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Multidimensional screening under nonlinear costs: Limits of standard approach
Economics Letters 107 (2010) 263–265
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Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Multidimensional screening under nonlinear costs: Limits of standard approach Sergey Kokovin a, Babu Nahata b,⁎, Evgeny Zhelobodko a a b
Department of Economics, Novosibirsk State University, Novosibirsk 630090, Russia Department of Economics University of Louisville, Louisville, Kentucky 40292, USA
a r t i c l e
i n f o
Article history: Received 13 August 2009 Received in revised form 7 January 2010 Accepted 22 January 2010 Available online 1 February 2010
a b s t r a c t We generalize the usual screening approach and conditions for efficiency-at-the-top and acyclic property from linear to fixed-plus-separable or concave costs and multidimensional commodities. But under nonconcave costs, like capacity constraints, an example shows a cycle in the solution graph. The cycle makes the standard screening solution non-implementable and approach inadequate. © 2010 Elsevier B.V. All rights reserved.
JEL classification: D82 L12 Keywords: Nonlinear pricing Screening Graphs Capacity constraints Increasing returns
1. Introduction This paper follows the recent screening or non-linear pricing literature that relaxed the restrictive single-crossing condition (see Armstrong (2006), Andersson (2008)). Several interesting properties of screening solutions still hold under this generalization, notably, existence of an undistorted package and some ordering among profit contributions from different agents (Andersson (2005)). By obtaining almost necessary-and-sufficient conditions, we extend these findings to situations with non-constant returns to scale and to multidimensional, possibly discrete, commodity characteristics, dropping other assumptions also. Non-constant returns are common in real life: fixed costs of starting a business, capacity constraints, etc., discussed in IO literature, but not in screening. Nonlinear costs could undermine the standard approach, specifically, the usual “no-type-partitioning,” and “friendly agent” assumptions. The standard assignment-optimization problem (SAOP) can become insufficient and its assignments non-implementable, because of so-called envy-cycles (see Section 2). Under concave costs, these complexities are avoidable, SAOP approach and the discussed properties of solutions remain valid, as shown in our Theorem. In
⁎ Corresponding author. Tel.: +1 502 852 4864; fax: +1 502 852 7672. E-mail addresses: [email protected] (S. Kokovin), [email protected] (B. Nahata), [email protected] (E. Zhelobodko). 0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.01.042
contrast, under decreasing returns or capacity constraints modelled by convex costs, the type-partitioning and non-implementation are probable, appealing therefore for a new model, further motivated and explained in Kokovin et al. (2009). 2. Model and definitions Consider a standard (except for costs) discrete screening or nonlinear pricing problem. A monopolist offers a product or service using a menu of several discrete packages, i.e., quantity or qualitiestariff bundles on take-it-or-leave-it basis. The consumer types indexed by i ∈ In = {1, ..., n} are known, but personal discrimination infeasible. The number of agents of type i is mi N 0; its outlay or tariff is ti, and xi ∈ X denotes characteristics chosen from some l-dimensional consumption set X ⊂ ℝl that can be discrete or continuous and Xn: = X × ... × X. Utility functions ui(xi, ti) = vi(xi) + ti are quasi-concave with no other restrictions except normalization vi(0) = 0. The cost function C(m, x) is of general form. Specially, it may have an aggregate form C(m, x) = c( ∑ imixi); or a fixed-plus-separable (per-consumer) cost C(m, x) = f0 + ∑ imic(xi), where f0 is fixed cost and c : ℝ → ℝ. The seller designs an assignment (x, t) = {(xi, ti)}i= 1...n, which becomes the quantity–tariff menu offered to consumers. This approach and implementation of this assignment depends on two standard crucial assumptions: (1) “one-type-one-package”; (2) the so-called “friendly-agent.” The former excludes type-partitioning, but does allow same packages for different agents. The latter assumes an
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S. Kokovin et al. / Economics Letters 107 (2010) 263–265
agent prefers the principal's choice when faced with multiple equivalent choices. Though we generalize costs, the resulting profit-maximization problem is labelled hereafter as “standard,” i.e., SAOP: n
πðx; t Þ := ∑ mi ti −C ðm; x1 ; :::; xn Þ→ i=1
i ≥ 1 ⇒ vi ðxi Þ−ti
≥
vi ðxk Þ−tk
max
; s:t:
ðx;t Þ∈ðX n ;ℝn Þ
∀k ≥ 0; ðx0 ; t0 Þ := ð0; 0Þ
