Dynamic stochastic general equilibrium models December 4, 2007
Dynamic stochastic general equilibrium models Random shocks to generate trajectories that look like the observed national accounts. Rational expectations; representative agents; often calibration and solution by numerical techniques. 1. The classical or real business cycle model 2. The new keynesian model Basic building blocks.
Description of consumer s behavior Discrete time, infinite horizon. Nominal riskless bonds of one period maturity, no bankruptcy, incomplete markets Non storable consumption good Endogenous labor supply Important notation: upper case letters are for levels; lower case letters for the natural logarithms
Description of consumer s behavior The household has a competitive behavior: s/he considers prices, wages and interest rates as given. Budget constraints: maxe 0 β t U(C t, L t ) t=0 P t C t + Q t B t = W t L t + B t 1 + lump sum transfers net of taxes End of game: In practice we shall take lim B T 0 T U(C, L) = C 1 σ 1 σ L1+φ 1 + φ
First order conditions or in logarithms U L U C = C σ L φ = W P w t p t = σc t + φl t labor supply and a straightforward computation yields Q t U C,t P t = βe t U C,t+1 P t+1 Euler equation (1)
Transformation of Euler equation Definitions: nominal interest rate i t = ln Q t discount rate ρ = ln β inflation rate π t = ln(p t+1 /P t ) (expected) real interest rate r t = i t E t π t+1 Approximation in a neighborhood of a perfect foresight constant growth path at rate γ: γ = ln(c t+1 /C t ) = c t+1 c t. Along the constant growth path, the Euler equation is i = ρ + π + σγ
Log linearization of Euler equation The Euler equation can be rewritten as: E t exp[ σ c t+1 π t+1 ρ + i t ] = 1 A first order Taylor expansion in a small neighborhood of the reference trajectory gives exp[ σ c t+1 π t+1 ρ + i t ] 1 σ c t+1 π t+1 ρ + i t, so that, at a first order approximation i t = ρ + E t π t+1 + σe t c t+1, or Euler equation (2) c t = E t c t+1 1 σ [i t E t π t+1 ρ]
Firms Exogenous stochastic technical progress, no capital (or fixed capital stock) Competitive representative firm with production function or in logarithm Y t = A t N 1 α t y t = a t + (1 α)n t Maximization of profit P t Y t W t N t in each period, given prices and wages W t P t = (1 α)a t N α t which yields the labor demand schedule in logarithm w t p t = a t αn t + ln(1 α) labor demand
Production and costs The cost of producing Y is Ψ t (Y ) = W t ( Y A t ) 1/(1 α). The real marginal cost is (the wage argument is implicitly in the index t) MC t (Y, P) = Ψ t(y ) P = W t P ( ) 1 Y 1/(1 α) 1 1 α A t Y = W t P 1 N(Y ) α 1 α A t Here N(Y ) stands for the quantity of labor required to produce Y.
Marginal costs In logarithm: mc t (n, p) = w t p ln(1 α) a t + αn. The labor demand schedule is defined by the first order condition that price is equal to marginal cost, that is real marginal cost equals 1, or the logarithm of real marginal cost is equal to zero.
Equilibrium allocation Equalizing labor supply with labor demand, and good demand with production yields w t p t = a t αn t + ln(1 α) = σy t + φn t, where y t = a t + (1 α)n t. A tedious straightforward computation gives 1 σ n t = σ(1 α) + φ + α a ln(1 α) t + σ(1 α) + φ + α y t = 1 + φ σ(1 α) + φ + α a (1 α) ln(1 α) t + σ(1 α) + φ + α Employment and output are determined independently of nominal quantities (dichotomy).
Real interest rate The Euler equation gives the real interest rate that supports the allocation: r t = i t E t π t+1 = ρ + σe t y t+1, so that 1 + φ r t = ρ + σ σ(1 α) + φ + α E t a t+1 The real interest rate is determined by the real path of consumption (or output) together with the impatience of the agents.
