Week 8: Fiscal Policy in the New Keynesian Model

Week 8: Fiscal policy in the New Keynesian Model

Bianca De Paoli

November 2008 1 Fiscal Policy in a New Keynesian Model

1.1 Positive analysis: the e¤ect of …scal shocks

How do …scal shocks a¤ect in‡ation? Current economic environment: US …scal stimulus. See Summers & Calvo discussion (blog FT.com) UK potential …scal stimulus. "Fiscal Policy should be used when constrained Monetary Policy cannot react to fall in activity"

"Treasury …scal stimulus would prevent central bank to lower rates" Can …scal policy be used as an instrument to boost economic activity? What are the implications for in‡ation? 1.2 Positive analysis: the e¤ect of government expenditure shocks

Suppose that government consumption is …nanced by lump sum taxes Household aggregate budget constraint (in real terms):

Yt = Ct + Tt where Tt denotes lump sum taxes

Government resource constraint: Gt = Tt

So economy-wide market clearing Yt = Ct + Gt

The economy-wide resource constraint is given by yt = (1 sg)ct + sggt where sg = G=Y . (slightlyb di¤erentb speci…cationb than in previous lec- tures)

If one wants to allow for zero steady-state government consumption - de…ne gt = (Gt G)=Y = Gt=Y -then obtain e yt = ct + gt

b b e The supply side e¤ect For any given level of output, …scal policy crowds out private consumption Lower consumption implies higher marginal utility of consumption ct = yt + gt Higher marginal utility implies higher labour supply and lower real marginal b b e cost

– Total cost (constant returns Yt = Nt)

WtNt = WtYt

– Marginal cost is Wt, so real marginal cost, in log terms

mct = wt pt = ct + 'nt = ( + ')yt gt d b b b b b e higher labour supply/lower marginal cost implies higher potential output – With ‡exible prices, marginal cost is constant

n y = gt t + ' With sticky prices, the Phillips curveb would bee given by the usual equation

t = yt + Ett+1

e The demand side e¤ect

As we have seen yt = ct + gt

The Euler equation b b e 1 ct = Etc (it Et ) t+1 t+1 becomes b b b 1 yt gt = Et(y g ) (it Et ) t+1 t+1 t+1 or b e b e b 1 n yt = Ety (it Et r ) t+1 t+1 t n ' where r = Etgt+1 e e t b+ ' b b e So, a …scal shock (gt) increases the natural interest rate. Fiscal shock increasese aggregate demand. ...But it also increases supply. The net e¤ect on in‡ation depends on the central bank response. – Can you already infer the policy prescription of a central bank that wants to maintain price stability? n – This is given by rt : the interest rate that is consistent with constant prices increases after an increase in g.exp. => so, the central bank that wants to maintainb in‡ation has to raise interest rates to contain the increase in demand and in‡ationary pressures c g i 0 2 0.1

•0.2 1 0.05 •0.4 0 0 10 20 30 40 10 20 30 40 10 20 30 40 pi r rn 0.04 0.05 0.1

0.02 0.05

0 0 0 10 20 30 40 10 20 30 40 10 20 30 40 y ygap yn 1 0.04 1

0.5 0.02 0.5

0 0 0 10 20 30 40 10 20 30 40 10 20 30 40

Taylor rule that responds to the output gap and in‡ation with = 1 and y = 1:5 :Real rate increase by less than the natural rate -> shock is in‡ationary Exercise: (A) Assume that the shock is iid. – A.1) Derive the reduce form solution for interest rates and in‡ation as a function of the shock – A.2) How does this interest rate rule compared with one which guar- antees price stability?

– A.3) Show that the larger the larger is the interest rate response to the shock and the lower is the in‡ation level.

(B) Assume that the shock is an AR(1) with coe¢ cient = 0:9 and code the model in Dynare or Matlab

– Is it still the case that the larger the larger the nominal interest rate response? What about in‡ation? – Can you explain this result? 1.3 Positive analysis: the e¤ect of income tax shocks

What are the e¤ects of income taxes? Suppose that the government taxes income and redistribute in lump sum transfers – Government budget constraint: tYt = Tt where t denote income taxes – Aggregate household budget constraint: (1 )Yt + Tt = Ct t – Aggregate resource constrain in log linear terms:

