The Smets-Wouters Model

The Smets-Wouters Model Monetary and Fiscal Policy 1 1 Humboldt Universität zu Berlin uhlig@wiwi.hu-berlin.de Winter 2006/07

Outline 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

Source and Impact Source: Smets, Frank and Raf Wouters, An estimated dynamic stochastic general equilibrium model of the Euro area, Journal of the European Economic Association, September 2003, 1(5), 1123-1175. Related: Christiano, Lawrence, Martin Eichenbaum and Charlie Evans, Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy, Journal of Political Economy, 2005, vol. 113, no. 1, 1-45. Impact: The Smets-Wouters model have become a modern workhorse and benchmark model for analyzing monetary and fiscal policy in European central banks, and is spreading to policy institutions in the US as well.

Overview This is an elaborate New Keynesian model. There is a continuum of households, who supply household-specific labor in monopolistic competition. They set wages. Wages are Calvo-sticky. There is a continuum of intermediate good firms, who supply intermediate goods in monopolistic competition. They set prices. Prices are Calvo-sticky. Final goods use intermediate goods and are produced in perfect competition. There is habit formation, adjustment costs to investment, variable capital utilization. The monetary authority follows a Taylor-type rule. There are many sources of shocks - enough to make sure the data can be matched to the model.

Outline 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

s Utility of household τ: U = E 0 β t Ut τ t=0 where U τ t = ǫ b t ( (C τ t H t ) 1 σc 1 σ c ǫ L t (l τ t )1+σ l 1 + σ l ) Ct τ : consumption. H t : external habit / catching up with the Joneses. l τ t : labor ǫ b t : intertemporal substitution shock ǫ L t : labor supply shock

Preference Shocks and Habit Preference Shocks: ǫ b t ǫ L t = ρ b ǫ b t 1 + ηb t = ρ L ǫ L t 1 + ηl t Habits: H t = hc t 1 where C t 1 is aggregate consumption in t 1.

Money... is left unmodelled, but implicitely assumed to be there. Easiest solution: assume that households have money in the utility, i.e. enjoy holding real balances,...... and the monetary authority influences nominal rates per helicopter drops of money on households,...... but that otherwise money does not influence budget constraints etc. See Woodford for details on how this can be done. Later on, we shall formulate an interest-rate setting rule for the monetary authority. Thus money does not need to be modelled explicitely (hopefully... but subtle and possibly crucial issues may be overlooked this way!!)

Intertemporal budget constraint and income Intertemporal budget constraint: b t B τ t P t = Bτ t 1 P t + Y τ t C τ t I τ t where B τ t are nominal discount bonds with market price b t. Real income Y τ t = (w τ t l τ t + A τ t ) + (r k t z τ t K τ t 1 Ψ(zτ t )K τ t 1 ) + Divτ t tax t wt τlτ t + A τ t : Labor income plus state contingent security payoffs. rt kzτ t K t 1 τ Ψ(zτ t )K t 1 τ : return on real capital stock minus costs from capital utilization zt τ. Assume Ψ(1) = 0. Div τ t : dividends from imperfectly competitive firms. tax t : real lump-sum tax.

Imperfect substitutability of labor Individual households supply different types of labor, which is not perfectly substitutable, ( 1 L t = (l τ t ) 1/(1+λw,t) dτ The degree of substitutability is random, 0 ) 1+λw,t λ w,t = λ w + ηt w In the flexible-wage economy, 1 + λ w,t will be the markup of real wages over the usual ratio of the marginal disutility of labor to the marginal utility of consumption. Thus, ηt w is a wage markup shock.

Wage setting s are monopolistically competitive suppliers of labor and wage setters, offering their labor in the quantity demanded at their current wage Wt τ. Wages are Calvo-sticky. Each period, the household has probability 1 ξ w that it is allowed to freely adjust its wage, choosing a new nominal wage W τ t P t 2 = w τ t If not, wages are adjusted according to the indexation rule ( ) γw Wt τ Pt 1 = Wt 1 τ γ w = 0: no indexation. γ w = 1: perfect indexation.

Perfect insurance markets Perfect insurance: labor income of an individual household equals aggregate labor income. Thus, the consumption of an individual household equals aggregate consumption, C τ t = C t,...... and marginal utility Λ τ t Λ t of consumption is equal across households. As a consequence, capital holdings Kt τ K t, bond holdings Bt τ = B t as well as firm dividends Div τ t Div t will be identical across different types of households. Also possible: these remain in constant proportions forever across different households.

