The Duel of the Dual Mandate

Transcription

1 The Duel of the Dual Mandate Jae Won Lee Yongseok Shin December 3, 23 Incomplete Note for a Paper) Abstract We study the distributional impact of monetary policy and ask how distributional considerations affect the optimal monetary policy. We introduce two ex-ante heterogenenous groups of agents workers and firm owners) into the work horse New Keynesian model Calvo pricing and flexible wages) and characterize the monetary policy that maximizes the welfare criterion of the monetary authority. In particular, we allow the monetary authority s welfare criterion to deviate from the utilitarian equal weight on each individual) welfare criterion and favor certain groups over others. We first show that the mapping between the monetary authority s welfare weights on the two groups and the optimal monetary policy depends on the forms of risk-sharing arrangements between workers and firm owners. In equilibria where no assets are traded, we find that a monetary authority that places a disproportionately high welfare weight on workers will more vigorously stabilize prices at the expense of output gap volatility. In the second part of the paper, we use these findings to reverse-engineer past monetary authorities social welfare criterion. Using the estimated policy rules of the Burns, Volcker and Greenspan eras, the model finds that the Volcker Fed placed percent more welfare weight on workers than the utilitarian criterion, while the Burns Fed placed 2 percent less welfare weight on workers. The Greenspan Fed is found to have had the almost exact utilitarian welfare criterion. Seoul National University and Rutgers University; jwlee7@snu.ac.kr. Washington University in St. Louis and Federal Reserve Bank of St. Louis; yshin@wustl.edu.

2 Households We extend the textbook New Keynesian model to consider distributional issues. As a first attempt, we consider the simplest heterogeneity: two groups of agents with complete withingroup risk sharing.. We have workers and firm owners. Both are infinitely lived and their type/occupation is fixed over time. Each group resides on a unit interval. However, the relative mass of the population is η for firm owners and η for workers. 2. All workers are identical and all firm owners are ex ante identical. 3. Workers supply labor and earn labor income. They are assumed to live hand to mouth. 4. Firm owners have no labor income but earn profits. Each firm owner owns one firm that produces a differentiated intermediate good. There is a unit measure of intermediate good firms. In what follows, we use subscripts w and e to denote workers and firm owners. The preference of the representative worker is C σ E β t w,t σ N ) +ϕ w,t, + ϕ where C w is the consumption of final good and N w is hours of work. The period budget constraint of the worker is P t C w,t = W t N w,t ) where P t is the final good price and W t is the nominal wage. Recall that the worker lives from hand to mouth. The preference of a firm owner is E β t C σ e,t σ, where C e is the consumption of final good. The firm owner s period budget constraint is P t C e,t + Q t B t B t + D t, 2) where B t is the quantity of nominal one-period bonds at a price Q t ) and D t is the dividends from firm ownership. The above sequential budget constraints come with a solvency condition of the form lim T E t {B T } for all t.

3 The first-order condition for the worker is N ϕ w,t Cw,t σ = W t, t, P t By combining ) and 3), we obtain N w,t = ) σ Wt P t ϕ+σ, Cw,t = Wt P t 3) ) +ϕ ϕ+σ. 4) The relevant first-order condition for the firm owner is { C σ } e,t+ P t Q t = βe t Ce,t σ, t. 5) P t+ Taking logs, equation 3) becomes w t p t = σc w,t + ϕn w,t, 6) where lowercase letters denote the logs of the original variables. The log-linearized version of equation 5) is c e,t = E t {c e,t+ } σ i t E t {π t+ } ρ), 7) where i t log Q t is the nominal interest rate, ρ log β is the discount rate, and π t+ log P t+ log P t is the inflation rate. 2 Firms 2. Final Good Producer There is a representative final good producer that behaves perfectly competitively. It takes the prices of the intermediate goods, P t i) for i, ], and the price of its output, P t as given. It operates a constant-returns-to-scale technology: ) Y t = X t i) di, 8) where is the elasticity of substitution and X t i) is the quantity of the intermediate input i. The first-order condition for the profit maximization problem is ) Pt i) X t i) = Y t. P t Free entry into the final good sector implies that the maximized profit is zero in the equilibrium, pinning down the price of the final good as a function of the intermediate good prices: P t Y t P t i)x t i)di = P t = 2 9) ) P t i) di. )

