Pick an answer to see whether it is right and why. Wrong options are common misconceptions, so read their feedback too. Your first-try score is shown at the top. 🟢 items check the core ideas; 🔵 and 🔴 items go deeper.
1. 🟢 According to the Euler equation \(\dot c / c = \sigma[f'(k) - \rho]\), when does consumption grow over time?
- Whenever income is growing. The Euler equation says nothing about current income. Consumption growth depends on the return to saving versus impatience.
- When the marginal product of capital exceeds the rate of time preference. ✓ Postponing consumption earns \(f'(k)\) and costs \(\rho\) in impatience. When the return wins, it pays to consume more later than now.
- When the household is very patient, whatever the return on capital. Patience alone is not enough: what matters is the gap between \(f'(k)\) and \(\rho\).
- When the elasticity \(\sigma\) is large. \(\sigma\) scales the response, but the sign of consumption growth is set by \(f'(k) - \rho\).
2. 🟢 Households become more willing to substitute consumption over time (σ rises). What happens to the steady-state capital stock \(k^*\)?
- It rises, because households save more. \(k^*\) solves \(f'(k^*) = \rho\), which does not contain \(\sigma\).
- It falls. The steady-state condition \(f'(k^*) = \rho\) is unchanged.
- It does not change; σ affects how fast the economy gets there, not where it goes. ✓ At the steady state \(\dot c = 0\), which requires \(f'(k^*) = \rho\) for any \(\sigma\). A larger \(\sigma\) speeds up adjustment along the saddle path.
3. 🟢 The economy is at its steady state when households unexpectedly and permanently become more patient (ρ falls). What does consumption do on impact?
- It jumps up, because the economy is now headed to a richer steady state. The destination is richer, but getting there requires more saving now.
- It falls, then rises with capital to a higher level than before. ✓ More patient households save more today. Capital builds up and consumption climbs along the new saddle path to a higher steady state.
- It stays put; only capital adjusts. Consumption is the jump variable: it must move onto the new saddle path immediately.
- It falls permanently: more saving means less consumption forever. This is the classic misconception. The new steady state has more capital and more consumption.
4. 🟢 Compared with the Solow golden rule, the optimal (modified golden rule) steady state has…
- more capital, because optimising households save more. Optimisation makes saving efficient, not maximal. Discounting tilts the choice towards consuming earlier.
- less capital, because future consumption is discounted (ρ > n). ✓ \(f'(k^*) = \rho > n = f'(k_G)\), and \(f'\) is decreasing, so \(k^* < k_G\). Building up to \(k_G\) would cost too much present consumption.
- the same capital: optimal saving is the golden rule. The golden rule maximises steady-state consumption. The optimum maximises discounted utility including the transition.
5. 🔵 Right after an unexpected permanent shock, which variable can change discontinuously?
- Capital per capita \(k\). Capital is a stock built by past saving: it can only change gradually through \(\dot k = f(k) - nk - c\).
- Consumption per capita \(c\). ✓ Consumption is the control variable. It is “jumpy”, and its jump puts the economy on the saddle path.
- Both. Only the control variable can jump. If capital could jump too, any point would be reachable and the saddle path would lose its role.
- Neither: both adjust smoothly. Then the economy could not reach the new saddle path, and would diverge.
6. 🔵 Many paths satisfy both the Euler equation and the resource constraint. Why is only the saddle path optimal?
- The others are not feasible from the initial capital stock. They are feasible at first; they start from the same \(k_0\) with a different \(c_0\).
- The others violate the transversality condition, or eventually force a jump in the price of capital. ✓ Paths starting too high run capital out and collapse consumption. Paths starting too low over-accumulate past the golden rule and leave valuable capital unused forever. The TVC rules both out.
- The others violate the Euler equation. By construction they satisfy it. The extra condition is the TVC.
7. 🔵 True or false: with one jump variable (\(c\)) and one predetermined variable (\(k\)), we would like the system to be a sink (two negative eigenvalues).
- True With a sink, every initial \(c_0\) would converge, so there would be infinitely many equilibria: indeterminacy.
- False ✓ Uniqueness needs as many positive (unstable) eigenvalues as jump variables. With one of each, the saddle is what pins down a unique \(c_0\).
8. 🔵 In the Hamiltonian, the co-state variable \(\lambda_t\) is best interpreted as…
- the household’s consumption. Consumption is the control variable; \(\lambda\) is a multiplier.
- the marginal value (shadow price) of one more unit of capital at time \(t\). ✓ Like a Lagrange multiplier, \(\lambda\) prices the constraint. Its law of motion is an asset-pricing arbitrage condition.
- the interest rate. The interest rate is \(f'(k)\) in the decentralised economy, while \(\lambda\) is a price level of capital in utility units.
9. 🔴 How do the no-Ponzi-game (NPG) condition and the transversality condition (TVC) differ?
- They are the same condition with different names. They look alike and end up equivalent in equilibrium, but they play different roles.
- The NPG is a constraint imposed on households by the market; the TVC is an optimality condition chosen by the optimiser. ✓ The NPG says households cannot borrow without end. The TVC says an optimiser does not leave valuable capital at infinity. In equilibrium they coincide.
- The TVC is a constraint and the NPG an optimality condition. The roles are the other way round.
10. 🟢 True or false: because saving is chosen optimally, income per capita grows forever in the neoclassical growth model.
- True On the balanced growth path \(k\) and \(y = f(k)\) are constant. Optimal saving does not overcome diminishing returns.
- False ✓ Like Solow, the model has no long-run growth in per-capita variables without exogenous technological progress.
11. 🔵 Piketty argues that \(r > g\) drives inequality up. Which point from the chapter can weaken that conclusion?
- In the model \(r < g\) on the balanced growth path. In the NGM, \(r = f'(k^*) = \rho > n = g\): the model does satisfy \(r > g\).
- If labour (including human capital) is hard to substitute (elasticity of substitution below one), the labour share can rise as the economy grows. ✓ With \(\varepsilon < 1\) labour becomes more essential over time. Add human capital and social mobility, and \(r > g\) need not mean exploding inequality.
- Interest rates have historically been below growth rates. Schmelzing’s eight centuries of data show real rates mostly above growth, until recently.
12. 🔴 At the steady state, \(\det \Omega = \sigma c^* f''(k^*)\). What does its sign tell us?
- It is positive, so both eigenvalues are positive: a source. \(f'' < 0\) makes the determinant negative.
- It is negative, so the eigenvalues have opposite signs: a saddle, with a unique converging path. ✓ The determinant is the product of the eigenvalues; a negative product means one stable and one unstable root.
- It is zero, so the system is degenerate. It is zero only if \(\sigma\), \(c^*\) or \(f''\) is zero, none of which holds.
13. 🔴 The planner learns today that ρ will rise temporarily, starting at a known future date. When can consumption jump?
- When the rise in ρ begins. An anticipated jump would offer predictable infinite capital gains, which optimal behaviour rules out.
- Only today, when the news arrives; afterwards it moves continuously so the economy lands on the saddle path when ρ returns to normal. ✓ New information can move consumption; anticipated events cannot. Consumption jumps once, on the news, and the dynamics do the rest.
- Twice: when the rise starts and when it ends. Both dates are known in advance, so a jump at either would be anticipated.
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