EconS 503 Advanced Microeconomics II 1 Adverse Selection Handout on Two-part tariffs (Second-degree price discrimination)

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1 EconS 503 Advanced Microeconomics II 1 Adverse Selection Handout on Two-part tariffs (Second-degree price discrimination) 1. Introduction Consider a setting where an uninformed firm is attempting to sell an item to a privately informed customer. The firm s profit function is FF cccc, where cc > 0 represents the firm s marginal costs, and F is the fee paid from the customer to the firm in exchange for q units of the good (price for the package of units, rather than a unit price). The customer s utility function is uu(qq, TT, θθ) = θθ vv(qq) FF, where uu > 0 and uu < 0. Parameter θθ is privately observed by the consumer, and takes on either θθ LL with probability ββ or θθ HH with probability 1 ββ, where θθ HH > θθ LL. 2. Complete Information [2 nd Stage] For a given pair of fee TT ii and quantity qq ii, (TT ii, qq ii ), consumers with valuation θθ ii purchase the good if and only if θθ ii vv(qq ii ) TT ii 0 [1 st Stage] Observing θθ ii (as we are in the complete-information version) and anticipating the buyers decision rule in the second stage, θθ ii vv(qq ii ) TT ii 0, the firm solves the PMP TT ii,qq ii TT ii ccqq ii subject to θθ ii uu(qq ii ) TT ii 0 PP. CC. The participation constraint (P.C.) must bind. Otherwise TT ii can be further increased, thus increasing profits. Hence, θθ ii vv(qq ii ) = TT ii, which simplifies the above problem to the following unconstrained imization problem qq ii θθ ii vv(qq ii ) ccqq ii Taking F.O.C with respect to qq ii, θθ ii vv (qq ii ) cc 0 (= 0 in interior solutions) MV = MC Hence, under complete information, qq ii is increased until the point in which the consumer s marginal utility of additional units coincides with the firm s marginal cost. As we next show, when the firm is uninformed about the customer s type, this result doesn t necesarily arise. 1 Felix Munoz-Garcia, Associate Professor, School of Economic Sciences, Washington State University, Pullman, WA , fmunoz@wsu.edu. 1

2 3. Incomplete Information The firm cannot observe the realization of θθ. The firm could offer contracts of the form (TT(qq), qq), with function TT(qq) being as general as you can imagine. For simplicity, let s consider three types of contracts: Linear pricing: TT(qq) = qq customers pay for every unit they buy. Nonlinear pricing (single two-part tariff for all types of customers) Nonlinear pricing (two two-part tariffs one for each type of customer) 3.1. Linear pricing, TT(qq) = qq [2 nd Stage] Every customer with type θθ ii pays a price p per unit of q purchased, thus obtaining a utility θθ ii uu(qq) for all ii = {HH, LL} In order to imize his utility (for every given p), he increases q until θθ ii uu 1 (qq) = Solving for q, we find θθ ii Walrasian demand qq ii = DD ii () Hence, θθ ii customer s utility is θθ ii uu(dd ii () DD ii () qq ii [1 st Stage] By backward induction, the monopolist anticipates the demand function DD ii () for θθ ii type buyer. Hence, the firm imizes expected profits: qq ii ( cc) [ββ DD LL () + (1 ββ) DD HH ()] Let DD() ββ DD LL () + (1 ββ) DD HH () denote the expected demand, which helps us simplify the above program to ( cc) DD() Taking FOC with respect to p yields Solving for p, we obtain a linear price of DD() + DD () cc = 0 2

3 LLLL = cc DD(LLLL ) DD ( LLLL ) where LLLL > cc if DD ( LLLL ) < 0. Depending on the parameter values, it might be profitable for the seller to only serve θθ HH buyers Single two-part tariff The firm sets a single two-part tariff (FF, ) to both types of customers, and each type of buyer decides to take it or leave it. Fee. From the UMP of each type of consumer, we obtained FOC of θθ ii vv (qq) =. Plotting them on the same figure, we find: $ p θ L v (q) θ H v (q) q L = D L (p) q H = D H (p) q where functions θθ ii vv (qq) are decreasing in qq by the concavity of vv( ), i.e., vv < 0 for all qq. Hence, DD HH () > DD LL (), thus implying that net surpluses, SS ii (), satisfy That is, SS HH () > SS LL (). SS HH () = θθ HH vv[dd HH ()] DD HH () > θθ LL vv[dd LL ()] DD LL () = SS LL () If the firm seeks the participation of both types of customers, we need the fee to satisfy More explicitly: FF SS LL () < SS HH () In the second stage, every customer with type θθ ii purchases the good if and only if FF SS ii (). In the first stage, the firm anticipates the customers decision rule of FF SS ii (), and chooses the single two part tariff that imizes profits. 3