Package #0 here means non-participation. Any SAOP solution ( x ̅ , t ̅) to the above problem can be characterized by its set of active constraints, viewed hereafter as an “envy-graph”or A-graph. Namely, any constraint (i, k) when active (i.e., becoming equality) is interpreted as “almost-envy”or just envy from i-th consumer having package i to some not-chosen package k (that can be k = 0 ). Any such constraint can be treated as the arc (i → k) of the A-graph G̿( x ̅, t ̅) — the set of nodes and arcs. Packages are represented as nodes, and active constraints as directed arcs, while #0 is the “sink”. We explore further possibilities of directed cycles–closed paths, called dicycles or just cycles in such A-graphs. In a package-menu ( x ̅, t ̅), a package x i̅ , t ̅i) is called undistorted when it maximizes (without incentive-compatibility constraints) the joint welfare of this agent and the principal, all other packages remaining fixed, i.e., xi ∈ arg maxzi ∈X;vi ðzi Þ≥0 ðvi ðzi Þ−C ðm; x1 ; :::; zi ; :::; xn ÞÞ: 3. Extended SAOP under separable or concave cost As mentioned in the Introduction, and in Claim below, under nonlinear cost the “one-type-one-package” and “friendly-agent” assumptions become questionable because of cycles. In the literature, the possibility of type-partitioning is not recognized, while cycles under linear costs are “reducible” without profit loss (see, Guesnerie and Seade (1982), Andersson (2005)), in the sense of our Theorem. It generalizes the named claims and states no-partitioning, assuming sort of concavity for C (decreasing average cost), as follows. Assumption C. The cost function C(m, x) can be estimated from above by some fixed-plus-separable cost function C ̅ such that C (̅ m, x) ≥ C(m, x) ∀ x Xn and at the solution point x ̅ they coincide: C ̅(m, x ̅) = C(m, x ̅).1 For a fixed-plus-separable cost function C(m, x) = f0 + ∑ ni= 1mici(xi), the per-package profit contribution π i̅ considering only variable cost is defined as: n
πi ðx; t Þ := ti −ci ðxi Þ; so that πðx; t Þ := ∑ mi πi ðx; t Þ−f0 :
procedure results in some feasible assignment. Under most general valuations v and costs C, the main solution properties follow. Theorem. Assume a solution ( x ̅, t ̅) and cost function being either fixedplus-separable, or satisfying Assumption C. Then: (A) A predecessor brings weakly higher per-package profit than all its successors in A-graph G̿( x ̅, t ̅), i.e., ( j → …i) ⇒ π j̅ (( x ̅, t ̅) ≥ π i̅ (( x ̅, t ̅). Any highest-profit node is undistorted. (B) Solution (( x ,̅ t )̅ either satisfies (i)–(iv) or can be modified through bunching into another optimal solution (x̂, t)̂ where: (i) different packages bring different profits (x̂i ≠ x̂j)⇒ t i̅ ≠ t ĵ ; (ii) strictly higher ̿ , t)̂ , i.e., (j → …i,i↛…j) brings strictly higher profit: position in G(x̂ π j̅ (x̂, t )̂ N πî (x̂, t )̂ ; (iii) there are no non-bunched cycles; and (iv) there is no type-partitioning.2 Proof. (A). When costs are separable, suppose a profit-ascending arc ( j → i): π i̅ = t ̅i − ci( x i̅ ) N π j̅ = t ̅j − c( x ̅j). Then through our transformation procedure, bunching j to i necessarily increases total profit ̅ , t ̂) N π (̅ x ̅, t ̅)), generating a feasible assignment (x̂, t ̂). So, the ( π(x̂ solution ( x ̅, t ̅) was not optimal, resulting in a contradiction for an adjacent ( j → i). Similarly, under non-separable concave cost for an adjacent couple ( j → i) assumed to contradict (A), we can construct an im̅ , t ̂) N π(̅ x ̅, t ̅) for artificial profit. What remains is to provement π(x̂ compare artificial profit π ̅ with the real profit π. By Assumption C and construction of π ̅(.), these two functions coincide at the solution: π(̅ x ̅, t ̅) =π( x ̅, t ̅), and artificial costs are everywhere weakly higher than real costs: C ̅(m, x̂) ≥ C(m, x̂), so the actual profit π is higher than the artificial one, yielding π(x̂, t ̂) ≥ π ̅(x̂, t̂) N π(̅ x ̅, t ̅). This contradiction to optimality of ( x ̅, t ̅) proves (A) for ( j → i). By induction, (A) is extended from any adjacent couple to any couple (j, i) of predecessor and successor. (B). To prove (i), it is sufficient to apply the bunching transformation to eliminate all the same-profit nodes. Due to feasibility of the bunching transformation, we thus get an optimal solution (x̂, t ̂) , and its properties (ii), (iii) and (iv) follow. □ Theorem extends the reducibility results by Guesnerie and Seade (1982), Andersson (2005) to multidimensional commodities and almost-separable or concave costs, adding also: (1) no-type-partitioning claim, (2) the claim that each solution can be simplified by bunching, (3) monotonicity of profit contributions along the entire A-graph. 4. Cycle and non-implementation under convex cost Claim below shows non-reducible cycles resulting solely from cost convexity (unlike cycles in Kokovin et al., 2009 connected with unusual type-partitioning).
