Nonlinear Pricing without Single Crossing Dmitri Blueschke Guilherme - - PowerPoint PPT Presentation

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Nonlinear Pricing without Single Crossing Dmitri Blueschke Guilherme - - PowerPoint PPT Presentation

Oct 12, 2022 14 likes •460 views

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

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

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

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

OUTLINE

1

NON-LINEAR PRICING IN MONOPOLY MARKET

2

AN EXAMPLE WITH SINGLE CROSSING

3

AN EXAMPLE WITHOUT SINGLE CROSSING

4

NUMERICAL EXPLORATIONS

Non-Uniform Distribution of Types Two Dimensional Types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

OUTLINE

1

NON-LINEAR PRICING IN MONOPOLY MARKET

2

AN EXAMPLE WITH SINGLE CROSSING

3

AN EXAMPLE WITHOUT SINGLE CROSSING

4

NUMERICAL EXPLORATIONS

Non-Uniform Distribution of Types Two Dimensional Types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

GENERAL SETUP

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 maximizeq,t θ

θ

t(θ) − C(q(θ))dF(θ) subject to v(q(θ), θ) − t(θ) 0 ∀θ ∈ Θ (IR) v(q(θ), θ) − t(θ) v(q(θ′), θ) − t(θ′) ∀θ, θ′ ∈ Θ (IC)

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

THE ROLE OF SINGLE-CROSSING

Definition: vq 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).

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

THE ROLE OF SINGLE-CROSSING

Definition: vq 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).

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

OUTLINE

1

NON-LINEAR PRICING IN MONOPOLY MARKET

2

AN EXAMPLE WITH SINGLE CROSSING

3

AN EXAMPLE WITHOUT SINGLE CROSSING

4

NUMERICAL EXPLORATIONS

Non-Uniform Distribution of Types Two Dimensional Types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

AN EXAMPLE WITH SINGLE CROSSING

Values: v(q, θ) = θ√q, with θ ∼ U[2, 3] and q 0. Cost: C(q) = cq, c > 0. Tariff: t 0.

maximizeq,t 3

2

t(θ) − C(q(θ))dF(θ) subject to v(q(θ), θ) − t(θ) 0 ∀θ (IR) v(q(θ), θ) − t(θ) v(q(θ′), θ) − t(θ′) ∀θ, θ′ (IC)

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

AN EXAMPLE WITH SINGLE CROSSING

Values: v(q, θ) = θ√q, with θ ∼ U[2, 3] and q 0. Cost: C(q) = cq, c > 0. Tariff: t 0.

maximizeq,t 3

2

t(θ) − C(q(θ))dF(θ) subject to v(q(θ), θ) − t(θ) 0 ∀θ (IR) v(q(θ), θ) − t(θ) v(q(θ′), θ) − t(θ′) ∀θ, θ′ (IC)

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

NUMERICAL APPROACH

We solve this constrained maximization problem numerically.

1

Discretize type space with N grid points, θ ∈ {θ1, . . . , θN}.

2

Reformulate the original problem to the discretized problem maximizeq,t 1 N

N

  • i=1

t(θi) − C(q(θi)) subject to v(q(θi), θi) − t(θi) 0 ∀i (IR) v(q(θi), θi) − t(θi) v(q(θj), θi) − t(θj) ∀i, j (IC)

3

Use KNITRO Active Set Algorithm to solve the discretized problem.

4

Increase N to improve the approximation.

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2 2.2 2.4 2.6 2.8 3 −0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

FIGURE: v(q, θ) under different discretization schemes

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2 2.2 2.4 2.6 2.8 3 0.05 0.1 0.15 0.2 0.25 Analytic Solution Approximation with 101 grid points Approximation with 21 grid points

FIGURE: q(θ) under different discretization schemes

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2 2.2 2.4 2.6 2.8 3 −7 −6 −5 −4 −3 −2 −1 1 x 10

−3

Type Deviation 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

FIGURE: Approx. error for q(θ) under different discretization schemes

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

OUTLINE

1

NON-LINEAR PRICING IN MONOPOLY MARKET

2

AN EXAMPLE WITH SINGLE CROSSING

3

AN EXAMPLE WITHOUT SINGLE CROSSING

4

NUMERICAL EXPLORATIONS

Non-Uniform Distribution of Types Two Dimensional Types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

AN EXAMPLE WITHOUT SINGLE CROSSING

Values: v(q, θ) = θq − θ2q2, with θ ∼ U[2, 3] and q 0. Cost: C(q) = 3q2, c > 0. Tariff: t 0.