Price indeterminacy, or the path of assets Rational expectations (or expectations are chosen to rationalize the equilibrium path) In the absence of stochastic shocks, there would be no motive to hold nominal assets in the long run (the aggregate stock of assets would have to be zero, and the general price level would be indeterminate). Possible precautionary motive when there are shocks: check that in the absence of bank interventions (Q t = 1 or i t = 0), there is an equilibrium with positive B 0.
How to have unemployment and a non neutral monetary policy, at least in the short run Equilibria with quantity rationing, with slow adjustment of nominal wages and prices (Walras tatonnement). Who sets prices? The only one with true unemployment. Confusion between real and nominal shocks may give scope for monetary policy. Equilibria with monopolistic competition (local monopoly power), underemployment compared with perfect competition. Some nominal inertia is added: Delayed arrival of information Menu costs in changing prices : the new keynesian model
Variety of goods: households Continuum of goods, designated with an index i which takes its valued in the interval [0, 1]. where Budget constraints: 1 0 maxe 0 β t U(C t, L t ) ( 1 C t = 0 t=0 ) ε/(ε 1) C t (i) 1 1/ε di P t (i)c t (i)di + Q t B t = W t L t + B t 1 + lump sum transfers
Allocation of total consumption among varieties Define ( 1 P t = 0 ) 1/(1 ε) P t (i) 1 ε di The choice among varieties follows from max 1 0 C t (i) 1 1/ε di subject to First order condition 1 0 P t (i)c t (i)di = Z C t (i) 1/ε = λp t (i)
Aggregation Define ( 1 P t = 0 ) 1/(1 ε) P t (i) 1 ε di We want to show that with this definition, whatever the individual prices, 1. P t C t = 1 2. The demand for good i satisfies 0 P t (i)c t (i)di = Z ε P t (i) C t (i) = C t P t
Allocation among varieties: aggregattion works Manipulating the FOC and the identities defining the aggregates: Z = λ ε 1 Z = λ 1 1 0 0 P t (i) 1 ε di = λ ε P 1 ε t C t (i) 1 1/ε di = λ 1 C 1 1/ε t Raising the first equation to the power 1/ε and multiplying with the second eliminates λ and gives the aggregate budget constraint that we are looking for. This in turn yields C 1/ε t = λp t, which gives the expression for the individual demand as a function of overall consumption abd the price ratio.
Households: concluded subject to maxe 0 β t U(C t, L t ) t=0 P t C t + Q t B t = W t L t + B t 1 + lump sum transfers First order conditions yield w t p t = σc t + φl t c t = E t c t+1 1 σ [i t E t π t+1 ρ].
Firms One firm per variety, all with identical production functions Y t (i) = A t N t (i) 1 α Common aggregate productivity shock. Firms are local monopolists who take the aggregates (C t, P t ) as given, and face identical isoelastic demand schedules. They choose optimally their price, given the possible rigidities.
Equilibrium with flexible prices Profit of the local monopolist is P(i)C(i) WN(i) = C(i) 1 1/ε W C 1/ε The first order condition yields ε 1 ε C(i) 1/ε P ( ) 1 C(i) 1 α A. P C 1/ε = 1 ( ) 1 C(i) 1 α W 1 α 1 A C(i). At the equilibrium, all agents produce the same quantity and choose the same price, so that taking logs ( ) ε 1 w t p t = y t n t + ln(1 α) + ln ε i.e. ( ) ε 1 w t p t = a t αn t + ln(1 α) + ln ε
Equilibrium with flexible prices: concluded The only difference with the classical labor demand schedule demand comes from the markup term (ε 1)/ε: firms reduce their output and labor demand to increase their profits. The dichotomy still is present and there are no real effects of monetary policy.
