/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Life-cycle saving. (Modigliani a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 1

Life-cycle saving. (Modigliani and Brumberg, 1954.) Consider an individual who lives from 0 to \(T\), and whose lifetime utility is given by \(U=\int_{t=0}^{T} u(C(t)) d t\) where \(u^{\prime}(\bullet) > 0, u^{\prime \prime}(\bullet) < 0 .\) The individual's income is \(Y_{0}+g t\) for \(0 \leq t < R\), and 0 for \(R \leq t \leq T\). The retirement age, \(R\), satisfies \(0 < R < T\). The interest rate is zero, the individual has no initial wealth, and there is no uncertainty. (a) What is the individual's lifetime budget constraint? (b) What is the individual's utility-maximizing path of consumption, \(C(t) ?\) (c) What is the path of the individual's wealth as a function of \(t\) ?

Short Answer

Expert verified
(a) \( \int_{0}^{R} (Y_0 + gt) \, dt = \int_{0}^{T} C(t) \, dt \). (b) \( C(t) = \frac{Y_0 R + \frac{g R^2}{2}}{T} \). (c) \( W(t)=\int_{0}^{t} (Y_0 + gt - C) \, dt \) for \( t < R \) and \( W(t) = W(R) - C(t-R) \) for \( t \geq R \).

Step by step solution

01

Set Up the Budget Constraint

The individual's lifetime income must equal lifetime consumption. The income is earned only until retirement, i.e., from age 0 to R. It is modeled as \( Y_0 + gt \). After retirement, income is zero. The budget constraint is given by:\[ \int_{0}^{R} (Y_0 + gt) \, dt = \int_{0}^{T} C(t) \, dt \]
02

Calculate Total Lifetime Income

Integrate the income function from 0 to \( R \):\[ \int_{0}^{R} (Y_0 + gt) \, dt = \left[Y_0 t + \frac{g t^2}{2}\right]_{0}^{R} \]This results in:\[ Y_0 R + \frac{g R^2}{2} \]
03

Establish Lifetime Budget Constraint

Since total income equals total consumption over the lifetime, the budget constraint becomes:\[ Y_0 R + \frac{g R^2}{2} = \int_{0}^{T} C(t) \, dt \]
04

Derive Consumption Path C(t)

The utility is maximized when consumption is smoothed over time, due to the assumption \( u''(\bullet) < 0 \) (concavity of utility function). Hence, consumption is constant: \[ C(t) = C \] The budget constraint implies:\[ C \times T = Y_0 R + \frac{g R^2}{2} \] Giving constant consumption:\[ C = \frac{Y_0 R + \frac{g R^2}{2}}{T} \]
05

Determine Wealth Path as a Function of Time

Wealth, \( W(t) \), changes over time depending on income and consumption. From time 0 to \( R \):\[ W'(t) = Y_0 + gt - C \]From \( R \) to \( T \), no income is earned:\[ W'(t) = -C \]Initially \( W(0) = 0 \), and wealth at time \( R \) is:\[ W(R) = \int_{0}^{R} (Y_0 + gt - C) \, dt \]Simplifying gives:\[ W(R) = Y_0 R + \frac{g R^2}{2} - CR \]After \( R \), wealth decreases due to constant consumption \( C \):\[ W(t) = W(R) - C(t - R) \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lifetime Utility
Lifetime utility is a way to measure how much satisfaction or happiness a person gains from consuming goods or services throughout their life. It is represented as an integral over time, suggested in the formula:
\[ U = \int_{t=0}^{T} u(C(t)) \, dt \]
This formula captures the total pleasure or utility of consumption, \(C(t)\), over a lifespan from time 0 to \(T\).
  • The function \(u'(\bullet) > 0\) means more consumption increases utility, so consuming more satisfies someone more.
  • The function \(u''(\bullet) < 0\) illustrates diminishing returns, meaning each additional unit consumed gives slightly less extra pleasure than the one before.
This diminishing returns aspect leads to a preference for a steady consumption rate over time to maintain a consistent level of satisfaction.
Budget Constraint
A budget constraint is a financial plan showing the total resources available for consumption must equal total lifetime income. It ensures that an individual won't spend more than they earn in their lifetime. In the given problem, this is shown as:
\[ \int_{0}^{R} (Y_0 + gt) \, dt = \int_{0}^{T} C(t) \, dt \]
The left side shows income received until retirement, and the right side represents total consumption over life:
  • Income is available from time 0 to \(R\), with income diminishing to zero after retirement.
  • The integral on the right calculates total consumption from age 0 to \(T\).
Simply put, a budget constraint balances between what is earned and what can be spent over a lifetime.
Consumption Path
The consumption path is the planned pattern or schedule of spending over a person鈥檚 life. It aims to optimize utility based on income and savings, reflecting decisions on when and how much to consume.
For utility maximization, consumption is smoothed over time due to diminishing returns 鈥� hence, it is constant:
\[ C(t) = C \]
To find "C", use:
\[ C \times T = Y_0 R + \frac{g R^2}{2} \]
This indicates consumption \(C\) is evenly distributed across the time \(T\), ensuring stable consumption without large interruptions, achieving constant utility.
Wealth Accumulation
Wealth accumulation describes how an individual's savings grow or shrink over time, influenced by their income, consumption habits, and timing. Initially, wealth is zero. The change in wealth over each period depends on whether income exceeds or falls below consumption.
  • From 0 to \(R\): Wealth increases as income \(Y_0 + gt\) is greater than consumption \(C\):
    \( W'(t) = Y_0 + gt - C \)
  • At \(R\), wealth is:
    \( W(R) = Y_0 R + \frac{g R^2}{2} - CR \)
  • From \(R\) to \(T\): No income is received, just expenditures:
    \( W'(t) = -C \)
  • Wealth then decreases as regular consumption reduces personal savings:

