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Advanced Macroeconomics

David Romer

Chapter 11

Budget Deficits and Fiscal Policy - all with Video Answers

Educators

Chapter Questions

01:46

Problem 1

The stability of fiscal policy. (Blinder and Solow, $1973 .$ ) By definition, the budget deficit equals the rate of change of the amount of debt outstanding:
$\delta(t)=\dot{D}(t) .$ Define $d(t)$ to be the ratio of debt to output: $d(t)=D(t) / Y(t)$ Assume that $Y(t)$ grows at a constant rate $g>0$
(a) Suppose that the deficit-to-output ratio is constant: $\delta(t) / Y(t)=a,$ where $a>0$
(i) Find an expression for $d(t)$ in terms of $a, g,$ and $d(t)$
(ii) Sketch $\dot{d}(t)$ as a function of $d(t) .$ Is this system stable?
(b) Suppose that the ratio of the primary deficit to output is constant and equal to $a>0 .$ Thus the total deficit at $t, \delta(t),$ is given by $\delta(t)=a Y(t)+$ $r(t) D(t),$ where $r(t)$ is the interest rate at $t .$ Assume that $r$ is an increasing function of the debt-to-output ratio: $r(t)=r(d(t)),$ where $r^{\prime}(\bullet)>0, r^{\prime \prime}(\bullet)>$
$0, \lim _{d \rightarrow-\infty} r(d)<g_{n} \lim _{d \rightarrow \infty} r(d)>g$
(i) Find an expression for $d(t)$ in terms of $a, g,$ and $d(t)$
(ii) Sketch $d(t)$ as a function of $d(t)$. In the case where $a$ is sufficiently small that $d$ is negative for some values of $d$, what are the stability properties of the system? What about the case where $a$ is sufficiently large that $d$ is positive for all values of $d ?$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 2

Precautionary saving, non-lump-sum taxation, and Ricardian equivalence. (Leland, 1968 , and Barsky, Mankiw, and Zeldes, 1986 .) Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts $Y_{1}$ and $Y_{2}$ in the two periods. $Y_{1}$ is known, but $Y_{2}$ is random; assume for simplicity that $E\left[Y_{2}\right]=Y_{1} .$ The government taxes income at rate $T_{1}$ in period 1 and $\tau_{2}$ in period $2 .$ The individual can borrow and lend at a fixed interest rate, which for simplicity is assumed to be zero. Thus second-period consumption is $C_{2}=\left(1-\tau_{1}\right) Y_{1}-C_{1}+\left(1-\tau_{2}\right) Y_{2} .$ The individual chooses $C_{1}$ to maximize expected lifetime utility, $U\left(C_{1}\right)+E\left[U\left(C_{2}\right)\right]$
(a) Find the first-order condition for $C_{1}$
(b) Show that $E\left[C_{2}\right]=C_{1}$ if $Y_{2}$ is not random or if utility is quadratic.
(c) Show that if $U^{\prime \prime \prime}(\bullet)>0$ and $Y_{2}$ is random, $E\left[C_{2}\right]>C_{1}$
(d) Suppose that the government marginally lowers $n$, and raises $\tau_{2}$ by the same amount, so that its expected total revenue, $\boldsymbol{T}_{1} Y_{1}+\boldsymbol{T}_{2} \boldsymbol{E}\left[Y_{2}\right],$ is un changed. Implicitly differentiate the first-order condition in part (a) to find an expression for how $C_{1}$ responds to this change.
(e) Show that $C_{1}$ is unaffected by this change if $Y_{2}$ is not random or if utility is quadratic.
(f) Show that $C_{1}$ increases in response to this change if $U^{\prime \prime \prime}(\bullet)>0$ and $Y_{2}$ is random.

