Consider An Economy Inhabited By Overlapping Generations ✓ Solved

Consider An Economy Inhabited By Overlapping Generations

1. Consider an economy inhabited by overlapping generations of two-period-lived individuals, each of whom receives an endowment of y in the first period of life and none in the second. The population of newborns grows as follows: Nt = nNt−1, where n > 1. The money stock (the only asset in this economy) is constant.

(a) Find the equation for the budget set of an individual in the monetary equilibrium. Graph it.

Show an arbitrary indifference curve tangent to the budget set and indicate the levels of consumption c1 and c2 that would be chosen by an individual in this equilibrium.

(b) On the figure from part a, draw the feasibility line. Label your graph carefully, distinguishing between the budget and feasibility constraints.

(c) Does the monetary equilibrium maximize the utility of future generations? Support your assertion with references to your figure.

2. Assume that fiat money and capital are perfect substitutes as assets and that individuals wish to hold the one with a higher rate of return, but that it takes time to adjust capital holdings.

In equilibrium individuals hold both assets. The real rate of return on money is n z, where n = 1 and z = Mt Mt−1. The gross real rate of return on capital is f ' (kt) = 100 kt.

(a) What is the equilibrium level of capital when the money stock is constant? What is it when the money stock is doubled in every period?

(b) What is the equilibrium nominal rate of return when the money stock is constant? What is it when the money stock is doubled in every period?

(c) What are the immediate and long-run effects on capital holdings of an unanticipated increase in the growth rate of the money stock?

(d) What does this model predict for the relationship between inflation and output? Does it matter whether inflation is anticipated or not?

(e) How does your answer in part d differ from what the Lucas model predicts?

3. Consider an overlapping-generations model with inside and outside money, where the rate of return on inside money is x, while the rate of return on outside money is 1 z.

There is a one-time transaction cost of ï† associated with inside money, independently of the purchase size.

(a) Show graphically how individuals optimally decide whether to use inside or outside money depending on the size si of their purchase.

Show who uses inside money and who uses outside money.

(b) What is the effect on the deposit-to-currency ratio, nominal money stock M1 and present and future GDP of an anticipated increase in z (anticipated inflation)?

(c) How would your answer in part b differ if the increase in z was unanticipated (unanticipated inflation)?

(d) How could this model explain positive co-movement between M1 and GDP without ever-increasing inflation?

Paper For Above Instructions

Introduction

This paper explores the dynamics of an overlapping generations economy, characterized by individuals living for two periods. Each individual receives an endowment in their first period and faces various economic choices, particularly regarding money as an asset. The analysis encompasses the budget set, the impact of monetary equilibrium, and the relationship between inflation, capital holdings, and output.

The Budget Set in Monetary Equilibrium

In an overlapping generations model, each individual’s budget can be formulated based on their lifetime consumption. Letting \( c_1 \) represent consumption in the first period and \( c_2 \) in the second period, the budget constraint can be expressed as:

Equation 1: \( c_1 + \frac{c_2}{(1 + r)} = y \)

Here, \( r \) denotes the nominal interest rate. In this constraint, the total value of first-period and discounted second-period consumption is equal to the individual's endowment. The monetary equilibrium occurs when the amount of currency held by individuals matches the demand for money.

Graphically, the budget set can be represented in a two-dimensional space where \( c_1 \) is on the x-axis and \( c_2 \) on the y-axis. The line representing the budget set will slope downwards, and the slope is given by the negative of the interest rate after adjusting for the time value of money. An indifference curve, which represents combinations of \( c_1 \) and \( c_2 \) that provide the same utility to an individual, can be drawn tangent to the budget line, indicating the optimal consumption choice.

Feasibility Line and Maximization of Future Generations' Utility

In part (b), to represent the feasibility line, we must consider the total resource constraint of the economy. This line typically encompasses the maximum output that can be consumed across generations based on the economy's production capabilities. The intersection of the feasibility line with the budget constraint will indicate the possible consumption combinations available to the current generation without compromising future utility.

To evaluate whether monetary equilibrium maximizes the utility of future generations (part c), we refer to the graphical representation. If the budget constraint allows the current generation to consume optimally while preserving resources for future generations—represented by the feasible set—the monetary equilibrium can be seen as utility-maximizing. However, if the budget constraint overshoots the feasibility line, this could imply an unsustainable path detrimental to future generations.

Capital Holdings and the Rate of Return

The second problem posits that fiat money and capital serve as perfect substitutes. Here, the equilibrium conditions hinge on interest rates where individuals favor the asset with a higher return. The return on money is given by \( n z \) (where \( n=1 \) and \( z = \frac{M_t}{M_{t-1}} \)), whereas the return on capital is described as \( f'(k_t) = 100 k_t \). For various scenarios, when the money stock is constant, the equilibrium level of capital can be derived from the equality of returns:

Equation 2: \( n z = f'(k_t) \rightarrow 1 \cdot z = 100 k_t \Rightarrow k_t = \frac{z}{100} \)

When the money stock doubles, \( z \) will adjust correspondingly, modifying capital returns:

Equation 3: \( k_t = \frac{2z}{100} \)

Furthermore, the nominal rate of return is similarly affected by the adjustments in money stock, where the stability scenario keeps returns constant across periods compared to when money fluctuates.

Inflation and Output Relationships

The model predicts that there exists a direct relationship between inflation and output. When examining anticipated versus unanticipated inflation, the latter often leads to immediate shifts in capital holdings as individuals scramble to adjust their portfolios. This result underscores the importance of expectations in monetary policy and economic modeling.

Comparative Analysis with Lucas Model

The Lucas model presents a stark contrast to the conclusions drawn about inflation and output. While the overlapping generations model suggests that individuals adjust to real rates of return based on capital allocations, the Lucas model emphasizes rational expectations and the neutrality of money in the long-run. Therefore, the disparate approaches highlight differing interpretations of inflation's role in shaping economic behavior.

Conclusion

Within the framework of overlapping generations, the interactions between money, capital, and consumption choices reveal essential insights into economic behavior over time. This analysis illustrates the fundamental balance policymakers must regard when considering monetary expansions and their potential ramifications on both current and future economic agents.

References

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