# Macroeconomic Effects of Financial Shocks Online Appendix

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3 ONLINE APPENDIX 3 First order conditions for the firm s problem Consider the optimization problem 3 in the paper and let λ and µ be the Lagrange multipliers for the budget and enforcement constraints. Differentiating we get: d : λϕ d d = 0 n : λf n z, k, n λw µf n z, k, n = 0 k : Em V k s ; k, b λ + µξ = 0 b : Em V b s ; k, b + λ R µξ + r = 0. The envelope conditions are V k s; k, b = λ δ + F k z, k, n µf k z, k, n, V b s; k, b = λ. Using the first condition to eliminate λ plus the envelope conditions we get equations 4-6 reported in the paper. Numerical solution We solve the model using a linear approximation under the assumption that the enforcement constraint is always binding. To check the accuracy of the linear solution and especially the assumption that the enforcement constraint is always binding, we also solve the model nonlinearly using a global approximation method. The linear and nonlinear methods are described below. Linear approximation If the enforcement constraint is always binding, we can solve the model by log-linearizing the dynamic system around the steady state. The equilibrium is characterized by the

4 4 ONLINE APPENDIX following system of dynamic equations: wu c c, n + U n c, n = 0, R τ U c c, n = β EU c c, n, τ wn + b b R + d c = 0, F n z, k, n = w, µϕ d d E mc, n, d, c, n, d δ + µ ϕ d d F k z, k, n + ξµϕ d d =, R τ RE mc, n, d, c, n, d + ξµϕ d d =, R τ δk + F z, k, n wn b + b R k ϕd = 0, ξ k b τ = F z, k, n. R τ The first three equations are the first order conditions and budget constraint for households. In equilibrium the tax payments of households is accounted by a lower interest earned on bonds, R, and the gross before tax interest rate is + r = R τ/ τ. The next three equations are the first order conditions for firms. The term mc, n, d, c, n, d = βu c c, n /U c c, nϕ d d/ϕ d d is the effective discount factor. The remaining two equations are the firms budget and enforcement constraints. We have eight dynamic equations. After linearizing around the steady state we can solve the system for the eight variables c t, d t, n t, w t, R t, µ t, k t+, b t+, as linear functions of the states, z t, ξ t, k t, b t. Nonlinear approximation The nonlinear approach approximates the three conditional expectations in the second, fifth and sixth equations with functions that interpolate linearly between the grid points of the four-dimensional state space z, ξ, k, b. Starting with initial guesses for the conditional expectations at the grid points, we can compute all variables of interest by solving a system of nonlinear equations. At each grid point, we first solve the system assuming that the enforcement constraint is binding. If the solution for the multiplier is negative, we set it equal to zero, and then solve the system ignoring the enforcement constraint. In doing so we essentially check the Kuhn-Tucker conditions at each grid point. Once we have solved for all of the grid points, we update the guesses for the conditional expectations and keep iterating until convergence. The new guesses are produced through Gauss-Hermite quadrature z t and ξ t are lognormal. We found that the values of the Lagrange multiplier obtained from the simulation of the model using the nonlinear solution are almost indistinguishable to those obtained with the linear solution, after an initial period of adjustment. The same holds true for output, hours and financial flows. The solution method described here is also used to solve the version of the model with an alternative specification of the enforcement constraint see the sensitivity section in

5 ONLINE APPENDIX 5 the paper. As shown in Figure 9, the Lagrange multiplier for this version of the model becomes zero in some of the simulation periods. Deriving the linearized wage equation in the extended model Differentiating the objective 6 with respect to w j,t we get: E t βω s γ t+s { U j,3,t+s υ t+s υ t+s w υ t+s υ t+s j,t W t+s υ t+s υ t+s nt+s λ t+s w υ υ t+s t+s } υ t+s υ t+s j,t nt+s = 0 υ t+s W t+s where the second subscript in the utility function denotes the derivative. Using the fact that λ t+s = U 2,t+s, this can be re-arranged as E t βω s γ t+s n t+s t U 2,t+s v t+s { } wt v t+s Λ t+s t = 0, +s where Λ j,t+s t U j,3,t+s t U 2,t+s is the marginal rate of substitution for workers that reset their wage in period t, and n j,t+s t is the labor supply of a worker that has changed the wage at time t. The j index can be dropped since all households that reset their wage at time t will make the same choice. Furthermore, because of separability in the utility function and the assumption that households can purchase contingent claims, U 2,t+s is the same for all agents independently of whether they are able to reset their wages. Linearizing the first order condition around the steady state we obtain E t βω s w P w j,t w w P +s P vλ v t+s v vλ Λ t+s t = 0, Λ which takes into account that in a steady state w/p = W/P = vλ. Re-arranging, the log-deviation of the wage, w j,t /w = ŵ t, can be written as ŵ t = βω E t βω s Pt+s + v t+s + Λ t+s t. Consider now the marginal rate of substitution in log form, lnλ t+s t = ln U 3,t+s lnu 2,t+s. Because U 3,t+s = An /ε t+s t we have ln Λ t+s t = ln U 2,t+s + ε ln n t+s t + ln A. Replacing n t+s t by the labor demand equation 5 in the paper, we obtain ln Λ t+s t = ln U 2,t+s + ε ln n t+s ε υt ln w t ln W t+s + ln A. υ t

