N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS Nonlinear Pricing without Single Crossing Dmitri Blueschke ∗ Guilherme Freitas † Martin Szydlowski ‡ Nan Yang § ∗ Klagenfurt University † California Institute of Technology ‡ Northwestern University § VU University Amsterdam & Tinbergen Institute ICE2009, Aug 12th 1 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS O UTLINE N ON -L INEAR P RICING IN M ONOPOLY M ARKET 1 A N E XAMPLE W ITH S INGLE C ROSSING 2 A N E XAMPLE W ITHOUT S INGLE C ROSSING 3 N UMERICAL E XPLORATIONS 4 Non-Uniform Distribution of Types Two Dimensional Types 2 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS O UTLINE N ON -L INEAR P RICING IN M ONOPOLY M ARKET 1 A N E XAMPLE W ITH S INGLE C ROSSING 2 A N E XAMPLE W ITHOUT S INGLE C ROSSING 3 N UMERICAL E XPLORATIONS 4 Non-Uniform Distribution of Types Two Dimensional Types 3 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS G ENERAL S ETUP A continuum of consumer with type θ ∈ Θ . Consumer with type θ values quantity q by v ( q, θ ) . Monopolist, without being able to observe consumers’ types, charges nonlinear tariff t ( q ) . Monopolist’s cost function C ( q ) . With Revelation Principle, monopolist solve the following problem � θ t ( θ ) − C ( q ( θ )) dF ( θ ) maximize q,t θ subject to v ( q ( θ ) , θ ) − t ( θ ) � 0 ∀ θ ∈ Θ (IR) ∀ θ, θ ′ ∈ Θ (IC) v ( q ( θ ) , θ ) − t ( θ ) � v ( q ( θ ′ ) , θ ) − t ( θ ′ ) 4 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS T HE R OLE OF S INGLE -C ROSSING Definition: v q monotonic in types θ . What does it mean? Ordering of demands. Incentive to lie “downwards”. Local incentive constraints imply global incentive constraints (F .O.C. is valid). 5 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS T HE R OLE OF S INGLE -C ROSSING Definition: v q monotonic in types θ . What does it mean? Ordering of demands. Incentive to lie “downwards”. Local incentive constraints imply global incentive constraints (F .O.C. is valid). 5 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS O UTLINE N ON -L INEAR P RICING IN M ONOPOLY M ARKET 1 A N E XAMPLE W ITH S INGLE C ROSSING 2 A N E XAMPLE W ITHOUT S INGLE C ROSSING 3 N UMERICAL E XPLORATIONS 4 Non-Uniform Distribution of Types Two Dimensional Types 6 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS A N E XAMPLE W ITH S INGLE C ROSSING Values: v ( q, θ ) = θ √ q , with θ ∼ U [2 , 3] and q � 0 . Cost: C ( q ) = cq , c > 0 . Tariff: t � 0 . � 3 t ( θ ) − C ( q ( θ )) dF ( θ ) maximize q,t 2 v ( q ( θ ) , θ ) − t ( θ ) � 0 ∀ θ (IR) subject to ∀ θ, θ ′ (IC) v ( q ( θ ) , θ ) − t ( θ ) � v ( q ( θ ′ ) , θ ) − t ( θ ′ ) 7 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS A N E XAMPLE W ITH S INGLE C ROSSING Values: v ( q, θ ) = θ √ q , with θ ∼ U [2 , 3] and q � 0 . Cost: C ( q ) = cq , c > 0 . Tariff: t � 0 . � 3 t ( θ ) − C ( q ( θ )) dF ( θ ) maximize q,t 2 v ( q ( θ ) , θ ) − t ( θ ) � 0 ∀ θ (IR) subject to ∀ θ, θ ′ (IC) v ( q ( θ ) , θ ) − t ( θ ) � v ( q ( θ ′ ) , θ ) − t ( θ ′ ) 7 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS N UMERICAL A PPROACH We solve this constrained maximization problem numerically. Discretize type space with N grid points, θ ∈ { θ 1 , . . . , θ N } . 1 Reformulate the original problem to the discretized problem 2 N 1 � maximize q,t t ( θ i ) − C ( q ( θ i )) N i =1 v ( q ( θ i ) , θ i ) − t ( θ i ) � 0 ∀ i (IR) subject to v ( q ( θ i ) , θ i ) − t ( θ i ) � v ( q ( θ j ) , θ i ) − t ( θ j ) ∀ i, j (IC) Use KNITRO Active Set Algorithm to solve the discretized 3 problem. Increase N to improve the approximation. 