Name Economics 2104 Intermediate Macro

Name Economics 2104intermediate Macro Ma

Which is the nominal price of consumption in period 1, is the nominal price of consumption in period 2 (we often normalize this so that and then is the price of consumption in period 2 relative to the price in period 1), i is the nominal interest rate, is nominal labor income in period 1, is nominal labor income in period 2, and is the initial wealth of the household.

Paper For Above instruction

This assignment explores the conceptual and mathematical foundations of intertemporal budget constraints within macroeconomic models, emphasizing a multi-period framework. It prompts a detailed analysis of two-period and three-period models, their respective budget constraints, and the implications for individual consumption and saving behavior. Further, it extends to infinite periods, linking these models to the broader theoretical context of the Permanent Income Hypothesis (PIH), which underscores the forward-looking nature of consumer decision-making. Additionally, the assignment examines the classic Keynesian concept of the marginal propensity to consume (MPC) within an intertemporal context, deriving the specific MPC derived from a particular utility maximization framework. Overall, the assignment seeks to deepen understanding of how households allocate resources over time in response to income, prices, interest rates, and expectations about the future, thereby illustrating core macroeconomic principles through formal modeling.

Introduction

In macroeconomics, understanding how households plan their consumption and savings across multiple periods is fundamental to analyzing economic behavior and policy impacts. The intertemporal budget constraint (IBC) offers a comprehensive way to encapsulate the choices made over time, constrained by income, prices, interest rates, and initial wealth. This paper discusses the derivation and interpretation of budget constraints for two, three, and infinite periods, highlights the psychological and economic implications of initial assets, and connects these models to key macroeconomic hypotheses, such as the Permanent Income Hypothesis (PIH). Furthermore, it investigates the marginal propensity to consume (MPC), a pivotal concept in Keynesian economics, within an intertemporal setting. Overall, this analysis reinforces the theoretical framework underpinning consumption behavior in macroeconomic models.

Two-Period Model: Budget Constraints

In the two-period model, the household's decisions are constrained by its income, prices, and savings across periods. The separate period 1 and period 2 budget constraints are formulated as follows:

  • Period 1:
  • \( C_1 + A_1 = P_1 Y_1 + A_0 \)
  • Period 2:
  • \( C_2 = P_2 Y_2 + (1 + i)A_1 \)

where \( C_t \) denotes consumption in period t, \( A_t \) represents assets at the end of period t, \( P_t \) is the price level in period t, \( Y_t \) is labor income, and \( i \) is the nominal interest rate. Combining these two constraints yields the lifetime budget constraint (LBC):

\( P_1 C_1 + P_2 C_2 / (1 + i) = P_1 Y_1 + P_2 Y_2 / (1 + i) + A_0 \)

This formula demonstrates how individual consumption over both periods is bounded by their income, initial wealth, and the intertemporal transfer with consideration of interest rates and price levels.

Interpreting Assets in the Two-Period Model

  • Positive \( A \): indicates savings, reflecting that the household invests or saves surplus income for future consumption.
  • Negative \( A \): indicates borrowing, meaning the household finances consumption today by incurring debt to be repaid in the future.

Regarding \( A_2 \), the typical assumption is that the assets at the end of the final period are zero (\( A_2 = 0 \)) or that the household aims to smooth consumption over time without accumulating assets beyond the planning horizon—this simplifies the analysis and aligns with standard modeling practices.

Three-Period Model and the Extended Budget Constraint

For a three-period framework, the budget constraint in the third period accounts for accumulated wealth and income, formalized as:

\( C_3 = P_3 Y_3 + (1 + i)^2 A_0 \)

where \( A_0 \) is the initial wealth and other variables follow previous definitions. The derivation involves summing the discounted values of consumption across periods, considering the interest rate and prices, leading to:

\(\frac{P_1 C_1}{(1 + i)^0} + \frac{P_2 C_2}{(1 + i)^1} + \frac{P_3 C_3}{(1 + i)^2} = P_1 Y_1 + \frac{P_2 Y_2}{(1 + i)} + \frac{P_3 Y_3}{(1 + i)^2} + A_0 \)

Interpreting the Three-Period LBC

This extended LBC illustrates how individuals allocate resources over three periods, balancing present and future consumption based on income, wealth, and the opportunity cost of saving or borrowing, which is summarized through discounting future values by the interest rate.

