Intertemporal Choice, Budget Constraints, and Consumption in Macroeconomics

Learning Objectives

• Translate a narrative scenario into a budget constraint equation

• Formulate an optimization problem for intertemporal consumption

• Solve for endogenous choice variables using calculus

• Interpret comparative statics in consumption models

Intertemporal Consumption and Budget Constraints

Setup

Intertemporal choice models analyze how households allocate consumption over multiple periods, typically present and future. The household faces income in two periods and must decide how much to consume and save in each, considering interest rates and initial wealth.

• Household Budget Constraint: The household can consume in period 0 (present) and period 1 (future). The budget constraint links consumption, income, savings, and interest rates.

• Variables: C_0 (consumption in period 0), C_1 (consumption in period 1), Y_0 (income in period 0), Y_1 (income in period 1), B (initial wealth), r (interest rate).

• Budget Constraint Equations:

  • Period 0: $C_0 + S_0 = Y_0 + B$

  • Period 1: $C_1 = Y_1 + (1 + r)S_0$

• Lifetime Budget Constraint: Combining both periods, the household's lifetime budget constraint is:

$C_1 + S_0 = Y_1 + (1 + r)S_0$

$C_0 + \frac{C_1}{1 + r} = Y_0 + B + \frac{Y_1}{1 + r}$

• Utility Function: The household seeks to maximize utility, typically represented as:

$u(C_0, C_1) = \frac{C_0^{1-\sigma}}{1-\sigma} + \beta \frac{C_1^{1-\sigma}}{1-\sigma}$

• Parameters: $\sigma$ is the coefficient of relative risk aversion, $\beta$ is the subjective discount factor.

Solving the Optimization Problem

Maximizing Utility Subject to the Budget Constraint

To find optimal consumption in each period, substitute the budget constraint into the utility function and maximize with respect to $C_0$.

• First Derivative of Utility with Respect to $C_0$:

$\frac{d}{dC_0} u(C_0, C_1) = C_0^{-\sigma}$

• First Derivative of Utility with Respect to $C_1$:

$\frac{d}{dC_1} u(C_0, C_1) = \beta C_1^{-\sigma}$

• Setting up the Lagrangian and First-Order Condition: Set the derivative of the utility function equal to zero and solve for $C_0$:

$C_0 = \left[ \beta (1 + r) \right]^{1/\sigma} C_1$

• Solving for $C_0$: Substitute back into the budget constraint and solve for $C_0$:

$C_0^* = \frac{\left[ \beta (1 + r) \right]^{1/\sigma} (Y_0 + B) + \frac{Y_1}{1 + r}}{1 + \left[ \beta (1 + r) \right]^{1/\sigma}}$

• Interpretation: This formula gives the optimal current consumption based on income, initial wealth, future income, interest rate, and preferences.

Comparative Statics and Marginal Propensity to Consume

Analyzing the Effects of Changes in Wealth and Interest Rates

Comparative statics examine how changes in parameters (e.g., initial wealth, interest rate) affect optimal consumption.

• Marginal Propensity to Consume (MPC): Measures the change in current consumption for each additional dollar of initial wealth or bequest received.

• Formula for MPC:

$\frac{\partial C_0^*}{\partial B} = \frac{\left[ \beta (1 + r) \right]^{1/\sigma}}{1 + \left[ \beta (1 + r) \right]^{1/\sigma}}$

• Example Calculation: If $\beta = 0.98$, $r = 0.04$, $\sigma = 0.5$:

$\left[ 0.98 \times 1.04 \right]^{2} = 1.039672$ $1 + 1.039672 = 2.039672$ $\frac{1.039672}{2.039672} \approx 0.5097$

• Interpretation: Each additional dollar of bequest increases current consumption by approximately 51 cents.

Log-Linearization of the Consumption Function

Approximating Nonlinear Functions for Analysis

Log-linearization simplifies nonlinear equations by taking logarithms and using Taylor expansions, making comparative statics easier.

• Consumption Function:

$C_0^* = \frac{\left[ \beta (1 + r) \right]^{1/\sigma} (Y_0 + B) + \frac{Y_1}{1 + r}}{1 + \left[ \beta (1 + r) \right]^{1/\sigma}}$

• Log-Linearization: Take the logarithm of both sides and expand using the product rule:

$\ln C_0^* = \ln \left( \left[ \beta (1 + r) \right]^{1/\sigma} (Y_0 + B) + \frac{Y_1}{1 + r} \right) - \ln \left( 1 + \left[ \beta (1 + r) \right]^{1/\sigma} \right)$

• Application: Log-linearization is useful for empirical work and for understanding percentage changes in consumption due to changes in parameters.

Summary Table: Key Variables and Their Roles

Variable	Description	Role in Model
C_0	Consumption in period 0 (present)	Choice variable to maximize utility
C_1	Consumption in period 1 (future)	Choice variable to maximize utility
Y_0	Income in period 0	Resource available for consumption/saving
Y_1	Income in period 1	Resource available for future consumption
B	Initial wealth/bequest	Additional resource for consumption/saving
r	Interest rate	Determines return on savings
\beta	Subjective discount factor	Measures time preference
\sigma	Relative risk aversion	Determines curvature of utility function

Additional info:

• Comparative statics are a key tool in macroeconomics for understanding how changes in parameters affect optimal choices.

• Log-linearization is a standard method for simplifying nonlinear models for empirical analysis.

• The marginal propensity to consume (MPC) is central to understanding the effects of wealth and income changes on consumption.