4 8 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS 0.35 0.03 0.04 0.3 0.05 0.25 0.06 0.07 0.2 0.08 0.09 0.15 0.1 0.1 0.05 0 −0.05 2 2.2 2.4 2.6 2.8 3 F IGURE : v ( q, θ ) under different discretization schemes 9 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS 0.25 0.2 0.15 0.1 0.05 Analytic Solution Approximation with 101 grid points Approximation with 21 grid points 0 2 2.2 2.4 2.6 2.8 3 F IGURE : q ( θ ) under different discretization schemes 10 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS −3 x 10 1 0.03 0.04 0 0.05 0.06 0.07 −1 0.08 0.09 −2 0.1 Deviation −3 −4 −5 −6 −7 2 2.2 2.4 2.6 2.8 3 Type F IGURE : Approx. error for q ( θ ) under different discretization schemes 11 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS O UTLINE N ON -L INEAR P RICING IN M ONOPOLY M ARKET 1 A N E XAMPLE W ITH S INGLE C ROSSING 2 A N E XAMPLE W ITHOUT S INGLE C ROSSING 3 N UMERICAL E XPLORATIONS 4 Non-Uniform Distribution of Types Two Dimensional Types 12 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS A N E XAMPLE W ITHOUT S INGLE C ROSSING Values: v ( q, θ ) = θq − θ 2 q 2 , with θ ∼ U [2 , 3] and q � 0 . Cost: C ( q ) = 3 q 2 , c > 0 . Tariff: t � 0 . � 3 t ( θ ) − C ( q ( θ )) dF ( θ ) maximize q,t 2 v ( q ( θ ) , θ ) − t ( θ ) � 0 ∀ θ subject to v ( q ( θ ) , θ ) − t ( θ ) � v ( q ( θ ′ ) , θ ) − t ( θ ′ ) ∀ θ, θ ′ 13 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS A N E XAMPLE W ITHOUT S INGLE C ROSSING Values: v ( q, θ ) = θq − θ 2 q 2 , with θ ∼ U [2 , 3] and q � 0 . Cost: C ( q ) = 3 q 2 , c > 0 . Tariff: t � 0 . � 3 t ( θ ) − C ( q ( θ )) dF ( θ ) maximize q,t 2 v ( q ( θ ) , θ ) − t ( θ ) � 0 ∀ θ subject to v ( q ( θ ) , θ ) − t ( θ ) � v ( q ( θ ′ ) , θ ) − t ( θ ′ ) ∀ θ, θ ′ 13 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.5000 Profit: 0.200057 0.35 quantity tariff 0.30 profit utility 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 14 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.1000 Profit: 0.200376 0.35 quantity tariff 0.30 profit utility 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 15 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.0800 Profit: 0.200316 0.35 quantity tariff 0.30 profit utility 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 Types θ 16 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.0500 Profit: 0.200269 0.35 quantity tariff 0.30 profit utility 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 17 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.0400 Profit: 0.200284 0.35 quantity tariff 0.30 profit utility 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 18 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.0300 Profit: 0.198063 0.40 quantity tariff 0.35 profit utility 0.30 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 Types θ 19 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.0200 Profit: 0.195707 0.45 quantity tariff 0.40 profit 0.35 utility 0.30 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 20 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS Discret: 0.0100 Profit: 0.191614 0.45 quantity tariff 0.40 profit 0.35 utility 0.30 0.25 0.20 0.15 0.10 0.05 0.00 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 21 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS S OLUTIONS q q NSC ( θ ) CS − q SC ( θ ) CS + q 0 1 √ 2 6 q 2 q 1 1 √ 6 2 6 3 θ Figure 2.8: The decision an the marginal tari F IGURE : Vieira’s Isoperimetric Approach 22 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS O UTLINE N ON -L INEAR P RICING IN M ONOPOLY M ARKET 1 A N E XAMPLE W ITH S INGLE C ROSSING 2 A N E XAMPLE W ITHOUT S INGLE C ROSSING 3 N UMERICAL E XPLORATIONS 4 Non-Uniform Distribution of Types Two Dimensional Types 23 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS N ON -U NIFORM D ISTRIBUTION OF T YPES Experiments: Variations of Uniform Discretization More mass (grid points) near the beginning. More mass near the end. More mass at both ends. More mass in the middle. 24 / 41
N ON -L INEAR P RICING S INGLE C ROSSING N O S INGLE C ROSSING E XTENSIONS I MPLEMENTATION coarse grid distance 0.04, fine grid distance 0.02. 64 -76 variables, 1024 - 1444 constraints Multistart option - 100 runs KNITRO with Active Set algorithm Best solution chosen 25 / 41
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