optimal interest rates and fiscal policy

7/28/2019 Optimal Interest Rates and Fiscal Policy

1/52

Optimal Interest Rates Under Endogenous Fiscal Policy

Mercedes Haga

Ponticia Universidad Catlica de Chile

March 17, 2009

Abstract

It is a fact that some politicians once they reach the power do not behave as benev-

olent planners but as self-interested bureaucrats, according to which, they make scaldecisions. On the other hand, central bank independence claims that the monetary

authority should set its control variable only in accordance to its loss function. This

paper studies wether this is still true in an environment where scal expenditure is

endogenously determined by a self-interested incumbent whose scal decisions might

be contrary to individuals welfare. I nd that the central bank should always care

about the way the scal authority sets its consumption path as a signaling eect. Fur-

thermore, the larger is the government relative to the private sector, the more sensitive

the central banks reaction should be. I also nd microfoundations for forward-looking

Taylor rules. Finally, even if the government would also be concerned with individuals

welfare; the central bank should still consider deviations of scal expenditure in its

reaction function.

1 Introduction

It is a fact that some politicians once they reach the power do not behave as benevolent

planners but as self-interested bureaucrats (sometimes with non-social objectives such as

personal enrichment or re-election), according to which, they make decisions. One of those

decisions is to specify scal expenditure. Fiscal expenditure has a high social impact

not only because of its direct eect on welfare, but also because it inuences the price

level determination and therefore ination. However, current general equilibrium modelsdo not account for this feature and consider scal choices as an exogenous process. On

the other hand, agency models do deal with political economy issues, but they are all

1


2/52

partial equilibrium models. In this way, many interactions between variables might not be

considered not only in general equilibrium models but also in partial equilibrium ones; for

example, how should the monetary authority react in the presence of a scal shock when

the incumbent cares only about its own welfare? How will households allocate consumptionand labor when they know that the government is not interested in soften shocks for them?

Will rms also behave in a dierent way? And more importantly, what can we expect from

the interaction between a self-interested scal authority and a central bank concerned with

consumerss welfare? In other words, what is gained in terms of interaction between scal

and monetary authorities when one of them behaves in a selsh way?

By the end of 2008 a group of twenty eight developed and developing countries have

adopted ination targeting regimes as a way to conduct their monetary policy.1 The rst

one to do it was New Zealand in 1990 while the last one was Ghana in May, 2007. It is

interesting to notice that even though countries are very dierent from each other as long astheir macro variables are concerned (for example current ination, interest rates, GDP per

capita, etc.), all targets uctuate between 0% and 5%.2 In this way; and precisely due to

their dierence in macro fundamentals, it is not expected for example, for the Reserve Bank

of Australia to react in the same way as the Central Bank of Chile or the Central Bank

of Peru whenever a shock occurs. Considering target interest rate, scal expenditure and

ination data for these three countries I nd that the percentage change of the interest rate

to a 1% change in scal expenditure is not statistically signicant in the case of Australia,

but equals 0:955 for Peru and 2:024 for Chile (regressions are shown in the appendix). This

tells us that there are some considerations underlying that makes central banks to behave

dierently in each country.

Keeping the previous discussion in mind, this paper investigates how political economy

aects the structure and dynamics of consumption, labor, ination and how do optimal

interest rate rules change when it is present the interaction between a self-interested in-

cumbent and a "benevolent" and independent central bank that wants to minimize its loss

function. For that, I use a closed-economy general equilibrium model to address the con-

ict of interest that arrises between voters and rent-seeking politicians who like to consume

for their own purposes the same type of goods as households. This approach will allow

us to obtain a general equilibrium model able to describe the behavior and interaction

1 Finland and Spain however, have abandoned the regime in 1999 when they joined the European Eco-nomic and Monetary Union.

2 Exceptions are Indonesia and Ghana with targets of 6% (1%) and 0 10% respectively.

2


3/52

of all variables (for example, the behavior of ination, not present in partial equilibrium

models), and at the same time incorporate distortions in scal expenditure manipulation

arising from political economy; all of which has a direct impact on ination and therefore

on the central banks best response to this environment. It is important to note that theintitutionality underlying this model is such that even though it can not prevent the incum-

bent from taking over a portion of total scal expenditure, it does guarantee the central

banks independence; that is to say, the government can not pressure the central bank to

deviate from its loss function.

This work is related to two dierent literatures: political economy and optimal targeting

rule. As long as political economy is concerned, this work is placed within "opportunistic-

rent-seeking" politicians models in line with Diermeier, Keane and Merlo [2005], Shi and

Svensson [2006] and Acemoglu, Golosov and Tsyvinski [2008] where re-election is pursued

for politicianss own welfare in terms of expected returns of being in power (i.e. ego rentsor self-enrichment). I will assume politicians want to hold oce because it enables them

to appropriate from a certain fraction of public expenditure for their own consumption.

On the other hand, the optimal targeting rule part of this paper is most closely related

to Clarida, Gal and Gertlers seminal contribution on the eld as I will analyze dierent

interest rate rules for monetary policy that come from the interest (or command) of the

central bank in minimizing a quadratic loss function.3 In particular, I will work with the

hybrid neokeynesian Phillips curve proposed by Gal and Gertler [1999] in order to capture

ination persistence that seems to characterize the data.4

In this way the model is derived by the interaction between a self-interested incumbent

concerned with holding oce and an independent central bank concerned with households

welfare. I study a monopolistic competitive closed economy with no capital and scal

shocks, equivalent to the framework used in Walsh [2003]. To capture the fact that politi-

cianss incentives are not always aligned with the citizenss ones, I will assume that the

policymaker would like to manipulate scal expenditure in a selsh way. In other words, the

incumbent will be willing to mislead the voters perception about his competence through

scal expenditure and have a higher probability of reappointment, as implied by Shi and

Svenssons [2006] work.

I nd that not only for the case with no government but also for the case where scal

expenditure is exogenously determined, the monetary authority should set the interest3 Woodford [2003] show that this is consistent with the maximization of the consumers welfare.4 See for example Henzel and Wollmershaeuser [2006] for a recent discussion.

3


4/52

rate considering deviations of expected ination with respect to its steady state level and

current and expected output gap. On the other hand, if scal expenditure is determined

by a self-interested incumbent, the central bank should react not only to expected ination

but also to scal deviations with respect to its steady state value. This is because evenin the case where the government is considered, if scal expenditure is exogenously given,

the central bank can do nothing to change this level; but this is not true if the incumbent

endogenously decide it. In this sense, changes in the interest rate when deviations of scal

expenditure around its steady state level occur, acts as a signal to the government which

reminds him how committed to ination the central bank is and looks forward to discourage

him from high expenditures levels by raising the interest rate. This signalling eect is, in

fact, bigger the bigger the public sector is relative to the private sector.

The paper is organized as follows. Section 2 briey summarizes the literature related

to this work. Section 3 describes the basic model. Section 4 solves it for the benchmarkcase of no government, for the case of a government with an exogenous expenditure path

and for the case of a self-interested incumbent (all of them considering an independent

central bank ). Section 5 shows the impulse response functions to a one percent scal

shock and section 6 extends the model for the case when not only the central bank but

also the government is concerned with individuals welfare. Finally section 7 concludes.

An appendix shows the estimated regressions and proves the main equations.

2 Related Literature

Two motivations are put forward in the political economy literature to capture politi-

cians incentives to move away from a benevolent behavior. On one hand, they might

be "ideologically-motivated" and in this way only care about the well-being of particular

groups of society; which means that they will choose policies in order to maximize a social

welfare function that puts a disproportionate weight on these groups. These are known

as partisan politicians. On the other hand, politicians can act in a purely self-interested

way if they choose policies as to hold oce because of its value per se; or because holding

oce allows them to extract tangible rents. These are known as opportunistic politicians,

and to be more precise, those who want to hold oce per se are called "oce-seeking",

while those who also want to extract tangible rents are called "rent-seeking". As it wassaid before, I shall focus on "opportunistic-rent-seeking" politicians in line with the ones

present in Diermeier, Keane and Merlo [2005], Shi and Svensson [2006] and Acemoglu,

4


5/52

Golosov and Tsyvinski [2007]. Next I will briey discuss these papers along with Clarida,

Gal and Gertler [1999].

