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Graduate Macro Theory II: Notes on New Keynesian Model

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Graduate Macro Theory II: Notes on New Keynesian Model Graduate Macro Theory II: Notes on New Keynesian Model Eric Sims University of Notre Dame Spring 2011 1 Introduction This note describes the simplest version of the New Keynesian model. The NK model takes a real business cycle model as its backbone and adds to it sticky prices. The sticky prices give rise to non-trivial monetary neutralities and also give rise to a serious role for active economic policy. 2 Households The household side of the model ends up looking very standard. Households consume goods, supply labor, and hold money. We assume that they receive utility from holding money. They can save through bonds, which pay nominal interest rate i. The household problem can be written: 1−ν 0 M 1 1 1−ξ t+1 − 1 X t (1 − nt) − 1 pt max E0 β Bln ct + θ + C ct;nt;Bt+1;Mt+1 @ 1 − ξ 1 − ν A t=0 s.t. Bt+1 − Bt Mt+1 − Mt Bt ct + + ≤ wtnt + Πt + it pt pt pt We can set the problem up using a Lagrangian: 1−ν 0 M 1 1−ξ t+1 − 1 X (1 − nt) − 1 pt L = E βt Bln c + θ + + ::: 0 @ t 1 − ξ 1 − ν t=0 Bt Bt+1 Mt+1 − Mt ··· + λt wtnt + Πt + (1 + it) − ct − − pt pt pt The first order conditions are: 1 @L 1 = 0 , = λt (1) @ct ct @L −ξ = 0 , θ(1 − nt) = λtwt (2) @nt @L pt = 0 , λt = βEt λt+1(1 + it+1) (3) @Bt+1 pt+1 −ν @L Mt+1 pt = 0 , = λt − βEt λt+1 (4) @Mt+1 pt pt+1 T Bt+T +1 TV : lim β Etλt+T = 0 (5) T !1 pt+T We can use (1) to simplify this to three first order conditions (plus the transversality condition): −ξ 1 θ(1 − nt) = wt (6) ct 1 1 pt = βEt (1 + it+1) (7) ct ct+1 pt+1 M −ν i 1 t+1 = t+1 (8) pt 1 + it+1 ct (6) is the standard static labor supply condition; (7) is the dynamic consumption Euler equation, with 1 + r = E (1 + i ) pt ; and (8) implicitly defines a demand for real balances: m ≡ Mt+1 . t+1 t t+1 pt+1 t pt 3 Firms As in our earlier discussion of monopolostic competition, we break production up into two sectors: final goods and intermediate goods. We discuss each in turn. 3.1 Final Goods Firm There is a single final goods firm which bundles intermediate goods into a final good available for consumption. There are a continuum of intermediate goods producers populating the unit interval; index these firms by j. The final good is a CES aggregate of these intermediate goods: " Z 1 "−1 "−1 " yt = yj;t 0 The profit maximization problem is: " Z 1 "−1 "−1 Z 1 " max pt yj;t − pj;tyj;t yj;t 0 0 The first order condition is: 2 " −1 Z 1 "−1 "−1 "−1 " " −1 pt yj;t yj;t = pj;t 0 This can be simplified to: 1 Z 1 "−1 "−1 1 " − " pj;t yj;t yj;t = 0 pt Simplifying further, and noting the definition of the final good, we get the demand function for each intermediate good j: −" pj;t yj;t = yt (9) pt The nominal value of the final good is the sum of prices times quantities of intermediates: Z 1 ptyt = pj;tyj;tdj 0 Plug in the demand curve for intermediate goods and simplify: Z 1 1−" " ptyt = pj;t pt ytdj 0 Z 1 1−" 1−" pt = pj;t dj 0 1 Z 1 1−" 1−" pt = pj;t dj (10) 0 3.2 Intermediate Goods Firms Intermediate goods use labor as the only input of production. They are affected by aggregated TFP. The production function is linear in labor input: yj;t = atnj;t (11) Given the downward sloping demand curve, firms have some market power and can set prices. It is assumed that they are not freely able to adjust prices in any period, however. In particular, firms face a constant hazard, 1 − φ, of being able to adjust their price in any period. With probability φ then pj;t = pj;t−1. This is a form of price stickiness due to Calvo (1983). Firms are nevertheless able to freely choose how much labor to use each period, whether they can adjust their price or not. Hence, let's first consider the problem of optimal labor choice for a given price. Since the firm can freely hire labor each period, we can write this as a static problem. Let's write it as the