A Modern Equilibrium Model. Jesús Fernández-Villaverde University of Pennsylvania

A Modern Equilibrium Model Jesús Fernández-Villaverde University of Pennsylvania 1

Household Problem Preferences: max E X β t t=0 c 1 σ t 1 σ ψ l1+γ t 1+γ Budget constraint: c t + k t+1 = w t l t + r t k t +(1 δ) k t, t>0 Complete markets and Arrow-Debreu securities. 2

First Order Conditions c σ t = λ t βe t c σ t+1 = λ t+1 ψl γ t = λ t w t λ t = λ t+1 (r t+1 +1 δ) 3

Problem of the Firm Neoclassical production function: y t = A t k α t l 1 α t By profit maximization: αa t kt α 1 lt 1 α = r t (1 α) A t kt α lt 1 α lt 1 = w t Investment x t induces a law of motion for capital: k t+1 =(1 δ) k t + x t 4

Evolution of the technology A t changes over time. It follows the AR(1) process: log A t = ρ log A t 1 + z t z t N (0, σ z ) Interpretation of ρ. 5

A Competitive Equilibrium We can define a competitive equilibrium in the standard way. Conditions: c σ t = βe t c σ t+1 (r t+1 +1 δ) ψl γ t = λ t w t r t = αa t kt α 1 lt 1 α w t = (1 α) A t k α t l 1 α t c t + k t+1 = A t kt α 1 lt 1 α log A t = ρ log A t 1 + z t lt 1 +(1 δ) k t 6

Behavior of the Model We have an initial shock: productivity changes. We have a transmission mechanism: intertemporal substitution and capital accumulation. We can look at a simulation from this economy. Whyonlyasimulation? 7

Comparison with US economy Simulated Economy output fluctuations are around 75% as big as observed fluctuations. Consumption is less volatile than output. Investment is much more volatile. Behavior of hours. 8

Assessment of the Basic Real Business Model It accounts for a substantial amount of the observed fluctuations. It accounts for the covariances among a number of variables. It has some problems accounting for the behavior of the hours worked. More important question: where do productivity shocks come from? 9

Negative Productivity Shocks I The model implies that half of the quarters we have negative technology shocks. Is this plausible? What is a negative productivity shocks? Role of trend. 10

Negative Productivity Shocks II s.d. of shocks is 0.01. Mean quarter productivity growth is 0.0048 (to give us a 1.9% growth per year). As a consequence, we would only observe negative technological shocks when ε t < 0.0048. This happens in the model around 33% of times. Ways to fix it. 11

Some Policy Implications ThebasicmodelisPareto-efficient. Fluctuations are the optimal response to a changing environment. Fluctuations are not a sufficient condition for inefficiencies or for government intervention. In fact in this model the government can only worsen the allocation. Recessions have a cleansing effect. 12

Extensions I We can extend our model in several directions. Examples we are not going to cover: 1. Fiscal Policy shocks (McGrattan, 1994). 2. Agents with Finite Lives (Ríos-Rull, 1996). 3. Indivisible Labor (Rogerson, 1988, and Hansen, 1985). 4. Home Production (Benhabib, Rogerson and Wright, 1991). 13

Extensions We Study (Cumulative) Money (Cooley and Hansen, 1989). 14

Money Money in the Utility function (MIU). Comparison with Cash-in-Advance. You can show both approaches are equivalent. Are we doing the right thing (Wallace, 2001)? 15

Households Utility function: X E 0 β t t=0 c 1 σ t 1 σ + υ 1 ξ Ã mt! 1 ξ ψ l1+γ t 1+γ. Budget constraint: c t + x t + m t + b t+1 = w t l t + r t k t + m t 1 + R t b t + T t + Π t 16

First Order Conditions I υ c σ t = λ t βe t c σ t+1 = λ t+1 ψlt γ = λ t w t! ξ 1 1 = λ t + λ t+1 +1 1 1 λ t = λ t+1 R t+1 Ã mt +1 λ t = λ t+1 (r t+1 +1 δ) 17