Diermeier, Keane and Merlo [2005] want to explore motivations and objectives of elected

politicians in a context of political career in order to quantify the returns of a United StatesCongress career. For this, they use the idea that politicians are forward-looking rational

individuals that make decisions over their career taking into account the expected returns

of dierent choices; and they also count on novel data base with information about post-

congressional employment of former members of Congress and their wages. In this way,

politicians might be interested in choosing a congressional career not only because of the

utility of being in oce, but also because of its monetary returns. Their key innovation is

that they explicitly model career opportunities of politicians outside Congress when they

leave it: they can choose between a job in the private sector or in the public sector. Their

salary depends on their age, education, number of terms in oce committee assignmentsand weather their exit was voluntary or due to electoral defeat. At the same time, politi-

cians can dier in their personal characteristics that may aect not only elections outcome

but also their wage out of Congress. In this way, the authors propose a dynamic problem

in which a Congress member must decide weather to run for re-election, run for a higher

oce or exit from Congress (either to work in the public or private sector or to retire)

every two years. To solve this problem, they assume there is a "minimum age" and a

"terminal" age to be a congressman (thirty and eighty years respectively); then by posing

the determinants of post-congressional payos, senators decisions, representatives deci-

sions, re-election probabilities and the evolution of exogenous state variables, they solve it

backwards. Their main results are the following: Congress experience has a positive and

signicant eect on post-congressional wages (in both, the public and the private sector);

second, non-pecuniary earnings from being in congress are as considerable as monetary ones

(in fact, monetary earnings alone are not enough to explain politicians behavior); third,

non obsevable skills like charisma helps politicians to increase their re-election probability,

but does not help them to get a higher wage out of Congress; fourth, there is no evidence of

selectivity bias about who runs for re-election but the contrary happens when a politician

must run for a higher oce; nally, to impose a limit for terms would rise signicantly

"voluntary exits", biased towards who have better political skills and the older.

Shi and Svensson [2006] contribute to the empirical cross-country literature on politicalbudget cycles and formulate a model that accounts for the evidence of the existence of

political budget cycles and their systematic dierences between developing and developed

5


6/52

countries. In a partial equilibrium model they focus on the incumbents optimizing behav-

ior, where they assume that the government likes to consume the same type of goods as

households; but its consumption path will be conditioned to being reelected. To achieve

this, the incumbent manipulates public expenditure in order to confuse uninformed voterswho can not see clearly weather a higher expenditure level means policy manipulation or

governments competence (that is to say, they work with a political career model). They

nd that, rst of all, the governments budget balance is inuenced by the timing of the

elections: prior to elections politicians have incentives to expand expenditure in order to

increase his re-election probability; which in turn will be stronger if there are high rents

from being in power and if there is a bigger portion of uninformed voters. This is very

intuitive as greater rents from being in power will make incumbents be willing to hold oce

no matter what and on the other hand, if by manipulating expenditure the incumbent can

mislead a greater share of voters, he will have incentives to do so; higher rents and higherportions of uninformed voters are expected to be consistent with less developed countries.

In this way, the authors can account for the mentioned dierence in political budget cycles

between developed and less developed countries.

Acemoglu, Golosov and Tsyvinski [2008] study optimal taxation also in a context where

the system is not operated by a benevolent planner but by a rent-seeking self-interested

politician. Political economy environment though, is dierent from the present paper as

they deal with an electoral accountability model; that is, politicians have complete discre-

tion once in oce, but voters can oust them from oce at the next elections; so voters

act retrospectively and deliberately punish bad behavior by replacing the bad government.

Their main contribution is to introduce political economy to a dynamic Mirrlees problem

and in that way, give conditions under which political economy distortions persist or dis-

appear in the long run. In particular, optimal tax structure will be the same as in the case

with no political economy distortions if politicians are as patient as individuals; otherwise

aggregate taxes on labor and income will persist even in the long run. To design the best

mechanism that might best avoid the distortions created by the presence of this type of

politicians and lack of commitment, contrary to this paper, they only need a partial equi-

librium model that species individual preferences and game present between the selsh

politician and the executioner voters to obtain the subgame perfect equilibrium path for

taxes.On the other hand, the other pillar of this paper and what makes it original is the

interaction of this self-interested government with the central bank. For this, I will rest on

6


7/52

the New Keynesian perspective of monetary policy pioneered by Clarida, Gal and Gertler

[1999]. In their paper they obtain the optimal monetary policy stressing the existence of

nominal rigidities and also incorporating micro-founded bases to their model. In this way,

they analyze the cases of a discretionary and a committed central bank that minimizes anination and output gap quadratic loss function taking into account the demand side of

the economy given by consumers optimal behavior and the supply side of the economy,

given by the rms optimal behavior. These cases allow them to show the gains from

commitment: it eliminates the inationary bias in the case the central banks target for

real output exceeds the market clearing level and more importantly, if the central bank

is to credibly ght ination in the future, commitment improves output-ination trade

o faced by the central bank. Even though it is not the aim of this paper to see the

gains of rules versus discretion (in fact, I will consider a committed central bank that can,

through this, aect ination and output expectations); it will be useful for the presentwork to make the central bank behave in the same way as theirs. Moreover, within the

New Keynesian perspective, I will use the hybrid concept of the Phillips curve proposed

by Gal and Gertler [1999]. To capture the ination inertia that seems to characterize US

ination time series, the authors set out a model with nominal rigidities and a fraction of

rms that set their prices not by maximizing the present value of future expected prots,

but by a backward-looking rule. They empirically test the dynamics of ination for US

data for the period 1960 - 1997 and use labor income share in the non-farm business sector

to approximate real marginal cost instead of using output gap. They actually nd that

"backwardness" though statistically signicant, it was of no quantitative importance and

so, the New Keynesian Phillips curve gives a good approximation to US ination dynamics.

They nally claim that this may be so because the source of ination inertia might be the

sluggish adjustment in real marginal costs and not backward-looking rms. Some authors

however, argue that this result is only due to estimation methods used in the paper.5

3 The Model

I will use the moral hazard model proposed by Shi and Svensson [2006] and put it to work

on a standard cashless neokeynesian general equilibrium model as the one used in Walsh

5 See Rudd and Whelan [2005] and Lind [2005] for a more recent discussion and Gal, Gertler andLpez-Salido [2005] for an answer to it.

7


8/52

[2003].6 In this way I will be able to derive an expression for the ination rate consistent

with an optimizing policymaker.

In this model there are four types of agents: households, rms, politicians and a central

bank. This section describes the decision problem for each of them.

3.1 Households

There is a continuum of innitely lived agents who drive utility from consumption and

leisure. Consumption is provided either by the public sector in a proportion of total

governments expenditure, or by the private sector. Individuals have the same preferences

over the public good, the composite consumption good and labor. However, they dier

over idiosyncratic preferences concerning candidates other policies (besides scal ones).

These idiosyncratic preferences are captured by the parameter s assumed to be uniformly

distributed on 12 ; 12 : Government is in charge of any of the two existing political parties,a or b; At is a binary variable that equals 1=2 if party a is elected and equals 1=2 if b

is elected. Preferences are concave in public and private consumption and in labor; in

particular, I assume this concave function is isoelastic and additive in its arguments (this

assumption has no impact on the results and greatly simplies the mathematics). The

instantaneous utility function for consumers is then:

Us (t) =(Gt)

1

1 +

C1t1

R10 N

1+jt dj

1 + + sAt

where Gt is the level of a Dixit-Stiglitz aggregate of public good; Ct is also a Dixit-Stiglitzaggregate of private consumption of each of the continuum of dierentiated goods cjt ,

Ct [

Z10

c1

jt dj]

1

with > 1 the price elasticity of individual goods and is the intertemporal elasticity of

substitution of the composite good. Njt is the labor supply of the representative agent to

sector j, for which she receives wt and is the corresponding elasticity.7

Households are the owners of the rms for which they receive prots t and hold

6

As Cochrane [1998] shows, provided the Fiscal Theory of the Price Level, cash can be deleted andthe price level still determined, if the government accepts maturating government bonds directly for taxliabilities or electronically convert them in dollars for a "nanosecond" before accepting them.

7 Note that wjt = wt 8j, this means perfect labor mobility among sectors.

8


9/52

one-period riskless nominal bonds issued by the central bank, Bt, which pay the nominal

interest rate it. Finally, let be the private sector discount factor and Pt the general price

level in period t; so, the consumer problem for citizen s is given by:

maxCt;Nt;Bt

Et

1Xi=0

i[(Gt+i)

1

1 +

C1t+i1

R10 N

1+jt+idj

1 + + sAt+i]

s:t : Ct +BtPt

=wtR10 Njt dj

Pt+ (1 + it1)

Bt1Pt

+ t (1)

3.2 Firms

There is also a continuum of prot-maximizing rms of measure one that operate under

monopolistic competition; so, they face their demand curve that will be given by FOC of

the consumers problem. The technology available is a CRS function with no capital and

a productivity shock Zt. The only thing that distinguishes one rm from another (besides

its output) is that they adjust their prices in dierent dates. In this way, output j is of the

form:

yjt = ZtNjt ; Zt iid

1; 2Z

and aggregate output is dened as:

Yt

Z10

pjtPt

yjt dj

where pjt is the price set for good j in period t. Prices are staggered la Calvo (1983); everyperiod, only a fraction 1 ! of all rms adjust prices. Ination persistence is introduced by

a fraction (1 ) of adjusting rms that set prices in a forward-looking way such that they

maximize the expected discounted value of current and future prots; and the remaining

fraction set prices in a backward-looking way. Let pjt be the optimal price set by all

adjusting rms that produce variety j; ination persistence comes from assuming that:

pjt = (1 )pf o

jt + pba

jt (2)

where pf ojt is the price set by a forward-looking rm and pba

jt is the price set by a backward-

looking rm. Lastly, let t be the income tax rate paid the rms.