4 Mathematically, (FF,) ββ[ff + ( cc) DD LL ()] + (1 ββ)[ff + ( cc) DD HH ()] = FF + ( cc)[ββ DD LL () + (1 ββ) DD HH ()] DD(), i.e., expected demand subject to FF SS ii () for all ii = {HH, LL} However, the seller can increase FF until FF = SS LL (). Raising it any further would lead the low-type customers to reject the purchase. Plugging FF = SS LL () into the above problem helps us obtain an unconstrained PMP (with only one choice variable, ), as follows SS LL () + ( cc) DD() Taking first-order conditions with respect to yields SS LL () + DD() + ( cc)dd () = 0 Solving for and rearranging, we obtain a price of the single two part tariff, SSSSSSSS, of SSSSSSSS = + cc DD() DD () LLLL, price under linear pricing Where the last term is positive since SS LL () < 0 and DD () < SS LL () DD () + Remark: SS ii () can be found by alying the Envelope Theorem on SS ii () = θθ ii vv[dd ii ()] DD ii () In particular, second-order effects are absent, so that DD ii () is unaffected by a price change. As a consequence Hence, prices in each setting are ranked as follows: The firm then sets a single two-part tariff SS ii () = 0 DD ii () = DD ii () < 0 SSSSSSSS > LLLL > cc (price under perfect competition) (FF, ) = (SS LL ( SSSSSSSS ), SSSSSSSS ) Practice: Considering a demand function DD ii () = θθ ii, where θθ ii = {1,2} and ββ = 1, find the profitimizing two-part 2 tariff.

5 In addition, qq HH = DD HH ( SSSSSSSS ) > DD LL ( SSSSSSSS ) = qq LL. We can depict this two-part tariff in the (FF, )- quadrant, as follows. F S L (p STPT ) + p STPT p STPT F = S L (p STPT ) q L q H q Graphical representation of the indifference curves using the same (FF, )-quadrant: F θ i -type indifference curve Same utility from: -Low F and low q -High F and high q q F IC i ICi Increasing utility 5 q

6 We can now superimpose IIII on top of the two-part tariff, obtaining: F A H IC H S L (p STPT ) + p STPT IC H A L IC L q L q H q Some points about equilibrium behavior in the case of a single two-part tariff that we just analyzed: Customer θθ HH is better off at AA HH than at AA LL Customer θθ LL is better off at AA LL than at AA HH Motivation to move to other contracts (in particular, toward two two-part tariffs, which we analyze in the following section): The seller could do better if he sets a contract that yields point AA HH to θθ HH -buyer (since this buyer is indifferent about accepting the contract meant for him or that of the θθ LL -customer) Several (or menu) two-part tariffs Consider a setting where the monopolist cannot observe the type of each consumer. Rather than offering a uniform price for all types of customers, or a single two-part tariff to all types of customers, the monopolist can design a menu of two-part tariffs, (FF LL, qq LL ) and (FF HH, qq HH ), with the property that the customer with type ii = {LL, HH} has the incentives to self-select the two-part tariff (FF ii, qq ii ) meant for him. In this setting, the monopolist must guarantee that: Both types of customers are willing to participate (i.e., the two-part tariff meant for each type of customer provides him with a weakly positive utility level), and Both types of customers do not have incentives to choose the two-part tariff meant for the other type of customer, that is, type ii customer prefers (FF ii, qq ii ) over (FF jj, qq jj ) where jj ii. 6

7 For compactness, the literature refers to the former conditions as participation constraints, as they guarantee the participation of all types of customers; whereas the latter conditions are referred to as incentive compatibility conditions. In particular, the participation constraints in this context are θθ LL uu(qq LL ) FF LL 0 θθ HH uu(qq HH ) FF HH 0, PPPP LL PPPP HH while the incentive compatibility conditions are θθ LL uu(qq LL ) FF LL θθ LL uu(qq HH ) FF HH θθ HH uu(qq HH ) FF HH θθ HH uu(qq LL ) FF LL IIII LL IIII HH We can rearrange the above four inequalities and insert them as constraints into the monopolist s profit imization problem, as follows: subject to [FF HH ccqq HH ] + (1 )[FF LL ccqq LL ] FF LL,qq LL,FF HH,qq HH θθ LL uu(qq LL ) FF LL θθ HH uu(qq HH ) FF HH θθ LL [uu(qq LL ) uu(qq HH )] + FF HH FF LL θθ HH [uu(qq HH ) uu(qq LL )] + FF LL FF HH Since both PPPP HH and IIII HH are now expressed in terms of the fee FF HH, we can easily see that the monopolist increases FF HH until such fee coincides with the lowest of θθ HH uu(qq HH ) and θθ HH [uu(qq HH ) uu(qq LL )] + FF LL, as depicted in figure 1, for all ii = {LL, HH}. Otherwise, one (or both) constraints will be violated, leading the high-demand customer to not participate (and/or select the two-part tariff meant for the low-demand customer). We examine this result more closely in the next discussion. Maximal F i that achives participation and self-selection PC i is binding θ i u(q i ) θ i [u(q i )- u(q j )] + F j F i Maximal F i that achives participation and self-selection IC i is binding θ i [u(q i )- u(q j )] + F j θ i u(q i ) F i 7

