TABLE IX Open Economy Relative Multiplier in Models with Variable Capital Output CPI Inflation

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1 Erratum on: Fiscal Stimulus in a Monetary Union: Evidence from U.S. Regions by Emi Nakamura and Jón Steinsson American Economic Review, 1043, , March There were errors in lines 303, 337, and 338 of Matlab file runhetcapitalghh.m, which produces some of the results in Table 9 of our paper. Correcting these errors slightly changes the open economy relative multiplier on output and inflation for the firm specific capital model. When prices are sticky, the output multiplier rises slightly from 1.47 to The interpretation of these robustness results in the paper is unaffected by this small change. We have reproduced an updated version of Table 9 below with the published version also included for reference. We would like to thank Bopjun Gwak and Paul Reimers, Ph.D. students at Goethe University in Frankfurt for finding these errors. TABLE IX Open Economy Relative Multiplier in Models with Variable Capital Output CPI Inflation Revised Version: Baseline Model Fixed Capital Firm-Specific Capital Model Regional Capital Market Model Firm-Specific Capital Model, Flexible Prices Published Version: Baseline Model Fixed Capital Firm-Specific Capital Model Regional Capital Market Model Firm-Specific Capital Model, Flexible Prices The table reports the open economy relative government spending multiplier for output and CPI inflation for our baseline model with GHH preferences and the two models with variable capital also with GHH preferences. Output is deflated by the regional CPI.

2 For Online Publication Appendix to: Fiscal Stimulus in a Monetary Union: Evidence from U.S. Regions Emi Nakamura and Jón Steinsson Columbia University September 2, 2013 A Constant Real and Nominal Rate Monetary Policies The paper considers specifications of monetary policy that hold the real or nominal interest rate constant in response to a government spending shock. Here, we illustrate the method used to solve for these monetary policy specifications. We do this for the case of separable preference. We use an analogous approach for the case of GHH preference. Consider the closed economy limit of our model. A log-linear approximation of the key equilibrium conditions of this model are where ζ = 1/1 + ψ ν θ and ψ ν = 1 + ν 1 /a 1. ĉ t = E t ĉ t+1 σˆr n t E tˆπ t+1, 1 ˆπ t = βe tˆπ t+1 + κζσ 1 ĉ t + κζψ ν ŷ t, 2 C ŷ t = ĉ t + ĝ t, 3 Using equation 3 to eliminate ŷ t from equations 1 and 2 yields ĉ t = E t ĉ t+1 σˆr n t E tˆπ t+1, 4 ˆπ t = βe tˆπ t+1 + κζ c ĉ t + κζ g ĝ t, 5 where ζ y = ζσ 1 + C/ ψ ν and ζ g = ζψ ν. Recall that ĝ t = ρ g ĝ t 1 + ɛ g,t. 1

3 A.1 Fixed Real Rate Monetary Policy An equilibrium with a fixed real interest rate must satisfy ĉ t = E t ĉ t+1, 6 ˆπ t = βe tˆπ t+1 + κζ c ĉ t + κζ g ĝ t. 7 We conjecture a solution of the form ĉ t = a c ĝ t, ˆπ t = a π ĝ t. Using the method of undetermined coefficients, it is easy to verify that such an equilibrium exists with a c = 0, a π = κ ζg 1 βρ g. This equilibrium can be implemented with the following policy rule ˆr n t = E tˆπ t+1 + φ π ˆπ t ˆπ t = a π ρ g ĝ t + φ πˆπ t a π φ π ĝ t = φ πˆπ t a π φ π ρ g ĝ t. 8 A.2 Fixed Nominal Rate Monetary Policy An equilibrium with a fixed nominal interest rate must satisfy ĉ t = E t ĉ t+1 + σe tˆπ t+1, 9 ˆπ t = βe tˆπ t+1 + κζ c ĉ t + κζ g ĝ t. 10 We again conjecture a solution of the form ĉ t = a c ĝ t, ˆπ t = a π ĝ t. Using the method of undetermined coefficients, it is easy to verify that such an equilibrium exists with a c = ρgκζg A c, a π = κ ζc 1 βρ g a c + κ ζg 1 βρ g. where A c = 1 ρ g 1 βρ g ρ g κζ c. This solution is however only valid when A c > 0. For 0 < ρ g < 1, A c is decreasing in ρ g. There is a critical point at which A c = 0. As ρ g rises and A c falls towards zero, a c. For values of ρ g that are above this point, our solution method breaks down since a c is infinite. Similar parameter restrictions arise in Eggertsson 2010 and Christiano, Eichenbaum, and Rebelo In the valid range, this equilibrium can be implemented with the following policy rule ˆr n t = φ π ˆπ t ˆπ t = φ πˆπ t a π φ π ĝ t. 11 2

4 B Persistence of the Government Spending Shocks We use annual data on aggregate military procurement spending to calibrate the persistence of the government spending shocks that we feed into our model. The first order autocorrelation of aggregate military procurement spending suggests a great deal of persistence. However, higher order autocorrelations suggest less persistence. To capture this behavior in a parsimonious way, we estimate a quarterly AR1 process by simulated method of moments using the first five autocorrelations as moments. More specifically, our procedure is the following. First, we take the log of aggregate military procurement spending and detrend it results are insensitive to this. We then estimate regressions of the following form: G agg t = α j + β j G agg t j + ɛ t, where G agg t is detrended log aggregate military procurement spending. We use β j for j = 1,..., 5 as the moments in our simulated methods of moments estimation. We then simulate quarterly data from AR1 processes with different degrees of persistence, time aggregate these data to an annual frequency, and run these same regressions on the simulated data. We then find the value of the quarterly AR1 parameter that minimizes the sum of the squared deviations of the empirical moments from the simulated moments. C Linear Approximation of Equation 1 for Model Regressions To calculate the open economy relative multiplier for simulated data from our model, we must take a linear approximation of the dependent and independent variables in equation 1 so as to be able to express the variables in the regression in terms of the variables in our model. For the output regression, we approximate the specification in which we deflate regional GDP by the regional CPI The second specification in Table 2. The linear approximation of the dependent variable is P Ht t P Ht 2 t 2 2 P Ht 2 t 2 = t Π Ht Π Ht 1 1 = ŷ t ŷ t 2 + ˆπ Ht + ˆπ Ht 1 ˆπ t ˆπ t 1 + h.o.t, 12 t 2 Π t Π t 1 2 3