9


10/52

To solve the pricing decision, a forward-looking rm will choose pf ojt to maximize

Et

1

Xi=0 !ii;t+i[(

pf ojt

Pt+i)yj;t+i(1 t+i) t+iyj;t+i]

where !i is the probability of not adjusting price between t and t+i; i;t+i is the subjective

discount factor; as households own rms, this discount factor is related to marginal utility

of private consumption and due to the utility form it equals i(Ct+i=Ct):8 Lastly, t

is the rms real marginal cost, which is obtained from the cost minimization problem

faced by the rm and equals wt=PtZt : On the other hand, I will use a very simple rule for

backward-looking rms. In fact, I will assume backward-looking rms set their price equal

to the optimal price set by adjusting rms the period before; i.e.:

pbajt = pjt1 (3)

It is worth noting that even though this very simple rule says that backward-looking rms

are completely short-sighted, they are actually incorporating information about future

expectations as pjt1 has a forward-looking part, pf o

jt1: So, backward-looking rms are

somehow naive, but not so much.

3.3 Government

Regardless from the ruling party, let gjt be the level of good j consumed by the govern-

ment in period t. Therefore, the level of composite public good Gt (assumed to have the

same elasticity of individuals goods as households), is Gt [R10 g

1

jt dj]

1 . Gt will be

exogenously given or endogenously obtained by solving a utility maximization problem

for a self-interested incumbent. Nevertheless, in both cases this level is, in every period,

constrained by tax revenues and a scal shock "t. The scal shock can be associated to,

for example, natural catastrophes and comes to be public information at the end of every

period. So, the government budget constraint can be expressed as:

Gt = tYt + "t; "t iid

0; 2"

(4)

8 In general, the stochastic discount factor between t and t + 1 equals u0

(ct+1)u0(ct)

:

10


11/52

3.4 Central Bank

Smoothing policies are in charge of an independent central bank that is not exclusively

concerned about stabilizing ination but also with the stability of the real economy. In

this way, the central bank acts with commitment and sets the nominal interest rate as

to minimize a quadratic intertemporal loss function, considering the eect of changes in

the interest rate on ination through its impact on private consumption and tax rates.

The central bank dislikes high ination, t and high levels of output gap xt (which has an

exogenous relative weight ), these targets are derived from an optimal policy problem that

wants to maximize a welfare measure given by the expected utility function of households. 9

The central banks loss function is then:

Et

1

Xi=0i(

1

22t+i +

2x2t+i)

It is important to note two things: rst, that because of the non existence of capital in

this economy, a central bank is needed so that the equilibrium interest rate is determined

and second, the central bank when it sales bonds, it collects goods, pays its previous debt

and throws away any possible excess.

3.5 Equilibrium

The previous description of all agentss behavior, yield to the following equilibrium deni-

tion:

Denition 1 An equilibrium in this economy is a sequencenCt; Gt; Nt; Bt; Yt; p

jt ; Pt; t; wt; t; ito1

t=0such that:

Consumers maximize expected utility,

Firms maximize prots,

Aggregate price level is a weighted average of individual prices and last periods

aggregate price level,

The government maximizes its expected utility;

9 For a formal derivation, see Benigno and Woodford [2003] or Woodford [2003], chapter. 4.

11


12/52

The central bank minimizes its loss function and

Markets clear.

4 Solving the Model

The timing of the events is as follows: rst, the central bank minimizes its loss function

by choosing the nominal interest rate, then the government (in the case of a self-interested

one) sets its expected consumption level and, as a residual from its budget constraint, the

tax rate needed to get a balanced budget. At this point is important to emphasize that the

government does not issue public debt and so, it does not aect the economys interest rate

through this channel (I made this assumption because with no default probability, this type

of bonds should be a perfect substitute of the ones issued by the central bank; this only

adds noise to my model and has no implications for the core of the paper). The third player

in this sequential game are rms which decide the optimal price level from which ination

can be obtained. Finally, consumers determine their optimal levels of private consumption,

labor and bond holdings. Solving backwards, I rst present consumers utility maximization

subject to their budget constraint, which will give us the private side of aggregate demand

curve; then rms maximize prots subject to consumers optimal choice and in this way

I obtain the aggregate supply curve; the Hybrid New Keynesian Phillips Curve. Thirdly,

the government solves its problem which will be trivial if scal expenditure is considered as

an exogenous process and will be microfounded for the case of a self-interested incumbent.

In the later case, the government will maximize its own utility subject to the re-electionprobability and the behavior of rms and households. Finally, the central bank solves its

problem subject to the demand and the supply side of the economy (the demand side will

consider scal behavior, whether it is exogenous or endogenous).

4.1 Benchmark Case: No Government

4.1.1 Households

As mentioned, I will start by solving the households problem. Like in standard procedures,

I divide the problem in two parts:

(a) First choose the optimal combination of individual goods that minimize the cost of

purchasing a certain level of composite good, given its denition. From the FOC, I

12


13/52


14/52

remaining fraction ! of all rms that set their price in earlier periods. To obtain Pt, from

(6) it can be shown that the average price in period t satises10

P

1

t = (1 !)(p

t )1

+ !P

1

t1 (11)

where pt is dened in equation (3) :

Let bkt denote the percentage change of a variable Kt around its steady state level K. Ialso assume that the steady state involves a zero ination rate, then (10) can be linearized

around the steady state to obtain aggregate ination as:

t = 2Ett+1 + 3t1 + 4bt (12)where the coecients s are dened in the appendix, are all positive and for the special case

of pure forward-looking rms scenario, it is true that 2 = ; 3 = 0 and 4 =1!

! (1 !)as in the traditional New Keynesian Phillips curve. Equation (12) is known as the Hybrid

New Keynesian Phillips Curve (HNKPC). Here, ination is forward-looking, it also has an

inertial component and real marginal cost is a very important determinant of the ination

rate. As standard results, ination rate does not directly depend on output gap, but it can

re related to deviations of real marginal cost; in fact, deviations of real marginal cost can

be expressed as: bt = ( + ) (byt byft )where the superscript f denotes the exible price equilibrium (i.e. ! = 0):

4.1.3 Central Bank

The central bank will minimize its loss function subject to the demand and supply side of

the economy. Let xt byt byft be the output gap with respect to the exible price case;linealization of the Euler equation results in the demand side, or the IS curve, whereas

the supply side is given by the HNKPC. In accordance with its steady state level, it was

assumed that the central bank has a zero ination target. The central banks problem is

10 See Walsh [2003], chapter 5.

14


15/52

presented as follows:

mint+i;xt+i;it+i

Et

1

Xi=0i(

1

22t+i +

2x2t+i)

st :

xt = Etxt+1 +1

Ett+1

i

bit + ut

t = 2Ett+1 + 3t1 + 4 ( + ) xt

where ut Etbyft+1 Etbyft is an exogenous disturbance. I will divide the problem intwo stages: rst, minimize the loss function subject to the ination equation considering

that, because of commitment, the central bank can eectively aect output and ination

expectations. Then, conditional on the optimal values of xt and t, nd the value of it

implied by the demand curve.11 The FOC of this problem entails the following dynamics

for the output gap:

xt Etxt+1 =4 ( + )

Ett+1 3Etxt+2 +

1

1

2

xt

The optimal policy rule for the interest rate that emerges from the demand side is then:

bit = 1i

1

4 ( + )

Ett+1 +

i3Etxt+2

i

1

1

2

xt +

iut (13)

So, from equation (13) we can see that interest rate will depend exclusively on expected

ination and, because of the existence of backward-looking rms, on actual and expectedoutput gap. Higher expected output gap will make the central bank to rise the interest rate

as expected and higher deviations of output today will also imply a higher interest rate for

a suciently low degree of stickiness (i.e. for a suciently high value of the probability

!). On the other hand, the central banks reaction to a higher expected ination rate

depends on the portion of rms adjusting prices through the parameter 4, on labor and

consumption elasticities and on the exogenous relative weigh for the output gap . If

there is a higher fraction of rms adjusting prices, output gap will be lower ( as can be

seen in the appendix, 4 is an increasing function of the fraction of rms adjusting prices)

and the central bank will not have to tighten so hard the interest rate if there is a risein the expected ination rate. The same eect will have a higher labor and consumption

11 This is because the demand side is not an active restriction in this problem.

15


16/52

elasticities. A higher sensibility of labor supply makes the output gap be lower due to

the eect of hours worked on consumption through the individuals budget constraint.