i=1
This notion π i̅ used for C or C ̅ enables eliminating cycles as follows. Bunching transformation. For any feasible plan ( x ̅, t ̅), where an agent j almost-envies agent i (i.e., j → i), we can replace j-th assignment ( x ̅j, t ̅j) with a new (envied) package (x̂j, t̂j) = ( x ̅i, t ̅i), keeping other components unchanged. This complete new assignment (x̂, t ̂): (( x ̅1, t ̅1)),...,(x j̅ − 1, t ̅j − 1), ( x ̅i, t ̅i), ( x j̅ + 1, t ̅j + 1),...,(x n̅ , t ̅n)) remains incentive compatible, because no new packages appear in the menu and all other agents, except for j, remain unaffected. This agent j also has exactly the same payoff as before: vj(x̂j) − t ĵ = vj( x j̅ ) − t j̅ , because of the envy arc ( j → i). So, the incentive-compatibility constraints hold completely: vj(x̂j) − t ĵ ≥ vj( x k̅ ) − t k̅ ∀ (k, j). Therefore, this bunching 1
In particular, when X = ℝ and C is differentiable, C ̅ can be taken linear: C ̅(m, x) =
c0 + mi c i̅ xi,where ci : =
1 ∂C ðm; xÞ , mi ∂xi
fixed-plus-separable, C ̅ = C.
c0: C ̅ (m, x ̅) = C(m, x ̅). Naturally, if initial C is
Claim. Under convex costs and concave utilities, a solution can contain a non-reducible cycle, which can occur at-the-top of the envy-graph, with different profit contributions from mutual successors. Proof. Consider an example with the aggregate-type cost function C(xΣ) = max{0, 3(x − 6.5)3}. Thus, high production xΣ: = ∑ ixi N 8 is prohibitively costly, while any production less than 6.5 is costless.3
2 Cycles are absent among distinct packages, whereas a mutual almost-envy cycle among agents bunched together with the same package is inevitable. No-typepartitioning means that agents with the same utilities take the same package, even when they are formally represented by different “types” in the model. 3 “Costless” means just a usual normalization. Instead, one can add a linear component Ax, A N 4 to the cost function and to all valuations, thereby getting rid off agents’ satiation and costless production, but keeping the example intact. This example is intentionally one-dimensional to show that it is convexity of C that matters, not dimensionality. Under l N 1 such effects are easier to get.