First Order Conditions II υ c σ t c σ t Ã mt = βe t {c σ t+1 (r t+1 +1 δ)} = βe t {c σ t+1 R t+1 } +1 ψlt γ = c σ t w t! ξ = c σ t + E t {c σ t+1 } +1 18

The Government Problem The government sets the nominal interest rates according to the Taylor rule: R t+1 R = µ Rt R γr µ Ã! πt γπ γy yt e ϕ t π y How? Through open market operations. How are those operations financed? Lump-sum transfers T t such that the deficit are equal to zero: T t = m t m t 1 + b t+1 R t b t 19

Interpretation π: target level of inflation (equal to inflation in the steady state), y: the steady state output R: steady state gross return of capital. ϕ t : random shock to monetary policy distributed according to N (0, σ ϕ ). The presence of the previous period interest rate, R t,isjustified because we want to match the smooth profile of the interest rate over time observed in U.S. data. 20

Advantages and Disadvantages of Taylor Rules Advantages: 1. Simplicity. 2. Empirical foundation. Problems: 1. Normative versus positive. 2. Empirical specification. 21

Behavior of the Model We have a second shock: interest shock. We have a transmission mechanism: intertemporal substitution and capital accumulation. For a standard calibration, the second shock buys us very little. 22

Extensions We Study (Cumulative) Money (Cooley and Hansen, 1989). Monopolistic Competition (Blanchard and Kiyotaki, 1987, and Horstein, 1993). 23

Monopolistic Competition Final good producer. Competitive behavior. Continuum of intermediate good producers with market power. Alternative formulations: continuum of goods in the utility function. Otherwise, the model is the same as the model with money. 24

The Final Good Producer Production function: y t = Ã Z 1 ε 1 y ε 0 it di! ε ε 1 where ε controls the elasticity of substitution. Final good producer is perfectly competitive and maximize profits, taking as given all intermediate goods prices i and the final good price. 25

Maximization Problem Thus, its maximization problem is: max y it y t Z 1 0 p ity it di First order conditions are: ε ε 1 Ã Z 1 ε 1 y ε 0 it di! ε ε 1 1 ε 1 ε ε 1 y ε 1 it p it =0 i 26

Working with the First Order Conditions I Dividing the first order conditions for two intermediate goods i and j, we get: or: p it p jt = p jt = Ã! 1 yit ε y jt Ã yit y jt!1 ε pit Interpretation. 27

Working with the First Order Conditions II Hence: 1 ε 1 p jt y jt = p it y ε it y ε jt Integrating out: Z 1 0 p jty jt dj = p it y 1 Z 1 ε 1 ε it y ε 0 jt 1 ε 1 dj = p it y ε it y ε t 28

Input Demand Function By zero profits ( y t = R 1 0 p jt y jt dj), we get: 1 ε 1 y t = p it y ε it y ε t 1 = p it y ε it y 1 ε t Consequently, the input demand functions associated with this problem are: y it = Ã pit! ε y t i Interpretation. 29

Price Level By the zero profit condition y t = R 1 0 p it y it di and plug-in the input demand functions: y t = Z 1 0 p it Ã pit! ε y t di p 1 ε t = Z 1 0 p1 ε it di Thus: = Ã Z 1 0 p1 ε it di! 1 1 ε 30

Intermediate Good Producers Continuum of intermediate goods producers. No entry/exit. Each intermediate good producer i has a production function y it = A t k α it l1 α it A t follows the AR(1) process: log A t = ρ log A t 1 + z t z t N (0, σ z ) 31

Maximization Problem I Intermediate goods producers solve a two-stages problem. First, given w t and r t,theyrentl it and k it in perfectly competitive factor markets in order to minimize real cost: min {w t l it + r t k it } l it,k it subject to their supply curve: y it = A t k α it l1 α it 32

