Cheat sheet
Chapter 3 — The neoclassical growth model on one page
Thesis. With optimal saving, consumption grows exactly when the return on capital beats impatience, and the economy follows a unique saddle path to a steady state with no oversaving, but no long-run growth per person.
Key equations
| Equation | In words | Use it when |
|---|---|---|
| \(\max \int_0^\infty u(c_t)e^{nt}e^{-\rho t}dt\), \(\ \rho > n\) | discounted utility of a growing population | setting up the problem |
| \(\dot k = f(k) - nk - c\) | capital grows by output minus consumption minus capital widening | the resource constraint; the \(\dot k = 0\) hump |
| \(H = u(c)e^{nt} + \lambda[f(k) - nk - c]\) | the current-value Hamiltonian; \(\lambda\) = shadow price of capital | deriving the FOCs |
| \(\rho + n = \dot\lambda/\lambda + f'(k)\) | arbitrage: opportunity cost = dividend + capital gain | interpreting the dynamic FOC |
| \(\dot c / c = \sigma[f'(k) - \rho]\) | consumption grows when the return on saving beats impatience | any intertemporal choice |
| \(f'(k^*) = \rho\) | the modified golden rule: optimal steady-state capital | finding the steady state |
| \(f'(k_G) = n\) | the golden rule: maximum steady-state consumption | comparing with Solow |
| \(\det\Omega = \sigma c^* f''(k^*) < 0\) | opposite-sign eigenvalues: a saddle | proving uniqueness |
| \(\lim k_t\lambda_t e^{-\rho t} = 0\) | transversality: no valuable capital left at infinity | ruling out diverging paths |
| \(\int_0^\infty c_s e^{-(r-n)s}ds = a_0 + \int_0^\infty w_s e^{-(r-n)s}ds\) | intertemporal budget constraint (under no-Ponzi) | the decentralised household |
Comparative statics and shocks
| Change | On impact | Long run (\(k^*\), \(c^*\)) | Why |
|---|---|---|---|
| \(\rho\) ↓ (permanent, unexpected) | \(c\) jumps down | both ↑ | more patience means more saving; \(\dot c = 0\) moves right |
| \(\rho\) ↑ (permanent, unexpected) | \(c\) jumps up | both ↓ | the mirror image |
| \(\sigma\) ↑ | faster adjustment | unchanged | \(\sigma\) is absent from \(f'(k^*) = \rho\) |
| \(n\) ↑ | \(c\) jumps down, straight to the new \(c^*\) (\(k\) is already at \(k^*\)) | \(k^*\) unchanged (\(f'(k^*) = \rho\)); \(c^*\) ↓ | more capital widening lowers the hump |
| \(\rho\) ↑, anticipated and transitory | \(c\) jumps only on the news | back to the old steady state | no anticipated jumps (no infinite capital gains) |
Facts to remember
| Fact | Source |
|---|---|
| \(k^* < k_G\) whenever \(\rho > n\): no dynamic inefficiency | modified golden rule, Derivations |
| \(k\) is sticky (state), \(c\) is jumpy (control) | resource constraint |
| one jump variable needs one unstable eigenvalue | counting rule, Derivations |
| \(r = f'(k^*) = \rho > n = g\) on the BGP | the NGM satisfies \(r > g\) |
| no long-run per-capita growth without technological progress | “Do we have growth after all?” |
Notation
\(k, c\): capital and consumption per capita · \(n\): population growth · \(\rho\): rate of time preference · \(\sigma\): elasticity of intertemporal substitution · \(\lambda\): co-state (shadow price of capital) · \(r, w\): interest rate and wage · \(a\): assets per capita · \(k_G\): golden-rule capital · \(\Omega\): Jacobian at the steady state.
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