5 where h.o.t. denotes higher order terms. The linear approximation of the independent variable is P Ht G Ht P W t P Ht 2G Ht 2 P W t 2 P Ht 2 t 2 P W t 2 = G Ht Π Ht Π Ht 1 G Ht 2 t 2 Π W t = ĝ t ĝ t 2 + Π W t 1 1 C t 2 ˆπHt + ˆπ Ht 1 ˆπ W t ˆπ W t 1 + h.o.t. 13 D Separable Preferences Model The model consists of two regions that belong to a monetary and fiscal union. We refer to the regions as home and foreign. Think of the home region as the region in which the government spending shock occurs a U.S. state or small group of states and the foreign region as the rest of the economy. The population of the entire economy is normalized to one. The population of the home region is denoted by n. D.1 Households The home region has a continuum of household types indexed by x. A household s type indicates the type of labor supplied by that household. Home households of type x seek to maximize their utility given by E 0 β t uc t, L t x, 14 t=0 where β denotes the household s subjective discount factor, C t denotes household consumption of a composite consumption good, L t x denotes household supply of differentiated labor input x. There are an equal large number of households of each type. The period utility function takes the form uc t, L t x = C1 σ 1 t 1 σ 1 χl tx 1+ν ν The composite consumption good in expression 14 is an index given by [ C t = φ 1 η 1 η H C η Ht 1 η + φ η 1 F C η F t ] η η 1, 16 where C Ht and C F t denote the consumption of composites of home and foreign produced goods, respectively. The parameter η > 0 denotes the elasticity of substitution between home and foreign goods and φ H and φ F are preference parameters that determine the household s relative preference 4

6 for home and foreign goods. It is analytically convenient to normalize φ H + φ F = 1. If φ H > n, household preferences are biased toward home produced goods. The subindices, C Ht and C F t, are given by [ 1 C Ht = 0 c htz θ 1 θ ] θ θ 1 dz [ 1 and C F t = 0 c ftz θ 1 θ ] θ dz θ 1 17 where c ht z and c ft z denote consumption of variety z of home and foreign produced goods, respectively. There is a continuum of measure one of varieties in each region. The parameter θ > 1 denotes the elasticity of substitution between different varieties. Goods markets are completely integrated across regions. Home and foreign households thus face the same prices for each of the differentiated goods produced in the economy. We denote these prices by p ht z for home produced goods and p ft z for foreign produced goods. All prices are denominated in a common currency called dollars. Households have access to complete financial markets. There are no impediments to trade in financial securities across regions. Home households of type x face a flow budget constraint given by C t + E t [M t,t+1 B t+1 x] B t x + 1 τ t W t xl t x Ξ ht zdz T t, 18 where is a price index that gives the minimum price of a unit of the consumption good C t, B t+1 x is a random variable that denotes the state contingent payoff of the portfolio of financial securities held by households of type x at the beginning of period t + 1, M t,t+1 is the stochastic discount factor that prices these payoffs in period t, τ t denotes a labor income tax levied by the government in period t, W t x denotes the wage rate received by home households of type x in period t, Ξ ht z is the profit of home firm z in period t and T t denotes lump sum taxes. 1 To rule out Ponzi schemes, household debt cannot exceed the present value of future income in any state of the world. Households face a decision in each period about how much to spend on consumption, how many hours of labor to supply, how much to consume of each differentiated good produced in the economy and what portfolio of assets to purchase. Optimal choice regarding the trade-off between current consumption and consumption in different states in the future yields the following consumption Euler equation: σ 1 Ct+j C t = M t,t+j β j +j 19 1 The stochastic discount factor M t,t+1 is a random variable over states in period t + 1. For each such state it equals the price of the Arrow-Debreu asset that pays off in that state divided by the conditional probability of that state. See Cochrane 2005 for a detailed discussion. 5

7 as well as a standard transversality condition. Equation 19 holds state-by-state for all j > 0. Optimal choice regarding the intratemporal trade-off between current consumption and current labor supply yields a labor supply equation: χl t x ν 1 C σ 1 t = 1 τ t W tx. 20 Households optimally choose to minimize the cost of attaining the level of consumption C t. This implies the following demand curves for home and foreign goods and for each of the differentiated products produced in the economy: C H,t = φ H C t PHt η and C F,t = φ F C t PF t η, 21 c ht z = C Ht pht z P Ht θ and c ft z = C F t pft z P F t θ, 22 where P Ht = [ ] p htz 1 θ 1 θ dz and P F t = [ ] p ftz 1 θ 1 θ dz, 23 and = [ φ H P 1 η Ht ] + φ F P 1 η 1 1 η F t. 24 The problem of the foreign household is largely analogous and we therefore refrain from describing all details for this problem. We use superscript to denote foreign variables. The utility function of the foreign households is analogous to that of home households except that the foreign conposite consumption index is [ Ct = φ 1 η H η 1 C η Ht + φ 1 η F ] η η 1 C η F t η 1, 25 where CHt is a composite index of foreign consumption of home goods and C F t is a composite index of foreign consumption of foreign goods. These indexes are given by analogous expressions to those in equation 17. We assume for convenience that φ H + φ F = 1. The foreign budget constraint is analogous to that of home households. This implies that the foreign Euler equation and labor supply equations are analogous to those of the home household. Foreign demand for home and foreign goods and for each of the differentiated products produced in the economy is given by C H,t = φ H C t PHt P t η and C F,t = φ F C t PF t P t η, 26 c ht z = C Ht θ pht z P Ht and c ft z = C pft z θ F t P F t, 27 6

8 where P t = [ φ HP 1 η Ht ] + φ F P 1 η 1 1 η F t. 28 It is useful to note that combining the home and foreign consumption Euler equations to eliminate the common stochastic discount factor yields C σ 1 t C t = Q t, 29 where Q t = P t / is the real exchange rate. This is the Backus-Smith condition that describes optimal risk-sharing between home and foreign households Backus and Smith, For simplicity, we assume that all households in both regions initially have an equal amount of financial wealth. D.2 The Government The economy has a federal government that conducts fiscal and monetary policy. Total government spending in the home and foreign region follow exogenous AR1 processes. Let G Ht denote government spending per capita in the home region. Total government spending in the home region is then ng Ht. For simplicity, we assume that government demand for the differentiated products produced in each region takes the same CES form as private demand. In other words, we assume that g ht z = G Ht pht z P Ht θ and g ft z = G F t pft z P F t θ. 30 The government levies both labor income and lump-sum taxes to pay for its purchases of goods. Our assumption of perfect financial markets implies that any risk associated with variation in lump-sum taxes and transfers across the two regions is undone through risk-sharing. Ricardian equivalence holds in our model. We consider two specifications for tax policy. Our baseline tax policy is one in which government spending shocks are financed completely by lump sum taxes. Under this policy, all distortionary taxes remain fixed in response to the government spending shock. The second tax policy we consider is a balanced budget tax policy. Under this policy, labor income taxes vary in response to government spending shocks such that the government s budget remains balanced throughout: np Ht G Ht + 1 np F t G F t = τ t W t xl t xdx, 31 This policy implies that an increase in government spending is associated with an increase in distortionary taxes. 7

