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Advanced Macroeconomics

David Romer

Chapter 8

Consumption - all with Video Answers

Educators

Chapter Questions

04:54

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 $t$

Natalie Britton
Natalie Britton
Numerade Educator
02:20

Problem 2

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

Jennifer Stoner
Jennifer Stoner
Numerade Educator
07:37

Problem 3

The time-averaging problem. (Working, 1960 .) Actual data do not give con sumption at a point in time, but average consumption over an extended period, such as a quarter. This problem asks you to examine the effects of this fact.
Suppose that consumption follows a random walk: $C_{t}=C_{t-1}+e_{t},$ where $e$ is white noise. Suppose, however, that the data provide average consumption over two-period intervals; that is, one observes $\left(C_{t}+C_{t+1}\right) / 2,\left(C_{t+2}+C_{t+3}\right) / 2,$ and so on
(a) Find an expression for the change in measured consumption from one twoperiod interval to the next in terms of the $e^{\prime}$ 's.
(b) Is the change in measured consumption uncorrelated with the previous value of the change in measured consumption? In light of this, is measured consumption a random walk?
(c) Given your result in part ( $a$ ), is the change in consumption from one two period interval to the next necessarily uncorrelated with anything known as of the first of these two-period intervals? Is it necessarily uncorrelated with anything known as of the two-period interval immediately preceding the first of the two-period intervals?
(d) Suppose that measured consumption for a two-period interval is not the average over the interval, but consumption in the second of the two periods. That is, one observes $C_{t+1}, C_{t+3},$ and so on. In this case, is measured consumption a random walk?

Heather Duong
Heather Duong
Numerade Educator
01:15

Problem 4

In the model of Section $8.2,$ uncertainty about future income does not affect con sumption. Does this mean that the uncertainty does not affect expected lifetime utility'

James Kiss
James Kiss
Numerade Educator

Problem 5

(This follows Hansen and singleton, 1983 .) Suppose instantaneous utility is of the constant-relative-risk-aversion form, $u\left(C_{t}\right)=C_{t}^{1-t} /(1-\theta), \theta>0 .$ Assume that the real interest rate, $r,$ is constant but not necessarily equal to the discount rate, $\rho$
(a) Find the Euler equation relating $C_{t}$ to expectations concerning $C_{t+1}$
(b) Suppose that the log of income is distributed normally, and that as a result the $\log$ of $C_{t+1}$ is distributed normally; let $\sigma^{2}$ denote its variance conditional on information available at time $t$. Rewrite the expression in part (a) in terms $\operatorname{of} \ln C_{t}, E_{t}\left[\ln C_{t+1}\right], \sigma^{2},$ and the parameters $r, \rho,$ and $\theta .$ (Hint: If a variable $x$ is distributed normally with mean $\left.\mu \text { and variance } V, E\left[e^{x}\right]=e^{\mu} e^{V / 2} .\right)$
(c) Show that if $r$ and $\sigma^{2}$ are constant over time, the result in part ( $b$ ) implies that the log of consumption follows a random walk with drift: $\ln C_{t+1}=$ $a+\ln C_{t}+u_{t+1},$ where $u$ is white noise.
(d) How do changes in each of $r$ and $\sigma^{2}$ affect expected consumption growth. $E_{t}\left[\ln C_{t+1}-\ln C_{t}\right] ?$ Interpret the effect of $\sigma^{2}$ on expected consumption growth in light of the discussion of precautionary saving in Section 8.6

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02:55

Problem 6

A framework for investigating excess smoothness. Suppose that $C_{t}$ equals
$[r /(1+r)]\left[A_{t}+\sum_{s=0}^{\infty} E_{t}\left[Y_{t+s]}\right] /(1+r)^{s}\right],$ and that $A_{t+1}=(1+r)\left(A_{t}+Y_{t}-C_{t}\right)$
(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 $\sum_{s=0}^{\infty} E_{t}\left[C_{t+s}\right] /(1+r)^{s}=A_{t}+$ $\sum_{s=0}^{\infty} E_{t}\left[Y_{t+s}\right] /(1+r)^{s}$
(b) Suppose that $\Delta Y_{t}=\phi \Delta Y_{t-1}+u_{t},$ where $u$ is white noise. Suppose that $Y_{t}$ ex. ceeds $E_{t-1}\left[Y_{t}\right]$ 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, $u_{t},$ 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

