Macro II. John Hassler. Spring John Hassler () New Keynesian Model:1 04/17 1 / 10

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1 Macro II John Hassler Spring 27 John Hassler () New Keynesian Model: 4/7 /

2 New Keynesian Model The RBC model worked (perhaps surprisingly) well. But there are problems in generating enough variation in labor supply. There is no role for stabilization policy. 2 A reasonable avenue to make a more realistic business cycle model is to take seriously that prices and perhaps wages are not continously adjusted. 3 To talk about price stickyness, we need to allow some price-setting power monopolistic competition. 4 Different monopolistic firms requires different goods with potential for price dispersion. 5 Otherwise, our model will build on the RBC model, i.e., being a stochastic general equilibrium model with forward looking rational agents. 6 Have become the central modeling approach in e.g., central banking. Will look at the simplest possible version. John Hassler () New Keynesian Model: 4/7 2 /

3 Bonds instead of capital As before, we assume a representative household that maximizes E t β s U (C t+s, L t+s ) s= In order to allow monetary policy to affect intertemporal tradeoff, we introduce government bonds, B t but disregard capital. Budget constraint of individual is then s.t. P t C t + Q t B t = B t + W t ( L t ) + T t, t where Q t is the price one-period nominal bonds and T t is a lump-sum transfer (firm profits, taxes...) In contrast to above, we now think of C t as a basket/index of differentiated goods C (i), i [, ], ( ( C t C t (i) ) di), where determines how substitutable the goods are. Continous version of Dixit & Stiglitz (977). John Hassler () New Keynesian Model: 4/7 3 /

4 Constructing a price index In the budget constraint, we used an aggregate price index, P t. Can we construct that from the underlying prices P t (i)? Consider the problem of minimizing the cost of getting a given amount of aggregate consumption C t min P t (i) C t (i) di λ t {C t (i)} i= FOC for C t (i) ( ) ( C t (i) ) di C t P t (i) ( = λ t ) ( ) ( C t (i) ) ( di ) C t (i) ( ) ( = λ t C t (i) ) di C t (i) John Hassler () New Keynesian Model: 4/7 4 /

5 Constructing a price index:2 ( ) Note that : C t (i) ( ) di = C t P t (i) = λ t C t C t (i). by definition, giving What is λ t in the FOC P t (i) = λ t C t C t (i)? It is the minimized cost of increasing aggregate consumption C t by one unit, i.e., λ t is the price index P t.thus, P t (i) = λ t C t C t (i) gives P t (i) = P t C t C t (i) ( ) Pt = C t (i) P t (i) One percent change in the relative price of good i, leads to percent decline in relative demand for that good. What happens with budget shares of different goods when prices increase if =, lower than one, higher than one? John Hassler () New Keynesian Model: 4/7 5 / C t

6 The exact price index Use ( ) Pt P t (i) = C t (i) C t in aggregate expenditure; P t C t = P t (i) C t (i) di = Dividing by C t, gives P t = P t ( ) Pt P t (i) C t di = C t Pt P t (i) di. P t (i) ( P t = P t (i) di, or ) P t (i) di This is an exact price index, defining the minized cost per unit of aggregate consumption. John Hassler () New Keynesian Model: 4/7 6 /

7 Discussion Price index P t = Note that; it is H. ( ) P t (i) di the larger is, the more dispersion reduces the price index. Discrete example with three goods. Suppose prices are, 2 and 3. Consider three cases, =, 2 and. With =, price index is = 2, i.e., the average price. [ ( With = 2, P t = [ ( With =, P t = minimum price. Explain! ) ] =.636 =2 ] =., close to the = ) John Hassler () New Keynesian Model: 4/7 7 /

8 Using the price index We can now conveniently treat the consumer problem in two stages; given distribution of prices, mininize cost of consuming a given consumption level. Gives P t. 2 decide how much to work, consume and save. In many applications we can forget about the first step. But recall that relative price differences have welfare costs. John Hassler () New Keynesian Model: 4/7 8 /

9 Individual agregate decisions - labor supply Given the two stage decision problem, the second yields optimality conditions as in RBC-model. U L (C t, L t ) = W t U C (C t, L t ) P t [ ] P t U C (C t, L t ) = βe t U C (C t+, L t+ ) Q t P t+ Let us use a utility function in terms of consumption and disutility of labor N t. U (C t, N t ) = C t σ σ φ N +ϕ t +ϕ where ϕ measures how inelastic labor supply is. = φn ϕ t Take log of the intratemporal condition W t P t and let lower case Ct σ variables denote logs and dropping the constant ln φ w t p t = σc t + ϕn t John Hassler () New Keynesian Model: 4/7 9 /

10 Individual aggregate decisions - NK Euler The Euler equation = βe t [ ( Pt Ct Q t P t+ ) σ ] C t+ can be written, = E t (exp ( ρ + i t π t+ σ c t+ )) where ρ ln β β, i t ln Q t Q t, π t+ ln P t+ ln P t P t+ P t. Note that in a perfect foresight steady state with constant inflation and constant consumption growth γ, we have ρ + i π σγ =. First-order Taylor approximation around this steady state exp ( ρ + i t π t+ σ c t+ ) + (i t i) (π t+ π) σ ( c t+ γ) = + i t π t+ σ c t+ i + π + σγ = + i t π t+ σ c t+ ρ According to the Euler equation the expected value of this should be unity, implying c t = E t c t+ σ (i t E t π t+ ρ) John Hassler () New Keynesian Model: 4/7 /