9 The federal government operates a common monetary policy for the two regions. This policy consists of the following augmented Taylor-rule for the economy-wide nominal interest rate: ˆr n t = ρ rˆr n t ρ i φ πˆπ ag t + φ y ŷ ag t + φ g ĝ ag t, 32 where hatted variables denote percentage deviations from steady state. The nominal interest rate is denoted ˆr n t. It responds to variation in the weighted average of consumer price inflation in the two regions ˆπ ag t ˆπ t = nˆπ t + 1 nˆπ t, where ˆπ t is consumer price inflation in the home region and is consumer price inflation in the foreign region. It also responds to variation in the weighted average of output in the two regions ŷ ag t = nŷ t + 1 nŷ t. Finally, it may respond directly to the weighted average of the government spending shock in the two regions ĝ ag t section D.4 below, for precise definitions of these variables. = nĝ t + 1 nĝ t. See D.3 Firms There is a continuum of firms indexed by z in the home region. Firm z specializes in the production of differentiated good z, the output of which we denote y ht z. In our baseline model, labor is the only variable factor of production used by firms. Each firm is endowed with a fixed, non-depreciating stock of capital. The production function of firm z is y ht z = L t z a. 33 The firm s production function is increasing and concave. It is concave because there are diminishing marginal returns to labor given the fixed amount of other inputs employed at the firm. Labor is immobile across regions. We follow Woodford 2003 in assuming that each firm belongs to an industry x and that there are many firms in each industry. The goods in industry x are produced using labor of type x and all firms in industry x change prices at the same time. Firm z acts to maximize its value, E t M t,t+j [p ht+j zy ht+j z W t+j xl t+j z]. 34 Firm z must satisfy demand for its product. The demand for firm z s product comes from three sources: home consumers, foreign consumers and the government. It is given by y ht z = nc Ht + 1 ncht pht z θ + ng Ht. 35 P Ht 8

10 Firm z is therefore subject to the following constraint: nc Ht + 1 ncht pht z θ + ng Ht fl t z. 36 Firm z takes its industry wage W t x as given. Optimal choice of labor demand by the firm is given by P Ht W t x = al t z a 1 S t z, 37 where S t z denotes the firm s nominal marginal cost the Lagrange multiplier on equation 36 in the firm s constrained optimization problem. Firm z can reoptimize its price with probability 1 α as in Calvo With probability α it must keep its price unchanged. Optimal price setting by firm z in periods when it can change its price implies p ht z = θ θ 1 E t α j M t,t+j y ht+j z E t k=0 αk M t,t+k y ht+j z S t+jz. 38 Intuitively, the firm sets its price equal to a constant markup over a weighted average of current and expected future marginal cost. Since all frims in industry x face the same wage, have the same production function and set price at the same time, they will all set the same price, produce the same amount, hire the same amount of labor and have the same marginal cost. We can therefore replace all indexes z by x. Combining labor supply 20 and labor demand 37, we get that S t x = χ a L t x ν 1 a+1 Ct σ 1. 1 τ t Using the production function to eliminate L t x from this equation yields S t x = χ a y ht x ν 1 a+1/a Ct σ τ t D.4 Linear Approximation We linearize the model around a zero-growth, zero-inflation steady state. We start by deriving a log-linear approximation for the home consumption Euler equation that relates consumption growth and the return on a one-period, riskless, nominal bond. This equation takes the form E t [M t,t r n t ] = 1. Using equation 19 to plug in for M t,t+1 and rearranging terms yields E t [β σ 1 Ct+1 C t +1 9 ] = rt n. 40

11 The zero-growth, zero-inflation steady state of this equation is β1 + r n = 1. A first order Taylor series approximation of equation 40 is ĉ t = E t ĉ t+1 σˆr n t E tˆπ t+1, 41 where ĉ t = C t C/C, ˆπ t = π t 1, and ˆr n t = 1 + r n t 1 r n /1 + r n. A linear approximation of the Backus-Smith equation 29 is where ˆq t = Q t 1. ĉ t ĉ t = σˆq t, 42 We next linearize equation 39 in period t + j of a firm that last changed its price in period t. Let ŷ t,t+j x denote the percent deviation from steady state in period t + j of output in industry x that last was able to change prices in period t. Let other industry level variables be defined analogously. We then get that ŝ ht,t+j x = ψ v ŷ ht,t+j x + σ 1 ĉ t+j + τ 1 τ ˆτ t+j, 43 where ŝ ht,t+j x denotes the percentage deviation from steady state of real marginal costs in period t + j of home output in industry x that last was able to change prices in period t, ˆτ t = τ t τ/τ, and the parameter ψ v = ν 1 + 1/a 1. The foreign analog of equation 43 is ŝ ft,t+j x = ˆq t+j + ψ v ŷ ft,t+j x + σ 1 ĉ t+j + Recall that we have assumed φ H + φ F = 1 and φ H + φ F τ 1 τ ˆτ t+j, 44 = 1. These assumptions imply that in steady state P = P H = P F = pz. We then have that in steady state C H = φ H C, C F = φ F C, CH = φ H C, CF = φ F C, c h z = C H, c f z = C F, c h z = C H, and c f z = C F. Since we assume that home and foreign households have equal initial wealth, C = C. We also assume that steady state government spending per capita is equal in the two regions, G H = G F = G. We furthermore assume that overall demand for home products as a fraction of overall demand for all products is equal to the size of the home population relative to the total population of the economy. In other words, nc H + 1 nc H = nc, which implies that φ H = n/1 nφ F. We take a linear approximation of all prices relative to the home price level. A linear approximation of the demand for home and foreign goods by home households equation 21 and its foreign counterpart equation 26 yield ĉ Ht = ĉ t ηˆp Ht and ĉ F t = ĉ t ηˆp F t 45 10

