Advanced Macroeconomics

David Romer

Chapter 6 Microeconimic Foundations of Incomplete Nominal Adjustment - all with Video Answers

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Chapter Questions

Problem 1

Consider the problem facing an individual in the Lucas model when $P_{i} / P$ is unknown. The individual chooses $L_{i}$ to maximize the expectation of $U_{i} ; U_{i}$ continues to be given by equation (6.3).
(a) Find the first-order condition for $L_{i}$, and rearrange it to obtain an expression for $L_{i}$ in terms of $E\left[P_{i} / P\right] .$ Take logs of this expression to obtain an expression for $\ell_{i}$.
(b) How does the amount of labor the individual supplies if he or she follows the certainty-equivalence rule in (6.17) compare with the optimal amount derived in part $(a) ?$ (Hint: How does $E\left[\ln \left(P_{i} / P\right)\right]$ compare with $\ln \left(E\left[P_{i} / P\right]\right) ?$ )
(c) Suppose that (as in the Lucas model) $\ln \left(P_{i} / P\right)=E\left[\ln \left(P_{i} / P\right) | P_{i}\right]+u_{i},$ where $u_{i}$ is normal with a mean of 0 and a variance that is independent of $P_{i}$ Show that this implies that $\ln \left[E\left[\left(P_{i} / P\right) | P_{i}\right]\right\}=E\left[\ln \left(P_{i} / P\right) | P_{i}\right]+C,$ where
$C$ is a constant whose value is independent of $P_{i}$. (Hint: Note that $P_{i} / P=$ $\exp \left\{E\left[\ln \left(P_{l} / P\right) | P_{i}\right]\right\} \exp \left(u_{i}\right),$ and show that this implies that the $\ell_{i}$ that maximizes expected utility differs from the certainty-equivalence rule in (6.17) only by a constant.

Ameer Said

Numerade Educator

19:17

Problem 2

(This follows Dixit and Stiglitz, $1977 .$ ) Suppose that the consumption index $C_{i}$ in equation (6.2) is $C_{i}=\left[\int_{j=0}^{1} Z_{j}^{1 / \eta} C_{i j}^{(n-1) / n} d j\right]^{n / i n-1},$ where $C_{i j}$ is the individual's consumption of good $j$ and $Z_{j}$ is the taste shock for good $j .$ Suppose the individual has amount $Y_{i}$ to spend on goods. Thus the budget constraint is $\int_{j=0}^{1} P_{j} C_{i j} d j=Y_{i}$.
(a) Find the first-order condition for $C_{i j}$ for the problem of maximizing $C_{i}$ subject to the budget constraint. Solve for $C_{i j}$ in terms of $Z_{j}, P_{j}$, and the Lagrange multiplier on the budget constraint.
(b) Use the budget constraint to find $C_{i j}$ in terms of $Z_{j}, P_{j}, Y_{i},$ and the $Z$ 's and $P^{\prime} s$.
(c) Substitute your result in part ( $b$ ) into the expression for $C_{i}$ and show that $C_{i}=Y_{i} / P,$ where $P=\left(\int_{j=0}^{1} Z_{j} P_{j}^{1-\eta} d j\right)^{1 / 1-\eta)}$.
(d) Use the results in part $(b)$ and part $(c)$ to show that $C_{i j}=Z_{j}\left(P_{j} / P\right)^{-n}\left(Y_{i} / P\right)$.
(e) Compare your results with (6.7) and (6.9) in the text.

