These problems are designed to test derivation, economic interpretation, and quantitative implementation. Students should show intermediate steps and state timing conventions.
Problem set 1 — Household optimization
- Derive the Euler equation for a household with utility \(E_0\sum_{t=0}^\infty\beta^tu(c_t)\) and budget \(c_t+a_{t+1}=y_t+(1+r_t)a_t\).
- Under CRRA utility, derive consumption growth for a deterministic return.
- Add a borrowing limit and derive the complementary-slackness condition.
- Explain the role of the transversality condition.
Solution guidance: Form the Lagrangian with one multiplier per period. Differentiate with respect to consumption and next-period assets, then eliminate the multiplier. Under CRRA, substitute \(u'(c)=c^{-\sigma}\). For a binding borrowing constraint, include a nonnegative multiplier and impose feasibility and complementary slackness.
Problem set 2 — Production and growth
- Derive factor prices for \(y=Ak^\alpha n^{1-\alpha}\).
- Derive the Solow steady state with effective labor growth.
- Derive the golden-rule condition.
- Explain saddle-path stability in the Ramsey model.
Solution guidance: Equate marginal products to competitive factor prices. Set capital accumulation to zero for the Solow steady state. Differentiate steady-state consumption with respect to capital for the golden rule. In Ramsey, classify capital as predetermined and consumption as a jump variable.
Problem set 3 — RBC equilibrium
- Write the household, firm, resource, and capital accumulation conditions.
- Derive the steady-state capital-output ratio.
- Explain how shock persistence changes impulse-response duration.
- Design a calibration table with parameter, value, source, and target.
Solution guidance: Combine the Euler equation with the steady-state condition. Keep productivity innovation variance distinct from persistence. Calibration must identify the empirical moment or external estimate supporting each parameter.
Problem set 4 — New Keynesian model
- Interpret the IS equation, Phillips curve, and policy rule.
- Explain the Taylor principle in the simplest benchmark.
- Compare the response to a natural-rate shock and a cost-push shock.
- Discuss how price stickiness affects the Phillips-curve slope.
Solution guidance: The IS relation links the output gap to the expected future gap and ex-ante real rate. The Phillips curve links inflation to expected inflation and real marginal cost/output gap. Do not infer determinacy from the Taylor coefficient alone in an extended model; check the full eigenvalue conditions.
Problem set 5 — Fiscal debt
- Starting from the nominal government budget constraint, derive the approximate debt-ratio transition.
- Calculate the contribution of the interest-growth differential.
- State the assumptions required for Ricardian equivalence.
- Define an impact and cumulative fiscal multiplier.
Solution guidance: Divide the nominal constraint by nominal GDP, express debt in real terms, and distinguish primary balances from interest payments. For small rates, the debt change contains approximately \((r-g)b_{t-1}\) plus the primary deficit.
Problem set 6 — Open economy
- Derive the saving-investment-current-account identity.
- State uncovered interest parity using domestic currency per unit of foreign currency.
- Explain the role of a risk premium.
- Compare fixed and flexible exchange-rate regimes.
Solution guidance: Use national income accounting and define the exchange-rate quote before interpreting appreciation or depreciation. Add a risk-premium term to UIP and maintain a consistent sign convention.
Problem set 7 — DSGE solution and estimation
- Define states and jump variables in a linear rational-expectations model.
- State the Blanchard–Kahn root-count condition.
- Write the state-space measurement system.
- Derive the Gaussian innovation log likelihood.
- Propose MCMC and posterior-predictive diagnostics.
Solution guidance: Report the generalized eigenvalue convention and timing. The Kalman filter supplies forecast errors and their covariance matrices; these determine the Gaussian likelihood. Posterior assessment requires more than reporting a point estimate.
Problem set 8 — Heterogeneous agents and HANK
- Derive the household Bellman equation and Euler condition.
- Explain how household policy functions induce a distribution transition.
- Define stationary recursive competitive equilibrium.
- Interpret the HJB and Kolmogorov forward equations.
- Explain why equal aggregate transfers can have different demand effects across household distributions.
Solution guidance: Separate individual state variables from aggregate distributions. Market clearing requires aggregation under the invariant distribution. In HANK, MPC heterogeneity explains why transfer incidence affects aggregate consumption.
Research extension
Select one published quantitative macroeconomic model. Reproduce its steady state and one central figure, document all data and parameter choices, then change one structural mechanism. Submit a short report distinguishing replication results, modifications, and new conclusions.