12 ĉ Ht = ĉ t ηˆp Ht ˆq t and ĉ F t = ĉ t ηˆp F t ˆq t, 46 where ĉ Ht = C Ht C H /C H, Ĉt = C t C/C, ˆp Ht = P Ht / 1 and other variables are defined analogously. It is useful to define per capita home and foreign output as and total output as Ht = 1 n 1 0 y htzdz and F t = n 0 y ftzdz, 47 t = n Ht + 1 n F t. In the steady state, we have H = y h z/n, F = y f z/1 n, and = n H + 1 n F. Next we take a linear approximation of demand for home and foreign variety s equation 35 and its foreign counterpart. The steady state of demand for home varieties is y h z = nc H + 1 nc H + ng H. Using C H = φ H C, C H = φ H C, C = C, G H = G, φ H + φ F = 1, φ H = n/1 nφ F, yields H = C + G. Similar manipulations for the demand for foreign varieties yield F = C + G. Since = n H + 1 n F, we thus have = H = F. Using these steady state relationships, we can take a linear approximation the demand in period t + j for home and foreign firms in industries that last changed their prices in period t. This yields C ŷ ht,t+j x = φ H ĉ Ht+j + 1 n C n φ H ĉ Ht+j +ĝ Ht+j θ ˆp ht x ˆp Ht+j ŷ ft,t+j x = n 1 n φ F C ĉ F t+j + φ F C ĉ F t+j + ĝ F t+j θ ˆp ft x ˆp F t+j where ĝ Ht+j = G Ht+j G/ and ĝ F t+j = G F t+j G/. as π t+k, 48 π t+k, 49 Given the way prices are set in the economy, home output from equation 47 can be expressed Ht = 1 n 1 αα j y ht j,t x. A linear approximation of this equation is ŷ Ht = 1 αα j ŷ ht j,t x. Using equation 48 to plug in for ŷ ht j,t x yields C ŷ Ht = φ H ĉ Ht + 1 n C n φ H ĉ Ht + ĝ Ht j 1 θ 1 αα ˆp j ht j x ˆp Ht π t k k=0

13 Using again the structure of price setting in the economy, equation 23 for the price index for home produced goods can be rewritten as P 1 θ Ht = A linear approximation of this equation is ˆp Ht = 1 αα j p ht j x 1 θ. j 1 1 αα ˆp j ht+j x ˆπ t k. Combining this equation and equation 50 yields ŷ Ht = φ H C ĉ Ht + 1 n n φ H k=0 C ĉ Ht + ĝ Ht. A similar set of manipulations for foreign output and foreign prices yields yields ŷ F t = n 1 n φ F C ĉ F t + φ F C ĉ F t + ĝ F t. Using equations 45 and 46 to eliminate ĉ Ht, ĉ Ht, ĉ F t, and ĉ F t C ŷ Ht = φ H ĉ t + 1 n C n φ H C η ŷ F t = n 1 n φ F C ĉ t C ĉ t + φ F C η φ H + 1 n n φ H ĉ t n 1 n φ F + φ F ˆp Ht + η ˆp Ht + η Combining equations 43, 48, and 51 and??, 49, and 52 yields ŝ ht,t+j x = ψ v ŷ Ht+j ψ v θ ŝ ft,t+j x = ˆq t+j + ψ v ŷ F t+j ψ v θ ˆp ht x ˆp Ht+j ˆp ft x ˆp F t+j A linear approximation of equations 23 yields ˆπ t+k ˆπ t+k from the last two equations C 1 n n φ H ˆq t + ĝ Ht, 51 C φ F ˆq t + ĝ F t σ 1 ĉ t+j + τ 1 τ ˆτ t+j, 53 + σ 1 ĉ t+j + τ 1 τ ˆτ t+j. 54 ˆπ Ht = 1 α α ˆp htx ˆp Ht and ˆπ F t = 1 α α ˆp ftx ˆp F t 55 12

14 yields A linear approxmation of equation 38 and its foreign counterpart yields ˆp ht x = 1 αβ αβ j E t ŝ ht,t+j x + αβ αβ j E tˆπ t+j, 56 ˆp ft x = 1 αβ αβ j E t ŝ ft,t+j x + αβ αβ j E tˆπ t+j. 57 Using equations 53 and 54 to substitute for ŝ ht,t+j x and ŝ ft,t+j x in the last two equations ˆp ht x = 1 αβζ ˆp ft x = 1 αβζ j=1 j=1 [ αβ j E t ψ v ŷ Ht+j + θψ v ˆp Ht+j + σ 1 ĉ t+j + τ ] 1 τ ˆτ t+j +αβ αβ j E tˆπ t+j, 58 [ αβ j E t ˆq t+j + ψ v ŷ F t+j + θψ v ˆp F t+j + σ 1 ĉ t+j + τ ] 1 τ ˆτ t+j +αβ j=1 αβ j E tˆπ t+j, 59 where ζ = 1/1 + ψ v θ. Quasi-differencing these equations yields [ ˆp ht x αβe t ˆp ht+1 x = 1 αβζ ψ v ŷ Ht + θψ v ˆp Ht + σ 1 ĉ t + τ ] 1 τ ˆτ t + αβe tˆπ t+1, 60 [ ˆp ft x αβe t ˆp ft+1 x = 1 αβζ ˆq t + ψ v ŷ F t + θψ v ˆp F t + σ 1 ĉ t + τ ] 1 τ ˆτ t + αβe tˆπ t+1, 61 Using equation 55 to eliminate ˆp ht x and ˆp ft x from these equations yields π Ht αβe t π Ht α α ˆp Ht αβe t ˆp Ht+1 = κζ [ ψ v ŷ Ht + θψ v ˆp Ht + σ 1 ĉ t + π F t αβe t π F t α α ˆp F t αβe t ˆp F t+1 = κζ [ ˆq t + ψ v ŷ F t + θψ v ˆp F t + σ 1 ĉ t + j=1 τ ] 1 τ ˆτ t + 1 αβe tˆπ t+1, 62 τ ] 1 τ ˆτ t + 1 αβe tˆπ t+1, 63 where κ = 1 α1 αβ/α. Notice that ˆp Ht+1 ˆp Ht = ˆπ Ht+1 ˆπ t+1. This implies that ˆp Ht αβ ˆp Ht+1 = 1 αβˆp Ht αβe tˆπ Ht+1 + αβe tˆπ t+1. Similarly, Using these in equations 62 and 63 yields ˆπ Ht = βe tˆπ Ht+1 + κζψ v ŷ Ht κζ ˆp Ht + κζσ 1 τ ĉ t + κζ 1 τ ˆτ t, 64 13