8 Figure 1 PC condition binds (uer panel) and IC condition binds (lower panel) High-demand customer. Let us first focus on the high-demand consumer and show that IIII HH is binding, (the lower panel of figure 1 arises for this type of customer). Proof. An indirect way to show that IIII HH binds, i.e., FF HH = θθ HH [uu(qq HH ) uu(qq LL )] + FF LL, is to demonstrate that FF HH < θθ HH uu(qq HH ) (i.e., as depicted be in the lower panel of Figure 1). By contradiction, consider that FF HH = θθ HH uu(qq HH ). If this condition holds, then IIII HH can be rewritten as FF HH θθ HH uu(qq LL ) + FF LL FF HH, which simplifies to FF LL θθ HH uu(qq LL ) In addition, we can combine this result with the property that θθ HH > θθ LL to obtain FF LL θθ HH uu(qq LL ) > θθ LL uu(qq LL ) That is, FF LL > θθ LL uu(qq LL ). This finding, however, violates the participation constraint of the low-demand customer, PPPP LL, indicating that we have reached a contradiction and, therefore, FF HH < θθ HH uu(qq HH ) (i.e., PPPP HH is not binding). Thus, IIII HH is binding but PPPP HH is not, confirming that for the high-demand customer the lower panel of Figure 1 alies (i.e., FF HH = θθ HH [uu(qq HH ) uu(qq LL )] + FF LL ). Q.E.D. Low-demand customer. Let us now use a similar aroach to show that the top panel of Figure 1 arises for the low-demand customer (i.e., PPPP LL binds since FF LL = θθ LL uu(qq LL )). Proof. Similarly as for high-demand customers, we can prove this result by instead showing that FF LL < θθ LL [uu(qq LL ) uu(qq HH )] + FF HH holds. Proving this result by contradiction, assume that FF LL = θθ LL [uu(qq LL ) uu(qq HH )] + FF HH. Plugging this expression into IIII HH (which binds, as shown in our discussion of the highdemand customer), we obtain This expression simplifies to θθ HH [uu(qq HH ) uu(qq LL )] + θθ LL [uu(qq LL ) uu(qq HH )] + FF HH = FF HH, θθ HH [uu(qq HH ) uu(qq LL )] = θθ LL [uu(qq LL ) uu(qq HH )] and ultimately reduces to θθ HH = θθ LL, violating the initial assumption θθ HH > θθ LL. Therefore, FF LL = θθ LL [uu(qq LL ) uu(qq HH )] + FF HH cannot hold, but instead FF LL < θθ LL [uu(qq LL ) uu(qq HH )] + FF HH must be true. As a consequence, the top panel of Figure 1 alies for the low-demand customer, ultimately implying that PPPP LL binds while IIII LL does not. Q.E.D. Summarizing, from the high-demand customer we have that θθ HH [uu(qq HH ) uu(qq LL )] + FF LL = FF HH whereas from the low-demand customer we obtained that θθ LL uu(qq LL ) = FF LL. We can now plug this information about FF HH and FF LL into the monopolist s expected PMP, which now becomes an unconstrained imization problem, as follows: qq LL,qq HH 0 [FF HH ccqq HH ] + (1 )[FF LL ccqq LL ] 8