maximizeq,t 3

2

t(θ) − C(q(θ))dF(θ) subject to v(q(θ), θ) − t(θ) 0 ∀θ v(q(θ), θ) − t(θ) v(q(θ′), θ) − t(θ′) ∀θ, θ′

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

AN EXAMPLE WITHOUT SINGLE CROSSING

Values: v(q, θ) = θq − θ2q2, with θ ∼ U[2, 3] and q 0. Cost: C(q) = 3q2, c > 0. Tariff: t 0.

maximizeq,t 3

2

t(θ) − C(q(θ))dF(θ) subject to v(q(θ), θ) − t(θ) 0 ∀θ v(q(θ), θ) − t(θ) v(q(θ′), θ) − t(θ′) ∀θ, θ′

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Discret: 0.5000 Profit: 0.200057 quantity tariff profit utility 14 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Discret: 0.1000 Profit: 0.200376 quantity tariff profit utility 15 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Discret: 0.0800 Profit: 0.200316 quantity tariff profit utility 16 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Discret: 0.0500 Profit: 0.200269 quantity tariff profit utility 17 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Discret: 0.0400 Profit: 0.200284 quantity tariff profit utility 18 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Discret: 0.0300 Profit: 0.198063 quantity tariff profit utility 19 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

Discret: 0.0200 Profit: 0.195707 quantity tariff profit utility 20 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

2.0 2.2 2.4 2.6 2.8 3.0 3.2 Types θ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

Discret: 0.0100 Profit: 0.191614 quantity tariff profit utility 21 / 41

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SOLUTIONS

q

CS+ CS− q 0 √ 6

1 2 √ 6 1 6 2

3 q1 q2 θ qSC(θ) qNSC(θ) Figure 2.8: The decision an the marginal tari

FIGURE: Vieira’s Isoperimetric Approach

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

OUTLINE

1

NON-LINEAR PRICING IN MONOPOLY MARKET

2

AN EXAMPLE WITH SINGLE CROSSING

3

AN EXAMPLE WITHOUT SINGLE CROSSING

4

NUMERICAL EXPLORATIONS

Non-Uniform Distribution of Types Two Dimensional Types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

NON-UNIFORM DISTRIBUTION OF TYPES

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.

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

IMPLEMENTATION

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

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

GENERAL FINDINGS

Similar total profit. Similar pooling equilibrium near the end. Non-monotonic q − θ relationship near the beginning. Monotonic u − θ relationship.

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NON-UNIFORM DISCRETIZATION OF TYPES

0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Uniform Left Right Twin Peaks Single Peak

θ = 2 θ1 = 2.28 θ2 = 2.72 θ = 3

FIGURE: q(θ) for different distributions of types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

NON-UNIFORM DISCRETIZATION OF TYPES

0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 Uniform Left Right Twin Peaks Single Peak

θ = 2 θ1 = 2.28 θ2 = 2.72 θ = 3

FIGURE: T(q(θ)) for different distributions of types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

NON-UNIFORM DISCRETIZATION OF TYPES

0.16 0.165 0.17 0.175 0.18 0.185 0.19 0.195 0.2 Uniform Left Right Twin Peaks Single Peak

θ = 2 θ1 = 2.28 θ2 = 2.72 θ = 3

FIGURE: T(q(θ)) − C(q(θ)) for different distributions of types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

NON-UNIFORM DISCRETIZATION OF TYPES

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Uniform Left Right Twin Peaks Single Peak

θ = 2 θ1 = 2.28 θ2 = 2.72 θ = 3

FIGURE: v(q(θ), θ) − T(q(θ)) for different distributions of types

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

TWO DIMENSIONAL TYPES

Similar setup Values: v(q, θ) = θq − θ2q2, with θ ∼ U[2, 3] and q 0. But now: v(q, a, b) = aq − bq2 with (a, b) ∼ U[2, 3]2 Implications:

No single crossing again Utility not monotone in types Representation results do not hold - no analytical solution

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

UTILITY FUNCTIONS

0.5 1 1.5 2 2.5 3 2 2.5 3 −8 −6 −4 −2 2 quantity b−Type

FIGURE: Utility

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

BENCHMARK: FULL INFORMATION

Principal maximizes social surplus ...and takes it all.

2 2.5 3 2 2.5 3 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 b−Types a−Types

FIGURE: Optimal Supply Schedule

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

COMPUTATION - ISSUES

Larger Dimension (d’oh!)

Number of possible policies explodes Cannot rely on local incentive compatibility Number of IC constraints explodes

Small feasible region relative to action space

Hard to find a feasible starting point Algorithm often gets stuck

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

COMPUTATION - AMPL TO THE RESCUE!

