Growth Theory I

The Ramsey-Cass-Koopmans Model, Decentralization, and Fiscal Policy

Alan Lujan

Johns Hopkins University

April 7, 2026

Why Study Growth?

Growth Is the Central Question

Small differences in growth rates, sustained over decades, produce enormous differences in living standards. A country growing at 2 percent per year doubles its income every 35 years; at 1 percent, doubling takes 70 years.

Two questions frame the theory:

1. What determines the transition path? If an economy starts below its steady state, how fast does it converge?
2. What determines the long-run growth rate? Is it an exogenous force (technology) or something the economy can influence?

The Ramsey-Cass-Koopmans model answers the first question. Next lecture tackles the second.

Kaldor’s Stylized Facts

Nicholas Kaldor (1961) identified six regularities of growing economies:

1. Output per worker grows at a roughly constant rate
2. The capital-to-output ratio is roughly constant
3. The real return on capital is roughly constant
4. The shares of capital and labor in national income are roughly constant
5. Real wages grow at the same rate as output per worker
6. Wide differences in growth rates across countries persist

A successful growth model should be consistent with all six facts in steady state.

The Ramsey-Cass-Koopmans Model

Production Technology

The economy has a perfectly competitive production sector with a Cobb-Douglas aggregate production function:

\[Y = K^\alpha (\Theta L)^{1-\alpha}\]

where \(K\) is capital, \(L\) is labor, and \(\Theta\) is labor productivity.

• Population grows at rate \(\dot{L}/L = n\); productivity grows at rate \(\dot{\Theta}/\Theta = g\)
• \(\Theta L\) is the number of efficiency units of labor
• This is the Harrod-neutral (labor-augmenting) form: \(\Theta\) multiplies labor, not capital or output. Harrod-neutral is the only specification consistent with balanced growth for general production functions

A Convention on Notation

All Roman variables (\(Y\), \(K\), \(C\), \(L\)) are functions of time, but putting time subscripts on everything clutters the notation. We include subscripts only where needed for clarity.

Greek letters like \(\alpha\) (capital share) and \(\delta\) (depreciation) are constants whose values never change.

Notation: \(\dot{X} \equiv dX/dt\) is the time derivative; \(\dot{X}/X = d(\ln X)/dt\) is the growth rate of \(X\).

The growth rate \(g\) is labor-augmenting (Harrod-neutral) progress.

Intensive Form

Define lowercase variables as the uppercase version divided by efficiency units. For output per efficiency unit:

\[y = Y / (\Theta L) = k^\alpha\]

where \(k = K / (\Theta L)\) is capital per efficiency unit.

Why normalize this way? In steady state, \(k\) is constant even though \(K\), \(Y\), and \(C\) all grow. All the action in the transition shows up as changes in \(k\).

Deriving the Law of Motion for \(k\)

Aggregate capital accumulates according to \(\dot{K} = Y - C - \delta K\). Since \(k = K/(\Theta L)\), apply the quotient rule:

\[\dot{k} = \frac{\dot{K}}{\Theta L} - k \left( \frac{\dot{\Theta}}{\Theta} + \frac{\dot{L}}{L} \right) = \frac{\dot{K}}{\Theta L} - (g + n) k\]

Dividing \(\dot{K} = Y - C - \delta K\) by \(\Theta L\) and collecting terms:

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

Capital Accumulation in Efficiency Units

The law of motion for capital per efficiency unit is:

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

A steady state is a point where \(\dot{k} = 0\).

The Golden Rule

Setting \(\dot{k} = 0\), steady-state consumption is \(c = f(k) - (g + n + \delta) k\). Maximizing over \(k\):

\[\frac{\partial c}{\partial k} = f'(k) - (g + n + \delta) = 0\]

\[\boxed{f'(k^{GR}) = g + n + \delta}\]

The golden rule capital stock, identified by Phelps (1961), maximizes steady-state consumption per efficiency unit. The marginal product of capital exactly covers dilution from growth, population, and depreciation.

The Social Planner’s Problem

A social planner maximizes the discounted sum of CRRA utility from per-capita consumption:

\[\max \int_0^\infty \frac{(C/L)^{1-\rho}}{1-\rho} \, e^{-\beta \, t} \, dt\]

where \(\beta\) is the time preference rate and \(\rho\) is the coefficient of relative risk aversion.

Reformulating in Efficiency Units

Since \(C/L = \Theta \cdot c\) and \(\Theta_t = \Theta_0 \, e^{g t}\), the objective becomes

\[\Theta_0^{1-\rho} \int_0^\infty u(c) \, e^{-\nu \, t} \, dt\]

where \(\nu = \beta - (1-\rho) g\) is the modified discount rate.