15 ˆπ F t = βe tˆπ F t+1 + κζψ v ŷ F t κζ ˆp F t + κζ ˆq t + κζσ 1 ĉ τ t + κζ 1 τ ˆτ t. 65 A linear approximation of equation 24 is φ H ˆp Ht + φ F ˆp F t = 0, 66 which implies First differencing equation 69 and rearranging terms yields ˆp F t = φ H φ F ˆp Ht. 67 ˆπ t = φ H ˆπ Ht + φ F ˆπ F t. 68 A linear approximation of equation28 is φ H ˆp Ht + φ F ˆp F t = ˆq t. 69 First differencing this equation and rearranging terms yields ˆπ t = φ H ˆπ Ht + φ F ˆπ F t. 70 E Model with GHH Preference The second model we consider in the paper is the same as the baseline model except that households have GHH preferences. The utility function of home households is, in this case, given by uc t, L t x = 1 σ 1 C t χl t x 1+ν 1 /1 + ν 1 1 σ 1, 71 where ν is the Frisch elasticity of labor supply. Home household labor supply is then u l C t, L t x u c C t, L t x = 1 τ t W tx, where the subscripts on the function u denote partial derivatives. Notice that these partial derivatives take the form u ct = u lt = C t χ L1+ν ν 1 C t χ L1+ν ν 1 14 σ 1, σ 1 χl 1/ν t. 72

16 This implies that home labor supply is χl 1/ν t = 1 τ t W tx. 73 Labor demand and the production function of firms are the same as in the baseline model. Combining labor supply, labor demand, and the production function of the firms yields S t x = 1 χ 1 τ t a yψv ht x 74 A linear approximation of this equation in period t + j for firms that last changed their price in period t yields ŝ ht,t+j x = ψ v ŷ ht,t+j x + τ 1 τ ˆτ t 75 Using this equation and its foreign analog, we can derive Phillips curves for home and foreign PPI inflation in the same way as we did for the baseline model. These derivations yield τ ˆπ Ht = βe tˆπ Ht+1 + κζψ v ŷ Ht κζ ˆp Ht + κζ 1 τ ˆτ t, 76 τ ˆπ F t = βe tˆπ F t+1 + κζψ v ŷ F t κζ ˆp F t + κζ ˆq t + κζ 1 τ ˆτ t. 77 With GHH preference, the consumption Euler equation of households becomes [ E t β u cc t+1, L t+1 x u c C t, L t x A linear approximation of u c C t+1, L t+1 x yields +1 ] = rt n. 78 u ct = u ccc ĉ t + u cll ˆlt x. 79 u c u c Notice that u cc = σ 1 u cl = σ 1 C χ L1+ν ν 1 C χ L1+ν ν 1 σ 1 1 σ 1 1, χl ν 1. This implies that u cl L u c u cc C u c = σ 1 C = σ 1 L C χ L1+ν ν 1 C χ L1+ν ν 1 1 = σ 1 1 χl ν 1 = u ccc 15 u c 1 aµ 1 C ν 1 1 1, L χl ν 1 = u ccc C u c C 1 aµ 1.

17 Let σc 1 = u cc C/u c and σ 1 l = u cl L/u c. We then have that a linear approximation of equation 78 yields Let ξ l = σ c σ 1 l ĉ t σ c σ 1 l ˆl t = E t ĉ t+1 σ c σ 1 l E tˆlt+1 σ c ˆr t n E t π t = C 1 aµ 1. Also, a linear approximation of the production function equation 33 yields ŷ Ht = aˆl t. Let ξ y = ξ l /a = C equation 80 as 1 µ 1. Using these conditions, we can rewrite ĉ t ξ y ŷ Ht = E t ĉ t+1 ξ y E t ŷ Ht+1 σ c ˆr n t E t π t With GHH preferences, the Backus-Smith condition becomes A linear approximation of this equation is u c C t, L t x u c C t, L t x = Q t. 82 ĉ t ξ y ŷ t ĉ t + ξ y ŷ t = σ cˆq t 83 F Model with Regional Capital Markets This section presents an extension of the baseline model in Section 1 that incorporates investment, capital accumulation and variable capital utilization. We adopt a specification for these features that mirrors closely the specification in Christiano, Eichenbaum, and Evans F.1 Households Household preferences in the home region are given by equations as in the baseline model. Household decisions regarding consumption, saving and labor supply are thus the same as before. However, in addition to these choices, households own the capital stock, they choose how much to invest and they choose the rate of utilization of the capital stock. Let K t denote the physical stock capital of capital available for use in period t and I t the amount of investment chosen by the household in period t. For simplicity, assume that I t is a composite investment good given by an index of all the products produced in the economy analogous to equations for consumption. The capital stock evolves according to K t+1 = 1 δ K t + ΦI t, I t 1, 84 where δ denotes the physical depreciation of capital and ΦI t, I t 1 = [ 1 φ It I t 1 ] I t, summarizes the technology for transforming current and past investment into capital. Households choose the 16

18 utilization rate u t of the capital cost. The amount of capital services provided by the capital cost in period t is then given by K t = u t Kt. The budget constraint of households in the home region is given by C t + I t + Au t K t + E t [M t,t+1 B t+1 x] B t x + 1 τ t W t xl t x + R k t u t Kt Ξ ht xdz T t. 85 The differences relative to the model presented in Section 1 are the following. First, households spend I t on investment. Second, they incur a cost Au t K t associated with utilizing the capital stock. Here Au t denotes a convex cost function. Third, they receive rental income equal to Rt k u t Kt for supplying u t Kt in capital services. Here Rt k denotes the rental rate for a unit of capital services. In addition to equations and a standard transversality condition, household optimization yields the following relevant optimality conditions. Optimal capital utilization sets the marginal cost of additional utilization equal to the rental rate on capital, Optimal investment and capital accumulation imply A u t = Rk t 86 D t Φ 1 I t, I t 1 + βe t [D t+1 Φ 2 I t+1, I t ] = u c C t, L t x 87 D t = β1 δe t D t+1 + βe t [A u t+1 u t+1 Au t+1 u c C t+1, L t+1 x] 88 where D t is the Lagrange multiplier on equation 84 and Φ j, denotes the derivative of Φ with respect to its j-th argument. F.2 Firms The production function of firms in industry x is y t x = f L t x, K t x 89 The demand for the output of firms in industry x is given by y t x = nc Ht + 1 ncht + ni H,t + 1 nih,t p t x θ + ng Ht 90 P Ht Optimal choice of labor and capital by firms implies W t x = f l L t x, K t xs t x, 91 17