Meanwhile, higher intertemporal elasticity of substitution means consumers can easily

transfer consumption to the future or vice versa, implying small changes in policy havebig eects on this variable and therefore on output gap. It is also true that if the output

gap has a high weigh in the central banks loss function, the monetary authority will

be willing to react largely to changes in the expected ination rate as this variable has an

indirect eect on output gap through the FOC of the consumers. Finally, it is worth noting

that, due to the backward-looking part of the model, I have obtained a forward-looking

Taylor rule; forward-looking not only because interest rate reacts to expected ination, but

also to expected output gap; so "backwardness" can be seen as a microfoundation for this

type of rules.

4.2 The Case of an Exogenous Fiscal Expenditure

Households should not change their behavior with respect to the previous section. Firms,

however, must now consider governments demand for goods and the central bank, must

now take into account the governments expenditure level before solving its problem. This

section describes these changes.

4.2.1 Firms

There are two major considerations in this setting. First of all, distorted sales taxes t levied

by the government must be considered and second, now equilibrium needs yjt = cjt + gjt ;where gjt is the symmetric denition for the government demand of individual good j.

Using this, it can be shown that yj;t+i = (pjt

Pt+i)CTt+i; where C

Tt is total consumption of

composite good (private: Ct and public: Gt). Then, the forward-looking rms problem

can be rewritten as:

maxpfojt

Et

1Xi=0

!ii;t+i[(pf ojtPt+i

)1(1 t+i) t+i(pf ojtPt+i

)]CTt+i

The FOC for this problem is now given by:

pf otPt

=

1

EtP

1

i=0 !iiCt+i (Ct+i + Gt+i)t+i(

Pt+iPt

)

EtP

1

i=0 !iiCt+i (Ct+i + Gt+i)(1 t+i)(

Pt+iPt

)1(14)

16


17/52

This is the same result as in the previous section, except that the optimal price is

now explicitly aected by the presence of the government consumption and the tax rate.

Considering a distorted tax rate makes the price charged by each forward-looking rm to

rise. This is a consequence of the type of tax assumed plus imperfect competition. If thepolicymaker decides to rise taxes on sales, producers will, due to their monopolistic power,

transfer some portion of it to the consumers and thereby rise the price.

From the previous equation, government consumption has not a clear cut eect on

prices. However intuitively, as higher public consumption is assumed to be nanced only

trough taxes, scal expenditure should have a direct eect on prices. That is to say, the

scal authority must contract public expenditure if it does not want to put pressure on

prices. Ination rate is now given by:

t = 1bt + 2Ett+1 + 3t1 + 4bt (15)where s are dened in the appendix.

As well as in equation (12), ination is forward-looking, real marginal cost is a very

important determinant of the ination rate, but so are taxes in a positive way because of

rms monopolistic power, as explained earlier.12 On the other hand, real marginal cost

can again be expressed as:

bt = P + P (b

yt

byft )

GP

(

bgt

bg ft ) (16)

where P is the private participation in total consumption, with P + G = 1: So, whenthe governments behavior is taken into consideration, the real marginal cost deviation is

a weighted average of the output gap and the "public gap". If we think of byt as composedby bct and bgt, the later has a lower weight. If for example there is a positive productivityshock and output increases, this pushes labor demand, which has a positive impact on real

wage and therefore on real marginal cost. However, higher private consumption has an

additional eect: higher consumption makes labor supply to contract, which means that

real wage must rise, as well as real marginal cost. This last eect is not present in public

consumption and that is why it has a lower weigh in the real marginal cost deviation.

12

Equation (15) is in line with the one found by Benigno and Woodford [2006] for the case of distortionarysales taxes.

17


18/52

4.2.2 Government

When scal expenditure is considered to be an exogenous process, I will claim it is of the

form:

Gt = G 8t

So deviations around its steady state will be equal to zero in every moment of time

(i.e. bgt = 0 8t) :4.2.3 Central Bank

The central bank will again minimize its loss function subject to the demand and supply

side of the economy. Demand side is obtained from the linelization of the Euler equation.

Recalling that the output gap is dened as xt

byt

byft where goods market equilibrium

imply byt = Pbct + Gbgt and exogenous scal expenditure implies bgt = 0; it will still be truethat the IS curve for this economy will be simply given by the linelization of the Euler

equation. Dierent from the previous section, deviations of real marginal cost do take into

account the presence of the government and therefore the central banks problem is now

given by:

mint+i;xt+i;it+i

Et

1Xi=0

i(1

22t+i +

2x2t+i)

st :

xt = Etxt+1 +P

Ett+1 Pi

bit + utt = 1bt + 2Ett+1 + 3t1 + 4 P +

Pxt

The FOC of this problem still needs the output gap to behave as before, except that

now it must be considered the private participation in total output P:

xt Etxt+1 =4 (P + )

PEtt+1 3Etxt+2 +

1

1

2

xt

The optimal policy rule for the interest rate is now:

bit = 1i1 4 ( + )

2P

Ett+1 + iP

3Etxt+2 iP

1 1

2 xt + iP

ut (17)

18


19/52

So, the deviations of the interest rate around its steady state will behave in the same

way as when no government was considered.

4.3 The Case of a Self-interested Incumbent

To see the behavior of economic variables when political issues are considered, I use the

former except for the governments nature. As mentioned before, I assume politicians

are self-interested and like to consume for their own purposes the same type of goods as

households do, but to do this, the incumbent must be re-elected. However, re-election only

happens if voters nd that the incumbent did well during its term in oce in terms of con-

sumption and hours worked. That is to say, economic performance signals the incumbents

competence and voters reward competence with reappointment. So, to understand how

the incumbent decides on scal decits under this environment, it is useful to introduce

rst the timing of the events and then present his maximization problem.

4.3.1 Timing

I shall focus the attention just on period t as it is not the aim of this paper to explain the

economys behavior due to the existence of a political cycle but only due to an optimally

driven scal decision, I therefore assume terms last only one period. The timing is the exact

one as the described in the previous section, except that as far as re-election is concerned,

the individuals will have to vote not only considering economic results but also inferring the

governments competence. That is to say, during period t three dierent shocks occur: the

productivity and scal shocks introduced earlier and also a government competence shockjt (j = a; b), which is private information for the government and may cause eective

expenditure to be dierent from what expected. That is to say, while the government can

commit to a tax code, it is uncertain (as well as citizens are) about the tax revenues it

will generate; and consequently on its expenditure level. Competence is a time persistent

variant process of the form:

jt = jt +

jt1; j = a; b

where jt and jt1 are i.i.d. random variables with zero mean, known variance, distribution

function F() and density function f() :

At the end of period t citizens can observe the tax rate, the eective level of public

19


20/52

expenditure and consequently the eective ination rate which can be dierent from the

one induced by the central bank because of the possibly dierent expenditure level. They

can also partially deduce governments competence; that is to say, they can guess t1

using information from the announced public expenditure. To see this, note that at thebeginning of period t; E(Zt) = 1; E("t) = 0; E(t) = 0; so it must be true that:

Gannouncedt = E(tYt) + t1

from where t1 can be obtained. On the other hand, competence is not fully reveled

because at the end of period t voters are not able to distinguish between the portion of

public expenditure due to the productivity shock and the portion due to the competence

shock t, and so, they must infer this component. This is the scenario under which elections

take place at the end of period t.

4.3.2 Incumbents Maximization Problem

Policymakers are drawn randomly from citizenry and, since politicians are citizens, their

preferences are derived from the specication of the preferences of voters.13 To capture

this idea, I assume that when a candidate wins the elections, it will be able to consume

the remaining fraction of public good by devoting the maximum amount of labor, N to

the public sector. In contrast to Shi and Svenssons work I will not assume that while

being in power, the policymaker is able to extract some "ego rents" (these ego rents could

capture not only altruistic perceived benets, but also monetary private benets obtained

each period). Furthermore, I suppose that if the incumbent is defeated in any re-election

or voluntarily leaves the power, he never becomes a citizen again, i.e. there is no possibility

of a "political career" (when politicians leave the oce to work again in the private sector);

in other words, I will assume that no re-election entails minus innity utility. In this way,

the incumbents utility function is,

Vt =1X

i=0

i[((1 ) Gt+i)

1

1

N1+

1 + ]di

13 In this sense, this model ts into the citizen-candidate framework in political economy. However in this

case politicians are "exogenous", that is to say, citizens do not choose weather to run for public oce ornot, it is random.