9 = θθ HH [uu(qq HH ) uu(qq LL )] + FF LL ccqq HH + (1 ) θθ LL uu(qq LL ) FF HH FF LL ccqq LL which ultimately simplifies to = θθ HH [uu(qq HH ) uu(qq LL )] + θθ LL uu(qq LL ) FF LL ccqq HH + (1 )[θθ LL uu(qq LL ) ccqq LL ] = [θθ HH uu(qq HH ) (θθ HH θθ LL )uu(qq LL ) ccqq HH ] + (1 )[θθ LL uu(qq LL ) ccqq LL ] Importantly, constraint PPPP LL binding implies that IIII LL also holds (recall the lower panel of Figure 1), and constraint IIII HH binding entails that PPPP HH is also satisfied. In other words, all four constraints hold. Taking first-order conditions with respect to qq HH yields [θθ HH uu (qq HH ) cc] = 0, or θθ HH uu (qq HH ) = cc, Therefore, the amount offered to high-demand customers, qq HH, is socially efficient (their demand coincides with marginal cost). As we discuss next, such efficient outcome does not arise for low-demand customers. In particular, taking first-order conditions with respect to qq LL we obtain which can be rewritten as and further simplified to [ (θθ HH θθ LL )uu (qq LL )] + (1 )[θθ LL uu (qq LL ) cc] = 0, uu (qq LL )[(1 )θθ LL (θθ HH θθ LL )] = (1 )cc uu (qq LL )[θθ LL θθ HH ] = (1 )cc Dividing both sides by (1 ), we obtain uu (qq LL ) θθ LL θθ HH = cc 1 Note that this expression can alternatively be written as 2 uu (qq LL ) θθ LL 1 (θθ HH θθ LL ) = cc Figure 2 separately depicts the left- and right-hand side of the last expression. For comparison purposes, it also plots uu (qq LL ) θθ LL, which helps identify the socially optimal output qq LL ssss (i.e., that arising under complete information). 2 In order to find an expression in which θθ LL stands alone inside the parenthesis, we set up the equation θθ LL θθ HH = θθ LL xx. Solving for the unknown xx, yields 1 (θθ HH θθ LL ). 9 1

10 c u'(q L ) θ L u'(q L ) [θ L p 1 p (θθ HH θθ LL )] q L q L SO q Figure 2 Output for the low-demand customer Summarizing, the amount offered to high-demand customers is socially efficient (recall that θθ HH uu (qq HH ) = cc). In other words, qq HH = qq HH ssss, and there is no output distortion for high-demand customers relative to complete information allocations. That is a common finding in principal-agent models where the principal (in this case the monopolist) cannot observe the private type of the agent (in this case the consumer). In contrast, the output offered to low-demand customers entails a distortion relative to complete information, qq LL < qq LL ssss, as depicted in Figure 2. Furthermore, this output distortion qq LL ssss qq LL is increasing in term 1 (θθ HH θθ LL ). Specifically, it increases in the frequency of high-type buyers,, and on the difference between high- and low-type buyers, (θθ HH θθ LL ). In addition, the fact that constraint PPPP LL binds while PPPP HH does not, entails that only the high-demand customer retains a positive utility level, i.e., θθ HH uu(qq HH ) FF HH > 0. In other words, the firm s lack of information provides the high-demand customer with an information rent. Intuitively, this information rent emerges from the seller s attempt to reduce the incentives of the high-type customer to select the contract meant for the low type. In particular, while the low-demand buyer pays a lower fee, the output that he receives is sufficiently low to make it unattractive for the high-demand buyer, qq LL < qq LL ssss. In other words, the output distortion qq LL ssss qq LL that we described above stems from the seller s purpose to reduce the information rent of the high-type buyer. Example. Consider a monopolist selling a textbook to two types of graduate students, low- and highdemand, with utility function UU ii (qq ii, FF ii ) = θθ ii qq ii qq ii 2 FF 2θθ ii, ii 10

11 where ii = {LL, HH} and θθ HH > θθ LL. In this context, we obtain the direct demand function qq ii = θθ ii. In addition, assume that the proportion of high-demand (low-demand) students is γγ (1 γγ, respectively). The monopolist s constant marginal cost is cc > 0, which satisfies θθ ii > cc for all ii = {LL, HH}. Consider for simplicity that θθ LL > θθ HH+cc, which implies that each type of student would buy the textbook, both when the 2 firm practices uniform pricing and when it sets two-part tariffs (as we next show). Uniform pricing. Consider first that the monopolist does not practice price discrimination (i.e., it sets a uniform price that induces both types of customers to purchase positive units). In this setting, the monopolist sets a unique price p that solves the expected PMP γγ[( cc)(θθ LL )] + (1 γγ)[( cc)(θθ HH )], where qq ii = θθ ii for every type-i customer. Taking first-order conditions with respect to yields γγ(θθ LL ) γγ( cc) + (1 γγ)(θθ HH ) (1 γγ)( cc) = 0 And solving for p we obtain the uniform price which yields monopoly profits of UUUUUUUUUUUUUU = γγθθ LL + (1 γγ)θθ HH + cc, 2 ππ UUUUUUUUUUUUUU = [γγθθ LL + (1 γγ)θθ HH cc] 2 4 Note that the monopolist could use a uniform price to only serve high-demand students. The price that would imize its profits in this case solves (1 γγ)( cc)(θθ HH ) thus ignoring the segment of low-demand students. Taking first-order conditions with respect to and solving for yields HH = θθ HH+cc. In this context, monopoly profits become 2 ππ UUUUUUUUUUUUUU HH = (1 γγ) (θθ HH cc) 2, 4 which are larger than those when serving both types of students (i.e., ππ UUUUUUUUUUUUUU HH > ππ UUUUUUUUUUUUUU ) if the proportion of low-demand customers, γγ, is sufficiently small, that is, γγ < (θθ HH cc)(θθ HH 2θθ LL + cc) (θθ HH θθ LL ) 2. Intuitively, the frequency of high-value customers is large, thus inducing the seller ignore low-value customers to focus on high-value customers alone. For instance, parameter values θθ HH = 5, θθ LL = 2, cc = 1 and γγ = 3 satisfy this condition since 4 γγ < (θθ HH cc)(θθ HH 2θθ LL + cc) (θθ HH θθ LL ) 2 = (5 1)( ) (5 2) 2 =