4 2.2 Intermediate Good Producer There is a continuum of firms indexed by i, ]. Each firm produces one differentiated good. The intermediate goods markets have the typical monopolistic competition structure. All the firms use a common technology given by Y t i) = A t N t i) α, ) where A t is the common technology level that evolves exogenously over time. All firms face identical isoelastic demand schedules given by equation 9), taking as given the price and the quantity of the final good. We follow Calvo 983) and assume that each intermediate good firm may reset its price only with probability θ in any given period. Appealing to the law of large numbers, in each period a measure θ of firms reset their prices and a fraction θ of them keeps their prices unchanged. The average duration of a price is θ). 2.3 Dynamics of Final Good Price In this economy, the final good price follows ) P Π t = θ + θ) t, 2) P t where Π t P t /P t is the gross inflation rate and Pt is the price set in period t by intermediate good firms resetting their price. All firms will choose the same price because they are solving the same problem.) Taking the log-linear approximation, we obtain π t = θ)p t p t ). 3) We now consider the intermediate firms price setting problem. 2.4 Price Setting for Intermediate Good Producers An intermediate good producer resetting its price in period t will choose P t i) to maximize max P t i) k= θ k E t { Qt,t+k P t i)y t+k t i) Ψ t+k Yt+k t i) )]}, 4) where Q t,t+k is the stochastic discount factor for nominal payoffs defined as ) σ Q t,t+k β k Ce,t+k P t, C e,t P t+k 3

5 and Ψ t ) is the cost function. We denote by Y t+k t i) the output of good i in period t + k with the price last chosen in period t. The first-order condition of equation 4) is { θ k E t Qt,t+k Y t+k t i) )} Pt i) Mψ t+k t = 5) k= where ψ t+k t Ψ t+k Y t+k ti)) is the nominal marginal cost in period t + k for the firm and M / ). Before log-linearizing 5), we rewrite it as )} P θ k E t {Q t,t+k Y t+k t i) t i) MMC t+k t Π t,t+k =, 6) P t k= where MC t+k t ψ t+k t /P t+k is the real marginal cost in period t + k for the firm. In the zero inflation steady sate, we have P t i)/p t =, Π t,t+k =, P t i) = P t+k, Y t+k t i) = Y, MC t+k t = MC, and Q t,t+k = β k, for all i, t, and k. In addition, we have MC = /M. The first-order Taylor expansion of 6) around the zero inflation steady state is { p t p t = βθ) βθ) k E t mct+k t + p t+k p t ) }, 7) k= where mc t+k t mc t+k t mc and mc = log M = µ is the steady state value of the log real marginal cost. If we rewrite 7) as p t = µ + βθ) { } βθ) k E t mct+k t + p t+k, 8) k= then we see that the firm chooses a price so that it has the desired markup over a weighted average of their current and expected nominal marginal costs. 3 Equilibrium 3. Flexible Prices With flexible prices, intermediate goods producers will set prices to be P t = W t α)a t N α t As a result, a fraction α) /) of the revenue goes to labor and the rest goes to firm owners. In the equilibrium, we have C w,t = α η Y t = χ w Y t 9) 4

6 C e,t = η α) ] Y t = χ e Y t. 2) Note that η)χ w + ηχ e = and N t = η)n w,t. Using the first-order condition of the worker equation 4) gives N t = η) ϕ+σ σ α) A t ) σ ϕ+σ+α σ) and Y t = A +ϕ ϕ+σ+α σ) t η) ϕ+σ σ α) ) σ) α) ϕ+σ+α σ). Taking logs, we have yt n + ϕ = ϕ + σ + α σ) a σ) α) t µ log α) ϕ + σ ) log η), ϕ + σ + α σ) σ 2) which shows that the natural or flexible-price output of our model is different from the natural output of the canonical New Keynesian model only in terms of the intercept. We can also derive the real wage, W t P t = η) αϕ+σ) ϕ+σ+α σ) 3.2 Efficient Allocation α) ) ϕ+σ ϕ+σ+α σ) A t. 22) We contrast the flexible price equilibrium with the efficient allocation. solution to the following planner s problem: max C w,t,c e,t,n ti) η)λ C σ w,t σ η) N ti)di + ϕ ) +ϕ + η C σ e,t σ, The latter is the subject to the resource constraint: η)c w,t + ηc e,t = ) At N t i) α] di. Here, λ is the relative welfare weight on the workers. The associated optimality conditions are: λc σ w,t = ζ 5