20


21/52

where di is an indicator function that equals di1 if the politician is in power in period t = i

and zero otherwise; with d0 = 1: Now, because of the competence shock, the governments

budget constraint is:

Gt = tYt + "t + t

The second idea to capture is that to hold oce it is necessary for the incumbent to

be re-elected. As long as re-election is concerned, suppose party a is in power in period

t, assuming that citizens do not vote strategically but sincerely, voter i will re-elect the

incumbent if

Et [Ua (t + 1)] Et [Ub (t + 1)] i

0

Dierence in expected utility is given by:

Et [Ua (t + 1)] Et [Ub (t + 1)] = (G)1 Etbgat+1 Etbgbt+1 + C1 Etbcat+1 Etbcbt+1

N1+

Etbnat+1 Etbnbt+1It must be noticed that while citizens can learn about the incumbents competence,

there is no way they can infer the challengers one and so Etbt+1 = 0. Taking expectation

to the governments budget constraint and using the fact that Etat+1 = Et

at ; the Euler

equation and the linearized labor market equilibrium condition, the expected share of

votes (ES V) can be expressed in terms of dierences of expected ination and tax rates

as follows:

ES V = Pr i Et [Ua (t + 1)] Et [Ub (t + 1)]= (G)1

1

P

Etbat+1 Etbbt+1 1 Etat+1 Etbt+1 + 1PG Etat

C1

Et

at+1 Et

bt+1

N1+

Et

at+1 Et

bt+1

+ 1=2

Therefore, the ES V depends on the expected dierence of tax rates between the two

candidates, on expected dierence of ination rates and on the expected incumbents com-

petence, all of which have a clear-cut eect. Higher taxes is the other side of higher public

expenditure and higher public expenditure raises utility because a fraction of it is re-

turned to households; this same logic applies for expected competence as higher publicexpenditure can also be due to the incumbentss competence. As long as expected ina-

tion is concerned, higher ination rate tomorrow means lower expected consumption which

21


22/52

lowers expected utility.

To obtain the re-election probability it is necessary to notice rst that the term (G)1

PG

will be always positive for positive values of scal expenditure. Therefore, if candidate j

wins the oce with simple majority, the incumbent will be re-elected if the expected shareof votes exceeds 1=2; in this way, using steady state relations, the probability of re-election

is:

= Pr (ES V 1=2)

= Prn

at G

Etbat+1 Etbbt+1 + Etat+1 Etbt+1o= 1 F

hG

Etbat+1 Etbbt+1 + Etat+1 Etbt+1i (18)

where (G)1

PG h(G)1

+C1

+N1+

i > 0: The rst thing to notice is that ifination and taxes are expected to be the same between the incumbent and the challenger,the incumbent will be reelected with probability one, this is because voters can learn

something about the incumbent but know nothing about the challenger. Also, we can see

that raising tax rates raises re-election probability but raising expected ination lowers this

probability for the reasons given above.

Now, to see the incumbents maximization problem, note that it will have no incentives

to manipulate expenditure to impress voters beyond period t + 1. This is because the

probability of re-election at the end oft + 1, which determines periods t + 2 outcome, will

be inuenced by the incumbents expected competence at t + 2. The fact that competence

is a MA(1) process, autocorrelation of order two or higher is zero. In other words, expectedcompetence in t+2 is independent of actual competence, i.e., Et

t+2=t

= Et

t+2

= 0:

So, the incumbent will be interested in maximizing its total expected utility only over

the next two periods by choosing the optimal tax rate, that is to say, the incumbents

22


23/52

maximization problem is given by:

maxt

Et[(1 ) Gt]

1

1 + Et

[(1 ) Gt+1]1

1

st :

= 1 Fh

G

Etbat+1 Etbbt+1 + Etat+1 Etbt+1iEtt+1 =

11 23

Etbt+1 + 21 23

Ett+2 +4

1 23Etbt+1

+13

1 23bt + 34

1 23bt + 231 23t1bt = P + P (byt byft ) GP (bgt bg ft )

byt = P

bct + G

bgt

and Gt = tYt + "t + t

holds 8t

The FOC for the former problem leads to the following optimal consumption path:

EtG

t = f ()34G

(1 ) (1 23) GEtG

1t+1 (19)

where G

Etbat+1 Etbbt+1+ Etat+1 Etbt+1 : This equation shows that a self-interested incumbent will also equalize marginal benet of public expenditure to marginal

cost. Marginal benet is independent from the fraction of public good that is returned to

households (this is because of the functional form assumed for the preferences and therefore

is not only aecting the marginal benet, but also the marginal cost; that is to say, the

incumbent does not care the percentage of public expenditure it can appropriate as long

it is positive (i:e:0 < < 1) and does not change over time) and equals the consumption

level the incumbent can have today. Marginal cost is equal, on the other hand, to the

consumption level he is giving up tomorrow times the change in the re-election probability

he causes when he raises scal expenditure today and therefore puts pressure on ination.

4.3.3 Central Bank

I will linearize the equation for optimal scal expenditure and use it as the public component

of the output gap; the private component will be given again by the Euler equation. The

23


24/52

central banks problem is, therefore,

mint+i;xt+i

Et

1

Xi=0i(

1

22t+i +

2x2t+i)

st :

xt = Etxt+1 +P

Ett+1 Pi

bit G

Etbgt+1 + ut

t = 1bt + 2Ett+1 + 3t1 + 4 P + P

xt GP

(bgt bg ft )The optimal policy rule for the interest rate is now:

bit =

Pi

P

4 (P + )

P

Ett+1 +

iP3Etxt+2

iP

1

1

2

xt

+

G

Pi (1 )bgt + Pi ut (20)While it is still true that this is a forward-looking type of interest rate rule, comparing

the two interest rules obtained (equations (17) and (20)); the rst thing to notice is that now

the best response of the central bank should explicitly consider governments deviations

with respect to is steady state. If the incumbent rises scal expenditure such that it

overheats the economy, the central bank, in response to his loss function, must rise the

interest rate to cool down the economy. This is a signaling eect for the government of

how committed is the central bank with ination; when scal expenditure is given by an

exogenous process, the central bank can do nothing to change its level; however, if scal

expenditure is handled with discretion, it can somehow threaten him in order not to put

so much pressure on ination. Furthermore, the bigger the public sector relative to the

private one, the more responsive the monetary authority should be.

5 Economys Response to Shocks

I will allow scal expenditure to have a one percent external shock (explained for example

with a natural catastrophe or the implementation of a public policy that required more

resources than expected); and see how should the central bank react in both scenarios: when

scal expenditure is given and when it is endogenously determined. I will use the Rotenbergand Woodford [1997] calibration for US data; a value of seventy percent for the fraction

of rms using the backward-looking rule (Gal and Gertler [1999] mention a value between

24


25/52

sixty and eighty percent of rms adjusting prices in a backward-looking way); a relative

low fraction of scal expenditure returned to households and a weigh of thirty percent for

the relative importance of output gap in the central banks loss function. In this way, the

set of parameters to be used are: = 0:99; ! = 0:3; = 0:157; = 0:2; = 0:7; = 0:1and = 0:1: Relevant equations for the impulse response functions are:

Linearized Euler equation;

Linearized labor supply;

Linearized labor demand,

Ination equation (HNKPC);

Linearized FOC for the government;

Linearized governments budget constraint and

Optimal interest rate rule.

The resulting policy and transition functions are as follows:

bit = 4:4455t1 + 0:0231"tt = 0:0921t1 + 0:0359"t

bgt = "t

bt = 0:3496t1 + 0:9749"tbct = 0:3714t1 0:0358"tbnt = bzt1 + 0:2916t1 + 0:0281"twhen scal expenditure is considered as an exogenous process, and:

bit = 0:0665bgt1 + 4:4455t1 + 0:3572"tt = 0:0069bgt1 + 0:0921t1 + 0:0373"tb

gt = 0:1862

bgt1 + "t

bt = 0:1774bgt1 + 0:3496t1 + 0:9525"tbct = 0:0022bgt1 0:3714t1 0:012"tbnt = bzt1 0:0017bgt1 + 0:2916t1 + 0:0094"t25


26/52

when scal expenditure is considered as an endogenous process.

Figure 1 presents the impulse response functions for a one percent scal shock; the

doted line represents the case where scal expenditure is an exogenous process while the

solid line represents the case where scal expenditure is manipulated by a self-interestedincumbent. The main dierence between the two lines, and my main focus, is how reactive

the interest rate is in each scenario. While in the case of an exogenous scal process the

interest rate must rise by 0:0359 to a scal shock, in the case of an endogenous scal policy

it must rise by 0:3572: That is to say, the central bank must react in a signicantly higher

way to a scal shock to give the government a suitable signal of how committed it is to

its ination target. This is purely signaling as, due to the economys structure, inations

reaction is not signicantly dierent in both scenarios. We can also see that a one percent

shock to scal expenditure makes tax rate to rise a little less than proportional, this is

because total output (the tax base) will be lower after the shock. This is because eventhough the increase of scal expenditure relative higher than the decrease in consumption,

the parameter set used implies a very low scal participation in total output, G: Thereby,

the increase of total output ends up being lower than 1% and so is the increase in the tax

rate. It is also true that, starting from the steady state, higher scal expenditure puts

inationary pressures and therefore the central bank must contract the economy. This

higher interest rate level has a negative impact on consumption and this has a positive

impact on labor; this is why consumption and labor react in a very similar way (though

26


27/52

opposite in sign).