Yt = Ct and yt = ct

b b Taxes will a¤ect agent’slabour leisure decision 1 1+' 1 s t Cs Ns max Ut = Et ; 21 1 + ' 3 sX=t subject to 4 5 1 Ct (j) Pt (j) dj + QtBt Bt 1 + (1 t)WtNt Tt Z0 it decreases the supply of labour for each unit of real wage Un(Ct; Nt) Wt = (1 t) Uc(Ct; Nt) Pt – This, ceteris paribus, increases real marginal cost

mct = wt pt = ct + 'nt + t for simplicity we alsod assumedb thatb taxesb areb zero in steady state – Thus: 1 yn = t + ' t – An increase in taxes worksb as a negative supply shock and does not have any e¤ect on demand 1.4 Positive analysis: …scal policy

What are the e¤ects of government spending …nance by income taxes? – Government budget constraint:

tYt = Gt maintaining the zero-state assumptions

t = gt

– Aggregate household budge constraint:e i (1 )Yt = Ct t – So economy-wide market clearing in log linear terms

yt = ct + gt

b b e – Marginal cost

mct = wt pt = ct + 'nt + t = ( + ')yt gt + t d b b b b ( + ')yt + (1 )gt b e – Potential output b e n (1 ) y = gt t + ' – This …scal policy has a dubiousb e¤ect one supply and a positive e¤ect on demand. 1.5 Normative analysis: Fiscal Policy and Welfare

Policy instruments: Public spending, Lump-sum taxes, Income taxes, In- ‡ation tax In‡ation may be viewed as a tax from di¤erent points of view – government can …nance their expenditure via seigniorage – or in‡ation can de‡ate the real value of public debt, improving its …nancing conditions But which tax is more costly? In‡ation tax, income tax, etc. Income taxes are costly because they a¤ect agents labour leisure decision (distort the incentives of agents to work an extra hour) Lump sum transfers are not costly, but are they feasible? Are they equi- table? In an NK model, in‡ation is costly due to nominal rigidities 1.6 Normative analysis: Fiscal Policy and Welfare

1.6.1 What have we learned?

Production Subsidy can improve welfare (Steady-state analysis) E¢ cient allocation Un = MPNt Uc We know that monopolistic competition may lead to lower production (suboptimal employment): W P = MMPN

Un W MPN = = Uc P M Production subsidy s can increase production towards e¢ cient level Un W MPN = = Uc P (1 s) M An income subsidy can increase labour supply towards it’se¢ cient level 1.7 Normative analysis: Fiscal Policy and Welfare

1.7.1 What have we learned?

Apart from the steady state analysis: changes in government spending may introduce a trade-o¤ between output and in‡ation when steady state is ine¢ cient

the policymaker’sproblem: 1 k 2 w 2 min E0 " (t) + (~yt ) kX=0 h i s.t

t = yt + Ett+1

e w t but the welfare relevant output gap is y~ = yt y , where t t dg yt = t = yn (1) t (' + ) 6 t Intuition: …scal shock can reduce monopolist distortion (because, as we’ve seen, it increases potential output) 1.8 Normative analysis: Fiscal Policy and Welfare

1.8.1 Optimal …scal policy

Up to now we have looked as …scal policy either in steady-state or as a shock

What about formulating an endogenous feedback rule for the …scal instrument: ie how should taxes respond to shocks

what …scal instrument: lump-sum taxes not realistic state contingent income taxes? But income taxes are discretionary The neoclassical literature on optimal …scal policy has suggested that, when taxes are discretionary, welfare would be maximized if taxes are smoothed over time and across states of nature (see Barro, 1979 and Lucas and Stokey, 1983).

In these models, if possible, taxes would be essentially invariant (see Lucas and Stokey, 1983 and Chari, Christiano and Kehoe, 1991)

or would follow a random walk (see Barro, 1979, Aiyagari et al. 2002) when the …scal authority is forced to move taxes to adjust its public …nances 1.9 Normative analysis: Fiscal Policy and Welfare

1.9.1 Benigno and Woodford (2003): Optimal …scal and monetary policy in a NK model

The …scal authority controls income taxes, issues nominal bonds (and one can also assume that the government faces exogenous expenditure streams) Households - as before 1 1+' 1 s t Cs Ns max Ut = Et ; 21 1 + ' 3 sX=t where: 4 5 " 1 " 1 " 1 Ct = Ct (j) " dj "Z0 # subject to 1 Ct (j) Pt (j) dj + Bt Bt 1(1 + it 1) + (1 t)WtNt Tt Z0 1. Optimal allocation of expenditures " pt(j) ct (j) = Ct Pt ! 2. Labour leisure decision