First Order Conditions The first order conditions for this problem are: w t = % (1 α) A t kit α l α it r t = %αa t kit α 1 lit 1 α where % is the Lagrangian multiplier or: k it = α 1 α w t r t l it Note that ratio capital-labor only is the same for all firms i. 33

Real Cost The real cost of optimally using l it is: µ w t l it + α 1 α w tl it Simplifying: µ 1 1 α w t l it 34

Marginal Cost I The firm has constant returns to scale. Then, we can findtherealmarginalcostmc t by setting the level of labor and capital equal to the requirements of producing one unit of good A t k α it l1 α it =1 Thus: A t k α it l1 α it = A t Ã α 1 α! α w t l it l r it 1 α t = A t Ã α 1 α w t r t! α l it =1 35

Marginal Cost II Then: mc t = µ 1 1 α w t 1 A t Ã α 1 α! α w t = r t µ 1 1 α 1 α µ 1 α α 1 wt 1 α A t r α t Note that the marginal cost does not depend on i. Also, from the optimality conditions of input demand, input prices must satisfy: k t = α w t l t 1 α r t 36

Maximization Problem II The second part of the problem is to choose price that maximizes discounted real profits, i.e., (Ã! ) pit max mc y it it subject to Ã! ε yit+τ pit = y t+τ, First order condition: 1 Ã pit! ε y t+τ ε Ã pit mc t!ã pit! ε 1 1 y t+τ =0 37

Mark-Up Condition From the fist order condition: Ã!Ã! 1 pit pit 1 ε mc t =0 p it = ε (p it mc t ) p it = ε ε 1 mc t Mark-up condition. Reasonable values for ε. 38

Behavior of the Model Presence of monopolistic competition is, by itself, pretty irrelevant. Why? Constant mark-up. Similar to a tax. Solutions: 1. Shocks to mark-up (maybe endogenous changes). 2. Price rigidities. 39

Extensions We Study (Cumulative) Money (Cooley and Hansen, 1989). Monopolistic Competition (Blanchard and Kiyotaki, 1987, and Horstein, 1993). Price rigidities (Calvo, 1983). 40

The Baseline Sticky Price Model The basic structure of the economy is as before: 1. A representative household consumes, saves, holds money, and works. 2. There is a monetary authority that fixes the one-period nominal interest rate through open market operations with public debt. 3. Final output is manufactured by a final good producer, which uses as inputs a continuum of intermediate goods manufactured by monopolistic competitors. Additional constraint: the intermediate good producers face the constraint that they can only change prices following a Calvo s rule. 41

Maximization Problem II The second part of the problem is to choose price that maximizes discounted real profits, i.e., subject to (Ã X max E (βθ p ) τ pit v t+τ mc t+τ it τ=0 +τ Ã! ε yit+τ pti = y t+τ, +τ! y it+τ ) Think about different elements of the problem. 42

Pricing I Calvo pricing. Parameter θ p. What if θ p =0? 43

Pricing II Alternative pricing mechanisms: 1. Time-dependent pricing: Taylor and Calvo. 2. State-dependent pricing. Advantages and disadvantages: 1. Klenow and Kryvstov (2005): intensive versus extensive margin of price changes. 2. Caplin and Spulber (1987), Dotsey et al. (1999). 44

Discounting v t is the marginal value of a dollar to the household, which is treated as exogenous by the firm. We have complete markets in securities, Then, this marginal value is constant across households. Consequently, β τ v t+τ is the correct valuation on future profits. 45

First Order Condition Substituting the demand curve in the objective function, we get: max p it E t X τ=0 (βθ p ) τ v t+τ! 1 ε Ã! ε pit pti Ã mc t+τ y it+τ +τ +τ The solution p it E t X τ=0 (βθ p ) τ v t+τ implies the first order condition: µ p 1 ε (1 ε) it p p 1 t+τ it +ε µ p it ε p p 1 t+τ it mc t+τ y it+τ =0 46

