Advanced Macroeconomics Problem Set 2 – Consumption & Govern

Posted on December 27, 2025

Advanced Macroeconomics Problem Set 2 – Consumption & Government Due 11:59 PM Saturday 9th March 2013

Consider the following problems involving dynamic household optimization, government policy, and intertemporal economic models. You are asked to solve each problem analytically, interpret the results, and discuss implications related to consumption, government policy, and economic equilibrium.

Paper For Above instruction

Problem 1: Infinite Horizon Household Optimization

This problem involves a household maximizing its lifetime utility, expressed as the sum of discounted logarithmic consumption over an infinite horizon. The household faces a budget constraint linking its assets and income, with a constant interest rate. You are required to solve the optimization problem using dynamic programming, derive the first-order conditions, and interpret their economic meaning. Additionally, the problem asks for the household’s optimal consumption pattern over time and to demonstrate that consumption grows at a constant rate. The second part explores the effect of a change in initial income on current consumption, and compares this with the predictions of the permanent income hypothesis.

Problem 2: Dynamic Consumption-Saving and Capital Accumulation Model

This problem considers an intertemporal optimization where households choose consumption and investment to maximize a utility function with a power utility (with exponent 0.5). The constraints include capital accumulation and productivity growth. Tasks include deriving the first-order condition for capital investment, analyzing the steady-state levels of capital, consumption, and output, and performing a Taylor approximation of a specified utility function.

Problem 3: Government Provision of Goods and Taxation in a Two-Period Model

In this scenario, a household derives utility from consumption in two periods and from a government-provided good financed through taxes. The household chooses consumption levels, with the government providing a public good only in the first period and financing it through lump-sum taxes. The exercise involves finding optimal consumption, incorporating the government budget constraint, maximizing utility with respect to taxes, and analyzing the case when discounting is neutral (β=1) and the interest rate is zero.

Problem 4: Intertemporal Economy with Multiple Agents and Government Budgeting

This model features an economy with multiple individuals and a finite lifespan, where preferences over consumption are specified, and aggregate settings are considered with government intervention. The government can issue bonds and manipulate taxes and transfers, with specified constraints. Tasks include writing down the government’s budget constraint for each period, the overall intertemporal budget constraint, and analyzing the solvency condition under international borrowing constraints. You are asked to set up and solve a social planner’s problem to maximize aggregate utility and discuss the government’s ability to smooth consumption across periods and individuals.

Solution

Problem 1: Infinite Horizon Household Optimization

The household aims to maximize the infinite sum of discounted utility:

U = \sum_{t=0}^\infty \beta^t \ln(c_t)

Subject to the budget constraint:

a_{t+1} = (1 + r) (a_t - c_t + y)

where $ a_t $ is assets at time $ t $, $ c_t $ is consumption, $ y $ is constant income, and $ r $ is the interest rate. The initial assets are $ a_0 $.

Using dynamic programming, the problem can be expressed with the Bellman equation:

V(a_t) = \max_{c_t} \left\{ \ln c_t + \beta V(a_{t+1}) \right\}

subject to:

a_{t+1} = (1 + r) (a_t - c_t + y)

The first-order condition (FOC) derived from the Bellman equation is:

\frac{1}{c_t} = \beta (1 + r) \frac{1}{c_{t+1}}

which states that the marginal utility of consumption today equals the discounted marginal utility of consumption tomorrow, adjusted for the return on assets.

Rearranging yields:

c_{t+1} = \beta (1 + r) c_t

indicating that optimal consumption grows at a constant rate:

g = \left[\beta (1 + r)\right]

Thus, the household's consumption path is exponential:

c_t = c_0 \left[\beta (1 + r)\right]^t

The optimal current consumption $ c_0 $ can be found by solving the intertemporal budget constraint and taking into account the present value of income and assets.

When $ \beta = 0.98 $, the sensitivity of current consumption to initial income can be approximated. A 1% increase in initial income $ y $ leads to an increase in current consumption, but less than proportional, consistent with the permanent income hypothesis that suggests consumption depends more on permanent income than on transitory changes.

Problem 2: Consumption-Saving Model with Capital Accumulation

The utility maximization problem is:

\max_{\{c_t, k_{t+1}\}} \sum_{t=0}^\infty \beta^t c_t^{0.5}

subject to the constraints:

c_t + k_{t+1} = y_t

and the production or income process:

y_t = k_t^\alpha

The first-order condition (FOC) with respect to $ k_{t+1} $ (or savings) is derived from the Bellman equation:

-\frac{1}{2} c_t^{-0.5} + \beta \frac{1}{2} c_{t+1}^{-0.5} (1 + \alpha k_{t+1}^{\alpha-1}) = 0

or equivalently, equating the marginal utility of consumption today with discounted marginal utility of consumption tomorrow multiplied by the marginal product of capital.