Interest Rate

-0,05

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

1 2 3 4 5 6 7 8 9 10

End Exo

Inflation

-0,01

-0,005

0

0,005

0,01

0,015

0,02

0,025

0,03

0,035

0,04

1 2 3 4 5 6 7 8 9 10

End Exo

Fiscal Expenditure

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1 2 3 4 5 6 7 8 9 10

End Exo

Tax Rate

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1 2 3 4 5 6 7 8 9 10

End Exo

Consumption

-0,04

-0,035

-0,03

-0,025

-0,02

-0,015

-0,01

-0,005

0

0,005

1 2 3 4 5 6 7 8 9 10

End Exo

Labor

-0,005

0

0,005

0,01

0,015

0,02

0,025

0,03

1 2 3 4 5 6 7 8 9 10

End Exo

Figure 1: IRF0s to a 1% F iscal S hock

27


28/52

6 Extension: The Case of a Benevolent Government

In the previous sections I have obtained the optimal interest rate rule that came up when

scal and monetary authorities are not coordinated and have opposite interests. That is to

say, I have answered the question of how monetary policy be conducted if the central bank

is concerned with individuals welfare but the government is only concerned with his own

welfare (in terms of consumption and re-election). Monetary authority should behave in a

dierent, and in particular in a more aggressive way, when the government has incentives

to manipulate scal expenditure in order to maximize his own consumption and re-election

probability, compared to the case where scal expenditure is an exogenous process. That

is to say, the fact that scal and monetary authorities incentives are not aligned makes

the central bank be willing to signal its commitment to ination and rise the interest rate

whenever the government rises scal expenditure beyond its steady state level.

This section on the other hand, poses the question of how monetary policy should be

conducted is scal and monetary authorities incentives are aligned. That is to say, if not

only the central bank but also the government is concerned with individuals welfare. This

assumption gives us a very interesting scenario where the institutionality underlying has

guaranteed not only the use of the interest rate but also os scal expenditure in favor

of individuals welfare; as can be expected to be in countries that actually use ination

targeting.

6.1 Government

To obtain the optimal interest rate rule, three things must be noticed. First, households

should not change their behavior with respect to the previous sections. Second, rms

must still consider governments demand for goods; and nally, there is no reason why a

benevolent incumbent should not be reappointed in oce. A benevolent incumbent should

always be reappointed because it is known by voters that he will have no incentives (or no

means) to appropriate from a portion of public expenditure; so the challenger should not be

any dierent from the incumbent.14 In this way, a benevolent planner will be interested in

setting scal expenditure as to maximize the representative households utility, aware that

the consumer will set its allocation eciently where the Euler equation holds and marginal

14 Another way to deal with this issue is to assume that the incumbent wins the elections with an exogenousprobability . Considering 8t is just a special case that has no impact in the optimal interest rate ruleobtained.

28


29/52

utility of private consumption and leisure are equal. Moreover, the problem will also be

restricted by the prot maximization of rms through the HNKPC (considered one period

ahead) and market clearing: a feasibility condition that private and public consumption

should not exceed output, and labor demand equalizing labor supply. So, the benevolentplanners problem will be:

maxGt

Et

1Xi=0

i[(Gt+i)

1

1 +

C1t+i1

R10 N

1+jt+idj

1 + ]

s:t: :

Ct = (1 + it)EtPt

Pt+1Ct+1 ;

NtCt

=wtPt

; (Labor Supply)

Ett+1 = 11 23

Etbt+1 + 21 23

Ett+2 + 41 23

Etbt+1+

131 23

bt + 341 23

bt + 231 23t1bt = P + P (byt byft ) GP (bgt bg ft )Yt Ct + Gt

Nt =YtwtPt

; (Labor Demand)

and Gt = tYt + "t holds 8t

Optimal expenditure must then satisfy:

1Gt +

Y

341 23

(1 + it) C1t Et

Ct+1

(1 + t+1)2

1 +

1 + N1t (Ct + Gt)

=

N1t1 +

Ct

"1 +

Y

341 23

(1 + it) EtCt+1

(1 + t+1)2

#(21)

So, a benevolent government equals marginal benet of public expenditure to its mar-

ginal cost. The former is given by the direct eect of scal expenditure on utility, the term1Gt and the indirect eect of scal expenditure on utility through expected ination.

Higher expenditure today means, ceteris paribus, higher expected ination for tomorrow,

29


30/52

which lowers real interest rate; the Euler equation says households will make intertemporal

substitution raising present consumption and therefore utility. The second part of this

indirect eect says that higher consumption today makes labor supply to contract, making

households to work fewer hours which raises utility even further. Marginal cost of publicconsumption has, likewise, a direct and indirect eect. Higher public consumption pushes

aggregate demand so that production must raise and so must labor demand. higher labor

demand means more hours devoted to work and therefore lower utility. The indirect eect

is again a consequence of the ination channel. As mentioned before, public expenditure

has a positive eect on current consumption through expected ination. This in turn has

a positive eect on aggregate demand that expands further with the consistent decrease in

utility.

6.2 Central Bank

The central bank will again minimize its loss function subject to the demand and supply

side of the economy. As in the previous section, let xt byt byft be the output gap withrespect to the exible price case; recalling that byt = Pbct+Gbgt, linealization of the FOC forthe incumbent (which already incorporates the linearization of the Euler equation) results

in the demand side, whereas the supply side is given by the HNKPC. The central banks

problem is now as follows:

mint+i;xt+i

Et

1

Xi=0i(

1

22t+i +

2x2t+i)

st :

xt = Etxt+1 + GB1Ett+1 + (GB2 + P)bct + GB3bit+GB4bnt + (GB5 P) Etbct+1 GEtbgt+1 + ut

t = 1bt + 2Ett+1 + 3t1 + 4 P + P

xt GP

(bgt bg ft )where coecients B0s are dened in the appendix. The FOC of this problem still needs

the output gap to behave as before:

xt Etxt+1 =

4 (P + )

P Ett+1 3Etxt+2 + 1 12xt

30


31/52

The optimal policy rule for the interest rate is now:

bit = 1Ett+1 2xt+2 + 3xt + 4Et

bct+1 + 5

bgt + 6Et

bgt+1 +

eut (22)

where the 0s and eut are dened in the appendix. In this way, I obtain an interest raterule that once again is dierent from the one found in Clarida et al (it = i (Ett+1; gt))

but not dierent from more general specications of forward looking Taylor rules.15 I

would like to highlight four things about this last equation. First, equation (22) says

that the monetary authority sets the interest rate using the information available in t to

forecast ination and public and private consumption for next period; so, this interest

rate rule that explicitly incorporates the optimal decision problem of a benevolent planner

also gives microfoundations to forward looking Taylor rules. However, in contrast to the

optimal interest rate rule found for the case of a self-interested incumbent, the forward-

looking part is independent of "backwardness"; that is to say, even in the special case

where all rms adjust prices in a forward-looking way (i.e. 2 = 3 = 0) I would still

obtain a microfounded forward-looking Taylor rule. Another important result is that (22)

tells us that the presence of a government (even a benevolent one) must alter the central

banks behavior. The central bank must explicitly consider scal policy not only because

this model assumes that a fraction of public expenditure is returned to households, but

also (and more relevant) because public expenditure aects expected ination a that has

a direct eect on households welfare through private consumption.16 In third place, and

closely related to the previous point, optimal interest rate rules for the case of a benevolent

planner diers from the one for the case of a self-interested incumbent in the fact that for theformer, the central bank is also concerned about expected deviations of scal expenditure

the next period. The intuition behind this result is that when a self-interested incumbent

is in charge of scal policy, he faces a trade-o between actual expenditure and re-election

probability. If the self-interested incumbent rises scal expenditure beyond its steady

state level, it is expected to be lower tomorrow so that his re-election probability does

not drop deeply. So, deviations of scal expenditure today have information about scal

expenditure tomorrow and in this way, bgt turns to be the relevant variable for the centralbanks decisions. This trade-o however, is no longer present when scal expenditure is

15

Of the form it = i + (et+1 ) + xx

et+1:16 This result does not change if the game between the government and the central bank would be

simultaneous.

31


32/52

in charge of a benevolent planner; so the central bank must also consider expected scal

expenditure tomorrow before setting the interest rate. Finally, it is possible to show that

for certain parameters values (in particular for and suciently low), contrary to what

the optimal interest rate rule tells us when the government is purely self-interested, interestrate is less reactive the bigger is the public sector relative to the private sector. This is

because incentives are now aligned and both authorities are looking forward individuals

welfare; so if the public sector is relatively large, the central bank can somehow rest on the

government and need not to have a so aggressive policy.