S. Kokovin et al. / Economics Letters 107 (2010) 263–265
There are five buyers-types (mi = 1) with valuations: v1(x1) = 4x1 − 3.325x21, v2(x2) = 1.32x2 − 131/360x22, v3(x3) = min{5x3; 1 + 0.01x3; 12.8−4x3}, v4(x4)=min{5x4;1+0.01x4;4−2x4}, v5(x5):=min{2x5;1+ 0.01x5;12.8−4x5}. We calculated an optimal assignment: ( x ̅1, t ̅1) = (0.6,1.203), ( x ̅ 2 , t ̅ 2 ) = (1.8,1.197), ( x ̅ 3 , t ̅ 3 ) = (0.25,1.0025), ( x ̅ 4 , t ̅ 4 ) = (1.0,1.01), x 5̅ = 173/60 ≈ 2.88333, t ̅5 = 6173/6000 ≈ 1.0288333 with total output xsum = 98/15 ≈ 6.533333. This assignment is really a SAOP solution because the entire consumer surplus goes to the principal and total welfare is maximized. Indeed, all derivatives v1'( x 1̅ ) = v2′( x 2̅ ) = … = C′(98/15) = 0.01 are equal, while the problem max(∑ivi(xi) − C( ∑ixi)) is convex (there are also nonessential different equivalent solutions, from permuting agents #3, #4 and #5 around the envy-cycle, or very small (±0.1) simultaneous changes in x3, x4, x5 keeping their sum). At ( x ̅, t ̅), one can find a cycled envy-graph G ̅( x ̅, t ̅):#i → 0∀i, #4 ← #5 ← #3#4, these nine equalities being active. To see that this 1 and bounds cycle is non-reducible, calculate total cost C ð98 = 15Þ = 9000 1 ≤π3 ≤ t 3 9000
on profit contributions from three agents: t 3 − 1 t4 − ≤π4 ≤ t 4 9000
=
265
takes her assignment (x4, t4) from what remains in the menu. Then comes #5 and goes away, since he values the remaining packages #1, #2 and #3 lower than non-participation. So, about $1.0287 of profit is lost! Thus, the expected profit here is less than the planned one and SAOP solution was naive. If not for cycles, infinitesimally small reductions in tariffs could break the envy-arcs making the menu strictly incentive-compatible, and almost-optimal. However, cycles generally cannot be broken by such tool and also here exact implementation cannot be guaranteed. Only the almost-planned profit can be achieved by making packages #3 and #5 cheaper than #4 by a small amount (at all realizations of the game). This goal cannot be achieved under slight modification of the third valuation to v3 = min{5x3; 1 + 0.01x3; 12.8 − 4x3; 0.95 + (x − 1)2}, because the cycle becomes symmetric: #3→ #4→ #5→#3. In this case, no rewards can help to get the almost-planned profit and avoid substantial loss. From this example and examples showing profitable typepartitioning in Kokovin et al. (2009) we conclude that all solution properties stated in Theorem do not hold under convex cost, hence, some other model, instead of SAOP, should be applied.
= 1:0025,
1 1:01, t 5 − ≤π5 ≤ t 5 ≈1:0288333. These intervals 9000
are non-intersecting, so, any attempt to bunch an agent with his successor like in Theorem's proof, either decreases profit, or exceeds the implicit capacity constraint — this is the essence of the example. □ Something like type-partitioning is inevitable here, because different packages should be offered to the couple of locally similar agents #3 and #4, and to the couple #4, #5 (see Kokovin et al. (2009) for more elaborate treatment of partitioning in such cases). To comprehend the impact of non-reducible cycles on implementation, consider the case when the seller cannot re-negotiate with all consumers simultaneously in a selling game. Conjecture. Under convex costs a cycle in the envy-graph can make the solution non-implementable. Without formally defining dynamic Nash implementation (for implementation theory see Maskin and Sjostrom (2002)), we explain this idea intuitively. Suppose on a given day, in the above example, the seller has produced the five optimal quantities x ̅ = (0.6,1.8,0.25,1, 2.8833) and waits for the five unrecognizable consumers. It is unimportant when consumers #1 and #2 come, but suppose that agent #3 comes first and takes the non-assigned package (x5, t5) for him, being indifferent between (x3, t3) and (x5, t5). Then comes consumer #4 and
Acknowledgement We are grateful for the financial support from the University of Louisville and from the Economics Education and Research Consortium, Inc. (EERC) grant No 06-056, with funds provided by the Eurasia Foundation (with funding from the US Agency for International Development), the World Bank Institution, the Global Development Network and the Government of Sweden. We are thankful for the assistance from Richard Ericson, Victor Polterovich and Alexei Savvateev and comments of the participants in 2009 Far Eastern and South Asia Meeting of Econometric Society at Tokyo University. References Andersson, T., 2005. Profit maximizing nonlinear pricing. Economics Letters 88, 135–139. Andersson, T., 2008. Efficiency properties of non-linear pricing schedules without the single-crossing condition. Economics Letters 99, 364–366. Armstrong, M., 2006. Recent developments in the economics of price discrimination. In: Blundell, R., Newey, W.K., Persson, T. (Eds.), Advances in Economics and Econometrics: Theory and Applications, Chapter 4 Ninth World Congress Econometric Society. Cambridge University Press, Cambridge, pp. 97–141. Guesnerie, R., Seade, J., 1982. Nonlinear pricing in a finite economy. Journal of Public Economics 17, 157–179. Kokovin, S., Nahata, B., Zhelobodko, E. 2009, Multidimensional screening under capacity constraint: Overall distortion and type-partitioning. Working paper. Maskin, E., Sjostrom, T., 2002. Implementation theory. In: Arrow, K.J., Sen, A.K., Suzumura, K. (Eds.), Handbook of Social Choice Theory, vol. 1. North Holland, Amsterdam, pp. 237–288.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Multidimensional screening under nonlinear costs: Limits of standard approach Sergey Kokovin a, Babu Nahata b,⁎, Evgeny Zhelobodko a a b
Department of Economics, Novosibirsk State University, Novosibirsk 630090, Russia Department of Economics University of Louisville, Louisville, Kentucky 40292, USA
a r t i c l e
i n f o
Article history: Received 13 August 2009 Received in revised form 7 January 2010 Accepted 22 January 2010 Available online 1 February 2010
a b s t r a c t We generalize the usual screening approach and conditions for efficiency-at-the-top and acyclic property from linear to fixed-plus-separable or concave costs and multidimensional commodities. But under nonconcave costs, like capacity constraints, an example shows a cycle in the solution graph. The cycle makes the standard screening solution non-implementable and approach inadequate. © 2010 Elsevier B.V. All rights reserved.
