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Pricing Problems in Adoption of New Technologies

Yijin Wang, Subhonmesh Bose

TL;DR

The paper develops a discrete-time, price-responsive diffusion model that extends Bass diffusion by incorporating price effects through a logistic adoption term, ensuring adoption fractions stay within [0,1]. It proves the model is compatible with utility-based agent behavior and demonstrates strong data fit across multiple product categories, including green technologies. Using dynamic programming, it derives a closed-form solution for the single-period optimal price and establishes structural properties for multi-period pricing under peer effects. It then formulates a Stackelberg game between a policymaker and a monopolist to study rebate design, proving the rebate is unique in the single-period case and showing how rebates influence adoption and pricing; these insights extend to multi-period scenarios through numerical simulations. Overall, the work provides a tractable framework for pricing and rebate design in diffusion-driven markets, with particular relevance to green-energy technologies.

Abstract

We propose a generalization of the Bass diffusion model in discrete-time that explicitly models the effect of price in adoption. Our model is different from earlier price-incorporated models and fits well to adoption data for various products. We then utilize this model to study two decision-making problems. First, we provide a series of structural results on optimal pricing strategies to maximize profits from product sales by a monopolist over a finite horizon. We fully characterize the optimal pricing strategy in the single-period problem, and establish several structural properties of the same for the multi-period counterpart. Second, we study a Stackelberg game between a policy-maker and a monopolist, where the former seeks to maximize adoption through rebates, while the latter focuses on profits. For this problem, we analytically characterize crucial properties of the equilibrium path of the single-period game, and demonstrate how they carry over to the multi-period variant.

Pricing Problems in Adoption of New Technologies

TL;DR

The paper develops a discrete-time, price-responsive diffusion model that extends Bass diffusion by incorporating price effects through a logistic adoption term, ensuring adoption fractions stay within [0,1]. It proves the model is compatible with utility-based agent behavior and demonstrates strong data fit across multiple product categories, including green technologies. Using dynamic programming, it derives a closed-form solution for the single-period optimal price and establishes structural properties for multi-period pricing under peer effects. It then formulates a Stackelberg game between a policymaker and a monopolist to study rebate design, proving the rebate is unique in the single-period case and showing how rebates influence adoption and pricing; these insights extend to multi-period scenarios through numerical simulations. Overall, the work provides a tractable framework for pricing and rebate design in diffusion-driven markets, with particular relevance to green-energy technologies.

Abstract

We propose a generalization of the Bass diffusion model in discrete-time that explicitly models the effect of price in adoption. Our model is different from earlier price-incorporated models and fits well to adoption data for various products. We then utilize this model to study two decision-making problems. First, we provide a series of structural results on optimal pricing strategies to maximize profits from product sales by a monopolist over a finite horizon. We fully characterize the optimal pricing strategy in the single-period problem, and establish several structural properties of the same for the multi-period counterpart. Second, we study a Stackelberg game between a policy-maker and a monopolist, where the former seeks to maximize adoption through rebates, while the latter focuses on profits. For this problem, we analytically characterize crucial properties of the equilibrium path of the single-period game, and demonstrate how they carry over to the multi-period variant.
Paper Structure (14 sections, 4 theorems, 80 equations, 12 figures, 1 table)

This paper contains 14 sections, 4 theorems, 80 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Product Adoption Model
  3. Adoption Dynamics Via Agent Utilities
  4. Model Evaluation Using Data
  5. The Optimal Pricing Problem
  6. Towards Solving $\mathcal{P}$
  7. Optimal Pricing with Low Peer Effects
  8. Optimal Price and Sales Paths
  9. Optimal Rebate Design via a Stackelberg Game
  10. Optimal Rebate in a One-Stage Game
  11. Numerical Simulations of the $T$-Period Game
  12. Conclusions
  13. Appendix: Proof of Proposition \ref{['prop:general']}
  14. Appendix: Proof of Theorem \ref{['thm:special case']}

Key Result

theorem 1

The optimal price at $t = T-1$ is where $W$ is the principal branch of the Lambert-W function.

Figures (12)

  • Figure 1: Trajectory of $F_t$ for a fixed price of $\pi = 3$, with $\alpha = 1$.
  • Figure 2: Model fit for GBM air conditioner data (top) and California solar prices (bottom).
  • Figure 3: Optimal Price for $T = 2$ for cases $q = 1.5$ (top) and $q = 5$ (bottom). The remaining parameters common to both cases are $p = 1, C = 1$.
  • Figure 4: Optimal pricing trajectories for $q = 1$ for different time periods.
  • Figure 5: Optimal pricing trajectories for $q = 5$ for different time periods.
  • ...and 7 more figures

Theorems & Definitions (4)

  • theorem 1
  • proposition 1
  • theorem 2
  • theorem 3