Price rigidities Calvo: each firm has probability (1 θ) to be allowed to reset its price in any period, independently of the time since last adjustment. The set of firms that do not change price has average price level P t 1. All the other (identical) firms choose an identical product price P t. The aggregation formula gives: P 1 ε t = θpt 1 1 ε 1 ε + (1 θ)p t Log linearizing around the zero inflation steady state yields (recall that Π t = P t /P t 1, π t = ln Π t ): π t = (1 θ)(p t p t 1 )
Optimal price setting The firm chooses the price level P which maximizes its expected discounted profits during the period where it will stay unchanged: max θ τ E t Q t,t+τ [P Y t+τ t Ψ t+τ (Y t+τ t )] τ=0 Here Q t,t+τ is the discount factor for nominal transfers between periods t and t + τ; Y t+τ t is the output at date t + τ of a firm whose price has been fixed at date t; from the consumer demand Y t+τ t = C t+τ (P /P t+τ ) ε ; Ψ t+τ (Y ) = W t+τ N t+τ = W t+τ (Y /A t+τ ) 1/(1 α is the cost of producing Y at date t + τ.
First order condition The first order condition for the optimal price is (using the equality Y / P = εy /P ): or θ τ E t Q t,t+τ [ (ε 1)Y t+τ t + εψ t+τ τ=0 θ τ E t Q t,t+τ Y t+τ t [ 1 + τ=0 Using the real marginal cost concept τ=0 ε ε 1 Ψ t+τ Y t+τ t P ] = 0 1 P ] = 0 θ τ E t Q t,t+τ Y t+τ t [ 1 + ε ε 1 MC(Y t+τ t, P )] = 0
Linearization close to a zero inflation steady state Along a zero inflation steady state, output and prices are constant, Q t,t+τ = β τ, and the first order condition implies MC(Y, P) = ε 1 ε Linearizing the first order condition yields = MC < 1 or (βθ) τ E t [ 1 + ε ε 1 MC t+τ (Y t+τ t, P )] = 0, τ=0 (βθ) τ E t [mc t+τ (Y t+τ t, P ) mc] = 0. τ=0
From individual marginal cost to aggregate marginal cost Individual marginal cost at date t + τ for a firm who last changed its price at t is mc t+τ (Y t+τ t, P t ). The macroeconomic aggregate marginal cost is defined as mc t+τ (Y t+τ, P t+τ ) = mc t+τ. mc t+τ (Y t+τ t, P t ) = mc t+τ p t + p t+τ + α 1 α [y t+τ t y t+τ ] Substituting production for prices using the price elasticity ε mc t+τ (Y t+τ t, P t ) = mc t+τ p t + p t+τ αε 1 α [p t p t+τ ] mc t+τ (Y t+τ t, P t ) = mc t+τ 1 α + αε 1 α [p t p t+τ ]
The first order condition revisited (βθ) τ E t [mc t+τ (Y t+τ t, P ) mc] = 0, τ=0 is equivalent in macroeconomic terms to { (βθ) τ E t mc t+τ mc 1 α + αε } 1 α [p t p t+τ ] = 0 τ=0 Writing the same expression for date t + 1, taking its expectation from date t, multiplying it by βθ and adding to the current equality, eliminates all the t + τ terms for τ > 1: mc t mc 1 α + αε 1 α [p t p t ]+(βθ) (βθ) τ 1 α + αε 1 α [E tpt+1 p t ] = 0 τ=0
Inflation and expected marginal costs Rearranging 1 α (1 βθ) 1 α + αε [mc t mc] (pt p t ) + βθ(e t pt+1 p t ) = 0. Recalling the Calvo relationship between inflation and the newly posted price, π t = p t p t 1 = (1 θ)(p t p t 1 ), one can eliminate the p s: i.e. 1 α (1 βθ) 1 α + αε [mc t mc] π t 1 θ + π t + βθ E tπ t+1 1 θ = 0 π t = βe t π t+1 + π t = (1 βθ)(1 θ) θ (1 βθ)(1 θ) θ 1 α 1 α + αε 1 α 1 α + αε [mc t mc]. β τ E t [mc t+τ mc]. τ=0
The output gap Recall that the expression for the marginal cost is mc t = w t p t ln(1 α) a t + αn t. The labor supply of the household satisfies w t p t = σy t + φn t Eliminating the real wage and employment y t = a t + (1 α)n t, or mc t = σy t ln(1 α) a t + α + φ 1 α (y t a t ), mc t = ( σ + α + φ ) y t α + φ 1 α 1 α a t ln(1 α) The natural level of output yt n is defined as the one that would prevail under flexibles prices, i. the value of production such that mc t = mc. The output gap ỹ t is the difference between output y t and its natural level yt n.