19 R k t = f k L t x, K t xs t x. 92 Optimal price setting by firms yields equation 38 as in the baseline model. Combining labor demand, labor supply and the production function yields S t x = 1 v l L t x 1 τ t u c C t f l f 1 L t x, K t x 93 F.3 Calibration We set the rate of depreciation of capital to δ = 0.025, which implies an annual depreciation rate of 10 percent. The investment adjustment cost function is given by [ ] It ΦI t, I t 1 = 1 φ I t, 94 where φ1 = φ 1 = 0 and κ I = φ 1 > 0. We set κ I = 2.5. This is the value estimated by Christiano et al We require that capital utilization u t = 1 in steady state, assume that the cost of utilization function A 1 = 0 and set σ a = A 1/A 1 = Again, this is the value estimated by Christiano et al We assume that the production function is Cobb-Douglas with a capital share of 1/3. I t 1 F.4 Linear Approximation A linear approximation of equation 93 and its foreign analog yields vl L ŝ t x = f lll ˆl t x f lkk ˆkt x u ccc ĉ t + τ v l f l f l u c 1 τ ˆτ t 95 ŝ t x = vll L v l f lll f l ˆl t x f lkk ˆk f t x u ccc ĉ t + τ l u c 1 τ ˆτ t + ˆq t 96 Linearize approximations of the firm production function and capital demand equation 92 and their foreign analogs yields ŷ t x = aˆl t x + 1 aˆk t x and ŷ t x = aˆl t x + 1 aˆk t x 97 ˆr k t = aˆl t x aˆk t x + ŝ t x and ˆr k t = aˆl t x aˆk t x + ŝ t x 98 Note that there is no ˆq t term in the above equation, because ˆr k t equation we can derive that also is divided by. Using these ˆk t x = ŷ t x ˆr k t + ŝ t x, ˆl t x = ŷ t x + 1 a a ˆrk t 1 a a ŝtx. 18

20 Using these equations and the functional form of the utility function equation 15 and the production function, we can rewrite equation 95 as ŝ t x = ν a ŷ t x + 1 a a where ψ v equation 96 as where 1 a ˆrk t 1 a a ŝtx ŷ t x ˆr k t + ŝ t x + σ 1 ĉ t + τ 1 τ ˆτ t = ψ v ŷ t x + σ 1 ψ c ĉ t + ψ kˆr k t + ψ c τ 1 τ ˆτ t, 99 aν 1 1 aν 1 +1, ψ k 1+ν 1 1 a a and ψ 1 aν 1 +1 c. Correspondingly, we can rewrite 1 aν 1 +1 ŝ t x = ψ v ŷ t x + σ 1 ψ c ĉ t + ψ kˆr k t + ψ cˆq t + ψ c τ 1 τ ˆτ t 100 A Linear approximation firm demand equation 118 yields ŷ Ht = φ H C ĉht + φ H I ŷ F t = φ F C ĉ F t + φ F ŷ t,t+j = ŷ Ht+j θ ŷ t,t+j = ŷ F t+j θ îht + φ 1 n C H n ĉ Ht + φ 1 n I H n î Ht + ĝ Ht, 101 I n C î F t + φ F 1 n ĉf t + φ n I F 1 n îf t + ĝ F t, 102 ˆp t ˆp Ht+j ˆp ft ˆp F t+j π t+k π t+k Here we have made use of the fact that in steady state I H = φ H I, I F I F = φ F I = φ F I, I H = φ H I, and Substituting these expressions into equations yields ŝ t,t+j = ψ v ŷ Ht+j ψ v θ ˆp t ˆp Ht+j π t+k + σ 1 ψ c ĉ t+j + τ 1 τ ψ cˆτ t+j + ψ kˆr t+j k ŝ t,t+j = ψ v ŷ F t+j ψ v θ ˆp t ˆp F t+j π t+k + σ 1 ψ c ĉ t+j + τ 1 τ ψ cˆτ t+j + ψ kˆr t+j k + ψ cˆq t+j Applying the same method as we did in Section 1 to derive Phillips curves yields π Ht = βe t π Ht+1 + κζ ψ v ŷ Ht ˆp Ht + σ 1 τ ψ c ĉ t + ψ c 1 τ ˆτ t + ψ kˆr t k π F t = βe t π F t+1 + κζ ψ v ŷ F t ˆp F t + σ 1 ψ c ĉ τ t + ψ c 1 τ ˆτ t + ψ kˆr t k + ψ cˆq t We also have that î t = φ H î Ht + φ F î F t and î t = φ Hî Ht + φ F î F t 19

21 î Ht = î t ηˆp Ht and î F t = î t ηˆp F t î Ht = î t ηˆp Ht ˆq t and î F t = î t ηˆp F t ˆq t Analogously to 51 and 52 in Section 1, aggregate output is ŷ Ht = C φ H ĉ t + 1 n n C + I η C φ H + 1 n n φ H I φ Hĉ t + φ H î t + 1 n I n C + I ˆp Ht + η φ Hî t 1 n n φ H ˆq t + ĝ Ht, 107 ŷ F t = C φ F ĉ t + n 1 n C + I η C n 1 n φ F + φ F I φ F ĉ t + φ F î t + n 1 n C + I ˆp Ht + η I φ F î t φ F ˆq t + ĝ F t. 108 A linear approximation of K t = u t Kt and its foreign counterpart yields ˆk t = û t + ˆ kt. 109 ˆk t = û t + ˆ k t. 110 A linear approximation of A u t = R k t / and its foreign counterpart yields σ a û t = ˆr k t. 111 where σ a = A /A σ a û t = ˆr k t ˆq t. 112 and we use the fact that u = 1 in steady state. Assume that φ1 = φ 1 = 0 and φ 1 = k I > 0. A linear approximation of equation 87 and its foreign counterpart yields ] ˆd t + k I [β E t î t+1 î t ît î t 1 + σc 1 ĉ t = ] ˆd t + k I [β E t î t+1 î t î t î t 1 + σc 1 ĉ t = Assume that in steady state Au = 0, A u = Rk P. A linear approximation of equation 88 and its foreign counterpart yields ˆd t = β1 δe t ˆdt β1 δ E tˆr t+1 k σc 1 E t ĉ t+1 ˆd t = β1 δe t ˆd t β1 δ E tˆr t+1 k + E tˆπ t+1 q t E tˆπ t+1 σc 1 E t ĉ t