Oluwadamilola Ameobi

Numerade Educator

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Problem 3

(Sargent, $1976 .$ ) Suppose that the money supply is determined by $m_{t}=c^{\prime} z_{t-1}+e_{t},$ where $c$ and $z$ are vectors and $e_{t}$ is an i.i.d. disturbance uncorrelated with $z_{t-1} \cdot e_{t}$ is unpredictable and unobservable. Thus the expected component of $m_{l}$ is $c^{\prime} z_{l-1},$ and the unexpected component is $e_{l}$ In setting the money supply, the Federal Reserve responds only to variables that matter for real activity; that is, the variables in $z$ directly affect $y$.
Now consider the following two models: (i) Only unexpected money matters, so $y_{t}=a^{\prime} z_{t-1}+b e_{t}+v_{i} ;(i i)$ all money matters, so $y_{t}=\alpha^{\prime} z_{t-1}+\beta m_{t}+$
$v_{t} .$ In each specification, the disturbance is i.i.d. and uncorrelated with $z_{t-1}$ and $e_{t}$.
(a) Is it possible to distinguish between these two theories? That is, given a candidate set of parameter values under, say, model ( $i$ ), are there parameter values under model (ii) that have the same predictions? Explain.
(b) Suppose that the Federal Reserve also responds to some variables that do not directly affect output; that is, suppose $m_{t}=c^{\prime} z_{t-1}+\gamma^{\prime} w_{t-1}+e_{t}$ and that models ( $i$ ) and (ii) are as before (with their disturbances now uncorrelated with $w_{i-1}$ as well as with $z_{i-1}$ and $e_{i}$ ). In this case, is it possible to distinguish between the two theories? Explain.

Oluwadamilola Ameobi

Numerade Educator

03:32

Problem 4

Suppose the economy is described by the model of Section 6.4. Assume, how ever, that $P$ is the price index described in part (c) of Problem 6.2 (with all the $\left.Z_{j}^{\prime} \text { s equal to } 1 \text { for simplicity }\right) .$ In addition, assume that money-market equilibrium requires that total spending in the economy equal $M .$ With these changes, is it still the case that in equilibrium, output of each good is given by (6.46) and that the price of each good is given by (6.47)$?$

Heather Zimmers

Numerade Educator

06:17

Problem 5

Consider an economy consisting of some firms with flexible prices and some with rigid prices. Let $p^{f}$ denote the price set by a representative flexible-price firm and $p^{r}$ the price set by a representative rigid-price firm. Flexible-price firms set their prices after $m$ is known; rigid-price firms set their prices before $m$ is known. Thus flexible-price firms set $p^{\prime}=p_{i}^{*}=(1-\phi) p+\phi m,$ and rigidprice firms set $p^{r}=E p_{i}^{*}=(1-\phi) E p+\phi E m,$ where $E$ denotes the expectation of a variable as of when the rigid-price firms set their prices.
d vartable as or when the righ irms ser rnelr $p$ Assume that fraction $q$ of firms have rigid prices, so that $p=q p^{r}+(1-q) p^{f}$.
(a) Find $p^{f}$ in terms of $p^{r}, m,$ and the parameters of the model ( $\phi$ and $q$ ).
(b) Find $p^{r}$ in terms of $E m$ and the parameters of the model.
(c) $(i)$ Do anticipated changes in $m$ (that is, changes that are expected as of when rigid-price firms set their prices) affect $y ?$ Why or why not?
(ii) Do unanticipated changes in $m$ affect $y ?$ Why or why not?

Jesse Neumann

Numerade Educator

04:19

Problem 6

Consider an economy consisting of many imperfectly competitive, pricesetting firms. The profits of the representative firm, firm $i,$ depend on aggregate output, $y,$ and the firm's real price, $r_{i}: \pi_{i}=\pi\left(y, r_{i}\right),$ where $\pi_{22} < 0$ (subscripts denote partial derivatives). Let $r^{*}(y)$ denote the profit-maximizing price as a function of $y ;$ note that $r^{*}(y)$ is characterized by $\pi_{2}\left(y, r^{*}(y)\right)=0$.
Assume that output is at some level $y_{0},$ and that firm i's real price is $r^{*}\left(y_{0}\right)$.
Now suppose there is a change in the money supply, and suppose that other firms do not change their prices and that aggregate output therefore changes to some new level, $y_{1}$.
(a) Explain why firm i's incentive to adjust its price is given by $G=\pi\left(y_{1}\right.$ $\left.r^{*}\left(y_{1}\right)\right)-\pi\left(y_{1}, r^{*}\left(y_{0}\right)\right)$.
(b) Use a second-order Taylor approximation of this expression in $y_{1}$ around $y_{1}=y_{0}$ to show that $G \simeq-\pi_{22}\left(y_{0}, r^{*}\left(y_{0}\right)\right)\left[r^{* \prime}\left(y_{0}\right)\right]^{2}\left(y_{1}-y_{0}\right)^{2} / 2$.
(c) What component of this expression corresponds to the degree of real rigidity? What component corresponds to the degree of insensitivity of the profit function?

