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A dynamic competitive equilibrium model of irreversible capacity investment with stochastic demand and heterogeneous producers

Constantinos Kardaras, Alexandros Pavlis, Mihail Zervos

TL;DR

The paper develops a continuous-time, dynamic competitive equilibrium model for irreversible capacity investment under stochastic demand with heterogeneous producers. By casting each firm’s problem as a three-dimensional singular stochastic control and solving a linked HJB equation with explicit free-boundary surfaces, it yields a Markovian equilibrium where prices are a nonlinear functional of exogenous base demand and investments follow robust, explicit feedback rules. The authors prove existence and uniqueness of the equilibrium, derive closed-form boundary characterizations, and establish asymptotic behavior to understand investment timing in high-demand regimes. This framework provides tractable, explicit strategies for capacity expansion in mean-field-like competitive settings with irreversible investment and stochastic demand.

Abstract

We formulate a continuous-time competitive equilibrium model of irreversible capacity investment in which a continuum of heterogeneous producers supplies a single non-durable good subject to exogenous stochastic demand. Each producer optimally adjusts both output and capacity over time in response to endogenous price signals, while investment decisions are irreversible. Market clearing holds continuously, with prices evolving endogenously to balance aggregate supply and demand through a constant-elasticity demand function driven by a stochastic base component. The model admits a mean-field interpretation, as each producer's decisions both influence and are influenced by the aggregate behaviour of all others. We show that the equilibrium price process can be expressed as a nonlinear functional of the exogenous base demand, leading to a three-dimensional singular stochastic control problem for each producer. We derive an explicit solution to the associated Hamilton-Jacobi-Bellman equation, including a closed-form characterisation of the free-boundary surface separating investment and waiting regions.

A dynamic competitive equilibrium model of irreversible capacity investment with stochastic demand and heterogeneous producers

TL;DR

The paper develops a continuous-time, dynamic competitive equilibrium model for irreversible capacity investment under stochastic demand with heterogeneous producers. By casting each firm’s problem as a three-dimensional singular stochastic control and solving a linked HJB equation with explicit free-boundary surfaces, it yields a Markovian equilibrium where prices are a nonlinear functional of exogenous base demand and investments follow robust, explicit feedback rules. The authors prove existence and uniqueness of the equilibrium, derive closed-form boundary characterizations, and establish asymptotic behavior to understand investment timing in high-demand regimes. This framework provides tractable, explicit strategies for capacity expansion in mean-field-like competitive settings with irreversible investment and stochastic demand.

Abstract

We formulate a continuous-time competitive equilibrium model of irreversible capacity investment in which a continuum of heterogeneous producers supplies a single non-durable good subject to exogenous stochastic demand. Each producer optimally adjusts both output and capacity over time in response to endogenous price signals, while investment decisions are irreversible. Market clearing holds continuously, with prices evolving endogenously to balance aggregate supply and demand through a constant-elasticity demand function driven by a stochastic base component. The model admits a mean-field interpretation, as each producer's decisions both influence and are influenced by the aggregate behaviour of all others. We show that the equilibrium price process can be expressed as a nonlinear functional of the exogenous base demand, leading to a three-dimensional singular stochastic control problem for each producer. We derive an explicit solution to the associated Hamilton-Jacobi-Bellman equation, including a closed-form characterisation of the free-boundary surface separating investment and waiting regions.

Paper Structure

This paper contains 10 sections, 7 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Individual optimality and market clearing
  3. Competitive production equilibrium
  4. The main result: a Markovian competitive production equilibrium
  5. The solution to equation (\ref{['eq.PbarP']})
  6. Asymptotic results
  7. A singular stochastic control problem and its HJB equation
  8. The solution to the control problem's HJB equation
  9. The solution to the control problem studied in Sections \ref{['sec:singular-HJB']} and \ref{['sec:HJB-sol']}
  10. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 4.2

Suppose that Assumptions A1 and A2 hold true. A competitive production equilibrium in the sense of Definition de:equil is given by (eq.PbarP*) and (eq.C*), or equivalently, by (eq-barX1) and (eq-barX2). Furthermore, $\mathscr I [\mathbb{\Psi}^\star]$ and $\mathscr I [\mathbb{\Phi}^\star]$, where $\m with initial condition

Figures (2)

  • Figure 1: Graph illustrating the surfaces $\mathcal{S}_i^1$ (purple lines) and $\mathcal{S}_i^2$ (red lines).
  • Figure 2: Graph illustrating the functions $G_i$ and $\Gamma_i$.

Theorems & Definitions (11)

  • Definition 3.2
  • Definition 3.3
  • Definition 4.1
  • Theorem 4.2
  • Remark 4.3
  • Lemma 5.1
  • Theorem 5.2
  • Lemma 6.1
  • Lemma 6.2
  • Lemma 8.1
  • ...and 1 more