ECON 4325 Monetary Policy and Business Fluctuations

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1 ECON 4325 Monetary Policy and Business Fluctuations Tommy Sveen Norges Bank January 28, 2009 TS (NB) ECON 4325 January 28, 2009 / 35

2 Introduction A simple model of a classical monetary economy. Perfect competition and exible prices in all markets. A useful benchmark for later analysis, but many of the predictions are at odds with the empirical evidence. Soon we will know:... what drives the real wage.... what is the relationship between the real interest rate and output. TS (NB) ECON 4325 January 28, / 35

3 Introduction Households: Firms: Complete nancial markets. Perfectly competitive labor market. Competitive rms (monopolistic competition and sticky prices later). Cobb-Douglas production function with labour as the only input. General equilibrium a so-called DSGE-model (dynamic, stochastic, general equilibrium). TS (NB) ECON 4325 January 28, / 35

4 Households The representative household chooses labor supply, consumption, and one-period bonds. Maximize discounted expected utility: E t β k U (C t+k, N t+k ), () k=0 where β is the discount factor. U period utility C and N consumption and employment ( leisure). TS (NB) ECON 4325 January 28, / 35

5 Households We use the following period utility function: U (C t, N t ) = C t σ σ N +ϕ t + ϕ, (2) where σ is the intertemporal elasticity of substitution and ϕ is the Frisch labor supply elasticity. σ measures how willing the household is to substitute consumption over time when the real interest rate changes. ϕ measures how labor supply increases when the real wage increases. TS (NB) ECON 4325 January 28, / 35

6 Households Sequence of budget constraints: P t C t + Q t B t B t + W t N t T t, (3) P price of consumption goods. W and T nominal wages and taxes (net of dividends from ownership of rms). Q and B price and quantity of one-period risk-free nominal bonds that pay one nominal unit on maturity. Solvency constraint. TS (NB) ECON 4325 January 28, / 35

7 Households Let us look at the Lagrangian: L t = E t ( k=0 First-order conditions: L t C t = Ct σ L t " C σ β k t+k σ N +ϕ t+k + ϕ Λ t+k (P t+k C t+k + Q t+k B t+k B t+k W t+k N t+k + T t+k )]g (4) Λ t P t = 0 ) Λ t = C t σ, P t N t = N ϕ t + Λ t W t = 0 ) Λ t = N ϕ t, W t L t B t = Λ t Q t + βe t fλ t+ g = 0 ) Q t = βe t Λt+ Λ t. TS (NB) ECON 4325 January 28, / 35

8 Households We then get: C σ t N ϕ t where Ω t is period t real wage. The optimality condition for labor supply: = W t P t Ω t, (5) The real wage increases with hours worked (compensating for increases in marginal disutility). Increases with consumption (which makes the "utility value" of the real wage lower). TS (NB) ECON 4325 January 28, / 35

9 Households Moreover, we have βe t ( Ct+ C t σ Π t+ ) = Q t, (6) where Π t+ P t+ P t is gross in ation (and π t log Π t is the rate of in ation). The consumption Euler-equation: Re ects the household s preferences for consumption smoothing. TS (NB) ECON 4325 January 28, / 35

10 Households Note that the nominal (risk-free) interest rate is de ned as i t log Q t, since Q t is the period t price of getting one monetary unit in all states in period t +. The bond yield is implicitly given by Q t = ( + yield). We therefore have i = log Q t = log ( + yield) ' yield. TS (NB) ECON 4325 January 28, / 35

11 Households More About the Consumption Euler-Equation Consider the following two-period model: ( Ct σ max C t,c t+ σ + β C ) σ t+ σ Let the household s budget constraints be: where S t is period t saving. P t C t = W t Q t B t = W t S t, P t+ C t+ = B t = ( + i t ) S t, TS (NB) ECON 4325 January 28, 2009 / 35

12 Households More About the Consumption Euler-Equation Household s problem FOC: max S t ( ((W t S t ) /P t ) σ C σ t σ C t ) + β (S t ( + i t ) /P t+ ) σ σ /P t + βct+ σ ( + i t) /P t+ = 0, " Ct+ # σ β Πt+ = where r t is the real interest rate. β Ct+ C t σ = + i t, + r t, TS (NB) ECON 4325 January 28, / 35