Normalizing \(\Theta_0 = 1\), the complete optimization problem is:

\[\max \int_0^\infty u(c) \, e^{-\nu t} \, dt \quad \text{s.t.} \quad \dot{k} = k^\alpha - c - (n + g + \delta) k\]

The Hamiltonian Approach

The Hamiltonian converts an infinite-horizon integral problem into a sequence of static optimality conditions at each instant. Form the discounted Hamiltonian with control \(c\), state \(k\), and costate \(\lambda\):

\[\mathcal{H}(k, c, \lambda) = u(c) + \lambda \left[ k^\alpha - c - (n + g + \delta) k \right]\]

Control condition (\(\partial \mathcal{H}/\partial c = 0\)): \(\quad u'(c) - \lambda = 0 \;\Longrightarrow\; c^{-\rho} = \lambda\)

Costate condition (\(\dot{\lambda} = \nu\lambda - \partial \mathcal{H}/\partial k\)): \(\quad \dot{\lambda}/\lambda = \nu + (n+g+\delta) - f'(k)\)

Combining (substitute \(\dot{\lambda}/\lambda = -\rho\,\dot{c}/c\) from the control condition) yields the Euler equation.

The Euler Equation

Substituting \(\nu = \beta - (1-\rho)g\) and simplifying:

\[\boxed{\frac{\dot{c}}{c} = \frac{1}{\rho} \left[ f'(k) - \delta - \beta - n - \rho \, g \right]}\]

This is the Keynes-Ramsey rule. It is identical to the Euler equation in the perfect foresight consumption model, except that the interest rate \(f'(k)\) is now endogenous and varies as \(k\) changes along the transition path.

The Steady State

Setting \(\dot{c}/c = 0\) determines the steady-state capital stock \(\check{k}\):

\[f'(\check{k}) = \beta + n + \delta + \rho \, g\]

With Cobb-Douglas, \(f'(k) = \alpha k^{\alpha - 1}\), so:

\[\boxed{\check{k} = \left( \frac{\alpha}{\beta + n + \delta + \rho \, g} \right)^{\frac{1}{1-\alpha}}}\]

Steady State vs. Golden Rule

The golden rule requires \(f'(k^{GR}) = g + n + \delta\), while the RCK steady state requires \(f'(\check{k}) = \beta + n + \delta + \rho g\).

When \(\rho > 1\), we have \(\rho g > g\), so \(\beta + \rho g > g\) for any \(\beta \geq 0\). Therefore \(f'(\check{k}) > f'(k^{GR})\), which implies:

\[\check{k} < k^{GR}\]

The optimizing economy accumulates less capital than the golden rule. Even perfectly patient consumers (\(\beta = 0\)) with \(\rho > 1\) fall short of the golden rule.

Comparative Statics of the Steady State

Parameter	Effect on \(\check{k}\)	Intuition
\(\alpha\) (capital share)	\(\uparrow\)	Capital is more productive
\(\beta\) (impatience)	\(\downarrow\)	Less willing to save
\(n\) (population growth)	\(\downarrow\)	More dilution of capital
\(\delta\) (depreciation)	\(\downarrow\)	More capital lost each period
\(g\) (tech progress)	\(\downarrow\)	More dilution; higher effective discount

Exploring the Steady State

Calibrating to the US Economy

Standard parameter values for the postwar United States:

Parameter	Value	Source
\(\alpha\)	0.33	Capital share of income (NIPA)
\(\delta\)	0.08	Standard calibration (aggregate average)
\(\beta\)	0.04	Estimated from asset pricing data
\(\rho\)	2.0	Consumption Euler equation estimates
\(n\)	0.01	Population growth (postwar average)
\(g\)	0.02	TFP growth (Solow residual, postwar)

With these values: \(\check{k} \approx 2.7\), \(r_{ss} = f'(\check{k}) - \delta \approx 0.09\), and the gross saving rate is about 24%.

Phase Diagram: The Two Loci

The dynamics of \(k\) and \(c\) are governed by two differential equations:

The \(\dot{k}=0\) locus (from capital accumulation):

\[c = f(k) - (g + n + \delta) k\]

This is a hump-shaped curve peaking at the golden rule \(k^{GR}\).

The \(\dot{c}/c = 0\) locus (from the Euler equation):

\[f'(\check{k}) = \beta + n + \delta + \rho \, g\]

This is a vertical line at \(\check{k}\).

Phase Diagram: Directions of Motion

To the left of \(\check{k}\): \(f'(k) > f'(\check{k})\), so \(\dot{c}/c > 0\) (consumption rises).