problem of cost-minimization. The reason for this is that the firm will always 3 choose labor to minimize costs regardless of the price chosen, but they are not able to choose price to maximize profits: min Wtnj;t nj;t s.t. −" pj;t atnj;t ≥ yt pt The problem is to minimize nominal costs subject to the restriction that the firm produces at least as much as is demanded at the given price. Wt is the nominal wage common to all firms. Since we take the price as given, the constraint is not a choice variable. Set the problem up as a Lagrangian: −" ! pj;t L = −Wtnj;t + 'j;t atnj;t − yt pt The first order condition is: @L = 0 , Wt = 'j;tat @nj;t 'j;t has the interpretation as (nominal) marginal cost { how much do costs go up if the firm has to produce one more unit of output. Hence the first order condition says to hire labor up until the point where the wage equals marginal cost times the marginal product of labor. Now let's consider the pricing decision. Recall that a firm can change its price in any period only with probability 1 − φ. In expectation it will be \stuck" with that price going forward into the future. Hence, the pricing problem of a firm that gets to update its price in period t is dynamic { the price it picks today will, in expectation, affect the profits it earns both today and in the future. t 0 The firm discounts future profits flows by the stochastic discount factor, Mt = β u (ct). In addition, it will discount future profit flows by φt. Profits as measured are in nominal terms; divide by pt+s to put them into real terms (since that is what the households care about). Conditional on choosing its price today, there is a φ probability of having that price in effect the next period, φ2 for the period after that, and so on. This profit maximization problem is conditional on (i) the demand function, (ii) the production function, and (iii) the first order condition for labor choice holding: 1 X 1 s max Et Mt+s φ (pj;tyj;t+s − Wt+snj;t+s) pj;t p s=0 t+s s.t. −" pj;t yj;t+s = yt+s pt+s yj;t+s = atnj;t+s Wt+s = 'j;t+sat+s 4 We can make this an unconstrained problem by substituting the constraints in to the objective function: 1 −" −" ! X s 0 1 pj;t pj;t max Et (βφ) u (ct+s) pj;t yt+s − 'j;t+s yt+s pj;t p p p s=0 t+s t+s t+s This follows because n = yj;t+s , and multiplying by W eliminates the a . Let's take j;t+s at+s t+s t+s the derivative with respect to pj;t: 1 X s 0 −" "−1 −"−1 "−1 Et (βφ) u (ct+s) (1 − ")pj;t pt+s yt+s + "pj;t 'j;t+spt+s yt+s = 0 s=0 We can simplify this by noting that we can pull the pj;t out of the sum since it does not depend on s: 1 1 −" X s 0 "−1 −"−1 X s 0 "−1 (" − 1)pj;t Et (βφ) u (ct+s) pt+s yt+s = "pj;t Et (βφ) u (ct+s) 'j;t+spt+s yt+s s=0 s=0 Simplifying further: 1 X s 0 "−1 Et (βφ) u (ct+s) 't+spt+s yt+s " p# = s=0 t " − 1 1 X s 0 "−1 Et (βφ) u (ct+s) pt+s yt+s s=0 I do two things in the final step. First, I remove the j subscript from 'j;t+s, which has the interpretation of marginal cost. From the first order condition for optimal labor choice given above, marginal cost is just the ratio of the wage to at. Since this ratio does not depend on j, marginal cost is the same for all firms. Hence, looking at the above, all firms able to update their price will choose the same price. I denote this with a # and call it the \optimal reset price". This formula essentially says that the optimal reset price is a markup over expected future marginal costs. It is easy to see that, if prices are not sticky, then φ = 0 and this reduces to a # " condition we've seen before: pt = "−1 't. With φ > 0, the pricing decision will be forward-looking. 4 Aggregate Conditions and Market-Clearing We assume the existence of a central bank that sets the money supply according to a money growth rule similar to what we've seen before: ∗ ln Mt+1 − ln Mt = (1 − ρm)π + ρm (ln Mt − ln Mt−1) + "m;t We want to write this in terms of real money balances (which will be stationary).
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