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): Y t = C t + T t where T t denotes lump sum taxes Government resource constraint: G t = T t So economy-wide market clearing
Y t = C t + G t The economy-wide resource constraint is given by by t = (1 s g )bc t + s g bg t where s g = G=Y. (slightly di erent speci cation than in previous lectures) If one wants to allow for zero steady-state government consumption - de ne eg t = (G t G)=Y = G t =Y -then obtain by t = bc t + eg t
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 bc t = by t + eg t Higher marginal utility implies higher labour supply and lower real marginal cost Total cost (constant returns Y t = N t ) W t N t = W t Y t Marginal cost is W t, so real marginal cost, in log terms dmc t = bw t bp t = bc t + 'bn t = ( + ')by t eg t
higher labour supply/lower marginal cost implies higher potential output With exible prices, marginal cost is constant by n t = + ' e g t With sticky prices, the Phillips curve would be given by the usual equation t = ey t + E t t+1
The demand side e ect As we have seen by t = bc t + eg t The Euler equation bc t = E t bc t+1 1 ( b i t E t t+1 ) becomes by t eg t = E t (by t+1 eg t+1 ) 1 ( b i t E t t+1 ) or ey t = E t ey t+1 1 ( b i t E t t+1 br t n ) where br t n ' = + ' E teg t+1
So, a scal shock (eg t ) increases the natural interest rate. Fiscal shock increases 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? This is given by br t n : the interest rate that is consistent with constant prices increases after an increase in g.exp. => so, the central bank that wants to maintain in ation has to raise interest rates to contain the increase in demand and in ationary pressures
0 c 2 g 0.1 i 0.2 1 0.05 0.4 0.04 10 20 30 40 0 pi 0.05 10 20 30 40 r 0 0.1 10 20 30 40 rn 0.02 0.05 0 1 10 20 30 40 0 y 0.04 10 20 30 40 ygap 0 1 10 20 30 40 yn 0.5 0.02 0.5 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Taylor rule that responds to the output gap and in ation with y = 1 and = 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 guarantees 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: ty t = T t where t denote income taxes Aggregate household budget constraint: (1 t)y t + T t = C t Aggregate resource constrain in log linear terms: Y t = C t and by t = bc t
Taxes will a ect agent s labour leisure decision max U t = E t 1 X s=t s t 2 4 C1 s 1 Ns 1+' 3 5 ; 1 + ' subject to Z 1 0 C t (j) P t (j) dj + Q t B t B t 1 + (1 t)w t N t T t it decreases the supply of labour for each unit of real wage U n (C t ; N t ) U c (C t ; N t ) = (1 t) W t P t This, ceteris paribus, increases real marginal cost dmc t = bw t bp t = bc t + 'bn t + t for simplicity we also assumed that taxes are zero in steady state
Thus: by n t = 1 + ' t An increase in taxes works 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: ty t = G t maintaining the zero-state assumptions t = eg t Aggregate household budge constraint: (1 i t)y t = C t So economy-wide market clearing in log linear terms by t = bc t + eg t
Marginal cost Potential output dmc t = bw t bp t = bc t + 'bn t + t = ( + ')by t eg t + t ( + ')by t + (1 )eg t by n t = (1 ) + ' e g t This scal policy has a dubious e ect on 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 equitable? 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 U n U c = MP N t We know that monopolistic competition may lead to lower production (suboptimal employment): P = M W MP N U n = W U c P = MP N M
Production subsidy s can increase production towards e cient level U n U c = W P = MP N (1 s )M An income subsidy can increase labour supply towards it s e 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 s problem: s.t X 1 min E 0 k h " ( t ) 2 + (~y t w ) 2i k=0 t = ey t + E t t+1
but the welfare relevant output gap is ~y w t = y t y t t, where y t t = dg t (' + ) 6= yn t (1) 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 max U t = E t 1 X s=t s t 2 4 C1 s 1 Ns 1+' 3 5 ; 1 + ' where: C t = " Z 1 C t (j) " 1 " dj 0 # " " 1
subject to Z 1 0 C t (j) P t (j) dj + B t B t 1 (1 + i t 1 ) + (1 t)w t N t T t 1. Optimal allocation of expenditures c t (j) = p t(j) P t! " C t 2. Labour leisure decision 3. Intertemporal decision U n (C t ; N t ) U c (C t ; N t ) = (1 t) W t P t Q t = E t U c;t+1 P t! U c;t P t+1
Firms- as before Optimal Price Setting X k Aggregate price dynamics E t k Q t:t+k Y t+k;t h P t M t+k;t i = 0 P t = h (P t 1 ) 1 " + (1 ) Pt 1 " i 1 1 "
1.10 Government budget constraint the government issues one period nominal risk free bonds and collects taxes, expressed in nominal terms, follows the law of mo- Government debt Dt n tion: Or, we can de ne D t = D t 1 (1 + i t 1 ) P t i t Y t d t D t(1 + i t ) P t ; in order to rewrite the government budget constraint as d t = d t 1 (1 + i t ) t i ty t (1 + i t ) In ation reduces the real value of government debt
2 A log-linear representation of the model Phillips Curve t = (bc t + 'by t +!b t (1 + ')ba t ) + E t t+1 as in the case of income taxes we ve talked about, but here we allow for non-zero steady state taxes, so! = =(1 ) Resource constraint IS curve by t = bc t bc t = E t bc t+1 1 ( b i t E t t+1 ) Government budget constraint (de ning e d t (d t d)=y and d ss = d=y ) ed t = e d t 1 d ss t + d ss b i t (b i t + by t )
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: min U c CE t0 X t 1 2 Y y e t w2 + t:i:p:
Alternatively, using the relationship between discretionary taxes and output dictated by the Phillips curve, it is possible to rewrite the objective function as where = min U c CE t0 X t! '+ Y 1 2 e w2 t + t:i:p + O(jjjj 3 ); 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 ey w t = e w t = 0:
Since in ation is not costly, optimal policy can induce unexpected variations 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. d ss = 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 E t ey w t+1 = E te w t = 0: (a similar rule would hold if d ss 6= 0 but the government issued real bonds) Result consistent with Barro (1979)
3.2 The case of sticky prices The policy problem min U c CE t0 X t 1 2 ey t w2 + 2 t + t:i:p + O(jjjj 3 ); Do not want to use in ation to 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 s not 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 permanently (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. d ss = 0, the optimal plan implies!e t e w t + k 1 t = 0:
3.3 Open economy Introduce another distortion... optimize the use of instruments... L i to = U cce X t0 t 1 2 Y y b t 2 + 1 2 RSrs c 2 t + 1 2 (b H t ) 2 + t:i:p; (2) If we further assume that debt is zero in steady state, i.e. d ss = 0, the optimal plan implies rs E t frs w t+1 + (1 + l) (1 ) Y E t ey w t+1 = 0; and E t b H t+1 = 0