7 C σ e,t λ = ζ η) ϕ N t i)di] = ζ where ζ is a multiplier. The last condition is simplified to ) At N t i) α] At di N t i) α] α) A t N t i) α λn ϕ w,t = ζ α) Y t Y t i) Nt i) 23) which implies that the efficient outcome is symmetric across i, ]: Y t i) = Y t N t i) = N t. We substitute out the multiplier ζ from 23) to obtain and MP N t Y t Cw,tN σ ϕ w,t = α). }{{} N t MRS t }{{} Compare the above to the corresponding equation in the flexible price equilibrium MRS t = M MP N t. To obtain the efficient level of output and consumption, work with the resource constraint to obtain: Y t = η)c w,t + ηc e,t = η)c w,t + η This implies where C w,t = χ w Y t and C e,t = χ e Y t, χ w = { η) + η ) σ Cw,t. λ ) } σ and χ e = η) χ w. λ η ) Note that χ w = χ w and χ e = χ e with λ χ σ. = w χ e We now compute the efficient level of output as follows: C σ w,tn ϕ w,t = α) Y t = χ σ wy σ t = χ σ wy σ t = Y e t N t ) ϕ N +ϕ t η ) ϕ Yt = A η +ϕ σ+ϕ+α σ) t A t = α) Y t ) +ϕ α = α) Yt χ σ w η) ϕ α) ] α σ+ϕ+α σ). 6

8 Note that the natural level of output is proportional to the efficient level. If χ w = χ w, then Y e t = A +ϕ σ+ϕ+α σ) t Taking logs, we have y e t = y n t Sticky Prices η) σ+ϕ σ) α) ] σ) α) σ+ϕ+α σ) M σ α) σ+ϕ+α σ). α) σ + ϕ + α σ) µ The final good market clearing requires η)c w,t + ηc e,t = Y t. For all i, ], the intermediate goods market clearing condition is X t i) = Y t i). Labor market clearing condition is η)n w,t = N t which can be rewritten as ) Yt i) α Yt N t = di = A t Taking logs, we obtain N t i)di, 24) A t ) α Pt i) P t ) α di. 25) α)n t = y t a t + d t 26) ) ) Pt i) α d t α) log di. P t In a neighborhood of the zero inflation steady state, d t is equal to zero up to a first-order approximation. We then write the following approximate relationship between aggregate output, employment and technology as y t = a t + α)n t. 27) We now derive an individual intermediate good producer s marginal cost in terms of the economy s average real marginal cost. The latter is defined in log as mc t = w t p t ) a t αn t ) log α). 28) Using equation 27), we obtain mc t = w t p t ) α a t αy t ) log α) 29) 7

9 Analogously, mc t+k t = w t+k p t+k ) α a t+k αy t+k t ) log α). 3) We obtain mc t+k t mc t+k = Substituting 3) into 7), we obtain p t p t = βθ)θ α α y t+k t y t+k ) = α α p t p t+k ) 3) βθ) k E t { mc t+k } + k= βθ) k E t {π t+k }, 32) k= where Θ α)/ α + α). difference equation This equation can be rewritten as the following where p t p t = βθe t {p t+ p t } + βθ)θ mc t + π t. 33) Combining 3) and 33), we obtain the inflation equation π t = βe t {π t+ } + ς mc t, 34) ς θ) βθ) Θ θ Solving 34) forward, inflation is expressed as the discounted sum of current and future deviations of real marginal costs from the steady state π t = ς β k E t { mc t+k }. 35) k= We now establish a relationship between the economy s real marginal cost and aggregate economic activity. Since the average marginal cost is mc t = w t p t ) mpn t, we use 4) and 27) to obtain mc t = ϕ + α + σ α) α) σ) y + ϕ t α) σ) a t log α) ϕ + σ log η). 36) σ Subtracting 2) from 36), we obtain mc t = ϕ + α + σ α) α) σ) y t y n t ), 37) where y t y n t ỹ t is the output gap. 8