49 grid points KNITRO with Active Set algorithm 98 variables, 2401 constraints Multistart option - 50 runs Best solution chosen

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

AMPL OUTPUT

Final objective value = 1.88400489913399e+01 Final feasibility error (abs / rel) = 5.50e-13 / 7.00e-14 Final optimality error (abs / rel) = 1.01e-08 / 5.78e-09

When things go well:

Iter Objective FeasError OptError ||Step|| CGits

  • ------- -------------- ---------- ---------- ---------- -------

446 1.883564e+01 3.600e-08 6.101e-01 3.056e-02 2 447 1.883601e+01 3.324e-10 6.422e-01 1.910e-03 6 448 1.883670e+01 1.220e-09 5.466e+00 5.759e-03 1 449 1.883723e+01 5.542e-09 1.527e+00 5.040e-03 4 450 1.883764e+01 9.788e-11 6.268e-01 5.039e-03 1 451 1.883828e+01 4.610e-09 3.196e-02 8.818e-03 3 452 1.883923e+01 1.554e-08 3.980e-02 1.764e-02 1 453 1.883979e+01 1.669e-08 4.290e-02 1.764e-02 1 454 1.883992e+01 2.584e-08 2.794e-02 1.180e-02 5 455 1.884004e+01 2.525e-08 2.867e-02 1.179e-02 2 456 1.884005e+01 8.475e-10 1.118e-02 4.480e-03 9 457 1.884005e+01 4.582e-12 3.906e-05 1.363e-05 7 458 1.884005e+01 2.220e-15 9.310e-08 2.196e-08 6

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WHEN THINGS GO BAD..

Iter Objective FeasError OptError ||Step|| CGits

  • ------- -------------- ---------- ---------- ---------- -------

0 -4.821871e+01 1.046e+01 1 -6.597676e+00 5.046e+00 9.596e+03 6.716e+00 2 2 6.938318e+00 4.733e+00 1.434e+05 9.334e+00 1 3 -5.364320e+01 3.293e+00 7.311e+04 1.598e+01 2 4 -7.743043e+01 1.711e+00 1.033e+05 6.115e+00 1 5 -5.054798e+01 1.481e-01 1.386e+02 2.865e+00 0 6 -4.105956e+01 7.316e-02 4.901e+01 8.087e-01 0 7 -3.547193e+01 1.743e-02 1.398e+01 4.493e-01 1 8 -3.303212e+01 4.191e-03 1.466e+01 2.151e-01 2 9 -3.110397e+01 5.802e-03 2.018e+01 2.646e-01 1 10 -2.808023e+01 2.913e-04 9.807e+00 2.665e-01 2 11 -2.663157e+01 5.511e-05 1.125e+01 1.436e-01 2 12 -2.593095e+01 1.710e-05 4.918e+01 7.349e-02 2 13 -2.591876e+01 1.687e-05 4.895e+00 1.148e-03 3 14 -2.591589e+01 9.421e-06 8.968e+00 2.871e-04 8 15 -2.590045e+01 3.179e-08 9.182e+00 1.426e-03 1 16 -2.590009e+01 3.119e-08 1.756e+00 4.975e-05 1 17 -2.589618e+01 7.424e-09 4.210e+00 3.809e-04 1 18 -2.589618e+01 2.366e-09 8.687e+00 1.387e-04 3 19 -2.589313e+01 2.970e-09 3.071e+00 2.774e-04 1 20 -2.588970e+01 4.187e-09 3.685e+00 3.334e-04 1 21 -2.588957e+01 4.226e-09 1.428e+00 1.248e-05 1 22 -2.588956e+01 4.412e-09 2.508e-02 5.990e-07 1 37 / 41

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MORE GRAPHS!

2 2.2 2.4 2.6 2.8 3 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 b−Types a−Types

FIGURE: Optimal Supply Schedule with Types

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DISTORTIONS

2 2.5 3 2 2.5 3 0.05 0.1 0.15 0.2 0.25 a−Types b−Types

FIGURE: Supply w. Types vs Supply with full info

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SENSE AND SENSITIVITY

2 2.2 2.4 2.6 2.8 3 2 2.5 3 −0.2 −0.15 −0.1 −0.05 b−Types a−Types

FIGURE: Type (2.6,2.6) - sample IC constraint

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NON-LINEAR PRICING SINGLE CROSSING NO SINGLE CROSSING EXTENSIONS

SENSE AND SENSITIVITY II

2 2.2 2.4 2.6 2.8 3 2 2.5 3 0.1 0.2 0.3 0.4 b−Types a−Types

FIGURE: Slack in IR constraint by type (mind the supply distortion!)

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