7 Conclusion

Governments do not always take their scal decisions thinking about individuals welfare

but thinking only in their owns. This has a great impact on the decisions an independent

and welfare-concerned central bank makes. In this paper I have used a general equilibrium

model with nominal rigidities (staggered prices) and ination inertia that comes from the

assumption that a fraction of rms behave in a backward-looking way. This model has

enabled me to study how does optimal interest rate rules should change when there is

no government and when government expenditure is considered as an exogenous process

or is in charge of a self-interested incumbent that reveals its competence only partially.

The main contribution of this paper is the interaction between the central bank and a

self-interested incumbent whose trade-o is given by the fact that he would like to rise

todays expenditure level in order to appropriate from a fraction of it but, starting from

the steady state, scal expenditure puts inationary pressures in the economy. Due to

ination inertia present, higher ination today implies higher ination tomorrow, and this

aects in a negative way to the incumbents re-election probability, which clearly curbs

scal decision.

At the beginning of the paper, in the introduction, I put forward regression results for

Australia, Chile and Peru that showed that target interest rates react dierently in each

country. The Federal Reserve Bank of Australia has a very little response (and even not

statistically signicant) to a 1% change in scal expenditure; while this is considerably

larger in Chile and Peru. In tune with this observation, the main result of the paper is

that if scal expenditure is given by an exogenous process, then the central bank can donothing to change it and therefore, the optimal interest rate rule says that deviations of the

nominal interest rate should not consider scal expenditure at all; as if no government was

32


33/52

present. When scal expenditure is, on the other hand, manipulated by a self-interested

incumbent, then the central bank should signal how much it is willing to defend ination

and rise the interest rate above its steady state level. This over-reaction is bigger the bigger

the public sector is relative to the private sector.Starting form the steady state, if I disturb the economy with a 1% scal shock, impulse

response functions tell us that the interest rate response is ten times larger with a self-

interested incumbent than with an exogenous process. This over-reaction is, as implied by

the model, purely a signaling eect as the change in ination does not dier signicantly

in both scenarios (this is clearly due to the economys structure).

Another result is that, considering a fraction of rms that set prices in a backward-

looking way results in a forward-looking Taylor rule. It is a forward-looking Taylor rule

not only because interest rate reacts to expected ination, as happens to be in standard

models, but also because interest rate reacts to expected output gap. In this way, thismodel gives a microfoundation to this type of rules.

Finally, if instead of a self-interested incumbent I assume the governments and central

banks incentives are aligned in such way that they both care about individuals welfare, I

nd that the optimal interest rate rule must consider not only deviations of scal expendi-

ture two periods ahead (which is a consequence of the backward-looking assumption of the

model), but must also consider deviations of scal expenditure for the next period. The

intuition behind this result is that when the government behaves as a benevolent planner

there no longer exists the trade-o between actual expenditure and re-election probability.

In this way, actual deviations of scal expenditure tells nothing to the central bank about

tomorrows level of public consumption (relative to its steady state level) and therefore the

central bank must use the information available in t to forecast scal expenditure in t + 1.

Furthermore incentives alignment enables the monetary authority to be less reactive to de-

viations of scal expenditure; that is to say, the central bank should response to deviations

of public consumption in a more friendly way if it knows the government is manipulating

his consumption not to maximize his own utility but to maximize individuals one.

33


34/52

A Appendix A: Regressions

For the regressions mentioned in the introduction I used target interest rates declared by

central banks, seasonally adjusted public consumption and the percentage change in CPI

index for Australia, Chile and Peru. Data was collected from the websites of the central

banks and national statistical oces of each country. Regressions were run in logs for

the target interest rates and scal expenditure so that the estimated coecients are the

elasticity of interest rate with respect to scal expenditure; ination was considered in

percentages. As an independent variable, scal expenditure was introduced with one lag

because information on GDP is available with some delay (in fact this is the reason why I

did not include the last quarter of 2008 in the data). The estimation method was simply

OLS. Output windows are the following:

Table 1 - Regressions

Dependent Variable: log(Target Interest Rate)

Country Australia Chile Peru

Method OLS OLS OLS

Observations 62 21 16

log(Fiscal Expenditure) 0.02171 2.02405* 0.95506*

(0.8796) (0.0000) (0.0035)

Ination 0.08658** 0.108*** -0.01873

(0.0176) (0.01018) (0.7967)

* Signicant at the 1% level

** Signicant at the 5% level

*** Signicant at the 10% levelp-values shown in parenthesis

34


35/52

B Appendix B: Solving the Model

B.1 Households

Consumers problem is solved in two stages: rst minimize the cost of buying a certainlevel of composite good Ct and then maximize their utility function subject to their budget

constraint. Minimization implies the solution of the following problem:

mincjt

Z10

pjt cjt dj

s:t : Ct [

Z10

c1

jt dj]

1

Let t be the lagrange multiplier associated to the constraint, the FOC with respect to cjt

is:

pjt = t[Z10

c1

jt dj]

11

c 1

jt

But [R10 c

1

jt dj]1

1 = C1

t ; so the former condition can be rewritten as:

pjt = tC1

t c

1

jt

cjt = Ct

pjtt

Using the denition of the composite good:

Ct =

Z10

c1

jt dj]

1

Ct = [

Z10

C1

t

pjtt

1dj]

1 )

t = [

Z10

p1jt dj]

1

t = [

Z10

p1jt dj]1

1 Pt

35


36/52

Substituting the denition of the lagrange multiplier in the FOC obtained earlier, I obtain

the demand for individual good j :

cjt = Ct pjtPt

The second step will give us the Euler equation and the labor supply:

maxCt;Nt;Bt

Et

1Xi=0

i[(Gt+i)

1

1 +

C1t+i1

R10 N

1+jt+idj

1 + + sAt+i]

s:t : Ct +BtPt

=wtR10 Njt dj

Pt+ (1 + it1)

Bt1Pt

+ t

Let t be the corresponding lagrange multiplier; then FOC with respect to Ct; Ct+1; Njt

and Bt are, respectively:

Ct = t

EtCt+1 = t+1

Njt = twtPt

tPt

= (1 + it)t+1Pt+1

Using the fourth FOC and combining the rst three, I obtain the Euler equation (equation

(8)) and labor supply (equation (9)):

CtEtC

t+1

= (1 + it)Pt

Pt+1)

Ct = (1 + it) EtPt

Pt+1Ct+1 and

Njt = Ct

wtPt

B.2 Firms

Let me rst obtain equation (15) and then show that (12) is just a special case when

no government is considered. Forward-looking rms, as well as consumers, also have atwo-step solution to set their optimal price; they rst minimize costs from where the real

marginal cost is obtained and then choose the price pjt that maximizes the present value

36


37/52

of expected utilities. The rst part of the problem implies solving:

minNjt

wtPt

Njt

s:t : yjt = ZtNjt

Let t be the lagrange multiplier, then, the FOC is:

t =wtPt

Zt

So, t is the real marginal cost. On the other hand, each rm must face the demand for its

dierentiated product, then total demand of product j

cTjt

is given by individual demand

from households and government:

cTjt = cjt + gjt

= Ct

pjtPt

+ Gt

pjtPt

=

pjtPt

(Ct + Gt)

=

pjtPt

CTt

Imposing market clear condition for goods, it must be true that:

yjt = cT

jt

=

pjtPt

CTt

Then the pricing decision for the forward-looking rm can be expressed as:

maxpfojt

Et

1Xi=0

!ii;t+i[(pf ojtPt+i

)1(1 t+i) t+i

pf ojt

Pt+i

!]CTt+i

37


38/52

The FOC with respect to the optimal price is:

Et

1

Xi=0 !ii;t+i[(1 ) (

pf ojt

Pt+i

)(1 t+i)1

Pt+i

+ t+i pf ojt

Pt+i!1

1

Pt+i

]CTt+i = 0

As all rms face the same problem, they will all set the same optimal price pf ot so that:

Et

1Xi=0

!ii;t+i[(1 ) (pf ot

Pt+i)(1 t+i) + t+i]C

Tt+iP

t+i = 0

Recall that the stochastic discount factor is i;t+i = i(Ct+i=Ct)

; so pf ot must satisfy:

Et

1

Xi=0 !iiCt+i (Ct+i + Gt+i)[(1 )

pf otPt

PtPt+i

(1 t+i) + t+i]Pt+i = 0

Then solving forpfotPt

:

pf otPt

=

1

EtP

1


Pt+iPt

)

EtP

1


Pt+iPt

)1

The optimal price set by each forward and backward-looking rm must be linearized

around its steady state in order to obtain the HNKPC, where ination is dened as t

ln

Pt

Pt1: To obtain equation (15) lets rst dene Qt as the relative price a rm sets

when it has to adjust prices, so Qt p

tPt ; note that in steady state, when all rms can

adjust prices it must be true that, QSS = 1; and also note that bqt = ln QtQSS : Secondly,we know that the general price level Pt is a weighted average between the price set by

adjusting rms and the general price level the period before; I also have an equation for

the optimal price set by adjusting rms (forward and backward-looking) and nally, the

FOC for forward-looking rms: So the relevant equations are:

Qt ptPt

; Qf ot pf otPt

; Qbat pbatPt

(B.1)

P1t = (1 !)(p

t )1 + !P1t1 (B.2)

38


39/52

pt = (1 )pf ot + p

bat (B.3)

pbat = p

t1 (B.4)

pf otPt

=

1

EtP

1


Pt+iPt

)

EtP

1


Pt+iPt

)1(B.5)

Expressing (B:2) in percentage deviations around the steady state, we can see that:

1 = (1 !)(ptPt

)1 + !