Outline 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

All sectors Final goods production: homogenous final good, produced with a continuum of imperfectly substitutable intermediate goods. A continuum of intermediate goods, produced with capital and labor. Labor, in turn, is a composite of individual household labor. New capital is produced with old capital and investment, subject to an investment adjustment cost. Depreciation varies with capital utilization.

Final goods production ( ) 1+λp,t 1 ( ) 1/(1+λp,t ) Y t = yj,t dj The degree of substitutability is random, 0 λ p,t = λ p + η p t It will turn out that 1 + λ p,t is the markup of prices over marginal costs at the intermediate goods level. Thus, η p t a goods markup shock or a cost-push shock. is

Intermediate goods production Intermediate goods production is y j,t = ǫ a K t j,t α L1 α j,t Φ Φ: a fixed cost. K j,t : effective utilization of the capital stock, ǫ a t K j,t = z t K j,t 1 : aggregate productivity shock, ǫ a t = ρ a ǫ a t 1 + ηa t The profits of intermediate goods firms are paid as dividends Div t.

Price setting Intermediate good firms are monopolistically competitive, offering their good in the quantity demanded at their current price P j,t. Prices are Calvo-sticky. Each period, the firm has probability 1 ξ p that it is allowed to freely adjust its price, choosing a new nominal price P j,t = p j,t If not, prices are adjusted according to the indexation rule ( ) γp Pt 1 P j,t = P j,t 1 P t 2 γ p = 0: no indexation. γ p = 1: perfect indexation.

Capital evolution New capital is produced from old capital and investment goods, ( ( )) I t K t = (1 τ)k t 1 + 1 S I t I t : gross investment τ: depreciation rate ǫ I t I t 1 S( ): cost for changing the level of investment, with S(1) = 0, S (1) = 0, S (1) > 0. ǫ I t : shock to investment cost, ǫ I t = ρ I ǫ I t 1 + ηi t

Outline 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

consumption: G t, following government spending rule, financed by lump sum taxation, G t = tax t Monetary authority: sets nominal interest rate R t = 1 + i t = 1/b t following some interesting setting rule. We shall specify these rules in the log-linearized version of the model.

Outline 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

Market clearing Labor market: ( 1 1 L j,t dj = L t = (l τ t ) 1/(1+λw,t) dτ 0 Final goods market: Capital rental market: 0 C t + I t + G t + I t + ψ(z t )K t 1 = Y t K j,t 1 dj = K t 1 ) 1+λw,t

Wage and price aggregation Dixit-Stiglitz aggregation: Aggregate wages are W t = Aggregate prices are ( 1 0 (W τ t ) 1/λ w,t dτ ) λw,t ( ) λp,t 1 ( ) 1/λp,t P t = Pj,t dτ 0 One can derive these formulas from first-order conditions of producing aggregate output or aggregate labor.

Equilibrium Definition Given policy rules for G t and R t and thus tax t, an equilibrium is an allocation (B t, C t, H t, (l τ t ) τ [0,1], (L i,t ) i [0,1], L t, ( K i,t ) i [0,1], (K i,t ) i [0,1], K t, z t, I t, (y i,t ) i [0,1], Y t, Div t ) and prices (b t, rt k, (Wt τ ) τ [0,1], W t, (P i,t ) i [0,1] ), so that 1 Given prices and the demand function for labor l τ t, the allocation maximizes the utility of the household, subject to the Calvo-sticky wages. 2 Given prices and the demand function for y t,i, the allocation maximizes the profits of the firms, subject to the Calvo-sticky prices. 3 Markets clear. 4 The policy rules are consistent with allocation and prices.