To the right of \(\check{k}\): \(f'(k) < f'(\check{k})\), so \(\dot{c}/c < 0\) (consumption falls).

Above the \(\dot{k}=0\) curve: consumption is too high, so \(\dot{k}<0\) (capital falls).

Below the \(\dot{k}=0\) curve: consumption is too low, so \(\dot{k}>0\) (capital rises).

Together, these create a saddle-point structure: stable in one direction (capital adjusts slowly) but unstable in the other (consumption jumps).

Why a Saddle Path?

Consider an economy starting at some \(k_0 < \check{k}\). Households must choose initial consumption \(c_0\):

• \(c_0\) too high: the economy moves northwest — capital shrinks and eventually hits zero (starvation)
• \(c_0\) too low: the economy moves northeast then southeast — capital overshoots and the household dies rich but miserable (wasted resources)
• \(c_0\) just right: the economy glides toward \((\check{k}, \check{c})\) — this is the saddle path

Only one initial \(c_0\) threads the needle. The transversality condition is what rules out the “too high” and “too low” paths.

Phase Diagram

The Transversality Condition

The Euler equation alone admits infinitely many solutions. What pins down the unique optimal path? The transversality condition (the continuous-time analog of the intertemporal budget constraint):

\[\lim_{t \to \infty} \lambda_t \, e^{(g+n)t - \int_0^t r_\tau \, d\tau} \, k_t = 0\]

This prevents the capital stock from becoming negative or exploding to infinity. It rules out all paths except the saddle path.

Three Types of Paths

1. Too much consumption: Households consume so much that capital declines to zero. The economy runs out of resources.
2. Too little consumption: Households over-save, capital grows without bound, and the marginal product falls below the effective discount rate. Resources are wasted.
3. The saddle path: The unique path converging to steady state, satisfying both the Euler equation and the TVC.

Simulating the Saddle Path

Convergence: Theory vs. Evidence

The RCK model predicts conditional convergence: countries with similar fundamentals (\(\alpha\), \(\beta\), \(n\), \(g\)) converge to the same steady state. Countries below their steady state grow faster.

Empirical findings (Barro 1991, Mankiw-Romer-Weil 1992):

• Unconditional convergence is weak: poor countries do not systematically grow faster
• Conditional convergence is strong: controlling for savings, population growth, and human capital, the convergence rate is about 2% per year
• The 2% rate implies a half-life of 35 years, consistent with the RCK model’s saddle-path dynamics

Decentralizing the RCK Model

The First Welfare Theorem

Each household maximizes \(\int_0^\infty u(c_{t,i}) e^{-\beta t} dt\) taking factor prices as given. Each firm maximizes \(f(k_{t,j}) - \hat{r}_t k_{t,j} - \hat{w}_t\) with FOC \(f'(k_{t,j}) = \hat{r}_t\).

In equilibrium, the household Euler equation \(\dot{c}/c = \rho^{-1}(f'(k) - \delta - \beta - n - \rho g)\) and the aggregate budget constraint \(\dot{k} = f(k) - c - (g+n+\delta) k\) are identical to the social planner’s equations. The competitive equilibrium is Pareto optimal.

When Does Decentralization Fail?

The First Welfare Theorem breaks down with:

• Externalities in production or consumption (e.g., knowledge spillovers in Romer’s model)
• Idiosyncratic risk without complete insurance markets (e.g., Aiyagari 1994, Krusell-Smith 1998)
• Heterogeneous agents with borrowing constraints
• Distortionary taxation (as we will see next)

Each violation creates a wedge between private and social returns, making the decentralized outcome suboptimal.

RCK with Government

Balanced Budget Government

For clarity, set \(n = g = 0\) throughout this section (the qualitative results are unchanged with positive growth).

Add a government that levies lump-sum taxes \(T\) per period and spends \(x\) per period, with \(x = T\). Government spending yields no utility.

The household’s budget constraint becomes:

\[\dot{a}_t = r_t \, a_t + w_t - c_t - T\]

The Euler equation is unchanged (lump-sum taxes don’t affect the return to saving):

\[\frac{\dot{c}_t}{c_t} = \frac{1}{\rho}(r_t - \beta)\]

Effect on the Phase Diagram

The \(\dot{c}/c = 0\) locus is unchanged: it depends on \(f'(k)\) and parameters, not on \(T\).

The \(\dot{k} = 0\) locus shifts down by \(T\):

\[c = f(k) - \delta k - T\]

Higher government spending crowds out consumption in steady state but does not affect the steady-state capital stock \(\check{k}\).

Government Spending and the Phase Diagram

Ricardian Equivalence

Now suppose the government issues bonds \(D\) instead of balancing the budget, financing interest payments with future lump-sum taxes.

