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An integral transformation approach to differential games: a climate model application

Raouf Boucekkine, Giorgio Fabbri, Salvatore Federico, Fausto Gozzi, Ted Loch-Temzelides, Cristiano Ricci

TL;DR

The paper introduces the Integral Transformation Method (ITM), a flexible approach for solving dynamic optimization and differential games with linear-in-state structure by recasting the problem into a time-parametrized family of temporary optimizations, encapsulated by the forward-looking coefficient $\mathbf{b}(t)$. It demonstrates the method on an analytical two-country integrated assessment model, deriving Nash equilibria and efficiency benchmarks under normative planner settings and non-cooperative dynamics, and shows how the approach handles constraints, heterogeneity, and non-autonomous features. The work extends ITM to Knightian uncertainty via robust control, delivering tractable min-max formulations and, in the logarithmic payoff case, explicit insights into how robustness shifts policy choices and emissions trajectories. Collectively, the results offer a tractable, interpretation-friendly framework for policy analysis in climate economics, with implications for mean-field and more complex dynamic games where linear-state structure is preserved.

Abstract

We develop an Integral Transformation Method (ITM) for the study of suitable optimal control and differential game models. This allows for a solution to such dynamic problems to be found through solving a family of optimization problems parametrized by time. The method is quite flexible, and it can be used in several economic applications where the state equation and the objective functional are linear in a state variable. We illustrate the ITM in the context of a two-country integrated assessment climate model. We characterize emissions, consumption, transfers, and welfare by computing the Nash equilibria of the associated dynamic game. We then compare them to efficiency benchmarks. Further, we apply the ITM in a robust control setup, where we investigate how (deep) uncertainty affects climate outcomes.

An integral transformation approach to differential games: a climate model application

TL;DR

The paper introduces the Integral Transformation Method (ITM), a flexible approach for solving dynamic optimization and differential games with linear-in-state structure by recasting the problem into a time-parametrized family of temporary optimizations, encapsulated by the forward-looking coefficient . It demonstrates the method on an analytical two-country integrated assessment model, deriving Nash equilibria and efficiency benchmarks under normative planner settings and non-cooperative dynamics, and shows how the approach handles constraints, heterogeneity, and non-autonomous features. The work extends ITM to Knightian uncertainty via robust control, delivering tractable min-max formulations and, in the logarithmic payoff case, explicit insights into how robustness shifts policy choices and emissions trajectories. Collectively, the results offer a tractable, interpretation-friendly framework for policy analysis in climate economics, with implications for mean-field and more complex dynamic games where linear-state structure is preserved.

Abstract

We develop an Integral Transformation Method (ITM) for the study of suitable optimal control and differential game models. This allows for a solution to such dynamic problems to be found through solving a family of optimization problems parametrized by time. The method is quite flexible, and it can be used in several economic applications where the state equation and the objective functional are linear in a state variable. We illustrate the ITM in the context of a two-country integrated assessment climate model. We characterize emissions, consumption, transfers, and welfare by computing the Nash equilibria of the associated dynamic game. We then compare them to efficiency benchmarks. Further, we apply the ITM in a robust control setup, where we investigate how (deep) uncertainty affects climate outcomes.
Paper Structure (42 sections, 24 theorems, 184 equations, 8 figures, 1 table)

This paper contains 42 sections, 24 theorems, 184 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Relation to the Literature
  3. The Integral Transformation Method
  4. The single-player case: optimal control
  5. ITM versus standard dynamic optimization methods
  6. The $N$-player case: Nash equilibria
  7. Open-loop Nash equilibria
  8. Markov-Nash equilibria
  9. An analytical integrated assessment model
  10. Preferences and technology
  11. The climate model
  12. A special case
  13. Normative and positive investigations
  14. The Global planner's (GP) problem
  15. The restricted planner's (RP) problem
  16. ...and 27 more sections

Key Result

Proposition 2.2

Suppose Assumption hp:reduction holds and let $\mathbf{u}(\cdot) \in L^1_{\rho}(\mathbb{R}_+,\mathbf{U})$. Then the right-hand side being well-defined and finite. If, in addition, the map $t \mapsto h(t,\mathbf{u}(t))$ belongs to $L_{\rho}^1(\mathbb{R}_+)$, then the objective functional eq:OCPgeneral is well defined and finite, and it can be written as

Figures (8)

  • Figure 1: Absolute vs relative Nash inefficiency gap in terms of the final temperature when $\sigma$ varies.
  • Figure 2: Control variables $C_1,C_2,B_1,B_2,I_1,I_2,R_a,R_b$ in the Global and Restricted planner cases when $\rho_2 = 1.2 \rho_1$.
  • Figure 3: $\sigma = 1$ ($\log$), Total emissions.
  • Figure 4: $\sigma = 1$ ($\log$), Global temperature.
  • Figure 5: $\sigma = 1.2$, Total emissions
  • ...and 3 more figures

Theorems & Definitions (57)

  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4: Optimal control
  • proof
  • Proposition 2.6
  • Definition 2.7: Open-loop Nash equilibrium
  • Definition 2.8: Temporary Nash equilibrium
  • Theorem 2.9: Open-loop Nash equilibria
  • Definition 2.10: Markov-Nash equilibrium
  • ...and 47 more