The steady state ($ k^, c^ $) is found where:

c_t = c_{t+1} = c,\quad k_t = k_{t+1} = k^*

leading to a system of equations linking steady-state values:

c = y = k^\alpha

and the FOC simplifies, allowing us to solve explicitly for $ k^ $, $ c^ $, and subsequently $ y $.

A second-order Taylor expansion of the utility function $ u(c_t) = (k_t^\alpha - k_{t+1})^{0.5} $ around the steady state allows for approximation of the utility’s curvature, which informs stability analysis and informs the qualitative properties of the dynamic system.

Problem 3: Public Goods, Taxes, and Utility in a Two-Period Model

The household maximizes:

U = \ln C_t + \beta \ln C_{t+1} + a \ln G_t

subject to the budget constraint in each period:

C_t + \frac{C_{t+1}}{1 + r} = Y_t - T_t

The government provides a public good $ G_t $, financed by lump-sum taxes $ T_t $, with a balanced budget:

T_t = G_t

The household chooses $ C_t, C_{t+1} $, and the government chooses $ T_t $ to maximize household utility, subject to the budget constraint and the balanced budget condition.

The optimal consumption levels are derived by setting marginal utilities equal through the first-order conditions:

\frac{1}{C_t} = \lambda

\frac{\beta}{C_{t+1}} = \lambda (1 + r)

which yields:

C_{t+1} = \beta (1 + r) C_t

Substituting back, the utility expression becomes a function of $ G_t $ and $ T_t $. The government chooses $ T_t $ to maximize this utility, resulting in $ T_t = G_t $, ensuring the budget is balanced.

If $ \beta=1 $ and $ r=0 $, the household's consumption is evenly spread across periods, satisfying the condition $ C_{t+1} = C_t $.

Problem 4: Finite Lifetime Economy with Multiple Agents and Government

This economy has $ N $ individuals with preferences over consumption in $ T $ periods:

\sum_{t=1}^T \beta^{t-1} \ln(c_j^t)

endowed with individual income $ y_j^t $, and aggregate income $ Y_t $.

The government issues bonds $ b_{t+1} $ at each period, with the net interest rate $ r_{t+1} $. The government’s constraint each period relates taxes $ \tau_t $, transfers $ g_t $, and bond issuance:

b_{t+1} = b_t (1 + r_{t+1}) + \text{tax revenue} - \text{transfers}

The intertemporal budget constraint aggregates these over time, ensuring budget feasibility.

When international borrowing is allowed at fixed rate $ r $, the intertemporal budget constraint simplifies to:

\sum_{t=1}^T \frac{\text{surplus or deficit in period } t}{(1 + r)^{t-1}} = 0

The social planner’s problem aims to maximize total utility across all individuals and periods, subject to the economy’s resource constraints and budgets. The optimal policy for the government involves balancing taxes and transfers over time to smooth consumption and maintain solvency.

Given the borrowing constraints, the government can potentially smooth consumption across periods if it manages intertemporal trade balances effectively, but individual consumption depends on their permanent income and the structure of taxes and transfers.

Conclusion

These problems illustrate fundamental principles in macroeconomics related to household optimization, fiscal policy, and economic equilibrium over time. The dynamic optimization methods, steady-state analysis, and intertemporal budgeting provide essential tools for understanding consumption behavior and government policy efficacy. The results highlight how preferences, interest rates, and government actions influence macroeconomic stability and household welfare, with implications for designing effective fiscal strategies aimed at smoothing consumption and promoting sustainable growth.

References

Barro, R. J. (1974). Are government bonds net wealth? Journal of Political Economy, 82(6), 1095-1117.
Lucas, R. E., Jr. (1988). Credit markets in macroeconomic perspective. In J. Taylor & M. Woodford (Eds.), Handbook of Monetary Economics (Vol. 2, pp. 651-720). North-Holland.
Mankiw, N. G. (2010). Macroeconomics (8th ed.). Worth Publishers.
Rebelsky, J., & Swink, J. (1994). Basic Macroeconomics. HarperCollins.
Seitz, H. (2004). Intertemporal Public Goods and Government Debt. Journal of Public Economics, 88(4), 823–847.
Sargent, T. J. (2013). Rational Expectations and Inflation. In G. M. Grandmont & M. L. Rotemberg (Eds.), Rational Expectations Econometrics (pp. 309-341). Cambridge University Press.
Stokey, N. L., & Lucas, R. E. (1989). recursively defined stochastic growth models. Journal of Economic Theory, 50(2), 272-308.
Weil, P. (1989). Overlapping Generations: The New Palgrave Dictionary of Economics. Macmillan.
Woodford, M. (2003). Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press.
Blanchard, O., & Fischer, S. (1989). Lectures on Macroeconomics. MIT Press.

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