Figuring out which incentive compatibility constraint binds is much more challenging, and actually can t be done before we solve the problem. To truly gure it out, we have to solve the problem both ways and see which way gives the highest pro ts for the monopolist. Almost every time (and every time in this class), the incentive compatibility constraint that binds is the opposite of the participation constraint. Thus, since PC 1 binds, IC 2 must bind. Eric Dunaway (WSU) EconS 425 Industrial Organization 40 / 46
Menu Pricing Since PC 2 and IC 1 don t bind in this case, we can get rid of them, leaving us, v(θ 1, q 1 ) T 1 = 0 (PC 1 ) v(θ 2, q 2 ) T 2 = v(θ 2, q 1 ) T 1 (IC 2 ) and we can directly solve for T 1 and T 2 from these expressions, T 1 = v(θ 1, q 1 ) T 2 = v(θ 2, q 2 ) v(θ 2, q 1 ) + v(θ 1, q 1 ) {z } T 1 Now we re ready to go back to our maximization problem! Eric Dunaway (WSU) EconS 425 Industrial Organization 41 / 46
Menu Pricing max N [γ(t 1 cq 1 ) + (1 γ)(t 2 cq 2 )] q 1,q 2,T 1,T 2 We can substitute in those values of T 1 and T 2 to get rid of those constraints alltogether, γ(v(θ max N 1, q 1 ) cq 1 ) q 1,q 2 +(1 γ)(v(θ 2, q 2 ) v(θ 2, q 1 ) + v(θ 1, q 1 ) cq 2 ) and from here, we can obtain rst-order conditions with respect to q 1 and q 2, π γ(v = N q1 (θ 1, q 1 ) c) = 0 q 1 +(1 γ)( v q1 (θ 2, q 1 ) + v q1 (θ 1, q 1 ) π = N(1 γ)(v q2 (θ 2, q 2 ) c) = 0 q 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 42 / 46
Menu Pricing From these two rst-order conditions, two relationships can be derived, v q1 (θ 1, q 1 ) = c + 1 γ γ [v q 1 (θ 2, q 1 ) v q1 (θ 1, q 1 )] v q2 (θ 2, q 2 ) = c These two relationships have very important interpretations, The rst relationship indicates that in equilibrium, the marginal gain to the low type consumer of consuming more, v q1 (θ 1, q1 ), is higher than marginal cost. This implies that the menu combination we design for the low type consumer will not be e cient. The second relationship is the opposite; since the marginal gain to the high type consumer in equilibrium, v q2 (θ 2, q2 ), is equal to marginal cost, they will consume the e cient quantity. Eric Dunaway (WSU) EconS 425 Industrial Organization 43 / 46
Summary While we cannot tell which consumers are in which groups in second-degree price discrimination, we can still design pricing schemes to extract surplus from those groups. Two-part tari s, while (relatively) simple to calculate, are not pro t maximizing. Eric Dunaway (WSU) EconS 425 Industrial Organization 44 / 46
Next Time Finishing up second-degree price discrimination. Quiz! See the results of our menu pricing as well as welfare e ects. Eric Dunaway (WSU) EconS 425 Industrial Organization 45 / 46
Homework 2-5 Using the functional form from our exercises today, v(θ i, q i ) = θ i q i Complete the menu pricing example by solving for values of T1, T 2, q1, and q 2. These values will be functions of γ and c. The book has parts of this solution already, but I am looking for your calculations. q 2 i 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 46 / 46