22 G A Model with Firm-Specific Capital This appendix presents a model of investment and capital accumulation that mirrors closely the specification in Woodford 2003, In this model, firms own their capital stock and face adjustment costs at the firm level in adjusting their capital stock. Household behavior is governed by the same equations as in our baseline GHH model presented in section E. As in our baseline model, firms belong to industries x, which make use of a specific type of labor. The production function of firms in industry x is y t x = fl t x, K t x. 117 The demand for the output of firms in industry x is given by y t x = nc Ht + 1 ncht + ni H,t + 1 nih,t pht x θ + ng Ht. 118 Firms optimal choice of labor demand implies P Ht W t x = f l L t x, K t xs t x. 119 Each firm faces convex adjustment costs for investment. A firm that would like to choose a capital stock K t+1 x for period t+1 must invest IK t+1 x/k t xk t x at time t. For simplicity, assume that I t is a composite investment good given by an index of all the products produced in the economy analogous to equations for consumption. We assume that I1 = δ, I 1 = 1, and I 1 = ɛ ψ. Optimal investment by firms implies [ I Kt+1 x +1 + E t M t,t+1 I K t x where Kt+2 x K t+1 x = E t M t,t+1 +1 I Kt+2 x K t+1 x ] Kt+2 x K t+1 x Rt+1 k x P H,t+1, 120 P H,t+1 +1 R k t x = f k L t x, K t xs t x. 121 Firm price setting is given by equation 38. This model has two new parameters relative to our baseline model. We follow Woodford 2003, 2005 in setting δ = and ɛ ψ = 3. Our results are virtually identical if we instead set δ = and ɛ ψ = 2.5. We assume that the production function is Cobb-Douglas with a capital share of 1/3. Combining labor demand and labor supply yields al t x a 1 K t x 1 a 1 τ t S tx P Ht P Ht = χl t x 1/ν

23 G.1 Linear Approximation Notice that K t+1 x = 1 δk t x + I t x 123 In steady state, we have I/K = δ. A linear approximation of equation 123 then yields ˆk t+1 x = 1 δˆk t x + δîtx. 124 Similarly, for the aggregate home capital stock we have ˆk Ht+1 = 1 δˆk Ht + δîht 125 A linear approximation of equation 122 yields ŝ t x = ν aˆl t x 1 aˆk t x ˆp Ht + A linear approximation of equation 121 yields τ 1 τ ˆτ t 126 ˆr k t x = ŝ t x + aˆl t x aˆk t x 127 Here, in contrast to earlier models, we are deflating nominal variables by P Ht. From equation?? we have that in steady state R k /P = δ + 1/β 1. A linear approximation of equation?? then yields û ct + ɛ ψ ˆkt+1 x ˆk t x + E t û ct+1 + βɛ ψ E tˆkt+2 x ˆk t+1 x +1 β1 δ E tˆr t+1x k + E t ˆp Ht A linear approximation of the production function yields Combining equations 126 and 129 yields ŝ t x = ν a a ŷ t x = aˆl t x + 1 aˆk t x 129 ŷ t x 1 aν ˆk t x ˆp Ht + a Adopting the notation used in Woodford 2005, this equation becomes ŝ t x = ωŷ t x ω νˆk t x ˆp Ht + τ 1 τ ˆτ t 130 τ 1 τ ˆτ t 131 where ω = ν a/a, ν = ν 1. Taking an average across all home firms we get ŝ Ht = ωŷ Ht ω νˆk Ht ˆp Ht + 22 τ 1 τ ˆτ t 132

24 and ŝ t x = ŝ Ht + ωŷ t x ŷ Ht ω νˆk t x ˆk Ht 133 Now using the demand curve for home firms we get that ŝ t x = ŝ Ht ωθˆp t x ω ν k t x 134 where k t x = ˆk t x ˆk Ht. A linear approximation of the firms price setting equation yields Ê x t αβ ˆp j t x π Ht+k ŝ t+j x = where following Woodford 2005 we use Êx t to denote an expectation conditional on the state of the world at date t, but integrating only over those future states in which firms in industry x have not reset their prices since period t. For aggregate variables Êx t x T = E t x T. For firm specific variables, this is not the case. Substituting for marginal costs in the above equation we get Ê x t ] αβ [ˆp j t x π Ht+k ŝ Ht+j x + ωθ ˆp t x π t+k + ω ν k t+j x = Thus, 1 + ωθˆp t x = 1 αβêx t ] αβ [ŝ j Ht+j ωθ π Ht+k ω ν k t+j x 137 Notice that unlike in the heterogeneous labor model ˆp t x is not independent of x. This is due to the presence of the k t+j x term on the right hand side. Notice also that Êx t k t+j x depends on ˆp t x, we need to determine this dependence to be able to solve for ˆp t x. Combining 127 and 131, we have ˆr k t x = ωŷ t x ω νˆk t x ˆp Ht + aˆl t x aˆk t x + Using the product function to eliminate ˆl t x yields ˆr k t x = ρ y ŷ t x ρ kˆkt x ˆp Ht + where ρ y = ω + 1 and ρ k = ρ y ν as in Woodford τ 1 τ ˆτ t 138 τ 1 τ ˆτ t

25 Thus, aggregating equation 128 yields û ct + ɛ ψ ˆk Ht+1 ˆk Ht = E t û ct+1 + βɛ ψ E tˆkht+2 ˆk Ht+1 +1 β1 δρ y E t ŷ Ht+1 ρ kˆkht+1 + τ 1 τ ˆτ t 140 Here our expression differs from Woodford 2005 in the coefficient on E t û ct+1. This difference arises because we are using GHH preferences. Combining this expression with the one on 128 yields ɛ ψ kt+1 x k t x = βɛ ψ E t kt+2 x k t+1 x + 1 β1 δ ρ y E t ŷ t+1 x ŷ Ht+1 ρ k kt+1 x 141 Rearranging and using the firm s demand curve 1 β1 δ ρ y θɛ 1 ψ E t ˆp t+1 x = βe t kt+2 x 1 + β + 1 β1 δρ k ɛ 1 ψ kt+1 x + k t x 142 or where QL = β ] ΘE t ˆp t+1 x = E t [QL k t+2 x Θ = 1 β1 δ ρ y θɛ 1 ψ [ 1 + β + 1 β1 δρ k ɛ 1 ψ ] L + L Notice that Q0 = β > 0, Qβ < 0, Q1 < 0 and Qn > 0 for large n. So, QL = β1 µ 1 L1 µ 2 L 144 with µ 1, µ 2 real and 0 < µ 1 < 1 < β 1 < µ 2. Using the argument on page 13 of Woodford 2005 we have ˆp t x = ˆp Ht ψ k t x 145 i.e. the choice of a price to set by firms in industry x is a function of aggregate variables and industry x s capital stock. Here, ψ is a coefficient to be determined below. Also, note that k t x = 0 on average for firms that reset their prices at time t because of the Calvo assumption. Thus ˆp t is the average relative price set by firms at time t. 24