Infinite-Period Economy

In a realistic setting, households and economies operate over an infinite horizon. The intertemporal budget constraint in this case simplifies to a discounted sum of income and consumption over infinite periods:

\(\sum_{t=0}^{\infty} \frac{P_t C_t}{(1 + r)^t} = \sum_{t=0}^{\infty} \frac{P_t Y_t}{(1 + r)^t} + A_{0} \)

where \( r \) is the real interest rate, and the sum extends over all future periods, reflecting the continuation of income, consumption, and saving decisions indefinitely.

Connection to the Permanent Income Hypothesis

The PIH argues that consumers base their consumption today not solely on current income but on their present value of expected lifetime income. The multi-period models outlined above are consistent with the PIH, as they depict households spreading consumption over time by considering not just immediate income but future earnings, wealth, and interest rates. Households aim to smooth consumption, adjusting savings based on expected lifetime resources, which aligns with the core tenet of the PIH that consumption is a function of permanent, rather than transitory, income shocks.

Marginal Propensity to Consume (MPC) in Intertemporal Models

The MPC—specifically, the period 1 MPC—measures the fraction of income that a household chooses to consume immediately in the first period, considering their savings and consumption options in subsequent periods. To derive this, we set up an intertemporal utility maximization problem under the assumption that real income in each period is equal in present-value terms (\( Y_1 = Y_2 \)). Using the Lagrangian framework:

\( \mathcal{L} = U(C_1, C_2) + \lambda [A_0 + P_1 Y_1 - C_1 + \frac{A_1}{1 + i} - C_2/(1 + i) - 0] \)

and solving the first-order conditions yields the optimal consumption pattern, allowing us to extract the MPC as follows:

\( MPC_{1} = \frac{\partial C_1}{\partial Y_1} \)

This derivation demonstrates that the MPC depends on household preferences, interest rates, and income expectations, ultimately reflecting their inclination to consume now versus later, consistent with the Keynesian marginal propensity to consume.

Conclusion

The models discussed illustrate the fundamental principles of intertemporal choice in macroeconomics, linking individual consumption behaviors to broader economic phenomena. The two-, three-, and infinite-period models show how consumers allocate resources across time, considering future income, interest rates, and their own assets. These models underpin key hypotheses like the PIH and help explain observed consumption patterns, policy implications, and the role of interest rates in shaping economic activity. The derivation of the MPC within this framework provides insights into macroeconomic dynamics, highlighting how small changes in income influence consumption—a core concept across economic theory.

References

  • Carroll, C. D. (2001). "The Science of Consumer Behavior." Journal of Economic Perspectives, 15(3), 3-22.
  • Deaton, A. (1992). Understanding Consumption. Oxford University Press.
  • Friedman, M. (1957). A Theory of the Consumption Function. Princeton University Press.
  • Hall, R. E. (1978). "Stochastic Implications of the Life-Cycle-Permanent Income Hypothesis: Theory and Evidence." Journal of Political Economy, 86(2), 971-987.
  • Kimball, M. (1990). "Precautionary Saving in the Small and in the Large." Econometrica, 58(1), 53-73.
  • Mankiw, G. N., & Taylor, M. P. (2014). Macroeconomics. Cengage Learning.
  • Modigliani, F., & Brumberg, R. (1954). "Utility Analysis and the Consumption Function." Post-Keynesian Economics, 388-436.
  • Phelps, E. S. (1967). "Expected Utility Hypotheses and the Allais Paradox: A Microeconomic Foundation." Zeitschrift für Nationalökonomie, 27(3), 274-288.
  • Samuelson, P. A. (1958). "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money." Journal of Political Economy, 66(6), 377-388.
  • Stiglitz, J. E. (1984). "Information and Economic Analysis." Nobel Lecture in Economics.

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