James Kiss

Numerade Educator

04:56

Problem 7

(Ball and D. Romer, $1991 .$ ) Consider an economy consisting of many imperfectly competitive firms. The profits that a firm loses relative to what it obtains with $p_{i}=p^{*}$ are $K\left(p_{i}-p^{*}\right)^{2}, K > 0 .$ As usual, $p^{*}=p+\phi y$ and $y=m-p$. Each firm faces a fixed cost $Z$ of changing its nominal price. Initially $m$ is 0 and the economy is at its flexible-price equilibrium, which is $y=0$ and $p=m=0 .$ Now suppose $m$ changes to $m^{\prime}$.
(a) Suppose that fraction $f$ of firms change their prices. since the firms that change their prices charge $p^{*}$ and the firms that do not charge $0,$ this implies $p=f p^{*}$. Use this fact to find $p, y,$ and $p^{*}$ as functions of $m^{\prime}$ and $f$.
(b) Plot a firm's incentive to adjust its price, $K\left(0-p^{*}\right)^{2}=K p^{* 2},$ as a function of $f .$ Be sure to distinguish the cases $\phi<1$ and $\phi > 1$.
(c) A firm adjusts its price if the benefit exceeds $Z$, does not adjust if the benefit is less than $Z$, and is indifferent if the benefit is exactly $Z$. Given this, can there be a situation where both adjustment by all firms and adjustment by no firms are equilibria? Can there be a situation where neither adjustment by all firms nor adjustment by no firms is an equilibrium?

Erwin Antoni

Numerade Educator

14:30

Problem 8

(This follows Diamond, $1982 .)^{49}$ Consider an island consisting of $N$ people and many palm trees. Each person is in one of two states, not carrying a coconut and looking for palm trees (state $P$ ) or carrying a coconut and looking for other people with coconuts (state $C$ ). If a person without a coconut finds a palm tree, he or she can climb the tree and pick a coconut; this has a cost (in utility units) of $c .$ If a person with a coconut meets another person with a coconut, they trade and eat each other's coconuts; this yields $\bar{u}$ units of utility for each of them. (People cannot eat coconuts that they have picked themselves.)
A person looking for coconuts finds palm trees at rate $b$ per unit time. A person carrying a coconut finds trading partners at rate $a L$ per unit time, where
$L$ is the total number of people carrying coconuts. $a$ and $b$ are exogenous. Individuals' discount rate is $r$. Focus on steady states; that is, assume that
$L$ is constant.
(a) Explain why, if everyone in state $P$ climbs a palm tree whenever he or she finds one, then $r V_{P}=b\left(V_{C}-V_{P}-c\right),$ where $V_{P}$ and $V_{C}$ are the values of being in the two states.
(b) Find the analogous expression for $V_{C}$.
(c) Solve for $V_{C}-V_{P}, V_{C},$ and $V_{P}$ in terms of $r, b, c, \bar{u}, a,$ and $L$.
(d) What is $L$, still assuming that anyone in state $P$ climbs a palm tree whenever he or she finds one? Assume for simplicity that $a N=2 b$.
(e) For what values of $c$ is it a steady-state equilibrium for anyone in state $P$ to climb a palm tree whenever he or she finds one? (Continue to assume $a N=2 b$.
(f) For what values of $c$ is it a steady-state equilibrium for no one who finds a tree to climb it? Are there values of $c$ for which there is more than one steady-state equilibrium? If there are multiple equilibria, does one involve higher welfare than the other? Explain intuitively.

