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The optimal growth model where a representative household maximizes discounted utility over an infinite horizon, endogenously determining the saving rate through an Euler equation.

Derivation

Step-by-step mathematical derivation with typeset equations and expandable detail.

Sections

The household optimization problem

The Ramsey-Cass-Koopmans model replaces the Solow model's exogenous saving rate with optimal saving by forward-looking households. A representative household maximizes lifetime utility $\int_0^\infty e^{-\rho t} u(c(t))\, dt$ subject to the economy's resource constraint, where $\rho > 0$ is the rate of time preference and $u(c)$ is a concave instantaneous utility function. The standard specification uses CRRA utility $u(c) = c^{1-\theta}/(1-\theta)$ with $\theta > 0$ governing the elasticity of intertemporal substitution ( $1/\theta$ ).

The economy has a neoclassical production function $f(k)$ with diminishing returns to capital per effective worker $k$ . Output is split between consumption and investment: $f(k) = c + \dot{k} + (n + g + \delta)k$ , where $n$ is population growth, $g$ is technology growth, and $\delta$ is depreciation. Unlike Solow, the household chooses the consumption path $c(t)$ optimally, trading off current enjoyment against future returns on saving. The transversality condition $\lim_{t \to \infty} e^{-\rho t} k(t) u'(c(t)) = 0$ ensures the household does not accumulate wealth without bound or die in debt.

\max_{\{c(t)\}} \int_0^{\infty} e^{-\rho t} \frac{c(t)^{1-\theta}}{1-\theta}\, dt

Household objective: maximize the discounted stream of CRRA utility over an infinite horizon, with $\rho$ as the discount rate and $\theta$ as the coefficient of relative risk aversion.

\dot{k} = f(k) - c - (n + g + \delta)k

Capital accumulation: output minus consumption and break-even investment. The household's saving decision determines how much of $f(k)$ goes to investment.

The Euler equation for consumption

Applying the Pontryagin maximum principle (or a Hamiltonian approach), the first-order conditions yield the Euler equation for consumption growth. The household equates the marginal rate of substitution between consumption at $t$ and $t + dt$ to the rate of return on capital. When the net marginal product of capital $f'(k) - \delta$ exceeds the effective discount rate $\rho + \theta g$ , the return to saving is high, and the household tilts consumption toward the future. When it falls below, consumption is front-loaded.

The parameter $\theta$ controls how responsive consumption growth is to the interest rate. A high $\theta$ (low intertemporal elasticity of substitution) means the household strongly dislikes consumption variability, so even a large gap between the interest rate and the discount rate produces only modest consumption tilting. A low $\theta$ means the household readily shifts consumption across time in response to return differentials. The Euler equation, together with the capital accumulation equation, forms a two-dimensional dynamical system in $(k, c)$ space.

\frac{\dot{c}}{c} = \frac{1}{\theta}\bigl(f'(k) - \delta - \rho\bigr)

Euler equation (per effective worker, with $g = 0$ for clarity): consumption grows when the net marginal product of capital exceeds the discount rate, scaled by the intertemporal elasticity $1/\theta$ .

\frac{\dot{c}}{c} = \frac{1}{\theta}\bigl(f'(k) - \delta - \rho - \theta g\bigr)

General Euler equation with technology growth: the effective discount rate includes the term $\theta g$ reflecting the desire to smooth consumption per effective worker.

Phase diagram and saddle-path stability

The system $\{\dot{k}, \dot{c}\}$ defines a phase diagram in $(k, c)$ space. The $\dot{k} = 0$ locus is the set of points where consumption exactly absorbs output net of break-even investment: $c = f(k) - (n + g + \delta)k$ . The $\dot{c} = 0$ locus is the vertical line at $k^*$ where $f'(k^*) = \delta + \rho + \theta g$ . These two loci divide the phase plane into four regions with distinct directional arrows.

The steady state $(k^*, c^*)$ is a saddle point: there is exactly one path (the saddle path) that converges to it. All other paths eventually violate either the transversality condition (capital grows without bound) or feasibility (consumption hits zero). The economy must jump to the saddle path at $t = 0$ , which uniquely pins down the initial consumption level $c(0)$ given the initial capital stock $k(0)$ . This saddle-path property is what makes the model determinate: the forward-looking household's optimal behavior selects the unique stable trajectory.

\dot{k} = 0 \;\Rightarrow\; c = f(k) - (n + g + \delta)k

The $\dot{k} = 0$ locus: a hump-shaped curve showing the consumption level that exactly absorbs net output at each $k$ .

\dot{c} = 0 \;\Rightarrow\; f'(k^*) = \delta + \rho + \theta g

The $\dot{c} = 0$ locus: a vertical line at the capital stock where the marginal product of capital equals the effective discount rate.

The modified golden rule

At the steady state $k^*$ , the marginal product of capital satisfies $f'(k^*) = \rho + \delta + \theta g$ . This is the modified golden rule: unlike the Solow golden rule ( $f'(k) = n + g + \delta$ ), the Ramsey steady state accounts for the household's impatience ( $\rho$ ) and its preference for smooth consumption growth ( $\theta g$ ). Because $\rho > 0$ , the modified golden rule capital stock is below the golden rule level, ensuring the economy is dynamically efficient.

Steady-state consumption per effective worker is $c^* = f(k^*) - (n + g + \delta)k^*$ , which is the vertical distance between the production function and the break-even line evaluated at $k^*$ . Since $k^*$ lies below the golden rule, consumption is not maximized but the economy is on a path consistent with optimal intertemporal trade-offs. The household voluntarily accepts lower steady-state consumption in exchange for higher consumption along the transition (because it is impatient). The key comparative statics are: higher $\rho$ lowers $k^*$ (more impatient households save less); higher $\theta$ also lowers $k^*$ (households that dislike consumption variability save less when growth is positive).

f'(k^*) = \rho + \delta + \theta g

Modified golden rule: the optimal steady-state capital stock equates the marginal product of capital to the sum of the discount rate, depreciation, and the consumption-smoothing adjustment $\theta g$ .

c^* = f(k^*) - (n + g + \delta)k^*

Steady-state consumption per effective worker: output net of break-even investment at the modified golden rule capital stock.

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