4.1 The Ramsey Model: Optimal Growth with Endogenizing Savings

While the Solow model provides a powerful explanation for the roles of capital and exogenous technology, its assumption of a constant saving rate is a significant theoretical shortcut. We now rectify this by developing the Ramsey-Cass-Koopmans (RCK) model, also known as the neoclassical growth model. This model addresses the primary limitation of the Solow framework: its reliance on a mechanical and exogenous saving rate. In reality, saving and consumption decisions are made by forward-looking individuals and households who weigh the trade-offs between present satisfaction and future well-being. The RCK model builds the theory of capital accumulation from first principles by replacing the constant saving rate with an explicit optimization problem solved by a forward-looking, infinitely-lived representative household.

The central agent in the Ramsey model is a representative household whose objective is to maximize a discounted stream of utility derived from per capita consumption. The household’s objective function is: ∫ exp(-(ρ-n)t) u(c(t)) dt Here, c(t) is consumption per capita, ρ is the rate of time preference (a measure of impatience), and n is the population growth rate. The instantaneous utility function u(c) is typically assumed to have a Constant Relative Risk Aversion (CRRA) form: u(c) = (c^(1-θ)-1)/(1-θ), where θ is the inverse of the intertemporal elasticity of substitution. The assumption of an “infinitely-lived” household is a theoretical convenience, representing a dynasty of altruistic individuals who care about the welfare of their descendants.

To find the optimal path for the economy, we can solve a social planner’s problem, which is equivalent to the competitive equilibrium in this environment. The planner maximizes the representative household’s utility subject to the economy’s aggregate resource constraint, which dictates how output is allocated between consumption and investment: k̇(t) = f(k(t)) – (n+δ)k(t) – c(t)

Using the tools of optimal control theory (specifically, a current-value Hamiltonian), we can derive the first-order necessary conditions that characterize the optimal path. This yields two key dynamic equations that govern the economy’s evolution:

1. The Euler Equation: This equation describes the optimal path of consumption over time. ċ(t)/c(t) = (1/θ)(f'(k(t)) – δ – ρ) This is the core consumption-smoothing equation of modern macroeconomics. It states that per capita consumption will grow if the net return to capital (f'(k)-δ) is greater than the household’s rate of time preference (ρ). If the return on saving exceeds the rate of impatience, the household will defer consumption, leading to a rising consumption path.
2. The Resource Constraint: This is the law of motion for the capital-labor ratio, k̇(t), as stated above.

The steady state of the Ramsey model is where both consumption and capital per capita are constant (ċ=0 and k̇=0). The condition ċ=0 from the Euler equation implies that the steady-state capital stock k* must satisfy the modified golden rule: f'(k*) = δ + ρ In the steady state, the marginal product of capital, net of depreciation, must equal the rate of time preference. This contrasts with the Solow model’s golden rule (f'(k) = n+δ in continuous time). The Ramsey economy avoids “dynamic inefficiency” (over-accumulation) precisely because of this forward-looking optimization. Impatient households (ρ > 0) are unwilling to sacrifice current consumption to push the capital stock to a point where its net marginal return, f'(k)-δ, falls below their rate of time preference ρ.

The transitional dynamics of the Ramsey model can be analyzed using a phase diagram in (k, c) space. This analysis reveals the existence of a unique saddle path. For any given initial capital stock k(0), there is only one initial level of consumption c(0) that places the economy on this stable path. This instantaneous “jump” of consumption onto the saddle path reflects the perfect foresight of the representative household. Any other initial level of consumption would set the economy on a trajectory that is either suboptimal (violating the transversality condition by accumulating useless capital in the long run) or infeasible (violating the resource constraint by depleting capital too quickly). Therefore, rational, forward-looking behavior requires that consumption immediately “jumps” to the saddle path, after which the economy converges smoothly to the steady state.

In summary, the Ramsey model provides a fully micro-founded and internally consistent theory of saving and capital accumulation. However, like the Solow model, it still cannot generate long-run per capita growth without relying on an unexplained, exogenous rate of technological progress (g). This shared limitation sets the stage for theories where the engine of growth is brought inside the model.

4.2 Endogenous Growth: The AK Model

The defining feature of neoclassical growth models like Solow and Ramsey is the assumption of diminishing returns to capital. As an economy accumulates more capital relative to labor, the marginal product of capital falls, eventually choking off growth that relies on capital accumulation alone. The simplest way to generate sustained, endogenous growth is to abandon this assumption. The AK model does precisely this by positing a linear production function that exhibits constant, rather than diminishing, returns to the accumulable factor.

The aggregate production function in the AK model is simply: Y = AK Here, K is interpreted as a broad measure of capital that includes not just physical capital but also human capital, knowledge, and public infrastructure. The parameter A is a constant that represents the total productivity of the economy. In this formulation, the marginal product of capital is constant and equal to A, which violates the standard Inada conditions.

Let’s embed this production function into a decentralized economy with a representative household whose preferences are of the CRRA form. In equilibrium, the competitive interest rate r will equal the marginal product of capital net of depreciation, so r = A – δ. Since A is constant, the interest rate is also constant. We can substitute this constant interest rate into the standard consumer Euler equation (g = (1/θ)(r – ρ)) to find the equilibrium growth rate of the economy: g* = (1/θ)(A – δ – ρ)

This result is remarkable for its simplicity and its profound implications. In stark contrast to the neoclassical models, the long-run growth rate g* in the AK model is endogenous. It depends on:

• Technology and efficiency (A)
• Household preferences (impatience ρ and desire for consumption smoothing θ)
• Economic policy (e.g., a tax on capital returns would lower A and thus reduce g*)

This framework provides a powerful mechanism for explaining large, persistent differences in income levels across countries. Small differences in policies or preferences can translate into small differences in growth rates, which, when compounded over long periods, lead to enormous disparities in economic outcomes.

The simple AK model has a significant counterfactual prediction: because labor is not in the production function, the model implies that the share of capital in national income should be 100%. To address this, more plausible two-sector endogenous growth models have been developed. One version features separate accumulation equations for physical and human capital, creating a structure that behaves like an AK model in the long run but allows for a positive labor share. Another, developed by Sergio Rebelo, uses two sectors—one for producing consumption goods and another for producing investment goods—with different capital intensities. These more sophisticated models retain the core insight of the AK framework—that constant returns to an accumulable factor can sustain growth—while generating more realistic predictions about income distribution.

While AK-style models demonstrate how endogenous growth is possible by eliminating diminishing returns, they do not explain the economic origin of these non-diminishing returns. This provides the motivation for the next generation of models, where growth is driven by the purposeful, profit-seeking pursuit of technological innovation.