Pt1

Pt

1)

0 = (1 !)bqt !tbqt = !1 !

t

From (B:4) ; I can express the price set by backward-looking rms as:

pbatPt

=pt1Pt1

Pt1Pt

)

Qbat = Qt1Pt1

Pt)

bqbat = bqt1 t

I can use this in (B:3), previously expressed in percentage deviations around the steady

state:

Qt = (1 ) Qf ot + Q

bat )bqt = (1 )bqf ot + bqbat )bqt = (1 )bqf ot + bqt1 t

Leaving this aside for a moment, note that if for simplicity I dene 1 ; (B:5) can

39


40/52

be expressed as:

Qt "Et1

Xi=0 !iiCt+i (Ct+i + Gt+i)(1 t+i)(

Pt+iPt

)1#= Et

1Xi=0

!iiCt+i (Ct+i + Gt+i)t+i(Pt+i

Pt)

So that the LHS (which I will denote by ) can be approximated by:

(SS) +@

@Qt+i

SS

Qf ot+i 1

+

@

@Ct+i

SS

(Ct+i C) +@

@Gt+i

SS

(Gt+i G)

+@

@t+i

SS

(t+i ) +@

@Pt+i

Pt

SS

Pt+i

Pt 1

(a) (SS) =C(C+G)(1)

1!

(b) @@Qt+i

SS

(Qt+i 1) =C(C+G)(1)

1! bqf ot(c) @@Ct+i

SS

(Ct+i C) = (1 )X

i

!ii

(1 ) C1 CG

Etbct+i(d) @@Gt+i

SS

(Gt+i G) = (1 )X

i

!iiCGEtbgt+i(e) @

@t+i SS (t+i ) = C(C+ G)Xi !iiEtbt+i(f ) @

@Pt+iPt

SS

Pt+i

Pt 1

= (1 ) C(C+ G)

Xi

!ii ( 1) (Etbpt+i bpt)On the other hand, denote the RHS by ; so that it can be approximated by:

(SS) +@

@Ct+i

SS

(Ct+i C) +@

@Gt+i

SS

(Gt+i G)

+@

@t+i SS t+i

+ +@

@Pt+i

Pt

SS

Pt+iPt

1

(a) (SS) =

C(C+G)1!

40


41/52

(b) @@Ct+i

SS

(Ct+i C) = X

i

!ii

(1 ) C1 CG

Etbct+i(c) @@Gt+i SS (Gt+i G) = Xi !

iiCGEtbgt+i(d) @@t+i

SS

t+i

= C(C+ G)

Xi

!iiEtbt+i(e) @

@Pt+iPt

SS

Pt+i

Pt 1

= C(C+ G)

Xi

!ii(Etbpt+i bpt)Equating both sides and canceling terms:

bpf ot + bq

f ot = (1 !)"

1 Xi !iiEtbt+i + Xi !

ii Etbt+i + Etbpt+i#I can then bring forward this expression, take the expected value in t and then replace

it again in the previous equation so that bpf ot and bqf ot will be expressed only in terms ofbpf ot+1 and bqf ot+1 :bqf ot = (1 !) 1 bt + bt

+ !Etbqf ot+1 + !Ett+1

It is also possible to express Et

bqf ot+1 in terms of Et

bqt+1;

bqt and Ett+1: Finally, recall

that

bqt = !1 !

t )

Etbqt+1 = !1 !

Ett+1

Substituting these two equations in the resulting one from the previous steps, I can

obtain an expression for the ination rate, which is equation (15) :

t = 1

bt + 2Ett+1 + 3t1 + 4

bt

41


42/52

where

0

!

1 !+

!

1 +

1

1 0 (1 )(1 !) 1

2 0!

(1 )

+

!

(1 !)

3 0

!

1 !4 0 (1 )(1 !)

Note that for the special case where no government is considered, = 0 and in that

way I obtain equation (12) :

B.3 Central Bank

In order to obtain the central banks best response function (13) ; I must solve the following

problem:

mint+i;xt+i;it+i

Et

1Xi=0

i(1

22t+i +

2x2t+i)

st :

xt = Etxt+1 +1

Ett+1

i

bit + ut

t = 2Ett+1 + 3t1 + 4 ( + ) xt

Let t and t be the lagrange multipliers associated to the IS and HNK PC curves

respectively; then the FOC whith respect to it+i says that:

t+i

= 0

42


43/52

which implies that t = 08t; that os to say, the IS curve is not an active restriction. In

this way, I can rewrite the central banks problem as:

mint+i;xt+i Et

1Xi=0

i(1

2 2t+i +

2 x2t+i)

st :

t = 2Ett+1 + 3t1 + 4 ( + ) xt

Let again t be the lagrange multiplier associated to the active restriction, the FOCs

with respect to t+i and xt+i respectively are:

t+i + t+i 2

t+i1 3t+i+1 = 0

xt+i t+i4 ( + ) = 0

Then, using the value of the lagrange multiplier from the second equation and replacing it

in the rst one, I obtain:

t+i =

4 ( + )xt+i

t+i1 =

4 ( + )xt+i1

t+i+1 =

4 ( + )xt+i+1

Therefore:

t+i =2

4 ( + )xt+i1

4 ( + )xt+i + 3

4 ( + )xt+i+1 )

Ett+1 =

4 ( + )

2

xt Etxt+1 + 3Etxt+2

)

xt Etxt+1 = 4 ( + ) Ett+1 3Etxt+2 +

1

2

xt

43


44/52

Now, replace this in the IS curve:

xt Etxt+1 =1

Ett+1 i

bit

+ ut (IS curve)

4 ( + ) Ett+1 3Etxt+2 + 1 2 xt = 1

Ett+1 ibit + ut

The optimal interest rule is:

bit = 1i

1

4 ( + )

Ett+1 +

i3Etxt+2

i

1

1

2

xt +

iut

If the government with an exogenous expenditure level is considered, the IS curve is

still not binding, so the central banks problem is as follows:

mint+i;xt+i

Et

1Xi=0

i( 12 2t+i + 2 x

2t+i)

st :

t = 1bt + 2Ett+1 + 3t1 + 4 P + P

xt

This is because as Gt = G8t, then bgt = 08t ) xt = byt byft = Pbct byft : The FOC are now:t+i =

P4 (P + )

xt+i

t+i1 = P4 (P + )

xt+i1

t+i+1 =P

4 (P + )xt+i+1

Which implies:

t+i =P

4 (P + )

2

xt Etxt+1 + 3Etxt+2

Ett+1 =

P4 (P + )

2

xt Etxt+1 + 3Etxt+2

xt Etxt+1 = 4 (P + )

PEtt+1 3Etxt+2 + 1 2

xt

44


45/52

Once again, I replace this condition in the IS curve:

xt Etxt+1 =P

Ett+1 i

bit

+ ut (IS curve) )

bit = 1i1 4 ( + )

2P

Ett+1 + iP

3Etxt+2 iP

1 1

2xt + iP

ut

For the case of a self-interested government the only dierence in the central banks

problem is given by the IS curve as Gt obeys equation (20):

xt Etxt+1 =P

Ett+1 ibit P

Etbgt+1 + ut (IS curve)

Therefore:

bit = Pi P 4 (P + )P Ett+1 + iP 3Etxt+2 iP 1 12xt+

G

Pi (1 )bgt +

Piut

B.4 Extension: The Case of a Benevolent Government

To obtain the interest rate rule in the case of a benevolent planner, it is necessary rst to

solve the governments problem as follows:

maxGt

Et

1

Xi=0i[

(Gt+i)1

1 +

C1t+i1

R10 N

1+jt+idj

1 + ]

s:t: :

Ct = (1 + it)EtPt

Pt+1Ct+1 ;

Nt =

(Ct + Gt) Ct

1

Ett+1 =1

1 23Etbt+1 + 2

1 23Ett+2 +

41 23

Etbt+1+

131 23

bt + 341 23

bt + 231 23t1b

t =P +

P(P

bct

optimal interest rates and fiscal policy

Documents