    • The balance of saving and spending leads to a cycle of accumulation and drawdown of wealth through phases of life, aligning with life-stage needs.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A framework for investigating excess smoothness. Suppose that \(C_{l}\) equals \([r /(1+r)]\left|A_{t}+\sum_{s=0}^{\infty} E_{t}\left[Y_{t+s}\right] /(1+r)^{s}\right|,\) and that \(A_{l+1}=(1+r)\left(A_{t}+Y_{t}-C_{t}\right)\) (a) Show that these assumptions imply that \(E_{t}\left[C_{t+1}\right]=C_{t}\) (and thus that consumption follows a random walk) and that \(\sum_{s=0}^{\infty} E_{t}\left[C_{t+s}\right] /(1+r)^{s}=\) \(A_{i}+\sum_{s=0}^{\infty} E_{t}\left[Y_{t+s}\right] /(1+r)^{s}\) (b) Suppose that \(\Delta Y_{i}=\phi \Delta Y_{i-1}+u_{l}\), where \(u\) is white noise. Suppose that \(Y_{i}\) exceeds \(E_{t-1}\left[Y_{t}\right]\) by 1 unit (that is, suppose \(u_{t}=1\) ). By how much does consumption increase? (c) For the case of \(\phi>0,\) which has a larger variance, the innovation in income, \(u_{l}\), or the innovation in consumption, \(C_{t}-E_{t-1}\left[C_{t}\right] ?\) Do consumers use saving and borrowing to smooth the path of consumption relative to income in this model? Explain.

Consider a stock that pays dividends of \(D_{i}\) in period \(t\) and whose price in period \(t\) is \(P_{r}\). Assume that consumers are risk-neutral and have a discount rate of \(r ;\) thus they maximize \(E\left[\sum_{t=0}^{\infty} C_{t} /(1+r)^{t}\right]\) (a) Show that equilibrium requires \(P_{t}=E_{t}\left[\left(D_{t+1}+P_{t+1}\right) /(1+r)\right]\) (assume that if the stock is sold, this happens after that period's dividends have been paid). (b) Assume that \(\lim _{s \rightarrow \infty} E_{t}\left[P_{t+s} /(1+r)^{s}\right]=0\) (this is a no-bubbles condition; see the next problem). Iterate the expression in part ( \(a\) ) forward to derive an expression for \(P_{1}\) in terms of expectations of future dividends.

The equity premium and the concentration of aggregate shocks. (Mankiw, 1986.) Consider an economy with two possible states, each of which occurs with probability \(\frac{1}{2} .\) In the good state, each individual's consumption is \(1 .\) In the bad state, fraction \(\lambda\) of the population consumes \(1-(\phi / \lambda)\) and the remain der consumes \(1,\) where \(0<\phi<1\) and \(\phi \leq \lambda \leq 1 . \phi\) measures the reduction in average consumption in the bad state, and \(\lambda\) measures how broadly that re duction is shared. Consider two assets, one that pays off 1 unit in the good state and one that pays off 1 unit in the bad state. Let \(p\) denote the relative price of the bad-state asset to the good-state asset. (a) Consider an individual whose initial holdings of the two assets are zero, and consider the experiment of the individual marginally reducing (that is, selling short) his or her holdings of the good-state asset and using the proceeds to purchase more of the bad-state asset. Derive the condition for this change not to affect the individual's expected utility. (b) since consumption in the two states is exogenous and individuals are ex ante identical, \(p\) must adjust to the point where it is an equilibrium for individuals' holdings of both assets to be zero. Solve the condition derived in part ( \(a\) ) for this equilibrium value of \(p\) in terms of \(\phi, \lambda, U^{\prime}(1)\) and \(U^{\prime}(1-(\phi / \lambda))\) (c) Find \(\partial p / \partial \lambda\) (d) Show that if utility is quadratic, \(\partial p / \partial \lambda=0\) (e) Show that if \(U^{\prime \prime \prime}(\bullet)\) is everywhere positive, \(\partial p / \partial \lambda<0\)

The average income of farmers is less than the average income of nonfarmers, but fluctuates more from year to year. Given this, how does the permanent income hypothesis predict that estimated consumption functions for farmers and nonfarmers differ?

Time-inconsistent preferences. Consider an individual who lives for three periods. In period \(1,\) his or her objective function is \(\ln c_{1}+\delta \ln c_{2}+\delta \ln c_{3}\) where \(0<\delta<1 .\) In period \(2,\) it is \(\ln c_{2}+\delta \ln c_{3} .\) (since the individual's period3 choice problem is trivial, the period-3 objective function is irrelevant.) The individual has wealth of \(W\) and faces a real interest rate of zero. (a) Find the values of \(c_{1}, c_{2},\) and \(c_{3}\) under the following assumptions about how they are determined: (i) Commitment: The individual chooses \(c_{1}, c_{2},\) and \(c_{3}\) in period 1 (ii) No commitment, naivete: The individual chooses \(c_{1}\) in period 1 to maximize the period- 1 objective function, thinking he or she will also choose \(c_{2}\) to maximize this objective function. In fact, however, the individual chooses \(c_{2}\) to maximize the period- 2 objective function. (iii) No commitment, sophistication: The individual chooses \(c_{1}\) in period 1 to maximize the period-1 objective function, realizing that he or she will choose \(c_{2}\) in period 2 to maximize the period- 2 objective function. (b) \(\quad\) (i) Use your answers to parts \((a)(i)\) and \((a)(i i)\) to explain in what sense the individuals' preferences are time-inconsistent. (ii) Explain intuitively why sophistication does not produce different behavior than naivete.
See all solutions

Recommended explanations on Economics Textbooks

Taxation Economics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.