JEL classification: D82 L12 Keywords: Nonlinear pricing Screening Graphs Capacity constraints Increasing returns
1. Introduction This paper follows the recent screening or non-linear pricing literature that relaxed the restrictive single-crossing condition (see Armstrong (2006), Andersson (2008)). Several interesting properties of screening solutions still hold under this generalization, notably, existence of an undistorted package and some ordering among profit contributions from different agents (Andersson (2005)). By obtaining almost necessary-and-sufficient conditions, we extend these findings to situations with non-constant returns to scale and to multidimensional, possibly discrete, commodity characteristics, dropping other assumptions also. Non-constant returns are common in real life: fixed costs of starting a business, capacity constraints, etc., discussed in IO literature, but not in screening. Nonlinear costs could undermine the standard approach, specifically, the usual “no-type-partitioning,” and “friendly agent” assumptions. The standard assignment-optimization problem (SAOP) can become insufficient and its assignments non-implementable, because of so-called envy-cycles (see Section 2). Under concave costs, these complexities are avoidable, SAOP approach and the discussed properties of solutions remain valid, as shown in our Theorem. In
⁎ Corresponding author. Tel.: +1 502 852 4864; fax: +1 502 852 7672. E-mail addresses: [email protected] (S. Kokovin), [email protected] (B. Nahata), [email protected] (E. Zhelobodko). 0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.01.042
contrast, under decreasing returns or capacity constraints modelled by convex costs, the type-partitioning and non-implementation are probable, appealing therefore for a new model, further motivated and explained in Kokovin et al. (2009). 2. Model and definitions Consider a standard (except for costs) discrete screening or nonlinear pricing problem. A monopolist offers a product or service using a menu of several discrete packages, i.e., quantity or qualitiestariff bundles on take-it-or-leave-it basis. The consumer types indexed by i ∈ In = {1, ..., n} are known, but personal discrimination infeasible. The number of agents of type i is mi N 0; its outlay or tariff is ti, and xi ∈ X denotes characteristics chosen from some l-dimensional consumption set X ⊂ ℝl that can be discrete or continuous and Xn: = X × ... × X. Utility functions ui(xi, ti) = vi(xi) + ti are quasi-concave with no other restrictions except normalization vi(0) = 0. The cost function C(m, x) is of general form. Specially, it may have an aggregate form C(m, x) = c( ∑ imixi); or a fixed-plus-separable (per-consumer) cost C(m, x) = f0 + ∑ imic(xi), where f0 is fixed cost and c : ℝ → ℝ. The seller designs an assignment (x, t) = {(xi, ti)}i= 1...n, which becomes the quantity–tariff menu offered to consumers. This approach and implementation of this assignment depends on two standard crucial assumptions: (1) “one-type-one-package”; (2) the so-called “friendly-agent.” The former excludes type-partitioning, but does allow same packages for different agents. The latter assumes an
264
S. Kokovin et al. / Economics Letters 107 (2010) 263–265
agent prefers the principal's choice when faced with multiple equivalent choices. Though we generalize costs, the resulting profit-maximization problem is labelled hereafter as “standard,” i.e., SAOP: n
πðx; t Þ := ∑ mi ti −C ðm; x1 ; :::; xn Þ→ i=1
i ≥ 1 ⇒ vi ðxi Þ−ti
≥
vi ðxk Þ−tk
max
; s:t:
ðx;t Þ∈ðX n ;ℝn Þ
∀k ≥ 0; ðx0 ; t0 Þ := ð0; 0Þ
Package #0 here means non-participation. Any SAOP solution ( x ̅ , t ̅) to the above problem can be characterized by its set of active constraints, viewed hereafter as an “envy-graph”or A-graph. Namely, any constraint (i, k) when active (i.e., becoming equality) is interpreted as “almost-envy”or just envy from i-th consumer having package i to some not-chosen package k (that can be k = 0 ). Any such constraint can be treated as the arc (i → k) of the A-graph G̿( x ̅, t ̅) — the set of nodes and arcs. Packages are represented as nodes, and active constraints as directed arcs, while #0 is the “sink”. We explore further possibilities of directed