Heather Duong
Heather Duong
Numerade Educator
02:49

Problem 7

Consider the two-period setup analyzed in Section $8.4 .$ Suppose that the government initially raises revenue only by taxing interest income. Thus the individual's budget constraint is $C_{1}+C_{2} /[1+(1-\tau) r] \leq Y_{1}+Y_{2} /[1+(1-\tau) r]$ where $\tau$ is the tax rate. The government's revenue is 0 in period 1 and $\tau r\left(Y_{1}-C_{1}^{0}\right)$ in period $2,$ where $C_{1}^{0}$ is the individual's choice of $C_{1}$ given this tax rate. Now suppose the government eliminates the taxation of interest income and instead institutes lump-sum taxes of amounts $T_{1}$ and $T_{2}$ in the two periods; thus the individual's budget constraint is now $C_{1}+C_{2} /(1+r) \leq\left(Y_{1}-T_{1}\right)+\left(Y_{2}-T_{2}\right) /(1+r) .$ Assume that $Y_{1}, Y_{2},$ and $r$ are exogenous.
(a) What condition must the new taxes satisfy so that the change does not affect the present value of government revenues?
(b) If the new taxes satisfy the condition in part (a), is the old consumption bundle, $\left(C_{1}^{0}, C_{2}^{0}\right),$ not affordable, just affordable, or affordable with room to spare?
(c) If the new taxes satisfy the condition in part ( $a$ ), does first-period consumption rise, fall, or stay the same?

Akash M
Akash M
Numerade Educator
01:23

Problem 8

Consider a stock that pays dividends of $D_{t}$ in period $t$ and whose price in period $t$ is $P_{t}$. 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 $P_{z}=E_{t}\left[\left(D_{t+1}+P_{t+1}\right) /(1+r)\right]$ (assume that if the stock is sold, this happens after that period's dividends have been paid).
(b) Assume that $\lim _{x \rightarrow \infty} E_{t}\left[P_{t+s} /(1+r)^{s}\right]=0$ (this is a no-bubbles condition; see the next problem). Iterate the expression in part (a) forward to derive an ex. pression for $P_{t}$ in terms of expectations of future dividends.

Nick Johnson
Nick Johnson
Numerade Educator
03:05

Problem 9

Bubbles. Consider the setup of the previous problem without the assumption that $\lim _{s \rightarrow \infty} E_{t}\left[P_{t+s} /(1+r)^{s}\right]=0$
(a) Deterministic bubbles. Suppose that $P_{t}$ equals the expression derived in part (b) of Problem 8.8 plus $(1+r)^{t} b, b>0$
(i) Is consumers' first-order condition derived in part (a) of Problem 8.8 still satisfied?
(ii) $\mathrm{Can} b$ be negative' (Hint: Consider the strategy of never selling the stock.)
(b) Bursting bubbles. (Blanchard, 1979 .) Suppose that $P_{\mathrm{r}}$ equals the expression derived in part (b) of Problem 8.8 plus $q_{l^{\prime}}$. where $q_{t}$ equals $(1+r) q_{t-1} / \alpha$ with probability $\alpha$ and equals 0 with probability $1-\alpha$
(i) Is consumers' first-order condition derived in part (a) of Problem 8.8 still satisfied?
(ii) If there is a bubble at time $t$ (that is, if $q_{t}>0$ ), what is the probability that the bubble has burst by time $t+s$ (that is, that $q_{t+s}=0$ )? What is the limit of this probability as $s$ approaches infinity?
(c) Intrinsic bubbles. (Froot and Obstfeld, 1991 ) Suppose that dividends follow
a random walk: $D_{\mathrm{t}}=D_{\mathrm{t}-1}+e_{t},$ where $e$ is white noise.
(i) In the absence of bubbles, what is the price of the stock in period $t$ ?
(ii) Suppose that $P_{t}$ equals the expression derived in $(i)$ plus $b_{t},$ where $b_{t}=$ $(1+r) b_{t-1}+c e_{t}, c>0 .$ Is consumers' first-order condition derived in part $(a)$ of Problem 8.8 still satisfied? In what sense do stock prices overreact to changes in dividends?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:30