10 Combining 34) and 37), we obtain the New Keynesian Phillips curve, π t = βe t {π t+ } + κỹ t, 38) where κ ςϕ + α + σ α))/ α)/ σ). We now log-linearize the final good market clearing condition, η)c w,t + ηc e,t = Y t, as follows y t = η)χ w c w,t + ηχ e c e,t + hot, 39) which implies y t η)χ w c w,t = c e,t + hot. ηχ e ηχ e Plug 39) into the firm owner s Euler equation to obtain y t η)χ { w c w,t = E t y t+ η)χ } w c w,t+ ηχ e ηχ e ηχ e ηχ e σ i t E t {π t+ }) + constant/hot. 4) From the worker s optimality condition and production function, we have assuming σ ) c w,t = + ϕ σ n w,t = + ϕ σ n t + constant. Using this relation, we rewrite 4) as: where y t = E t {y t+ } Θ n E t { n t+ } Θ r i t E t {π t+ }) + constant/hot 4) = Θ n n t Θ r E t {i t+j π t+j+ } + constant/hot, j= Θ n η)χ w + ϕ σ and Θ r ηχ e σ. Some features of this equilibrium condition a version of the dynamic IS equation) are worth mentioning. First, there is a direct effect of employment on output. The elasticity, Θ n, perhaps not surprisingly, depends on the relative importance of workers η and χ w ) in the economy. To the extent that the substitution effect dominates the income effect i.e. σ < ), employment has a positive effect on worker s consumption, increasing aggregate demand. Second, as is well known, aggregate demand depends negatively on the real interest rate due to the presence of households that optimize intertemporally firm owners in our model). The elasticity, Θ r, depends on the relative importance of firm owners η and χ e ). 9

11 Although it is not necessary, we can derive an alternative representation of 4) using the production function as follows: y t = E t {y t+ } Θ Θ n r α Θ n ) i t E t {π t+ }) + α Θ n )E t { a t+ } + constant/hot. α 42) According to 42), in equilibrium, the overall effect of real interest rate on output is ambiguous. In fact, if Θ n is sufficiently large, the model may generate a positive effect of real rate on output. 4 Welfare Function We use the following notations for the agents utility functions: V N w,t ) N +ϕ w,t + ϕ and UC j) C σ j,t σ for j = {w, e}. We follow Woodford 23) and take a second order Taylor expansion of the utility function. We first obtain UC j,t ) = U C j ) + U jc j,t C j ) + 2 U j C j,t C j ) 2 + tip + O ξ 3) 43) where O ξ 3 ) represents all relevant terms that are of third or higher order and tip denotes all the terms independent of monetary policy. We also take a second order Taylor expansion of C j,t to obtain C j,t C j = C j ĉ j,t + 2 Cĉ 2 j,t + O ξ 3), 44) where ĉ j,t log C j,t log C j. Substituting 44) into 43) gives UC j,t ) = U C j ) + U j C j ĉ j,t + 2 U j C j ĉ 2 j,t + 2 U j C 2 j ĉ 2 j,t + tip + O ξ 3). 45) Because U C j ) is independent of monetary policy, we rewrite 45) as { UC j,t ) = U j C j ĉ j,t + 2ĉ2 j,t + U C } j j ĉ 2 2 U j j,t + tip + O ξ 3) 46) and then as UC j,t ) = U j C j {ĉ j,t + } 2 σ)ĉ2 j,t + tip + O ξ 3).