Outline s Intermediate goods firms 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

Intertemporal optimization 1 s Intermediate goods firms Lucas asset pricing equation for bonds: [ E t β Λ ] t+1 R t P t = 1 Λ t P t+1 where Λ t = U C,t = ǫ b t (C t H t ) σc Lucas asset pricing equation for capital: [ Q t = E t β Λ ( ) ] t+1 Q t+1 (1 τ) + z t+1 rt+1 k Λ ψ(z t+1) t

Intertemporal optimization 2 s Intermediate goods firms Optimal investment: ( ǫ I )) ( Q t (1 S t I t ǫ = Q t S I ) t I t ǫ I t I t + 1 I t 1 I t 1 I [ ( )( t 1 ) E t β Λ t+1 ǫ I Q t+1 S t+1 I t+1 ǫ I t+1 I t+1 Λ t I t I t I t+1 I t copied from Smets-Wouters, equation (16). Is it correct? For capital utilization: r k t = ψ (z t ) ]

Wage setting s Intermediate goods firms Demand curve for labor: ( W l τ τ ) (1+λw,t )/λ w,t t = t L t W t Optimality condition for setting a new wage w t : [ w t ( ) ] E t β i ξw i (Pt /P t 1 ) γw l τ t+i U C,t+i P t P t+i /P t+i 1 1 + λ w,t+i i=0 [ ] = E t β i ξwl i τ t+i U l,t+i i=0 where U C,t, U l,t denote marginal utility of consumption and marginal disutility of labor.

Evolution of wages s Intermediate goods firms Per aggregation of wages, ( ( ) γw ) 1/λw,t (W t ) 1/λ w,t Pt 1 = ξ w W t 1 +(1 ξ w )( w t ) 1/λ w,t P t 2

Outline s Intermediate goods firms 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

Labor and capital s Intermediate goods firms Cost minimization gives W t L j,t r k t K j,t = 1 α α Thus, marginal costs for producing one extra unit of intermediate goods output is MC t = 1 ǫ a t ( W 1 α t r k t ) α (α α (1 α) (1 α))

Demand and profits s Intermediate goods firms The demand function y j t = D(P j,t ; P t, Y t ) is given by y j,t = Nominal profits are ( Pj,t P t ) (1 λp,t )/λ p,t Y t π j,t = ( ) ( ) P (1 λp,t )/λ p,t j,t P j,t MC t Y t MC t Φ P t

Price setting s Intermediate goods firms Optimality condition for setting a new price p t : [ ( ( )) ] E t β i ξpλ i pt (Pt 1+i /P t 1 ) γp t+i y j,t+i P t P t+i /P t i=0 [ ] = E t β i ξpλ i t+i y j,t+i (1 + λ p,t+i )mc t+i where i=0 are the real marginal costs. mc t = MC t P t

Evolution of prices s Intermediate goods firms Per aggregation of prices, ( ( ) γp ) 1/λp,t (P t ) 1/λ p,t Pt 1 = ξ p P t 1 + (1 ξ p )( p t ) 1/λ p,t P t 2

Outline A list of equations Calibration 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

Remark A list of equations Calibration The equations in the published version of the paper do not appear to be entirely correct (this can easily happen),...... and they do not appear to be consistent with the code either, even once corrected. We therefore spent considerable time clarifying the differences. We believe we got everything correct now. Very special thanks go to Wenjuan Chen and Matthieu Droumaguet for doing this and to Stefan Ried for supervising it! This was a lot of work... Details are in a "SmetsWouters-"Manual.

Equations 1,2 and 3 A list of equations Calibration The capital accumulation equation: The labour demand equation: K t = (1 τ) K t 1 + τît 1 (1) L t = ŵ t + (1 + ψ) r k t + K t 1 (2) The goods market equilibrium condition: Ŷ t = (1 τk y g y )Ĉt + τk y Î t + ǫ G t (3)

Equations 4,5 A list of equations Calibration The production function: Ŷ t = φǫ a t + φα K t 1 + φαψ r k t + φ(1 α) L t (4) The monetary policy reaction function: a Taylor-type rule { } R t = ρ R t 1 + (1 ρ) π t + r π ( π t 1 π t ) + r Y (Ŷt Ŷ t P ) +r π ( π t π t 1 ) + r Y [Ŷt Ŷ P t (Ŷt 1 Ŷ P t 1 ) ] + η R t (5) where Yt P refers to a hypothetical frictionless economy and potential output. The difference Ŷt Ŷ t P is the output gap.