Problem 10

The Lucas asset-pricing model. (Lucas, 1978 ) Suppose the only assets in the economy are infinitely lived trees. Output equals the fruit of the trees, which is exogenous and cannot be stored; thus $C_{t}=Y_{t},$ where $Y_{t}$ is the exogenously determined output per person and $C_{t}$ is consumption per person. Assume that initially each consumer owns the same number of trees. since all consumers are assumed to be the same, this means that, in equilibrium, the behavior of the price of trees must be such that, each period, the representative consumer does not want to either increase or decrease his or her holdings of trees.

Let $P_{\mathrm{r}}$ denote the price of a tree in period $t$ (assume that if the tree is sold, the sale occurs after the existing owner receives that period's output). Finally, assume that the representative consumer maximizes $E\left[\sum_{t=0}^{\infty} \ln C_{t} /(1+\rho)^{t}\right]$
(a) Suppose the representative consumer reduces his or her consumption in period $t$ by an infinitesimal amount, uses the resulting saving to increase his or her holdings of trees, and then sells these additional holdings in period $t+1$ Find the condition that $C_{t}$ and expectations involving $Y_{t+1}, P_{t+1},$ and $C_{t+1}$ must satisfy for this change not to affect expected utility. Solve this condition for $P_{t}$ in terms of $Y_{t}$ and expectations involving $Y_{t+1}, P_{t+1},$ and $C_{t+1}$
(b) Assume that $\lim _{x \rightarrow \infty} E_{t}\left[\left(P_{t+s} / Y_{t+s}\right) /(1+\rho)^{s}\right]=0 .$ Given this assumption,
iterate your answer to part $(a)$ forward to solve for $P_{z}$. (Hint: Use the fact that
\[
C_{t+s}=Y_{t+s} \text { for all } s .
\]
(c) Explain intuitively why an increase in expectations of future dividends does not affect the price of the asset.
(d) Does consumption follow a random walk in this model?

M Hassan Anwar
M Hassan Anwar
Numerade Educator
03:56

Problem 11

The equity premium and the concentration of aggregate shocks. (Mankiw, 1986 .) Consider an economy with two possible states, each of which occurs with probability one-half. 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 re mainder 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 reduction 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, $\partial p / \partial \lambda<0$

Sana Riaz
Sana Riaz
Numerade Educator

Problem 12

Consumption of durable goods. (Mankiw, 1982 ) Suppose that, as in Section $8.2,$ the instantaneous utility function is quadratic and the interest rate and the discount rate are zero. Suppose, however, that goods are durable; specifically, $C_{t}=(1-\delta) C_{t-1}+X_{t},$ where $X_{t}$ is purchases in period $t$ and $0 \leq \delta<1$
(a) Consider a marginal reduction in purchases in period $t$ of $d X_{r}$. Find values of $d X_{t+1}$ and $d X_{t+2}$ such that the combined changes in $X_{t}, X_{t+1},$ and $X_{t+2}$ leave the present value of spending unchanged $\left(\text { so } d X_{t}+d X_{t+1}+d X_{t+2}=0\right)$ and leave $C_{r+2}$ unchanged $\left(s o(1-\delta)^{2} d X_{t}+(1-\delta) d X_{t+1}+d X_{t+2}=0\right)$
(b) What is the effect of the change in part ( $a$ ) on $C_{t}$ and $C_{t+1}$ ? What is the effect on expected utility?
(c) What condition must $C_{t}$ and $E_{t}\left[C_{t+1}\right]$ satisfy for the change in part $(a)$ not to affect expected utility? Does $C$ follow a random walk?
(d) Does $X$ follow a random walk? (Hint: Write $X_{t}-X_{t-1}$ in terms of $C_{t}-C_{t-1}$ and $C_{t-1}-C_{t-2}$.) Explain intuitively. If $\delta=0$, what is the behavior of $X$ ?