The New Keynesian Phillips curve It follows from expressing the marginal cost in terms of the output gap: ( (1 βθ)(1 θ) 1 α π t = βe t π t+1 + σ + α + φ ) ỹ t. θ 1 α + αε 1 α }{{} k
The dynamic IS equation From the intertemporal first order condition of the consumer s program: y t = E t y t+1 1 σ [i t E t π t+1 ρ]. Define the natural rate of interest ρ n t as the rate that would prevail on the reference constant price trajectory: y n t = E t y n t+1 1 σ [r n t ρ]. The difference between these two equalities gives the dynamic IS equation ỹ t = E t ỹ t+1 1 σ [i t E t π t+1 r n t ].
The equilibrium Two equations to determine the output gap and the inflation rate. One needs to make precise the monetary policy to determine the nominal interest rate i t and close the model.
Equilibrium under an interest rate rule Assume i t = ρ t + φ π π t + φ y ỹ t + v t The coefficients of inflation and the output gap are non negative. The constant term is chosen so as to allow a zero inflation rate. The random term v t is exogenous with zero mean. Substituting into the dynamic IS equation gives ( 1 + φ ) y ỹ t + φ π σ σ π t = E t ỹ t+1 + 1 σ [E tπ t+1 + rt n ρ v t ]
The system of equilibrium equations In matrix form, the Phillips curve and IS equations become ( ) ( ) ( ) ( ) ( k 1 ỹt 0 β Et ỹ = t+1 0 + σ + φ y φ π π t σ 1 E t π t+1 rt n ρ v t ) This yields ỹt π t 1 = σ+φ y +kφ π φ π 1 σ + φ y k 0 β σ 1 E tỹ t+1 E t π t+1 + 0 rt n ρ v t ỹt π t 1 = σ+φ y +kφ π σ 1 βφ π σk β(σ + φ y ) + k E tỹ t+1 E t π t+1 + 1 k (r t n ρ vt)
The matrix eigenvalues The eigenvalues are the roots of characteristic polynomial f (λ) = λ 2 Sλ + P The sum of the roots S is equal to the trace S = σ + β(σ + φ y ) + k σ + φ y + kφ π while their product is equal to the determinant βσ P = σ + φ y + kφ π
Position of the eigenvalues relative to the unit circle The determinant is smaller than 1: by assumption the coefficients of the interest rate rule are non negative and k is non negative. If the roots are complex, they are both inside the unit circle. If they are real, at least one of them is inside the unit circle. A necessary and sufficient condition for both roots to be inside the unit circle then is P( 1) > 0, P(1) > 0. Now P( 1) = 1+S+P > 0, P(1) = 1 S+P = (1 β)φ y + k(φ π 1) σ + φ y + kφ π A necessary and sufficient condition for both roots to be inside the unit circle is (1 β)φ y + k(φ π 1) > 0
The rational expectations equilibrium: monetary shocks Suppose v t is AR(1) v t = ρ v v t 1 + ε v t One looks for a linear solution in v t, ỹ t = ψ y v t π t = ψ π v t where the unknowns are the coefficients psi y and ψ π. E t ỹ t+1 = ψ y ρ v v t E t π t+1 = ψ π ρ v v t
Finding the rational expectations equilibrium Substituting into the IS and Phillips curves yield two equations in the two unknowns ψ π = βρ v ψ π + kψ y ψ y = ρ v ψ y 1 σ (φ πψ π + φ y psi y + 1 ψ π ρ v A similar computation can be done for the impact of technology shocks, supposing a AR(1) structure for technical progress a t = ρ a a t 1 + ε a t
Calibration β = 0.99 Period: quarter σ = 1 log utility φ = 1 unit Frisch elasticity of labor supply θ = 2/3average price duration of three quarters ρ v = 0.5 ρ a = 0.9 Greenspan time φ π = 1.5, φ y = 0.5/4
Figure 3.1: Effects of a Monetary Policy Shock (Interest Rate Rule))
Figure 3.2: Effects of a Technology Shock (Interest Rate Rule)