Oluwadamilola Ameobi

Numerade Educator

04:04

Problem 9

This problem follows Ball, $1988 .$ ) Suppose production at firm $i$ is given by $Y_{i}=S L_{i}^{\alpha},$ where $S$ is a supply shock and $0<\alpha \leq 1 .$ Thus in logs, $y_{i}=s+\alpha \ell_{i} .$ Prices are flexible;
thus (setting the constant term to 0 for simplicity), $p_{i}=w_{i}+(1-\alpha) \ell_{i}-s$. Aggregating the output and price equations yields $y=s+\alpha \ell$ and $p=w+$ $(1-\alpha) \ell-s .$ Wages are partially indexed to prices: $w=\theta p$, where $0 \leq \theta \leq 1$.
The production function and the pricing equation then imply that $y_{i}=$ $y-\phi\left(w_{i}-w\right),$ where $\phi \equiv \alpha \eta /[\alpha+(1-\alpha) \eta]$.
(i) What is employment at firm $i, \ell_{i},$ as a function of $m, s, \alpha, \eta, \theta,$ and $\theta_{i} ?$
(ii) What value of $\theta_{i}$ minimizes the variance of $\ell_{i} ?$
(iii) Find the Nash equilibrium value of $\theta$. That is, find the value of $\theta$ such that if aggregate indexation is given by $\theta,$ the representative firm minimizes the variance of $\ell_{i}$ by setting $\theta_{i}=\theta .$ Compare this value with the value found in part $(b)$.

Wen Zheng

Numerade Educator

10:52

Problem 10

Consider the Taylor model. Suppose, however, that every other period all the individuals set their prices for that period and the next. That is, in period $t$ prices are set for $t$ and $t+1 ;$ in $t+1,$ no prices are set; in $t+2,$ prices are set for $t+2$ and $t+3 ;$ and so on. As in the Taylor model, prices are both predetermined and fixed, and individuals set their prices according to $(6.87) .$ Finally, assume that $m$ follows a random walk.
(a) What is the representative individual's price in period $t, x_{t},$ as a function of $m_{i}, E_{i} m_{i+1}, p_{i},$ and $E_{l} p_{r+1} ?$
(b) Use the fact that synchronization implies that $p_{t}$ and $p_{t+1}$ are both equal to $x_{t}$ to solve for $x_{t}$ in terms of $m_{t}$ and $E_{t} m_{t+1}$.
(c) What are $y_{t}$ and $y_{t+1} ?$ Does the central result of the Taylor model-that nominal disturbances continue to have real effects after all prices have been changed-still hold? Explain intuitively.

Sajay Krishnan Paruthiyil

Numerade Educator

01:42

Problem 11

Consider the Taylor model with the money stock white noise rather than a random walk; that is, $m_{t}=\varepsilon_{t},$ where $\varepsilon_{t}$ is serially uncorrelated. Solve the model using the method of undetermined coefficients. (Hint: In the equation analogous to $(6.90),$ is it still reasonable to impose $\lambda+v=1 ?$)

Adriano Chikande

Numerade Educator

06:00

Problem 12

Repeat Problem 6.11 using lag operators.