12 We now take a second order Taylor expansion of V N w,t ): V N w,t ) = V N w ) + V N w,t N w ) + 2 V N w,t N w ) 2 + tip + O ξ 3). 47) With ˆn w,t log N w,t log N w, the second order approximation of N w,t is: N w,t N w = N wˆn w,t + 2 N wˆn 2 w,t + O ξ 3). 48) Substituting 48) into 47) gives V N w,t ) = V Nw { ˆn w,t + 2 ˆn2 w,t + 2 } V Nw ˆn 2 V w,t + tip + O ξ 3) 49) and eventually V N w,t ) = V Nw {ˆn w,t + } 2 + ϕ)ˆn2 w,t + tip + O ξ 3). 5) Recall that η)n w,t = Taking logs, we obtain N t i)di = Yt i) A t ) α di = Yt α)n w,t = y t a t + d t α) log η). Alternatively, α)ˆn w,t = ŷ t â t + d t. A t ) α Pt i) P t ) α di. We then use the well-known proposition that d t = 2Θ V ar i p t i)] + O ξ 3 ) and rewrite 5) as V N w,t ) = V { Nw ŷ t â t + α 2Θ V ar i p t i)] + 2 ) } + ϕ ŷ t â t ) 2 + tip + O ξ 3). α The weighted sum of each type s utility can be written as: C σ w,t u t λ η) σ N ] +ϕ w,t + η C σ e,t + ϕ σ } { + ηu e C e } 5) = ū + λ η) U w C w {ĉ w,t + 2 σ)ĉ2 w,t λ η) V Nw α { ŷ t â t + 2Θ V ar i p t i)] + 2 ĉ e,t + 2 σ)ĉ2 e,t ) } ŷ t â t ) 2 + tip + O ξ 3) + ϕ α

13 In the steady state, we have V U w = M MP N = Ȳ α) M N = = V Nw = η V N = ) α η Ȳ U w. ) α) Ȳ N 52) Note that / represents the wedge between the marginal product of labor and the marginal rate of substitution between consumption and hours in the steady state; in other words, it measures the size of the steady state distortion due to imperfect competition. In addition, we have U w U e = χ σ w χ σ e = U e C e = U wλ χ e Ȳ and U w C w = U wχ w Ȳ. 53) 4. Case with λ = λ We first consider the case with λ = λ ; in other words, the monetary authority assigns the same relative welfare weight to workers that a social planner would assign if he were to implement the market outcome without nominal rigidities. Using 52) and 53), we rewrite 5) as: u t ū U wȳ = λ η) χ w {ĉ w,t + } { 2 σ)ĉ2 w,t + λ ηχ e ĉ e,t + } 2 σ)ĉ2 e,t 54) λ ) { ŷ t â t + 2Θ V ar i p t i)] + ) } + ϕ ŷ t â t ) 2 + tip + O ξ 3). 2 α Taking a second-order approximation of the resource constraint yields: ŷ t + 2ŷ2 t = η) χ w ĉ w,t + ηχ e ĉ e,t + 2 Equation 54) thus can be written as: u t ū U wȳ ) η) χw ĉ 2 w,t + ηχ e ĉ ) 2 e,t + tip + O ξ 3. = λ {ŷ t + 2ŷ2 t 2 σ η) χ wĉ 2 w,t 2 σηχ eĉ 2 e,t ŷ t + â t 2Θ V ar i p t i)] ) } + ϕ ŷ t â t ) 2 2 α + λ { ŷ t â t + 2Θ V ar i p t i)] + ) } + ϕ ŷ t â t ) 2 + tip + O ξ 3). 2 α Under the small steady-state distortion assumption so that the product of / with a second order term can be treated as a third-order term), u t ū U wȳ = λ 2 { σ η) χ w ĉ 2 w,t + σηχ e ĉ 2 e,t ŷ 2 t + Θ V ar i p t i)] + 2 ) } + ϕ ŷ t â t ) 2 α