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07:18

Problem 3

Consider the Barro tax-smoothing model. Suppose that output, $Y$, and the real interest rate, $r,$ are constant, and that the level of government debt outstanding at time 0 is $0 .$ Suppose that there will be a temporary war from time 0 to time $\tau .$ Thus $G(t)$ equals $G_{H}$ for $0 \leq t \leq \tau,$ and equals $G_{L}$ thereafter, where $G_{H}>G_{L} .$ What are the paths of taxes, $T(t),$ and government debt outstanding. $D(t) ?$

Alex Loukas
Alex Loukas
Numerade Educator
02:49

Problem 4

Consider the Barro tax-smoothing model. Suppose there are two possible values of $G(t)-G_{H}$ and $G_{L}-$ with $G_{H}>G_{L} .$ Transitions between the two values follow Poisson processes (see Section 9.4). Specifically, if $G$ equals $G_{H}$, the probability per unit time that purchases fall to $G_{L}$ is $a ;$ if $G$ equals $G_{L}$, the probability per unit time that purchases rise to $G_{H}$ is $b$. Suppose also that output, $Y$, and the real interest rate, $r,$ are constant and that distortion costs are quadratic.
(a) Derive expressions for taxes at a given time as a function of whether $G$ equals $G_{H}$ or $G_{L}$, the amount of debt outstanding, and the exogenous parameters. (Hint: Use dynamic programming, described in Section 9.4 to find an expression for the expected present value of the revenue the government must raise as a function of $G,$ the amount of debt outstanding. and the exogenous parameters.)
(b) Discuss your results. What is the path of taxes during an interval when $G$ equals $G_{N} ?$ Why are taxes not constant during such an interval? What happens to taxes at a moment when $G$ falls to $G_{t} ?$ What is the path of taxes during an interval when $G$ equals $G_{L} ?$

Akash M
Akash M
Numerade Educator
06:16

Problem 5

If the tax rate follows a random walk (and if the variance of its innovations is bounded from below by a strictly positive number), then with probability
1 it will eventually exceed 100 percent or be negative. Does this observation suggest that the tax-smoothing model with quadratic distortion costs is not useful as either a positive or normative model of fiscal policy, since it has an implication that is both clearly incorrect as a description of the world and clearly undesirable as a prescription for policy? Explain your answer briefly.

Karl Schaefer
Karl Schaefer
University of Chicago
01:17

Problem 6

The Condorcet paradox. Suppose there are three voters, $1,2,$ and $3,$ and three possible policies, $A, B,$ and $C .$ Voter 1 's preference ordering is $A, B, C ;$ voter $2^{\prime} \sin B, C, A ;$ and voter $3^{\prime}$ s is $C, A, B,$ Does any policy win a majority of votes in a two-way contest against each of the alternatives? Explain.

Aman Gupta
Aman Gupta
Numerade Educator
00:27

Problem 7

Consider the Tabellini-Alesina model in the case where $\alpha$ can only take on the values 0 and $1 .$ Suppose that there is some initial level of debt, $D_{0} .$ How, if at all, does $D_{0}$ affect the deficit in period $1 ?$

AG
Ankit Gupta
Numerade Educator
07:24

Problem 8

Consider the Tabellini-Alesina model in the case where $\alpha$ can only take on the values 0 and $1 .$ Suppose that the amount of debt to be issued, $D,$ is determined before the preferences of the period-1 median voter are known. Specifically, voters vote on $D$ at a time when the probabilities that $\alpha_{1}^{\mathrm{NAD}}=1$ and that $\alpha_{2}^{\mathrm{MH}}=1$ are equal. Let $\pi$ denote this common value. Assume that the draws of the two median voters are independent.
(a) What is the expected utility of an individual with $\alpha=1$ as a function of $D, \pi,$ and $W ?$
(b) What is the first-order condition for this individual's most preferred value of $D ?$ What is the associated value of $D ?$
(c) What is the most preferred value of $D$ of an individual with $\alpha=0 ?$
(d) Given these results, if voters vote on $D$ before the period- 1 median voter is known, what value of $D$ does the median voter prefer?
(e) Explain briefly how, if at all, the question analyzed in part (d) differs from the question of whether individuals will support a balanced-budget
requirement if it is proposed before the preferences of the period-1 median voter are known.

