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Subsidizing a New Technology: An Impulse Stackelberg Game Approach

Utsav Sadana, Georges Zaccour

TL;DR

This paper develops an impulse dynamic Stackelberg game (iDSG) between a government and a profit-maximizing firm to subsidize zero-emission vehicle adoption. Subsidies are adjusted at discrete dates while the firm continuously optimizes price, with learning-by-doing shaping costs and a diffusion-like adoption dynamic. A verification theorem characterizes the Feedback Stackelberg equilibrium, and Riccati-type equations determine the time-varying value functions and pricing paths; numerical experiments reveal that subsidies boost adoption and consumer surplus but can enable strategic price increases by the firm, and learning-speed and diffusion parameters markedly influence policy costs and outcomes. The work provides a realistic, tractable framework for designing discrete, cost-aware subsidy programs that steer technology adoption while accounting for firm incentives and cost dynamics.

Abstract

Governments are motivated to subsidize profit-driven firms that manufacture zero-emission vehicles to ensure they become price-competitive. This paper introduces a dynamic Stackelberg game to determine the government's optimal subsidy strategy for zero-emission vehicles, taking into account the pricing decisions of a profit-maximizing firm. While firms have the flexibility to change prices continuously, subsidies are adjusted at specific time intervals. This is captured in our game formulation by using impulse controls for discrete-time interventions. We provide a verification theorem to characterize the Feedback Stackelberg equilibrium and illustrate our results with numerical experiments.

Subsidizing a New Technology: An Impulse Stackelberg Game Approach

TL;DR

This paper develops an impulse dynamic Stackelberg game (iDSG) between a government and a profit-maximizing firm to subsidize zero-emission vehicle adoption. Subsidies are adjusted at discrete dates while the firm continuously optimizes price, with learning-by-doing shaping costs and a diffusion-like adoption dynamic. A verification theorem characterizes the Feedback Stackelberg equilibrium, and Riccati-type equations determine the time-varying value functions and pricing paths; numerical experiments reveal that subsidies boost adoption and consumer surplus but can enable strategic price increases by the firm, and learning-speed and diffusion parameters markedly influence policy costs and outcomes. The work provides a realistic, tractable framework for designing discrete, cost-aware subsidy programs that steer technology adoption while accounting for firm incentives and cost dynamics.

Abstract

Governments are motivated to subsidize profit-driven firms that manufacture zero-emission vehicles to ensure they become price-competitive. This paper introduces a dynamic Stackelberg game to determine the government's optimal subsidy strategy for zero-emission vehicles, taking into account the pricing decisions of a profit-maximizing firm. While firms have the flexibility to change prices continuously, subsidies are adjusted at specific time intervals. This is captured in our game formulation by using impulse controls for discrete-time interventions. We provide a verification theorem to characterize the Feedback Stackelberg equilibrium and illustrate our results with numerical experiments.
Paper Structure (21 sections, 2 theorems, 65 equations, 13 figures, 1 algorithm)

This paper contains 21 sections, 2 theorems, 65 equations, 13 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model
  4. Feedback Stackelberg equilibrium
  5. After the end of the subsidy program
  6. Before the subsidy ends
  7. Numerical example
  8. Firm's strategic behavior.
  9. Cost and benefits.
  10. Sensitivity analysis
  11. Impact of the target value.
  12. Impact of learning speed.
  13. Impact of word of mouth.
  14. Impact of the discount factor
  15. Impact of subsidy adjustments.
  16. ...and 6 more sections

Key Result

Proposition 1

The price charged by the firm for $t\in (\tau_i, \tau_{i+1}), i\leq N$ is as follows: when $\rho ^{2}-2\rho w_{2}\geq 0$, we have

Figures (13)

  • Figure 1: Cumulative sales, equilibrium producer and consumer price, and equilibrium subsidy for the benchmark case.
  • Figure 2: Cumulative sales, producer and consumer price when target is varied from the benchmark case.
  • Figure 3: Equilibrium subsidy when target is varied from the benchmark case.
  • Figure 4: Cumulative sales, producer and consumer price when $b_2$ is varied from the benchmark case.
  • Figure 5: Equilibrium subsidy plan when $b_2$ is varied from the benchmark case.
  • ...and 8 more figures

Theorems & Definitions (3)

  • Definition 1
  • Proposition 1
  • Theorem 1