26 A linear approximation of the home price index yields ˆp Ht = α 1 α π Ht 146 Let s now introduce the notation p t x for a generic relative price. This contrasts ˆp t x, which we have been using to denote the optimal price set at time t. Notice that E t p t+1 x = α [ p t x E t π Ht+1 ] + 1 αe t ˆp t+1 x 147 and using the last three equations we get E t p t+1 x = α p t x E t π Ht αe t [ˆp Ht+1 x ψ k t+1 x] = α p t x 1 αψe t kt+1 x 148 Again, Woodford 2005 argues that k t+1 x = λ k t x γ p t x 149 where λ and γ are to be determined. The algebra on pages in Woodford 2005 applies to our model. We plug the equation 148 into the last equation and get E t kt+2 x = [λ + 1 αγψ] k t+1 x αγ p t x 150 Using this to substitute for E t kt+2 x in equation 142, and again using 148 to substitute for E t p t+1 x, we obtain a linear relation of k t x and p t x. For convenience, denote A 1 + β + 1 β1 δρ k ɛ 1 ψ, thus QL = β AL + L2. Equation 142 becomes, Θα p t x Θ1 αψe t kt+1 x = λ 1 E t kt+1 x + λ 1 γ p t x A k t+1 x + β[λ + 1 αγψ] k t+1 x αβγ p t x For the conjectured solution 149 to satisfy this equation, we need the coefficient in front of p t x to satisfy 1 αβλγ = Θαλ. 151 Using this equation the coefficient in front of E t kt+1 x becomes Θ1 αψλ + 1 Aλ + βλ αβλψγ = 0 => Θ1 αψλ + 1 αβλ1 Aλ + βλ 2 = 0 => β 1 αλqβλ + 1 αθψλ =

27 Now, returning to optimal price setting, we focus on the term Ê x t αβ j kt+j x 153 Since k t+1 x = λ k t x γ p t x 154 we have Êt x k t+j+1 x = λêx k t t+j x γ p t x E t π Ht+k for all j 0, and using p t x E t j π Ht+k = Êx t p t+j x. Notice that 155 k t+1 x = λ k t x γ p t x k t+2 x = λ 2 kt x γλ p t x γ p t+1 x k t+3 x = λ 3 kt x γλ 2 p t x γλ p t+1 x γ p t+2 x so Ê x t Ê x t αβ j kt+j x = αβ j p t+j x = k t x 1 αβλ γαβ 1 αβλêx t αβ j p t x E t αβ j p t+j x π Ht+k In addition using the fact that, Ê x t αβj j π Ht+k = 1 1 αβ E t j=1 αβj π Ht+j, we have Ê x t αβ j k t+j x = k t x 1 αβλ γαβ 1 αβ1 αβλ p tx + γαβ 1 αβ1 αβλ E t Noting that for firms reoptimizing their price at time t, p t x = ˆp t x. equation 137 and the last equation yields Thus, 1 + ωθˆp t x = 1 αβe t αβ j ŝ Ht+j + 1 αβ1 + ωθe t αβ j 1 αβ ω ν γαβ ω ν k t x + 1 αβλ 1 αβλ ˆp γαβ ω ν tx 1 αβλ E t j=1 αβ j π Ht+j j=1 156 Therefore, combining αβ j π t+j j=1 π Ht+k φˆp t x = 1 αβe t αβ j ŝ Ht+j + φ αβ j E t π Ht+j ω ν 1 αβ 1 αβλ k t x

28 where φ = 1 + ωθ ω ν γαβ 1 αβλ. and For this last equation to be consistent with our conjecture 145, we must have φˆp Ht = 1 αβ αβ j E t ŝ Ht+j + φ αβ j E t π Ht+j 158 φψ = ω ν 1 αβ 1 αβλ j=1 159 This last equation along with 151 and 152 comprise a system of three equation in the three unknown coefficients λ, γ, and ψ. Woodford 2005, pages describes an algorithm for solving these three equations. The following explains how to reduce this system of equations to a single equation for λ. For λ 0, 151 can be solved for ψ. Similarly, 152 defines a function ψλ = β 1 αλqβλ 1 αθλ γλ = Θαλ 1 αβλ Substituting these functions for ψ and γ in 159, we get the equation to solve for λ: V λ = [ 1 + ωθ1 αβλ 2 α 2 β ω νθλ ] Qβλ + β1 α1 αβ ω νθλ = Quasi-differencing the expression for ˆp Ht, equation 158, yields ˆp Ht αβe t ˆp Ht+1 = 1 αβφ 1 ŝ Ht + αβe t π Ht Using equation 146 to plug in for ˆp t yields α 1 α π Ht α2 β 1 α E tπ Ht+1 = 1 αβφ 1 ŝ Ht + αβe t π Ht π Ht = kφ 1 ŝ Ht + βe t π Ht

29 References Backus, D. K., and G. W. Smith 1993: Consumption and Real Exchange Rates in Dynamic Economies with Non-Traded Goods, Journal of International Economics, 353 4, Calvo, G. A. 1983: Staggered Prices in a Utility-Maximizing Framework, Journal of Monetary Economics, 12, Christiano, L., M. Eichenbaum, and S. Rebelo 2011: When is the Government Spending Multiplier Large?, Journal of Political Economy, 119, Christiano, L. J., M. Eichenbaum, and C. L. Evans 2005: Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy, Journal of Political Economy, 1131, Cochrane, J. H. 2005: Asset Pricing. Princeton University Press, Princeton, NJ, second edn. Eggertsson, G. B. 2010: What Fiscal Policy Is Effective at Zero Interest Rates?, in NBER Macroeconomics Annual 2010, ed. by D. Acemoglu, and M. Woodford, Chicago IL. University of Chicago Press. Woodford, M. 2003: Interest and Prices. Princton University Press, Princeton, NJ. 2005: Firm-Specific Capital and the New-Keynesian Phillips Curve, International Journal of Central Banking, 12,