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08:23

Problem 13

Habit formation and serial correlation in consumption growth. Sup pose that the utility of the representative consumer, individual $i,$ is given by $\sum_{t=1}^{T}\left[1 /(1+\rho)^{t}\right]\left(C_{i t} / Z_{i t}\right)^{1-\theta} /(1-\theta), \rho>0, \theta>0,$ where $Z_{i t}$ is the "reference" level
of consumption. Assume the interest rate is constant at some level, $r,$ and that there is no uncertainty.
(a) External habits. Suppose $Z_{i t}=C_{i-1}^{\phi}, 0 \leq \phi \leq 1 .$ Thus the reference level of consumption is determined by aggregate consumption, which individual $i$ takes as given.
(i) Find the Euler equation for the experiment of reducing $C_{i t}$ by $d C$ and increasing $C_{t, t+1}$ by $(1+r) d C .$ Express $C_{1, t+1} / C_{i, t}$ in terms of $C_{t} / C_{t-1}$ and $(1+r) /(1+\rho)$
(ii) In equilibrium, the consumption of the representative consumer must equal aggregate consumption: $C_{i t}=C_{t}$ for all $t .$ Use this fact to express current consumption growth, $\ln C_{t+1}-\ln C_{t},$ in terms of lagged consumption growth, $\ln C_{t}-\ln C_{t-1}$, and anything else that is relevant. If $\phi>0$ and $\theta=1,$ does habit formation affect the behavior of consumption? What if $\phi>0$ and $\theta>1 ?$ Explain your results intuitively.
(b) Internal habits. Suppose $Z_{t}=C_{i, k-1}$. Thus the reference level of consump. tion is determined by the individual's own level of past consumption (and the parameter $\phi$ is fixed at 1 ).
(i) Find the Euler equation for the experiment considered in part $(a)(i)$. (Note that $C_{i t}$ affects utility in periods $t$ and $t+1$, and $C_{i, t+1}$ affects utility in $t+1 \text { and } t+2 .)$
(ii) Let $g_{t} \equiv\left(C_{t} / C_{t-1}\right)-1$ denote consumption growth from $t-1$ to $t$. As sume that $\rho=r=0$ and that consumption growth is close to zero (so that we can approximate expressions of the form $\left(C_{t} / C_{t-1}\right)^{r}$ with $1+\gamma g_{i},$ and can ignore interaction terms). Using your results in $(i),$ find an approximate expression for $g_{t+2}-g_{t+1}$ in terms of $g_{t+1}-g_{t}$ and anything else that is relevant. Explain your result intuitively.

Heather Duong
Heather Duong
Numerade Educator
10:08

Problem 14

Precautionary saving with constant-absolute-risk-aversion utility. Consider an individual who lives for two periods and has constant-absoluterisk-aversion utility, $U=-e^{-\gamma C_{1}}-e^{-\gamma C_{2}}, \gamma>0 .$ The interest rate is zero and the individual has no initial wealth, so the individual's lifetime budget constraint is $C_{1}+C_{2}=Y_{1}+Y_{2}, Y_{1}$ is certain, but $Y_{2}$ is normally distributed with mean $\bar{Y}_{2}$ and variance $\sigma^{2}$
(a) With an instantaneous utility function $u(C)=-e^{-\gamma c}, \gamma>0,$ what is the sign of $U^{\prime \prime \prime}(C) ?$
(b) What is the individual's expected lifetime utility as a function of $C_{1}$ and the exogenous parameters $Y_{1}, \bar{Y}_{2}, \sigma^{2},$ and $\gamma$ ? (Hint. See the hint in Problem 8.5 part (b).)
(c) Find an expression for $C_{1}$ in terms of $Y_{1}, \bar{Y}_{2}, \sigma^{2},$ and
$\gamma .$ What is $C_{1}$ if there is no uncertainty? How does an increase in uncertainty affect $C_{1} ?$

Heather Duong
Heather Duong
Numerade Educator
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Problem 15

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 period-3 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.