Matt Just
Matt Just
Numerade Educator
07:24

Problem 9

Consider the Tabellini-Alesina model in the case where $\alpha$ can only take on the values 0 and $1 .$ Suppose, however, that there are 3 periods. The period- 1 median voter sets policy in periods 1 and $2,$ but in period 3 a new median voter sets policy. Assume that the period-1 median voter's $\alpha$ is $1,$ and that the probability that the period-3 median voter's $\alpha$ is 1 is $\pi$
$(a)$ Does $M_{1}=M_{2} ?$
(b) Suppose that after choosing purchases in period 1 , the period- 1 median voter learns that the probability that the period-3 median voter's $\alpha$ will be 1 is not $\pi$ but $\pi^{\prime},$ where $\pi^{\prime}<\pi .$ How does this news affect his or her choice of purchases in period $2 ?$

Matt Just
Matt Just
Numerade Educator
06:21

Problem 10

The Persson-Svensson model. (Persson and Svensson, $1989 .$ ) Suppose there are two periods. Government policy will be controlled by different policymakers in the two periods. The objective function of the period-t policymaker is $U+\alpha_{t}\left[V\left(G_{1}\right)+V\left(G_{2}\right)\right],$ where $U$ is citizens" utility from their private consumption; $\alpha_{l}$ is the weight that the period- $t$ policymaker puts on public consumption; $G_{t}$ is public consumption in period $t ;$ and $V(\bullet)$ satisfies $V^{\prime}(\bullet)>0, V^{\prime \prime}(\bullet)<0 .$ Private utility, $U,$ is given by $U=W-C\left(T_{1}\right)-C\left(T_{2}\right)$ where $W$ is the endowment; $T_{i}$ is taxes in period $t ;$ and $C(\bullet),$ the cost of raising revenue, satisfies $C^{\prime}(\bullet) \geq 1, C^{\prime \prime}(\bullet)>0 .$ All government debt must be paid off at the end of period 2. This implies $T_{2}=G_{2}+D,$ where $D=G_{1}-T_{1}$ is the amount of government debt issued in period 1 and where the interest rate is assumed to equal 0
(a) Find the first-order condition for the period- 2 policymaker's choice of $G_{2}$ given $D$. (Note: Throughout, assume that the solutions to the policymakers' maximization problems are interior.)
(b) How does a change in $D$ affect $G_{2} ?$
(c) Think of the period-1 policymaker as choosing $G_{1}$ and $D$, Find the firstorder condition for his or her choice of $D$
(d) Show that if $\alpha_{1}$ is less than $\alpha_{2}$, the equilibrium involves inefficiently low taxation in period 1 relative to tax-smoothing (that is, that it has $T_{1}<T_{2}$ ). Explain intuitively why this occurs.
(e) Does the result in part (d) imply that if $\alpha_{1}$ is less than $\alpha_{2}$, the period-1 policymaker necessarily runs a deficit? Explain.

Heather Duong
Heather Duong
Numerade Educator
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Problem 11

Consider the Alesina-Drazen model. Describe how, if at all, each of the following developments affects workers' proposal and the probability of reform:
(a) A fall in $T$.
$(b)$ A rise in $B$
(c) An equal rise in $A$ and $B$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:05

Problem 12

Crises and reform. Consider the model in Section 11.7 . Suppose, however. that if there is no reform, workers and capitalists both receive payoffs of $-C$ rather than $0,$ where $C \geq 0$
(a) Find expressions analogous to (11.37) and (11.38) for workers' proposal and the probability of reform.
(b) Define social welfare as the sum of the expected payoffs of workers and capitalists. Show that an increase in $C$ can raise this measure of social welfare.

Jiali Liu
Jiali Liu
Numerade Educator
03:45

Problem 13

Conditionality and reform. Consider the model in Section 11.7 . Suppose an international agency offers to give the workers and capitalists each an amount $F>0$ if they agree to reform. Use analysis like that in Problem 11.12 to show that this aid policy unambiguously raises the probability of reform and the social welfare measure defined in part
(b) of that problem.