Nicholas Sacco

Numerade Educator

Problem 13

Consider the model of Section 6.10 with the assumption that each period, fraction $\alpha$ of firms can set new prices, with the firms chosen at random. $\alpha$ is assumed to satisfy $0<\alpha \leq 1 .$ Thus if a firm sets a price in period
$t,$ the probability that it will be in effect in period $t+j$ is $(1-\alpha)^{j}$.
(a) Show that (6.73) implies that the price set by firms that adjust in period $t, x_{t},$ is $\alpha \sum_{j=0}^{\infty}(1-\alpha)^{j} E_{t} p_{t+j}^{*} .$ Express $x_{t}$ in terms of $p_{t}^{*}$ and $E_{t} x_{t+1} .$ Subtract $p_{t}$ from both sides to find an expression for the relative price charged by firms that set prices in $t, x_{1}-p_{1},$ in terms of $p_{t}^{*}-p_{t}, E_{t}\left[x_{t+1}-p_{t+1}\right],$ and $\left.E_{t} \pi_{t+1} \text { (where } \pi_{t+1}=p_{t+1}-p_{t}\right)$.
(b) Show that the average (log) price in $t, p_{t},$ is $\alpha \sum_{j=0}^{\infty}(1-\alpha)^{j} x_{l-j}$. Express $p_{t}$ in terms of $x_{t}$ and $p_{t-1} .$ Use these results to express the inflation rate, $\pi_{t}=p_{t}-p_{t-1},$ in terms of $x_{t}-p_{t}$.
(c) Use your result in part ( $b$ ) to substitute for $x_{t}-p_{t}$ and $E_{t}\left[x_{t+1}-p_{t+1}\right]$ in your answer for part $(a),$ and solve for the inflation rate, $\pi_{t},$ in terms of $E_{t} \pi_{t+1}$ and $p_{t}^{*}-p_{t} .$ Use the fact that $p_{t}^{*}-p_{t}=\phi y_{t}$ to express $\pi_{t}$ in terms of $E_{t} \pi_{t+1}$ and $y_{t}$.

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10:52

Problem 14

Consider an economy like that of the Caplin-Spulber model. Suppose, however, that $m$ can either rise or fall, and that firms therefore follow a two-sided Ss policy: if $p_{i}-p_{i}^{*}(t)$ reaches either $S$ or $-S$, firm $i$ changes $p_{i}$ so that $p_{i}-p_{i}^{*}(t)$ equals $0 .$ As in the Caplin-Spulber model, changes in $m$ are continuous.
Assume for simplicity that $p_{i}^{*}(t)=m(t) .$ In addition, assume that $p_{i}-$ $p_{i}^{*}(t)$ is initially distributed uniformly over some interval of width $S ;$ that is, $p_{l}-p_{i}^{*}(t)$ is distributed uniformly on $[X, X+S]$ for some $X$ between $-S$ and 0.
(a) Explain why, given these assumptions, $p_{i}-p_{i}^{*}(t)$ continues to be distributed uniformly over some interval of width $S$.
(b) Are there any values of $X$ for which an infinitesimal increase in $m$ of $d m$ raises average prices by less than $d m ?$ by more than $d m ?$ by exactly $d m ?$ Thus, what does this model imply about the real effects of monetary shocks?

Sajay Krishnan Paruthiyil

Numerade Educator

01:11

Problem 15

(This follows Ball, 1994 a.) Consider a continuous-time version of the Taylor model, so that $p(t)=(1 / T) \int_{\tau=0}^{T} x(t-\tau) d \tau,$ where $T$ is the interval between each individual's price changes and $x(t-\tau)$ is the price set by individuals who set their prices at time $t-\tau .$ Assume that $\phi=1,$ so that $p_{i}^{*}(t)=m(t) ;$ thus
$x(t)=(1 / T) \int_{\tau=0}^{T} E_{t} m(t+\tau) d \tau.$
(a) Suppose that initially $m(t)=g t(g>0),$ and that $E_{t} m(t+\tau)$ is therefore $(t+\tau) g .$ What are $x(t), p(t),$ and $y(t)=m(t)-p(t) ?$
(b) Suppose that at time 0 the government announces that it is steadily reducing money growth to 0 over the next interval $T$ of time. Thus $m(t)=$ $t[1-(t / 2 T)] g$ for $0<t<T,$ and $m(t)=g T / 2$ for $t \geq T .$ The change is unexpected, so that prices set before $t=0$ are as in part $(a)$.

Carson Merrill

Numerade Educator