Un(Ct; Nt) Wt = (1 t) Uc(Ct; Nt) Pt 3. Intertemporal decision

Uc;t+1 Pt Qt = Et Uc;t Pt+1! Firms- as before

Optimal Price Setting

k Et Q Y P = 0 t:t+k t+k;t t M t+k;t Xk h i Aggregate price dynamics

1 1 " 1 " 1 " Pt = (Pt 1) + (1 ) Pt h i 1.10 Government budget constraint

the government issues one period nominal risk free bonds and collects taxes n Government debt Dt , expressed in nominal terms, follows the law of mo- tion: i Dt = Dt 1(1 + it 1) Pt tYt Or, we can de…ne Dt(1 + it) dt ; Pt in order to rewrite the government budget constraint as

(1 + it) i dt = dt 1 tYt(1 + it) t In‡ation reduces the real value of government debt 2 A log-linear representation of the model

Phillips Curve t = (ct + 'yt + ! (1 + ')at) + Et t t+1 as in the case of income taxes we’ve talked about, but here we allow for b b b b non-zero steady state taxes, so ! = =(1 ) Resource constraint yt = ct IS curve b b 1 ct = Etc (it Et ) t+1 t+1 Government budget constraint (de…ning dbt (dt d)=Y and dss = d=Y ) b b i dt = dt 1 dsst + dsse it ( t + yt) e e b b b 3 Optimal Policy

First: exploring the policy implication of discretionary taxation (ignoring the sticky price distortion)

3.1 The case of Flexible Prices

When prices are ‡exible (that is, = 0), the loss function derived in the previous section simpli…es to:

t 1 w2 min UcCE t y + t:i:p: 0 2 Y t X e Alternatively, using the relationship between discretionary taxes and output dictated by the Phillips curve, it is possible to rewrite the objective function as

t 1 w2 3 min UcCEt0 t + t:i:p + O( ); 2 jj jj ! X where = e '+ Y Under this speci…cation, domestic producer in‡ation is not costly (the assumption that = 0 implies that = 0), and policymakers’incentives are only a¤ected by tax distortions

The constraints of the policy problem are given by the equilibrium conditions presented before

Thus, the …rst order conditions implies that w w yt = t = 0:

e e Since in‡ation is not costly, optimal policy can induce unexpected varia- tions in domestic prices in order to restore …scal equilibrium.

This result is consistent with the …ndings of Bohn (1990), Chari, Christiano and Kehoe (1991) and Benigno and Woodford (2003). Show that if debt is zero in steady state, i.e. dss = 0 the government cannot fully stabilize taxes

Taxes vary across states but they remain constant after the shock hits the economy.

The best policy available entails a "jump" in the tax rate in order to adjust the level of primary surplus after the shock.

Subsequently, taxes are kept constant as to minimize distortions in the consumption/leisure trade-o¤. That is, the optimal plan implies

w w Etyt+1 = Et t = 0:

(a similar rule would hold if dss = 0 but the government issued real bonds) e 6 e Result consistent with Barro (1979) 3.2 The case of sticky prices

The policy problem

t 1 w2 2 3 min UcCE t y + + t:i:p + O( ); 0 2 t t jj jj X Do not want to use in‡ation toe adjust the …scal conditions because in‡ation is costly

Although there are two policy incentives and two policy instruments - that is, an active …scal and monetary policy - the …rst best cannot be achieved.

It’snot possible to keep simultaneously in‡ation and taxes constant across states and over time.

Nor is it possible, as in the ‡exible price case, to move tax rates perma- nently (and smooth them in subsequent periods). By inspection of the Phillips curve we note that, when prices are sticky, a permanent change in taxes would imply a non stationary process for in‡ation (and an explosive path for the domestic price level).

If we further assume that debt is zero in steady state, i.e. dss = 0, the optimal plan implies

w 1 !Et t + k t = 0:

e 3.3 Open economy

Introduce another distortion... optimize the use of instruments...

i t 1 2 1 2 1 H 2 L = UcCE t y + rs + ( ) + t:i:p; (2) to 0 2 Y t 2 RS t 2 t X If we further assume that debt isb zero in steadyc state, i.e.bdss = 0, the optimal plan implies

w (1 + l) w rsEtrs + Ety = 0; t+1 (1 ) Y t+1 and f e

H Ett+1 = 0