cycles–closed paths, called dicycles or just cycles in such A-graphs. In a package-menu ( x ̅, t ̅), a package x i̅ , t ̅i) is called undistorted when it maximizes (without incentive-compatibility constraints) the joint welfare of this agent and the principal, all other packages remaining fixed, i.e., xi ∈ arg maxzi ∈X;vi ðzi Þ≥0 ðvi ðzi Þ−C ðm; x1 ; :::; zi ; :::; xn ÞÞ: 3. Extended SAOP under separable or concave cost As mentioned in the Introduction, and in Claim below, under nonlinear cost the “one-type-one-package” and “friendly-agent” assumptions become questionable because of cycles. In the literature, the possibility of type-partitioning is not recognized, while cycles under linear costs are “reducible” without profit loss (see, Guesnerie and Seade (1982), Andersson (2005)), in the sense of our Theorem. It generalizes the named claims and states no-partitioning, assuming sort of concavity for C (decreasing average cost), as follows. Assumption C. The cost function C(m, x) can be estimated from above by some fixed-plus-separable cost function C ̅ such that C (̅ m, x) ≥ C(m, x) ∀ x Xn and at the solution point x ̅ they coincide: C ̅(m, x ̅) = C(m, x ̅).1 For a fixed-plus-separable cost function C(m, x) = f0 + ∑ ni= 1mici(xi), the per-package profit contribution π i̅ considering only variable cost is defined as: n
πi ðx; t Þ := ti −ci ðxi Þ; so that πðx; t Þ := ∑ mi πi ðx; t Þ−f0 :
procedure results in some feasible assignment. Under most general valuations v and costs C, the main solution properties follow. Theorem. Assume a solution ( x ̅, t ̅) and cost function being either fixedplus-separable, or satisfying Assumption C. Then: (A) A predecessor brings weakly higher per-package profit than all its successors in A-graph G̿( x ̅, t ̅), i.e., ( j → …i) ⇒ π j̅ (( x ̅, t ̅) ≥ π i̅ (( x ̅, t ̅). Any highest-profit node is undistorted. (B) Solution (( x ,̅ t )̅ either satisfies (i)–(iv) or can be modified through bunching into another optimal solution (x̂, t)̂ where: (i) different packages bring different profits (x̂i ≠ x̂j)⇒ t i̅ ≠ t ĵ ; (ii) strictly higher ̿ , t)̂ , i.e., (j → …i,i↛…j) brings strictly higher profit: position in G(x̂ π j̅ (x̂, t )̂ N πî (x̂, t )̂ ; (iii) there are no non-bunched cycles; and (iv) there is no type-partitioning.2 Proof. (A). When costs are separable, suppose a profit-ascending arc ( j → i): π i̅ = t ̅i − ci( x i̅ ) N π j̅ = t ̅j − c( x ̅j). Then through our transformation procedure, bunching j to i necessarily increases total profit ̅ , t ̂) N π (̅ x ̅, t ̅)), generating a feasible assignment (x̂, t ̂). So, the ( π(x̂ solution ( x ̅, t ̅) was not optimal, resulting in a contradiction for an adjacent ( j → i). Similarly, under non-separable concave cost for an adjacent couple ( j → i) assumed to contradict (A), we can construct an im̅ , t ̂) N π(̅ x ̅, t ̅) for artificial profit. What remains is to provement π(x̂ compare artificial profit π ̅ with the real profit π. By Assumption C and construction of π ̅(.), these two functions coincide at the solution: π(̅ x ̅, t ̅) =π( x ̅, t ̅), and artificial costs are everywhere weakly higher than real costs: C ̅(m, x̂) ≥ C(m, x̂), so the actual profit π is higher than the artificial one, yielding π(x̂, t ̂) ≥ π ̅(x̂, t̂) N π(̅ x ̅, t ̅). This contradiction to optimality of ( x ̅, t ̅) proves (A) for ( j → i). By induction, (A) is extended from any adjacent couple to any couple (j, i) of predecessor and successor. (B). To prove (i), it is sufficient to apply the bunching transformation to eliminate all the same-profit nodes. Due to feasibility of the bunching transformation, we thus get an optimal solution (x̂, t ̂) , and its properties (ii), (iii) and (iv) follow. □ Theorem extends the reducibility results by Guesnerie and Seade (1982), Andersson (2005) to multidimensional commodities and almost-separable or concave costs, adding also: (1) no-type-partitioning claim, (2) the claim that each solution can be simplified by bunching, (3) monotonicity of profit contributions along the entire A-graph. 4. Cycle and non-implementation under convex cost Claim below shows non-reducible cycles resulting solely from cost convexity (unlike cycles in Kokovin et al., 2009 connected with unusual type-partitioning).
