Nonlinear Pricing with Finite Information

38 Pages Posted: 24 Jan 2015

See all articles by Dirk Bergemann

Dirk Bergemann

Yale University - Cowles Foundation - Department of Economics; Yale University - Cowles Foundation

Ji Shen

London School of Economics & Political Science (LSE)

Yun Xu

Yale University

Edmund M. Yeh

Northeastern University

Date Written: January 22, 2015

Abstract

We analyze nonlinear pricing with finite information. A seller offers a menu to a continuum of buyers with a continuum of possible valuations. The menu is limited to offering a finite number of choices representing a finite communication capacity between buyer and seller.

We identify necessary conditions that the optimal finite menu must satisfy, either for the socially efficient or for the revenue-maximizing mechanism. These conditions require that information be bundled, or "quantized" optimally. We show that the loss resulting from using the n-item menu converges to zero at a rate proportional to 1 = n^2.

We extend our model to a multi-product environment where each buyer has preferences over a d dimensional variety of goods. The seller is limited to offering a finite number n of d-dimensional choices. By using repeated scalar quantization, we show that the losses resulting from using the d-dimensional n-class menu converge to zero at a rate proportional to d = n^{2/d}. We introduce vector quantization and establish that the losses due to finite menus are significantly reduced by offering optimally chosen bundles.

Keywords: Mechanism design, Nonlinear pricing, Multi-Dimension, Multi-product, Private information, Limited information, Quantization, Information theory

Suggested Citation

Bergemann, Dirk and Shen, Ji and Xu, Yun and Yeh, Edmund M., Nonlinear Pricing with Finite Information (January 22, 2015). Cowles Foundation Discussion Paper No. 1981. Available at SSRN: https://ssrn.com/abstract=2553980 or http://dx.doi.org/10.2139/ssrn.2553980

Dirk Bergemann (Contact Author)

Yale University - Cowles Foundation - Department of Economics ( email )

28 Hillhouse Ave
New Haven, CT 06520-8268
United States
203-432-3592 (Phone)
203-432-2128 (Fax)

HOME PAGE: http://www.econ.yale.edu/~dirk/

Yale University - Cowles Foundation

Box 208281
New Haven, CT 06520-8281
United States

Ji Shen

London School of Economics & Political Science (LSE) ( email )

Houghton Street
London, WC2A 2AE
United Kingdom

Yun Xu

Yale University ( email )

New Haven, CT 06520
United States

Edmund M. Yeh

Northeastern University ( email )

Boston, MA 02115
United States

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