12 Otherwise, if γγ > , the proportion of low-value customers is large enough to induce the seller to 9 not ignore this type of buyers, and thus serve both types. Two-part tariffs. Let us now consider that the monopolist offers a menu of two-part tariffs to each type of student (i.e., (FF LL, qq LL ) and (FF HH, qq HH )). From the previous discussion, we know that IIII HH and PPPP LL bind. Therefore, FF LL = θθ LL qq LL qq LL 2 and FF 2θθ HH = θθ HH qq HH qq 2 HH qq LL 2θθ LL qq 2 LL + θθ HH 2θθ LL qq LL qq LL LL Hence, the monopolist s PMP becomes γγ[ff HH ccqq HH ] + (1 γγ)[ff LL ccqq LL ] qq LL,qq HH θθ LL = γγ θθ HH qq HH qq HH 2 qq 2θθ LL qq 2 LL + θθ HH 2θθ LL qq LL qq 2 LL ccqq LL 2θθ HH + (1 γγ) θθ LL qq LL qq LL 2 LL 2θθ LL FF HH FF LL = θθ HH γγ qq HH qq HH 2 + (θθ 2θθ LL θθ HH γγ) qq LL qq 2 LL ccccqq HH 2θθ HH cc(1 γγ)qq LL LL Taking first-order conditions with respect to qq HH and qq LL yields γγθθ HH 1 qq HH θθ HH = cccc qq HH = θθ HH cc (θθ LL θθ HH γγ) 1 qq LL θθ LL = cc(1 γγ) qq LL = θθ LL cc θθ LL(1 γγ) (θθ LL θθ HH γγ) = θθ LL cc 1 γγ 1 θθ, HH γγ θθll ccqq LL which are both positive since θθ LL > cc 1 γγ, given that θθ 1 θθ HH HH > cc by definition. On the other hand, socially γγ θθ LL optimal outputs can be solved by setting marginal utility equals to marginal cost (i.e., uu ii (qq ii ) = θθ ii 1 qq ii = cc), thus yielding qq SSSS θθ HH = θθ HH cc and qq SSSS LL = θθ LL cc. ii We can then compare qq ii against qq ii SSSS for every type-i customer obtaining that qq HH = qq HH SSSS, and that qq LL < qq LL SSSS, since θθ LL cc 1 γγ 1 θθ HH θθll γγ < θθ LL cc This reduces to 1 γγ > 1 θθ HH θθ LL γγ, which is true given that θθ HH > θθ LL by definition. The monopolist can obtain larger profits by practicing second-degree price discrimination (two-part tariffs) than by setting a uniform price (either to attract both or only one type of customer). Using the same parameter values as under uniform pricing, θθ HH = 5, θθ LL = 2, cc = 1 and γγ = 3 4, we obtain we obtain output levels qq HH = 4, 12

13 qq LL = 16, and fees of FF 7 HH = and FF 49 LL = As a consequence, expected profits from twopart tariffs 49 are In contrast, those under uniform pricing become ππ TTTTTT = γγ[ff HH ccqq HH ] + (1 γγ)[ff LL ccqq LL ] = ππ UUUUUUUUUUUUUU = [γγθθ LL+(1 γγ)θθ HH cc] 2 4 = , and 64 ππ UUUUUUUUUUUUUU HH = (1 γγ) (θθ HH cc) 2 = 1 4 Hence, practicing two-part tariffs is profit-enhancing for the monopolist since ππ TTTTTT > ππ UUUUUUUUUUUUUU HH > ππ UUUUUUUUUUUUUU. 13