Santosh Raj
Santosh Raj
Numerade Educator
07:24

Problem 14

Status-quo bias. (Fernandez and Rodrik, 1991.) There are two possible policies, A and
B. Each individual is either one unit of utility better off under Policy A or one unit worse off. Fraction $f$ of the population knows what its welfare would be under each policy. Of these individuals, fraction $\alpha$ are better off under Policy A and fraction 1 - $\alpha$ are worse off. The remaining individuals in the population know only that fraction $\beta$ of them are better off under Policy A and fraction 1 $-\beta$ are worse off.

A decision of whether to adopt the policy not currently in effect is made by majority vote. If the proposal passes, all individuals learn which policy makes them better off; a decision of whether to revert to the original policy is then made by majority vote. Each individual votes for the policy that gives him or her the higher expected utility. But if the proposal to revert to the original policy would be adopted in the event that the proposal to adopt the alternative policy passed, no one votes for the alternative policy. (This assumption can be justified by introducing a small cost of changing policies.)
(a) Find an expression for the fraction of the population that prefers Policy A (as a function of $f, \alpha,$ and $\beta$ ) for the case where fraction $1-f$ of the population knows only that fraction $\beta$ of them are better off under Policy A.
(b) Find the analogous expression for the case where all individuals know their welfare under both policies.
(c) Given your answers to parts $(a)$ and $(b),$ can there be cases when whichever policy is initially in effect is retained?

Matt Just
Matt Just
Numerade Educator
05:50

Problem 15

The common-pool problem in government spending. (Weingast, Shepsle, and Johnsen, $1981 .$ ) Suppose the economy consists of $M>1$ congressional districts. The utility of the representative person living in district $i$ is $E+$ $V\left(G_{i}\right)-C(T) . E$ is the endowment, $G_{i}$ is the level of a local public good in district $i,$ and $T$ is taxes (which are assumed to be the same in all districts). Assume $V^{\prime}(\bullet)>0, V^{\prime \prime}(\bullet)<0, C^{\prime}(\bullet)>0,$ and $C^{\prime \prime}(\bullet)>0 .$ The government
budget constraint is $\sum_{i=1}^{N} G_{i}=M T$. The representative from each district
dictates the values of $G$ in his or her district. Each representative maximizes the utility of the representative person living in his or her district.
(a) Find the first-order condition for the value of $G_{j}$ chosen by the representative from district $j,$ given the values of $G_{i}$ chosen by the other representatives and the government budget constraint (which implies $T=\left(\sum_{l=1}^{M} G_{i}\right) / M$. CNote: Throughout, assume interior solutions.)
(b) Find the condition for the Nash equilibrium value of $G$. That is, find the condition for the value of $G$ such that if all other representatives choose that value for their $G_{i},$ a given representative wants to choose that value.
(c) Is the Nash equilibrium Pareto-efficient? Explain. What is the intuition for this result?

Jesse Neumann
Jesse Neumann
Numerade Educator
05:03

Problem 16

Debt as a means of mitigating the common-pool problem. (Chari and Cole,
1993.) Consider the same setup as in Problem 11.15. Suppose, however, that there is an initial level of debt, $D$. The government budget constraint is therefore $D+\sum_{i=1}^{M} G_{i}=M T$
(a) How does an increase in $D$ affect the Nash equilibrium level of $G ?$
(b) Explain intuitively why your results in part (a) and in Problem 11.15 suggest that in a two-period model in which the representatives choose $D$ after the first-period value of $G$ is determined, the representatives would choose $D>0$
(c) Do you think that in a two-period model where the representatives choose $D$ before the first-period value of $G$ is determined, the representatives would choose $D>0 ?$ Explain intuitively.

Rochelle Stein
Rochelle Stein
Numerade Educator
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Problem 17

Consider the model of crises in Section 11.10 , and suppose $T$ is distributed uniformly on some interval $[\mu-X, \mu+X],$ where $X>0$ and $\mu-X \geq 0$ Describe how, if at all, each of the following developments affects the two curves in $(R, \pi)$ space that show the determination of $R$ and $\pi$ :
(a) A rise in $\mu$.
(b) A fall in $X$

Lainey Roebuck
Lainey Roebuck
Numerade Educator