Equation 6 A list of equations Calibration The consumption equation: Ĉ t = h 1 + h Ĉt 1 + 1 1 + h E tĉt+1 1 h ( R t E t π t+1 ) (1 + h)σ c + 1 h (1 + h)σ c ǫ b t (6)

Equations 7,8 A list of equations Calibration The investment equation: Î t = 1 1 + β Ît 1 + β 1 + β E tît+1 + ϕ 1 + β Q t + ǫ I t (7) The Q equation: Q t = ( R t E t π t+1 )+ 1 τ 1 τ + r k E t Q t+1 + 1 τ + r k E t r k r k t+1 +ηq t (8)

Equation 9 A list of equations Calibration The inflation equation: π t = β 1 + βγ p E t π t+1 + γ p 1 + βγ p π t 1 (9) + 1 (1 βξ p )(1 ξ p ) [α r t k 1 + βγ p ξ p + (1 α)ŵ t ǫ a t ] + ηp t

Equation 10 A list of equations Calibration The wage equation is given as follows. Pay attention that the sign before the labour supply shock shall be positive instead of negative, which is confirmed by the authors. ŵ t = β 1 + β E tŵt+1 + 1 1 + β ŵt 1 + β 1 + β E t π t+1 1 + βγ w 1 + β π t + γ w 1 + β π t 1 1 (1 βξ w )(1 ξ w ) 1 + β (1 + (1+λw)σ... (10) L λ w )ξ w σ ] c 1 h (Ĉt hĉt 1) + ǫ L t + ηt w [ ŵ t σ L Lt

A list of equations Calibration Differences to published version Equation in Smets-Wouters (2003) Here 28 take out ǫ b t+1 29 take out ǫ I t+1 29 ǫ b t rescaled to equal 1 35a ǫ G t rescaled to equal 1 32 η p t rescaled to equal 1 33 η w t rescaled to equal 1

Calculating results A list of equations Calibration In order to calculate results, we have to create two systems. One is the flexible system where there is no price stickiness, wage stickiness or three cost-push shocks. The other one is the sticky system where prices and wages are set following a Calvo mechanism. We use the potential output produced in the flexible system to calculate the output gap in the Taylor rule. In each system, there are 8 endogenous variables and 2 state variables. The 8 endogenous variables are capital, consumption, investment, inflation, wages, output, interest rate, and real capital stock. The 2 state variables are labour and return on capital.

Outline A list of equations Calibration 1 2 3 s Intermediate goods firms 4 A list of equations Calibration

Calibration 1 A list of equations Calibration Parameters Value Description β 0.99 discount factor τ 0.025 depreciation rate of capital α 0.3 capital output ratio ψ 1/0.169 inverse elasticity of cap. util. cost γ p 0.469 degree of partial indexation of price γ w 0.763 degree of partial indexation of wage λ w 0.5 mark up in wage setting ξs p 0.908 Calvo price stickiness ξs w 0.737 Calvo wage stickiness σ L 2.4 inverse elasticity of labor supply

Calibration 1 A list of equations Calibration Parameters Value Description σ c 1.353 coeff. of relative risk aversion h 0.573 habit portion of past consumption φ 1.408 1 + share of fixed cost in prod. ϕ 1/6.771 inverse of inv. adj. cost r k 1/β 1 + τ steady state return on capital k y 8.8 capital output ratio inv y 0.22 share of investment to GDP c y 0.6 share of consumption to GDP k y inv y /τ capital income share, inv. share g y 1 c y inv y government expend. share in GDP rπ 0.14 inflation growth coeff. r y 0.099 output gap coeff

Calibration 1 A list of equations Calibration Parameter Value Description ry 0.159 output gap growth coefficient ρ 0.961 AR for lagged interest rate r π 1.684 inflation coefficient ρ ǫl 0.889 AR for labour supply shock ρ ǫa 0.823 AR for productivity shock ρ ǫb 0.855 AR for f preference shock ρ G 0.949 AR for government expenditure shock ρ π 0.924 AR for inflation objective schock ρ ǫi 0.927 AR for investment shock ρ ǫr 0 AR for interest rate shock,iid ρ λw 0 AR for wage markup,iid

Calibration 1 A list of equations Calibration Parameter Value Description ρ q 0 AR for return on equity,iid ρ λp 0 AR for price mark-up schock,iid σ ǫl 3.52 stand. dev. of labour supply shock σ ǫa 0.598 stand. dev. of productivity shock σ ǫb 0.336 stand. dev. of preference shock σ G 0.325 stand. dev. of goverment expenditure shock σ π 0.017 stand. dev. inflation objective shock σ ǫr 0.081 stand. dev. of interest rate shock σ ǫi 0.085 stand. dev. of investment shock σ λp 0.16 stand. dev. of mark-up shock σ λw 0.289 stand. dev. of wage mark-up shock σ ǫq 0.604 stand. dev. of equity premium shock