Oluwadamilola Ameobi
Oluwadamilola Ameobi
Numerade Educator
01:24

Problem 16

Consider the dynamic programming problem that leads to Figure $8.4 .$ This problem asks you to solve the problem numerically with one change: preferences are logarithmic, so that $u(C)=\ln C .$ Specifically, it asks you to approximate the value function by value-function iteration, along the lines of equation $(8.73),$ with $V^{\circ}(X)$ assumed to equal zero for all $\bar{X}$
(a) As a preliminary, explain why $V^{1}(X)=\ln X$
(b) since it is not literally possible to find $V^{\mathrm{m}}(X)$ for every $X$ from 0 to infin ity, proceed by discretizing the problem. Choose an $N$, and define $e=100 / N$ Now, assume that $Y$ can take on only the values $e, 3 e, 5 e, \ldots, 200-e,$ each with probability $1 / N .$ Likewise, assume that $C$ can only take on the values
$e, 3 e, 5 e, \ldots,$ and find the $V^{n}(X)^{\prime}$ s only for $X$ equal to $e, 3 e, 5 e, \ldots,$ up to some upper bound $B$ that you choose (and assume that $V^{n}(X)=V^{n}(B)$ for $X>B$ ). Finally, only do some finite number of iterations. (Use whatever programming language or software you wish; MATLAB is a natural candidate.) Plot or sketch the resulting $V(\bullet)$
(c) Comment briefly on the process of solving the problem numerically. For ex. ample, explain why you chose the values of $N, B,$ and the number of iterations that you did. Did you encounter anything unexpected?
(d) Using the value function you found, find $C(\bullet),$ and plot or sketch that.
(e) Compare the $C(\bullet)$ you found with that in Figure $8.4 .$ What are the main similarities? The main differences?

Manik Pulyani
Manik Pulyani
Numerade Educator
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Problem 17

Consider the following seemingly small variation on part ( $b$ ) of Problem 8.16 . Choose an $N$, and define $e \equiv 200 / N .$ Now, assume that $Y$ can take on only the values $0, e, 2 e, 3 e, \ldots, 200,$ each with probability $1 /(N+1) .$ Likewise, assume that $C$ can only take on the values $0, e, 2 e, 3 e, \ldots,$ and find the $V^{\prime \prime}(X)^{\prime}$ s only for $X$ equal to $0, e, 2 e, \ldots$ up to some upper bound $B$ that you choose (and assume that $\left.V^{n}(X)=V^{n}(B) \text { for } X>B\right)$
Show (analytically, not numerically) that value-function iteration using this numerical algorithm converges to $V(X)=-\infty$ for all $X$. (Hint: If you get stuck, try it with $N=2, B=300 .)^{24}$

Victor Salazar
Victor Salazar
Numerade Educator
06:08

Problem 18

This problem asks you to use your analysis in Problem 8.16 to see how a onetime income shock affects the path of consumption starting from different situations. Specifically, under the same assumptions about the household's preferences and the distribution of $Y$ as in Problem $8.16,$ plot, as a function of time, the difference between the paths of $C$ for a household with a realized path of income of $\{10,100,100,100, \ldots\}$ and a household with a realized path of income of $\{100,100,100,100, \ldots\}$
(a) In the case where both households enter the initial period with $A=10$. (Recall that $A_{t}$ is the household's wealth at the start of period $t$ -that is, before $Y_{t}$ is realized and $C_{t}$ is chosen.)
(b) In the case where both households enter the initial period with $A=200$.
(c) Discuss your results. Are there any noteworthy similarities between the results in parts $(a)$ and $(b)$ ? Any noteworthy differences?

Kim Matthews
Kim Matthews
Numerade Educator
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Problem 19

Consider Problem $8.16 .$ Change something about the model (the natural candidates are the utility function, the value of $\beta$, the value of $r,$ and the distribution of $Y$ ) and find the new $V(\bullet)$ and $C(\bullet)$ functions. Discuss how the change in as sumptions changes the results, and explain the intuition.

AP
Andreas Papavassiliou
Numerade Educator
01:52

Problem 20

Problem 8.16 had you use a very primitive way of tackling the problem numeri cally. How might one do better' (Some candidates might involve interpolation or extrapolation, or not making the points you consider equally spaced.)

Santana Scott
Santana Scott
Numerade Educator