14 + λ {ŷ t + )â t } + tip + O ξ 3). 55) Note that η) χ w ĉ 2 w,t + ηχ e ĉ 2 e,t = η) χ w ĉ w,t + ηχ e ĉ e,t ] 2 + η) ηχ w χ e ĉ w,t ĉ e,t ) 2 Plug 56) into 55) to obtain u t ū U wȳ = λ 2 = λ 2 = ŷ 2 t + η) ηχ w χ e ĉ w,t ĉ e,t ) 2. 56) { σ )ŷ 2 t + σ η) ηχ w χ e ĉ w,t ĉ e,t ) 2 + Θ V ar i p t i)] + + λ {ŷ t + )â t } + tip + O ξ 3) { σ + ϕ + α α + λ + tip + O ξ 3) { ŷt σ + ϕ + α ) α = λ 2 + λ ŷt + tip + O ξ 3), where the last equality follows from ) + ϕ ŷt e = σ + ϕ + α ) â t = α α ) ŷ 2 t + σ η) ηχ w χ e ĉ w,t ĉ e,t ) 2 + Θ V ar i p t i)] 2 ) } + ϕ ŷ t â t ) 2 α ŷ t ŷt e ) 2 + σ η) ηχ w χ e ĉ w,t ĉ e,t ) 2 + Θ V ar i p t i)] + ϕ ϕ + σ + α σ)ât. Therefore, the welfare of the economy as a fraction of steady-state output up to a second-order approximation) is ) W = E β t ut ū U wȳ = λ 2 E Note that β t V ar i p t i)] = We therefore have where W = λ 2 E β t { Θ V ar i p t i)] + θ θ) θβ) ) } + ϕ ŷ t â t α σ + ϕ + α ) ŷ t ŷt e ) 2 + σ η) ηχ w χ e ĉ w,t ĉ e,t ) 2 2 } α ŷt β t πt 2 + tip + O ξ 3) β {φ t π πt 2 + φ y ŷ t ŷt e ) 2 + φ c ĉ w,t ĉ e,t ) 2 2 } ŷt φ π =, φ y = σ + ϕ + α ς α, and φ c = σ η) ηχ w χ e. 3 57) }

15 4.2 Case with λ λ Let λ = λ + λ w >. With λ w, the weighted utility will have the following terms appended to equation 54). } λ w η) χ w {ĉ w,t + 2 σ)ĉ2 w,t λ w ) { ŷ t â t + 2Θ V ar i p t i)] + 2 ) } + ϕ ŷ t â t ) 2 α 58) We further use the following relations: ˆn t = σ + ϕĉw,t, 59) α)ˆn t = ŷ t â t + d t. 6) In particular, α) σ) ŷ t â t = ĉ w,t + ϕ 2Θ V ar i p t i)] + O ξ 3), 6) ŷ t â t ) 2 = α)2 σ) 2 ĉ 2 + ϕ) 2 w,t + O ξ 3). 62) We then rewrite 58) as λ w α) ) ϕ + σ + ϕ { ĉ w,t + } 2 σ)ĉ2 w,t 63) This implies that, with λ w, W = λ 2 E { β t φ π πt 2 + φ y ŷ t ŷt e ) 2 + φ c ĉ w,t ĉ e,t ) 2 + φ w ĉ 2 w,t 2 2φ } w, ŷt σ ĉw,t where φ w = λ w σ ) α) ) σ + ϕ λ + ϕ. where We can equivalently write W as, suppressing the additional constant term, W = λ 2 E c w = σ. β t { φ π π 2 t + φ y ŷ t ŷ e t ) 2 2 ŷt + φ c ĉ w,t ĉ e,t ) 2 + φ w ĉ w,t c w) 2 }, 4

16 5 Optimal Monetary Policy The central bank chooses {π t, ŷ t, ĉ w,t, ĉ e,t } to maximize W subject to π t = βe t {π t+ } + κ ŷ t ŷt n ), ĉ w,t = + ϕ α) σ) ŷ t â t ), ηχ e ĉ e,t = ŷ t η)χ w ĉ w,t. The other variables will be determined as residuals for a given optimal time path of {π t, ŷ t, ĉ w,t, ĉ e,t }. Write the Lagrangian as { L = λ 2 E β t φ π πt 2 + φ y ŷ t ŷt e ) 2 + φ c ĉ w,t ĉ e,t ) 2 + φ w ĉ 2 w,t 2 2φ } w ŷt σ ĉw,t +λ E β t q,t {π t βπ t+ κ ŷ t ŷt n )} } +λ E β t + ϕ q 2,t {ĉ w,t α) σ) ŷ t â t ) +λ E β t q 3,t { η)χ w ĉ w,t + ηχ e ĉ e,t ŷ t }, where {q,t }, {q 2,t } and {q 3,t } are the sequences of scaled) multipliers. First order conditions are given as: = φ π π t + q,t q,t = φ y ŷ t ŷ e t ) κq,t + ϕ α) σ) q 2,t q 3,t = φ c ĉ w,t ĉ e,t ) + φ w ĉ w,t φ w σ + q 2,t + η)χ w q 3,t = φ c ĉ w,t ĉ e,t ) + ηχ e q 3,t Simplify the conditions as φ π π t = q,t q,t ) q,t = φ y κ ŷ t ŷt e ) + where + ϕ α) σ) κ ςϕ + α + σ α))/ α)/ σ) φ w = λ w σ ) α) ) σ + ϕ λ + ϕ ] 2 φ w κ ŷ t â t ) + κ 5 + ϕ α) σ) ] φc ηχ e ĉ w,t ĉ e,t ) + Γ