13 Households More About the Consumption Euler-Equation Consumption period t+ ( + r t ) Consumption period t TS (NB) ECON 4325 January 28, / 35

14 Firms There is a representative rm that has access to the following production technology: Y t = A t Nt α, (7) where Y t and N t are production and labor input, and log A t a t = ρ a a t + ε a t. Price-taker in all markets (the labor market and the goods market). TS (NB) ECON 4325 January 28, / 35

15 Firms The rm maximizes pro ts: max [P t Y t W t N t ], (8) Y t,n t s.t. Y t = A t Nt α (9) First-order condition: MPL t ( α) Y t N t = Ω t. (0) TS (NB) ECON 4325 January 28, / 35

16 Market Clearing All markets clear: Y t = C t, () Nt s = Nt d = N t. (2) In addition B t = 0 (zero net savings). TS (NB) ECON 4325 January 28, / 35

17 Log-Linearized Model Households Labor supply: We start by looking at the steady state: Ω = C σ N ϕ, Then we log-linearize labor supply: e ω t = e σc t +ϕn t e 0 + e 0 (ω t 0) = e 0 + e 0 σ (c t 0) + e 0 ϕ (n t 0) ω t = ϕn t + σc t and ϕ measures how much labor supply increases when the real wage increases. TS (NB) ECON 4325 January 28, / 35

18 Log-Linearized Model Households Consumption Euler-equation: ( Ct+ ) σ = βe t. C t Q t Π t+ Steady state C σ QΠ = β = β C ) i + π = log β ρ, where ρ is household s discount rate. This implies a steady state real rate, r i π = ρ. TS (NB) ECON 4325 January 28, / 35

19 Log-Linearized Model Households Consumption Euler-equation (cont d): Log-linearizing: n o = E t e σ(c t+ c t ) e log Q t log Π t+ +log β n o = E t e σ(c t+ c t )+(i t π t+ ρ) = e 0 e 0 E t fσ (c t+ c t ) ((i t i) (π t+ π))g c t = E t c t+ σ (i t E t π t+ ρ). TS (NB) ECON 4325 January 28, / 35

20 Log-Linearized Model Households The consumption Euler-equation: c t = E t c t+ σ (r t ρ), (3) where we have used r t i t E t π t+ Consumption smoothing: parameter σ measures by how much consumption increases when the real interest rate drops. Consumption is a pure forward-looking or jump variable. TS (NB) ECON 4325 January 28, / 35

21 Log-Linearized Model More one the Consumption Euler-Equation Solving the equation forward gives: c t = σ (r t+k ρ) = rt L σ k=0 ρ, (4) where r L is related to long real rates. Given the expectation hypothesis the relationship between short and long (real) rates with maturity T, r T t, is given by: r T t = ρ + T T (r t+k ρ). k=0 Therefore: r L t Tr T t TS (NB) ECON 4325 January 28, / 35

22 Log-Linearized Model Firms and Market Clearing Firms Output y t = a t + ( α) n t, (5) Labor demand ω t = y t n t, (6) and we also have productivity a t = ρ a a t + ε a t. Consumption equals output: Market clearing y t = c t. (7) TS (NB) ECON 4325 January 28, / 35

23 Log-Linearized Model Firms and Market Clearing Money demand and monetary policy We assume that (log-linearized) money demand is given by m t p t = y t ηi t, (8) where η is the semi interest rate elasticity. Let monetary policy be given by: m t = ρ m m t + ε m t. (9) TS (NB) ECON 4325 January 28, / 35

24 Results Let us solve the model, i.e. show how the endogenous variables depend the exogenous variables. To this end, let us repeat the complete set of equations c t = E t c t+ σ (r t ρ), r t = i t E t π t+, ω t = φn t + σc t, w t p t = ω t, y t = ( α) n t + a t, m t p t = y t ηi t, ω t = y t n t, m t = ρ m m t + ε m t. y t = c t, a t = ρ a a t + ε a t TS (NB) ECON 4325 January 28, / 35

