Existence and Calculation of Optimal Monetary Equilibria on Overlapping Generations Economies

Technical Summary: Existence and Calculation of Optimal Monetary Equilibria on Overlapping Generations Economies

1. The Problem
Overlapping generations (OLG) economies are characterized by a "double infinity" of households and dated commodities, leading to features distinct from finite general equilibrium models. Specifically, the First Welfare Theorem often fails, allowing for inefficient equilibria. While the Cass (1972) criterion provides a necessary and sufficient condition for efficiency (based on the asymptotic behavior of price sequences), it does not guarantee the existence of such efficient equilibria; indeed, Cass, Okuno, and Zilcha (1979) provided examples where they do not exist.

Closely related is the "Hahn problem" (Hahn, 1965), which concerns the existence of monetary equilibria (equilibria where the price of money is positive). Existing literature often resolves existence issues by imposing restrictive assumptions, such as stationarity, cash-in-advance constraints, transaction costs, or the presence of durable, dividend-paying assets (e.g., land). This paper addresses the gap in understanding the existence and calculation of optimal monetary equilibria in non-stationary, consumption-loan OLG economies that lack these auxiliary mechanisms (no durable assets, no transaction costs, no wealth transfers, and no cash-in-advance constraints).

2. Methodology and Model Setup
The paper constructs a discrete-time consumption-loan OLG economy $E$ with heterogeneous households living for two periods. A fundamental departure from classical models (e.g., Balasko and Shell, 1980; Okuno and Zilcha, 1980) is the treatment of the economy's inception. Instead of assuming the economy starts at $t=1$ with an "old" generation born at $t=0$ having no prior history, the author models the economy as a continuation of long-standing economic activity.

• Time Labeling: The model explicitly includes the younger period of life for generation $G_0$ at $t=0$.
• Equilibrium Definition: Market clearing is imposed for all periods $t \ge 1$. At $t=0$, market clearing is not required; instead, the aggregate demand of generation $G_0$ is bounded by demographic and productivity constraints. This setup allows the model to endogenize saving/borrowing decisions for $G_0$ without a regressum ad infinitum.
• Pareto Optimality: The set of Pareto optimal equilibria ($H^{PO}$) is defined via the Cass (1972) criterion: $\sum_{t=0}^{\infty} (H_t \|p_t\|)^{-1} = +\infty$.

3. Key Contributions and Results

• Existence of Optimal Monetary Equilibria:
  The primary theoretical contribution is the establishment of sufficient conditions for the existence of Pareto optimal monetary equilibria in non-stationary OLG economies without durable assets. The central assumption is that the economy must be "prone to savings" (Definition 5).

  • Prone-to-Savings Condition: An economy is prone to savings if low real savings per capita by a generation imply a low real return rate relative to the demographic growth rate. This condition is linked to skewed endowment distributions (favoring youth), high demographic growth, and preferences valuing old-age consumption.
  • Proposition 7: In prone-to-savings economies, efficient equilibria are fully characterized by non-vanishing real savings per capita. If real savings per capita converge to zero, the equilibrium is inefficient.
  • Theorems 8 and 9: Under Assumptions 1–5 (including the prone-to-savings condition), the set of Pareto optimal equilibria $H^{PO}$ is non-empty and compact. Furthermore, Theorem 9 demonstrates that these equilibria are monetary (the price of money is positive), thereby providing a solution to the Hahn problem for a broad class of non-stationary economies. This result circumvents the non-existence examples of Cass, Okuno, and Zilcha (1979) by excluding the specific assumptions (e.g., completely inflexible tastes or completely skewed endowments toward old age) that generate those counterexamples.

• Calculation Algorithm:
  The paper addresses the computational challenge of finding these equilibria in infinite-horizon, non-stationary settings.

  • Finitely Replicating Sequences: The author introduces a sequence of auxiliary economies $\{E_k\}_{k \ge 1}$ that coincide with the reference economy $E$ up to period $k$ but are replaced by "well-behaved tail economies" ($T_k$) thereafter.
  • Backward Calculation (Theorem 12): The paper proposes an algorithm that starts from the known set of Pareto optimal equilibria of the tail economies ($H^{PO}_{\tau_k}$) and uses equilibrium equations to move backward to period 0.
  • Convergence: Theorem 12 proves that the limit of the union of Pareto optimal sets from these finite approximations ($\lim_{j \to \infty} \bigcup_{k \ge j} H^{PO}_k$) is a non-empty compact subset of the reference economy's optimal equilibria ($H^{PO}$). This provides a rigorous method to approximate efficient monetary transition paths.

4. Significance and Claims
The paper claims to provide a theoretical foundation for modeling monetary policy and social security systems in economies undergoing demographic transitions (e.g., plummeting fertility rates) without relying on stationarity or artificial constraints like cash-in-advance rules.

• Theoretical Significance: It bridges the gap between the existence of efficient equilibria and the existence of monetary equilibria in non-stationary settings. It establishes that "prone-to-savings" economies naturally support efficient monetary equilibria, linking the Cass criterion directly to the dynamics of real savings per capita.
• Methodological Significance: The backward calculation algorithm offers a practical approach for computing efficient transition paths in non-stationary OLG models. It justifies the common practice in applied economics of truncating infinite horizons with well-behaved tails, showing that such approximations converge to efficient equilibria of the full model.
• Policy Relevance: The results suggest that monitoring real savings per capita is a critical indicator of economic efficiency. In prone-to-savings economies, a plummeting of real savings signals a departure from Pareto optimality.

The paper remains modest regarding its scope, noting that it focuses on deterministic, consumption-loan models and does not address stochastic or production economies. It does not propose specific policy prescriptions but rather provides the necessary existence and calculation tools to evaluate such systems within the OLG framework.