17 κγ = + ϕ) φ w α) σ) 2 It is perhaps instructive to consider the discretionary case first. This can be obtained by setting q,t =. Furthermore, consider the special case where φ c = φ w =. This corresponds to the basic NK model. Then combining the two FOCs yields φ π π t = φ y κ ŷ t ŷ e t ) Γ. This is the standard targeting rule : the central bank adjusts its instrument i.e. nominal rate) until the two target variables, π t and ŷ t, satisfy the condition. We use the following relations. the η)χ w ĉ w,t + ηχ e ĉ e,t = ŷ t 64) + ϕ ĉ w,t = σ) α) ŷ t â t ) 65) Because η)χ w + ηχ e =, we subtract 64) from 65) to obtain ĉ w,t ĉ e,t = ] + ϕ ηχ e σ) α) ŷ t â t ) ŷ t We now use ŷ e t = and obtain and 66) + ϕ ϕ + σ + α σ)ât 67) ĉ w,t ĉ e,t = ηχ e κ ς ŷ t ŷ e t ) 68) ŷ t â t = ŷ t ŷ e t + α) σ) ϕ + σ + α σ)ât. 69) Replacing ĉ w,t ĉ e,t and ŷ t â t using 68) and 69), we rewrite q,t as follows. q,t = φ y κ + + ϕ α) σ) ] 2 φ w κ + κφ c ςηχ e ) 2 Alternatively, we write 7) as ) φy q,t = κ + Λ w + Λ c ŷ t ŷt e ) + Λ a â t + Γ 6 ) ŷ t ŷ e t ) + + ϕ α) σ) ] 2 ςφ w κ 2 ât + Γ 7)

18 Λ w + ϕ α) σ) Λ c κφ c ςηχ e ) 2 + ϕ Λ a α) σ) ] 2 φ w κ ] 2 ςφ w κ 2. Then the targeting rule under discretion is ) φy φ π π t = κ + Λ w + Λ c ŷ t ŷt e ) Λ a â t Γ. 7) }{{} Φ y Note that Λ w = Λ a = if λ w =. In that case, as in the textbook NK model, there is no policy tradeoff between inflation and the output gap. This property is often referred to as the divine coincidence Blanchard and Gali, 25). Assuming that κ is positive i.e., σ < ), Λ c is always positive but Λ w and Λ a inherit the sign of φ w. Finally, if λ w, there is a policy tradeoff, even though technology shock is the only shock in the economy. 5. Policy Tradeoff and Near-Divine Coincidence We solve for π t and ŷ t ŷ e t. We assume that â t is an AR) process with ρ a as the persistence parameter. First, substitute the targeting rule 7) into the NKPC to obtain Φy ŷ t ŷt e ) + Λ a â t + Γ ] { Φy = βe t ŷ t+ ŷ φ π φ π φ π φ t+) e + Λ a â t+ + Γ } + κŷ t ŷt e ), π φ π φ π which implies ŷ t ŷ e t = βφ y } E t {ŷt+ ŷ e Λ a βρ a ) t+ κφ π + Φ y κφ π + Φ y â t β) Γ κφ π + Φ y. Assuming κ is positive, we iterate the previous equation forward to obtain ŷ t ŷt e = Λ a βρ a ) β) Γ â t. 72) κφ π + βρ a )Φ y κφ π + β) Φ y Consequently, inflation is obtained as κλ a κγ π t = â t. 73) κφ π + βρ a )Φ y κφ π + β) Φ y The equations above, 72) and 73), reveal a somewhat surprising result: a change in Φ y and/or φ π ) has a symmetric effect on the response of inflation and that of output gap to 7