25 Results Classical Dichotomy : Real variables determined independently of monetary policy (neutrality). Optimal policy: undetermined. Speci cation of monetary policy needed to determine nominal variables. TS (NB) ECON 4325 January 28, / 35

26 Results Let us rst solve for the monetary part of the model (and we now assume that a t = 0). Substitute for the nominal interest rate in the money demand function using the Fisher equation: which can be written as m t p t = 0 ηe t fp t+ p t g, p t = This can be solved forward to yield: p t = + η E t η + η E tp t+ + + η m t. k=0 η + η k E t m t+k. TS (NB) ECON 4325 January 28, / 35

27 Results We want to rewrite the relationship in terms of changes in nominal money: We can then write: p t = m t m t + + η m t + η + η E η tm t+ + η E tm t+ + η + η + η E tm t p t = m t η + η m t + η + η + η + η + η E tm t+ E t m t TS (NB) ECON 4325 January 28, / 35

28 Results We can then write: η k p t = m t + E t E t m t+k + η k=0 ηρ = m t + m + η ( ρ m ) m t If ρ m > 0 (the parameter is often calibrated to 0.5 based on empirical evidence), the price level should respond more than one-for-one with the increase in the money supply. This prediction is in stark contrast to the sluggish response of the price level observed in empirical estimates of the e ects of monetary policy shocks. TS (NB) ECON 4325 January 28, / 35

29 Results We can solve the (real side of the) model explicitly as follows: Use the labor supply and demand equations and combine it with the aggregate resource constraint: y t n t = ϕn t + σy t, Next, combine the above equation with the production function ( σ) y t = ( + ϕ) α (y t a t ) y t = ψ ya a t which only depend on productivity and ψ ya = +ϕ σ( α)+ϕ+α. TS (NB) ECON 4325 January 28, / 35

30 Results The solution to the (real side of the) model is then: y t = ψ ya a t where ψ ya = σ ( n t = ψ na a t where ψ na = σ ( 9 + ϕ = α) + ϕ + α ; 9 σ = α) + ϕ + α ; σ + ϕ ω t = ψ ωa a t where ψ na = σ ( α) + ϕ + α r t ρ = σψ ya E t f a t+ g = ( ρ a ) σψ ya a t > if σ < = if σ = < if σ > > 0 if σ < = 0 if σ = < 0 if σ > TS (NB) ECON 4325 January 28, / 35

31 Results - A Permanent Increase in Productivity in Cashless Economy Parameters: σ =, β = 0.99, α = 0, ϕ = a c r y n ω TS (NB) ECON 4325 January 28, / 35

32 Results - A Temporary Increase in Productivity in Cashless Economy Parameters: σ =, β = 0.99, α = 0, ϕ =, ρ a = 0.95 a y c n r ω TS (NB) ECON 4325 January 28, / 35

33 Money in the utility function So far, money has only served as a unit of account (often referred to as cashless economies). We now assume that money generate utility: U C t, M t, N t P t The budget constraint becomes: (20) P t C t + Q t B t + M t B t + M t + W t N t T t. (2) TS (NB) ECON 4325 January 28, / 35

34 Money in the utility function The optimality conditions are: U N,t = Ω t, (22) U C,t UC,t+ βe t Πt+ = Q t (23) U C,t U M,t = Q t ' i t (24) U C,t + i t Interpretation of the latter: LHS: increased utility from holding more money (in consumption units). RHS: alternative cost (one monetary unit, minus the cost of buying a bond that gives one monetary unit in period t +, as is the case when holding money). TS (NB) ECON 4325 January 28, / 35

35 Money in the utility function Two cases: Utility is separable in real balances: neutrality. 2 Utility is non-separable in real balances: non-neutrality. Even in the case of non-neutrality there are very small real e ects from monetary shocks. Optimal policy: Friedman rule (zero nominal interest rate and π = ρ). TS (NB) ECON 4325 January 28, / 35

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