i=1
This notion π i̅ used for C or C ̅ enables eliminating cycles as follows. Bunching transformation. For any feasible plan ( x ̅, t ̅), where an agent j almost-envies agent i (i.e., j → i), we can replace j-th assignment ( x ̅j, t ̅j) with a new (envied) package (x̂j, t̂j) = ( x ̅i, t ̅i), keeping other components unchanged. This complete new assignment (x̂, t ̂): (( x ̅1, t ̅1)),...,(x j̅ − 1, t ̅j − 1), ( x ̅i, t ̅i), ( x j̅ + 1, t ̅j + 1),...,(x n̅ , t ̅n)) remains incentive compatible, because no new packages appear in the menu and all other agents, except for j, remain unaffected. This agent j also has exactly the same payoff as before: vj(x̂j) − t ĵ = vj( x j̅ ) − t j̅ , because of the envy arc ( j → i). So, the incentive-compatibility constraints hold completely: vj(x̂j) − t ĵ ≥ vj( x k̅ ) − t k̅ ∀ (k, j). Therefore, this bunching 1
In particular, when X = ℝ and C is differentiable, C ̅ can be taken linear: C ̅(m, x) =
c0 + mi c i̅ xi,where ci : =
1 ∂C ðm; xÞ , mi ∂xi
fixed-plus-separable, C ̅ = C.
c0: C ̅ (m, x ̅) = C(m, x ̅). Naturally, if initial C is
Claim. Under convex costs and concave utilities, a solution can contain a non-reducible cycle, which can occur at-the-top of the envy-graph, with different profit contributions from mutual successors. Proof. Consider an example with the aggregate-type cost function C(xΣ) = max{0, 3(x − 6.5)3}. Thus, high production xΣ: = ∑ ixi N 8 is prohibitively costly, while any production less than 6.5 is costless.3
2 Cycles are absent among distinct packages, whereas a mutual almost-envy cycle among agents bunched together with the same package is inevitable. No-typepartitioning means that agents with the same utilities take the same package, even when they are formally represented by different “types” in the model. 3 “Costless” means just a usual normalization. Instead, one can add a linear component Ax, A N 4 to the cost function and to all valuations, thereby getting rid off agents’ satiation and costless production, but keeping the example intact. This example is intentionally one-dimensional to show that it is convexity of C that matters, not dimensionality. Under l N 1 such effects are easier to get.
S. Kokovin et al. / Economics Letters 107 (2010) 263–265
There are five buyers-types (mi = 1) with valuations: v1(x1) = 4x1 − 3.325x21, v2(x2) = 1.32x2 − 131/360x22, v3(x3) = min{5x3; 1 + 0.01x3; 12.8−4x3}, v4(x4)=min{5x4;1+0.01x4;4−2x4}, v5(x5):=min{2x5;1+ 0.01x5;12.8−4x5}. We calculated an optimal assignment: ( x ̅1, t ̅1) = (0.6,1.203), ( x ̅ 2 , t ̅ 2 ) = (1.8,1.197), ( x ̅ 3 , t ̅ 3 ) = (0.25,1.0025), ( x ̅ 4 , t ̅ 4 ) = (1.0,1.01), x 5̅ = 173/60 ≈ 2.88333, t ̅5 = 6173/6000 ≈ 1.0288333 with total output xsum = 98/15 ≈ 6.533333. This assignment is really a SAOP solution because the entire consumer surplus goes to the principal and total welfare is maximized. Indeed, all derivatives v1'( x 1̅ ) = v2′( x 2̅ ) = … = C′(98/15) = 0.01 are equal, while the problem max(∑ivi(xi) − C( ∑ixi)) is convex (there are also nonessential different equivalent solutions, from permuting agents #3, #4 and #5 around the envy-cycle, or very small (±0.1) simultaneous changes in x3, x4, x5 keeping their sum). At ( x ̅, t ̅), one can find a cycled envy-graph G ̅( x ̅, t ̅):#i → 0∀i, #4 ← #5 ← #3#4, these nine equalities being active. To see that this 1 and bounds cycle is non-reducible, calculate total cost C ð98 = 15Þ = 9000 1 ≤π3 ≤ t 3 9000
on profit contributions from three agents: t 3 − 1 t4 − ≤π4 ≤ t 4 9000
=
265