19 â t. This implies that a shift in the central bank preference towards one type i.e. a change in λ w ) has no effect on the relative volatility of inflation and output gap: sd π t ) sd ŷ t ŷ e t ) = κ and βρ a sdπ t) sdŷ t ŷ e t ) λ w =. 74) We thus have a near-divine coincidence result: While the central bank cannot achieve full stabilization of the output gap and inflation simultaneously because of the presence of â t, this shock does not create a meaningful tradeoff since 74) always holds regardless of the central bank preference. We can determine the equilibrium interest rate under optimal discretionary policy by plugging 72) and 73) into the IS equation. where î t = ˆr e t + Φπ t + Γ i, Φ = ρ a ρ a) βρ a ) κ Γ ρ a ) Γ i = κφ π + β)φ y α Θ n α)θ r, α Θ n) βρ a ) α)θ r κ and ˆr e t is the efficient real interest rate that would prevail if ŷ t ŷ e t =. Note that Φ does not depend on the central bank preference parameters. This once again represents our divine coincidence result. ), 5.2 Policy Tradeoff with Real Imperfections Following the literature, we here allow for short-run discrepancies between the natural and the efficient levels of output due to some real imperfections: û t κ ŷ e t ŷ n t ). While we do not take a stand on the exact source of such deviations at least for now), as Woodford 23) discussed, û t may arise due to the time-varying market power of the supplier of each differentiated good and the existence of distorting taxes on output, consumption, employment, or wage income. As before, we substitute the targeting rule 7) into the NKPC Φy ŷ t ŷt e ) + Λ a â t + Γ ] { Φy = βe t ŷ t+ ŷ φ π φ π φ π φ t+) e + Λ a â t+ + Γ } π φ π φ π which implies ŷ t ŷ e t = βφ y } E t {ŷt+ ŷ e Λ a βρ a ) t+ κφ π + Φ y κφ π + Φ y 8 â t φ π β) Γ û t. κφ π + Φ y κφ π + Φ y + κŷ t ŷ e t ) + û t,

20 Assuming κ is positive and û t is an AR) process with a persistence parameter ρ u, we iterate forward to obtain ŷ t ŷ e t = Λ a βρ a ) κφ π + βρ a )Φ y â t Consequently, inflation is obtained as κλ a π t = â t + κφ π + βρ a )Φ y The relative standard deviation is sdπ t ) sdŷ t ŷ e t ) = κ Λ a κφ π+ βρ a)φ y σ a + Λ a βρ a) κφ π+ βρ a)φ y σ a + φ π κφ π + βρ u )Φ y û t Φ y κφ π + βρ u )Φ y û t Φ y κφ π+ βρ u)φ y σ u φ π κφ π+ βρ u)φ y σ u β) Γ κφ π + β) Φ y. 75) κγ κφ π + β) Φ y. 76) It can be shown that the relative volatility is decreasing in λ w in the relevant parameter space. The central bank, with a larger weight on workers, will stabilize inflation more relative to the output gap. We can determine the equilibrium interest rate under optimal discretionary policy by plugging 75) and 76) into the IS equation. After some derivation, we obtain ) î t = ˆr t e + Φ a Φ y ŷ t ŷt e κ ) + Φ a φ π π t Φ u û t + Γ Φ a, 77) κφ π + β)φ y where Φ a = βρ a )ρ a ) α Θ ] n + κρ a κφ π + βρ a )Φ y α)θ r Φ u = φ π ρ u ) α Θ ] n Φ y ρ u κφ π + βρ u )Φ y α)θ r In the relevant parameter spaceφ a is positive. Also, note that if ρ a = ρ u =, then Φ u = Φ a φ π. It can be shown, again, in the relevant parameter space, that Φ a Φ y is decreasing in λ w but Φ a φ π is increasing in λ w. References Blanchard, O. and J. Gali 25): Real Wage Rigidities and the New Keynesian Model, Working Paper 86, National Bureau of Economic Research. Calvo, G. A. 983): Staggered Prices in a Utility-Maximizing Framework, Journal of Monetary Economics, 2, Woodford, M. 23): Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton, New Jersey: Princeton University Press. 9