# An Empirical Analysis of Optimal Nonlinear Pricing in Business-to-Business Markets

## TL;DR

This paper develops an empirical method to estimate optimal nonlinear pricing strategies in B2B markets, demonstrating significant profit gains over linear pricing and analyzing the impact of various market factors.

## Contribution

It introduces a novel empirical approach for estimating optimal nonlinear price schedules considering multi-dimensional consumer heterogeneity in B2B markets.

## Key findings

- Optimal nonlinear pricing increases profit by at least 8.2% over linear pricing.
- Second-degree price discrimination recovers 7.1% of the profit gap to first-degree pricing.
- Demand and cost factors significantly influence the shape of the optimal price schedule.

## Abstract

In continuous-choice settings, consumers decide not only on whether to purchase a product, but also on how much to purchase. Thus, firms optimize a full price schedule rather than a single price point. This paper provides a methodology to empirically estimate the optimal schedule under multi-dimensional consumer heterogeneity with a focus on B2B applications. We apply our method to novel data from an educational-services firm that contains purchase-size information not only for deals that materialized, but also for potential deals that eventually failed. We show that this data, combined with identifying assumptions, helps infer how price sensitivity varies with "customer size". Using our estimated model, we show that the optimal second-degree price discrimination (i.e., optimal nonlinear tariff) improves the firm's profit upon linear pricing by at least 8.2%. That said, this second-degree price discrimination scheme only recovers 7.1% of the gap between the profitability of linear pricing and that of infeasible first degree price discrimination. We also conduct several further simulation analyses (i) empirically quantifying the magnitude by which incentive-compatibility constraints impact the optimal pricing and profits, (ii) comparing the role of demand- v.s. cost-side factors in shaping the optimal price schedule, and (iii) studying the implications of fixed fees for the optimal contract and profitability.