Working on the Expression I Then: E t X τ=0 (βθ p ) τ v t+τ (Ã(1 ε) p it p 1 p it t+τ E t X τ=0 (βθ p ) τ v t+τ (Ã p it +τ! ) + εp 1 it mc t+τ y it+τ ε ε 1 mc t+τ! y it+τ ) = 0 = 0 Thus, in a symmetric equilibrium, in every period, 1 θ p of the intermediate good producers set p t as their price policy function, while the remaining θ p do not reset their price at all. 47

Working on the Expression II Note, if θ p =0(allfirms move), all terms τ > 0 are zero: v t (Ã p it ε ε 1 mc t! y it ) =0 Then p it = ε ε 1 mct theresultwehadfromthebasicmonopolisticcase. 48

Price Level The price index evolves: = h θ p p 1 ε t 1 +(1 θ p) pt 1 ε i 1 1 ε Dividing by 1 : π t = = h θ p +(1 θ p ) π 1 ε i 1 1 ε t 1 where π t = p t 1. 49

Aggregation I Now, we derive an expression for aggregate output. Remember that: k it l it = α 1 α w t r t Since this ratio is equivalent for all intermediate firms, it must also be the case that: k t = α w t l t 1 α r t 50

Aggregation II We get: k it l it = k t l t If we substitute this condition in the production function of the intermediate good firm y it = A t kit αl1 α it we derive: Ã! α Ã! α kit kt y it = A t l it = A t l it l it l t 51

Aggregation III The demand function for the firm is: y it = Ã pit! ε y t i, Thus, we find the equality: Ã! ε Ã pit kt y t = A t! α l it l t 52

Aggregation IV If we integrate in both sides of this equation: Z Ã! 1 ε Ã! α Z pit kt 1 y t di = A t l itdi = A t kt α lt 1 α 0 0 l t Then: where v t = Z Ã 1 pit 0 y t = A t kt α lt 1 α v t! ε di = j ε t p ε t = jt ε p ε t and j t = ³ R 10 p ε it di 1ε. 53

Interpretation of j t What is j t? Measure of price disperssion. Measure of efficiency losses implied by inflation. Important to remenber for welfare analysis. 54

Equilibrium A definition of equilibrium in this economy is standard. The symmetric equilibrium policy functions are determined by the following blocks: 1. The first order conditions of the household. 55

The First Order Conditions of the Household υ c σ t Ã mt c t + x t + m t + b t+1 = βe t {c σ t+1 (r t+1 +1 δ)} c σ t = βe t {c σ R t+1 t+1 } π t+1 ψl γ t = c σ! ξ = c σ t t w t + E t {c σ t+1 = w t l t + r t k t + m t 1 1 } π t+1 + R t b t + T t + Π t 56

Pricing Decisions by Firms. The firms that can change prices set them to satisfy where: and E t X τ=0 (βθ p ) τ v t+τ (Ã p t +τ mc t = ε ε 1 mc t+τ Ã! ε yit+τ pti = y t+τ, +τ µ 1 1 α 1 α µ 1 α α 1 wt 1 α A t! y it+τ ) r α t =0 58

Input and Output Prices Input prices satisfy: k t = α 1 α w t r t l t The price level evolves: π t = = h θ p +(1 θ p ) π 1 ε i 1 1 ε t 1 Equilibrium 60

A definition of equilibrium in this economy is standard. The symmetric equilibrium policy functions are determined by the following blocks: 1. The first order conditions of the household. 2. Pricing decisions by firms. 3. Pricing of inputs and outputs. 4. Taylor rule and market clearing. 61

Taylor Rule and Market Clearing Taylor rule R t+1 R = µ Rt R γr µ Ã! πt γπ γy yt e ϕ t π y Markets clear: c t + x t = y t = A t kt α lt 1 α v t where v t = R 1 0 ³ pit ε di and kt+1 =(1 δ) k t + x t. 62

Further Extensions Investment-specific technological change. Sticky wages. Open economy. 63