takes her assignment (x4, t4) from what remains in the menu. Then comes #5 and goes away, since he values the remaining packages #1, #2 and #3 lower than non-participation. So, about $1.0287 of profit is lost! Thus, the expected profit here is less than the planned one and SAOP solution was naive. If not for cycles, infinitesimally small reductions in tariffs could break the envy-arcs making the menu strictly incentive-compatible, and almost-optimal. However, cycles generally cannot be broken by such tool and also here exact implementation cannot be guaranteed. Only the almost-planned profit can be achieved by making packages #3 and #5 cheaper than #4 by a small amount (at all realizations of the game). This goal cannot be achieved under slight modification of the third valuation to v3 = min{5x3; 1 + 0.01x3; 12.8 − 4x3; 0.95 + (x − 1)2}, because the cycle becomes symmetric: #3→ #4→ #5→#3. In this case, no rewards can help to get the almost-planned profit and avoid substantial loss. From this example and examples showing profitable typepartitioning in Kokovin et al. (2009) we conclude that all solution properties stated in Theorem do not hold under convex cost, hence, some other model, instead of SAOP, should be applied.
= 1:0025,
1 1:01, t 5 − ≤π5 ≤ t 5 ≈1:0288333. These intervals 9000
are non-intersecting, so, any attempt to bunch an agent with his successor like in Theorem's proof, either decreases profit, or exceeds the implicit capacity constraint — this is the essence of the example. □ Something like type-partitioning is inevitable here, because different packages should be offered to the couple of locally similar agents #3 and #4, and to the couple #4, #5 (see Kokovin et al. (2009) for more elaborate treatment of partitioning in such cases). To comprehend the impact of non-reducible cycles on implementation, consider the case when the seller cannot re-negotiate with all consumers simultaneously in a selling game. Conjecture. Under convex costs a cycle in the envy-graph can make the solution non-implementable. Without formally defining dynamic Nash implementation (for implementation theory see Maskin and Sjostrom (2002)), we explain this idea intuitively. Suppose on a given day, in the above example, the seller has produced the five optimal quantities x ̅ = (0.6,1.8,0.25,1, 2.8833) and waits for the five unrecognizable consumers. It is unimportant when consumers #1 and #2 come, but suppose that agent #3 comes first and takes the non-assigned package (x5, t5) for him, being indifferent between (x3, t3) and (x5, t5). Then comes consumer #4 and
Acknowledgement We are grateful for the financial support from the University of Louisville and from the Economics Education and Research Consortium, Inc. (EERC) grant No 06-056, with funds provided by the Eurasia Foundation (with funding from the US Agency for International Development), the World Bank Institution, the Global Development Network and the Government of Sweden. We are thankful for the assistance from Richard Ericson, Victor Polterovich and Alexei Savvateev and comments of the participants in 2009 Far Eastern and South Asia Meeting of Econometric Society at Tokyo University. References Andersson, T., 2005. Profit maximizing nonlinear pricing. Economics Letters 88, 135–139. Andersson, T., 2008. Efficiency properties of non-linear pricing schedules without the single-crossing condition. Economics Letters 99, 364–366. Armstrong, M., 2006. Recent developments in the economics of price discrimination. In: Blundell, R., Newey, W.K., Persson, T. (Eds.), Advances in Economics and Econometrics: Theory and Applications, Chapter 4 Ninth World Congress Econometric Society. Cambridge University Press, Cambridge, pp. 97–141. Guesnerie, R., Seade, J., 1982. Nonlinear pricing in a finite economy. Journal of Public Economics 17, 157–179. Kokovin, S., Nahata, B., Zhelobodko, E. 2009, Multidimensional screening under capacity constraint: Overall distortion and type-partitioning. Working paper. Maskin, E., Sjostrom, T., 2002. Implementation theory. In: Arrow, K.J., Sen, A.K., Suzumura, K. (Eds.), Handbook of Social Choice Theory, vol. 1. North Holland, Amsterdam, pp. 237–288.