The time path of taxes \(T_t\) does not appear in the Euler equation (which depends only on \(r_t\)). And the PDV of taxes is pinned down by the government’s IBC regardless of timing.

Therefore, the path of consumption is unaffected by the timing of taxes. This is Ricardian equivalence: a tax cut financed by debt is not a tax cut at all, just a rescheduling.

Capital Income Tax

Further assume \(\delta = 0\) (zero depreciation) to isolate the distortion.

Consider a tax rate \(\tau\) on capital income, with revenue rebated as a lump-sum transfer \(z_t = \tau \, r_t \, k_t\).

The household’s budget constraint:

\[\dot{a}_t = r_t (1 - \tau) a_t + w_t - c_t + z_t\]

The Euler equation now uses the after-tax return:

\[\frac{\dot{c}_t}{c_t} = \frac{1}{\rho}(r_t(1-\tau) - \beta)\]

Effect of the Capital Income Tax

The new steady state requires:

\[f'(\bar{k})(1 - \tau) = \beta \quad \Longrightarrow \quad f'(\bar{k}) = \frac{\beta}{1 - \tau}\]

Since \(\beta / (1-\tau) > \beta\), the marginal product must be higher, which means the capital stock is lower:

\[\bar{k} < \check{k}\]

The capital income tax reduces the steady-state capital stock and output per worker. Even though the revenue is rebated, the tax distorts the return to saving.

Why Does the Rebate Not Help?

The aggregate resource constraint is unchanged (the tax and rebate cancel):

\[\dot{k}_t = r_t k_t + w_t - c_t = f(k_t) - \delta k_t - c_t\]

But the household sees \(r_t(1-\tau)\) as the return to saving. The rebate goes to everyone equally, so an individual saver does not capture the tax revenue generated by their own extra saving. This is a positive externality from individual saving that the tax prevents agents from capturing.

Capital Tax and the Phase Diagram

Saving and Growth

The Gross Saving Rate in Steady State

Set \(n = 0\) to isolate the saving-growth relationship. In steady state (\(\dot{k}=0\)), the capital accumulation equation gives:

\[c = k^\alpha - (\delta + g) k\]

The gross saving rate \(s = (y - c)/y\) then equals:

\[s = \frac{(\delta + g) k}{k^\alpha} = (\delta + g) k^{1-\alpha}\]

Substituting the steady-state \(\check{k}\):

\[\boxed{s = \frac{\alpha(\delta + g)}{\delta + \beta + \rho \, g}}\]

Does Faster Growth Raise the Saving Rate?

Taking the derivative:

\[\frac{ds}{dg} = \frac{\alpha(\delta(1 - \rho) + \beta)}{(\delta + \beta + \rho \, g)^2}\]

This is positive only if \(\rho < 1 + \beta / \delta\). With typical values \(\beta = 0.04\) and \(\delta = 0.08\), the threshold is \(\rho < 1.5\).

Since we usually assume \(\rho \in [2, 5]\), the model predicts a negative relationship: faster growth lowers the saving rate. This is in tension with empirical evidence.

Saving Rate and Risk Aversion

Why Are Some Countries Poor?

The RCK model points to three channels through which countries can differ:

1. Low saving (high \(\beta\)): impatient consumers accumulate less capital
2. Rapid population growth (high \(n\)): capital is diluted across more workers
3. Slow productivity growth (low \(g\)): the long-run engine is weaker

Mankiw, Romer, and Weil (1992) showed that saving rates, population growth, and human capital explain roughly 80% of cross-country income variation, holding \(g\) constant. The RCK model provides the optimizing microfoundation for why saving rates differ.

Key Takeaways (1/2)

• The Ramsey-Cass-Koopmans model endogenizes the saving rate via the Keynes-Ramsey rule: \(\dot{c}/c = \rho^{-1}[f'(k) - \delta - \beta - n - \rho g]\).
• The steady-state capital stock \(\check{k}\) falls short of the golden rule because impatient consumers discount the future.
• The phase diagram reveals saddle-path dynamics; the transversality condition selects the unique convergent path.
• The First Welfare Theorem holds: the decentralized equilibrium replicates the planner’s solution absent externalities or distortions.

Key Takeaways (2/2)

• Ricardian equivalence holds with lump-sum taxes: debt-financed tax cuts do not affect consumption.
• A capital income tax reduces the steady-state capital stock, even when revenue is rebated lump-sum.
• For typical risk aversion (\(\rho > 1.5\)), faster growth lowers the saving rate.
• All long-run per-capita growth comes from exogenous technology at rate \(g\). The model cannot explain why \(g > 0\).

Next lecture: Endogenous growth models that explain where \(g\) comes from.