The Solow Growth Model

The Solow model is presented as the foundational “workhorse model of much of macroeconomics.” Its simplicity allows for a clear exposition of the mechanics of capital accumulation and its role in growth.

Key Features:

• Environment: A one-good economy where output is produced using capital and labor via a constant-returns-to-scale production function satisfying standard neoclassical assumptions (positive and diminishing marginal products, Inada conditions).
• Exogenous Saving Rate: The model’s key simplifying assumption is a constant, exogenous saving rate (s). Households do not optimize their consumption-saving decisions.
• Law of Motion: The fundamental law of motion for the capital stock is K(t+1) = sF[K(t), L(t)] + (1-δ)K(t). In per-worker terms, this describes the evolution of the capital-labor ratio, k.
• Steady State: The economy converges to a unique, stable steady-state capital-labor ratio (k*) where investment (sf(k)) exactly offsets the capital “replenished” due to depreciation (δk), population growth (nk), and technological progress (gk).
• Transitional Dynamics: If an economy starts below its steady-state capital-labor ratio (k < k*), it experiences a period of positive growth as capital accumulates. This growth slows as the economy approaches the steady state. This phenomenon predicts “conditional convergence.”
• Source of Sustained Growth: The basic model cannot generate sustained per-capita growth. Long-run growth in living standards is only possible with the introduction of exogenous, continuous technological progress.

Uzawa’s Balanced Growth Path Theorem: A critical result is that for a model to have a balanced growth path (where capital, output, and consumption grow at constant rates and factor shares are constant), technological progress must be purely labor-augmenting (Harrod-neutral). This provides a strong theoretical justification for focusing on this specific form of technological change.

The Neoclassical (Ramsey-Cass-Koopmans) Model

This model provides microfoundations for the savings decisions that are taken as exogenous in the Solow model.

Key Features:

• Representative Household: The economy is modeled as having an infinitely-lived representative household that maximizes an intertemporal utility function, typically ∫ e^(-ρt) * [c(t)^(1-θ) – 1]/(1-θ) dt. This setup is justified by assumptions of intergenerational altruism or by modeling households with a constant probability of death (the “perpetual youth” model).
• Optimal Savings: The household’s optimal consumption path is characterized by the Euler equation: ċ(t)/c(t) = (1/θ)(r(t) – ρ). This equation dictates that consumption grows when the interest rate (r) is higher than the rate of time preference (ρ).
• Transversality Condition: In addition to the Euler equation, the optimal plan must satisfy a no-Ponzi-game or transversality condition, which prevents the household from accumulating infinite debt.
• Equilibrium Dynamics: The economy is described by a system of two differential equations for per-capita capital (k) and consumption (c). The equilibrium path involves consumption immediately jumping to a “stable arm” and then monotonically converging to the steady state.
• Modified Golden Rule: The steady state is characterized by the modified golden rule f'(k*) = δ + ρ, which differs from the Solow model’s golden rule (f'(k*) = δ + n) because of discounting (ρ). This state is always dynamically efficient.
• Source of Sustained Growth: Like the Solow model, the neoclassical model requires exogenous, labor-augmenting technological progress to generate sustained per-capita growth. The long-run growth rate is determined entirely by this exogenous rate of technological progress (g).

Overlapping Generations (OLG) Models

OLG models depart from the infinite-horizon representative household assumption by modeling an economy composed of agents with finite, overlapping lifetimes.

Key Features:

• Finite Lives: The defining feature is that agents live for a finite period (e.g., two periods: young and old). The economy’s population consists of overlapping cohorts.
• No Intergenerational Link (in baseline): In the basic Samuelson-Diamond model, there is no altruism, so each generation saves only for its own old-age consumption. This breaks the intergenerational link present in the Ramsey model.
• Potential for Dynamic Inefficiency: A key result is that OLG models can produce equilibria that are dynamically inefficient, meaning the steady-state capital stock is above the golden rule level (k* > k_gold). This corresponds to a situation where the steady-state interest rate is less than the population growth rate (r* < n). In such cases, a Pareto improvement is possible.
• Role for Social Security: Unfunded social security systems (where taxes on the young are transferred to the old) can solve the problem of overaccumulation. By providing an alternative means for old-age consumption, they reduce private savings, lower the capital stock, and can generate a Pareto improvement if the economy is dynamically inefficient.
• Perpetual Youth Model: A continuous-time variant (Yaari-Blanchard model) where agents face a constant probability of death (ν) at each instant is a tractable alternative that shares features of both the Ramsey and OLG models.