6 6 ONLINE APPENDIX The linearized version is then Λ t+s t = Û2,t+s + ε n t+s υ υ ε ŵ t + υ υ ε Ŵ t+s. We can now substitute this term in the linearized equation for the wage. After collecting terms and re-arranging we get ŵ t = ΦE t where Φ = ευ βω ευ +υ. βω s +s + v t+s Û2,t+s + ε n t+s + υ υ ε The linearized wage equation can be written recursively as follows: ŵ t = Φ + Φ v t ΦÛc,t + Φ ε n t + υφ υ εŵt + βωe t ŵ t+ Ŵ t+s, Using the functional form of the utility function, the marginal utility of consumption in linearized form can be written as hσ σ Û 2,t = ĉ t ĉ t. h h Thus, the final expression for the linearized wage equation is hσφ σφ ŵ t = ĉ t + ĉ t + Φ h h + Φ v t + Φ ε n t + υφ + βωe t ŵ t+. υ εŵt Extended model This section of the appendix describes the full set of equations that characterize the equilibrium of the extended model studied in Section IV of the paper. This extends the model estimated by Smets and Wouters 2007 by adding financial frictions and financial shocks. The log-linearized version of the model is estimated structurally using Bayesian methods. We first derive the first order conditions of the firm and then we provide the full list of equations. First order conditions for the firm s problem Equation 25 in the paper reports the optimization problem faced by the firm. Denoting by λ, µ, χ, Q the Lagrange multipliers associated with the four constraints, the

7 ONLINE APPENDIX 7 first order conditions are: λ t = ϕ d d t, µ t λ t µ t λ t F l,t = W t + χ td l,t λ t, F u,t = Ψ u,tk t + χ td u,t, λ t λ t λ t G p,t + E m t+ λ t+ + G p,t+ = χ t Q t Υ i,t + Em t+ Q t+ Υ i,t+ = λ t, { } Q t = Em t+ δq t+ + λ t+ + F k,t+ Ψu t µ t+ F k,t+ χ t+ D k,t+ + ξ t µ t, = R t Em t+ λ t+ λ t + µ tξ t Rt. λ t + r t Dynamic system We can now list the complete set of dynamic equations. We also provide the list of variables that enter each individual equation. In reporting the first order conditions for the firm we eliminate the Lagrange multiplier λ t using the condition λ t = / ϕ d d t. Households euler equation for bonds: 2 Capital utilization: 3 Euler equation for capital: + r t Em t+ + = 0 fγ t, c t, R t,, c t, γ t+, +, c t+ = 0 µ t ϕ d,t F u,t Ψ u,t k t ϕ d,t χ t D u,t ϕ d,t = 0 fz t, η t, k t, n t, u t, d t, µ t, χ t = 0 { Em t+ δq t+ + F } k,t+ Ψ t µ t+ F k,t+ χ t+ D k,t+ + ξ t µ t Q t = 0 ϕ d,t+ fγ t, ξ t, c t, c t, µ t, Q t, z t+, γ t+, η t+, k t+, n t+, u t+, d t+, c t+, µ t+, χ t+, Q t+ = 0 4 Price of capital: Q t Υ i,t + Em t+ Q t+ Υ i,t+ ϕ d,t = 0

8 8 ONLINE APPENDIX fζ t, γ t, i t, c t, d t, c t, i t, Q t, ζ t+, γ t+, c t+, i t+, Q t+ = 0 5 Law of motion for capital: 6 New wage linearized: δk t + Υ t k t+ = 0 fζ t, i t, k t, i t, k t+ = 0 hσφ σφ c t + c t +Φ +Φv t + Φ h h ε n t+ υφ υ ε W t+βωe t w t+ w t = 0 where Φ = ευ βω ευ +υ. fυ t, c t, w t, W t,, n t, c t, w t+ = 0 7 Wage index: ωw υ t υ t t + ωwt υt W t = 0 fw t, w t, W t = 0 8 Labor demand: µ t ϕ d,t F n,t W t χ t ϕ d,t D n,t = 0 fz t, η t, k t, W t,, n t, u t, d t, µ t, χ t = 0 9 Bond demand: R t E m t+ Pt ϕ d,t + ϕ d,t+ Rt + ξ t µ t ϕ d,t = 0 + r t fγ t, ξ t, c t, R t,, d t, c t, µ t, γ t+, +, d t+, c t+ = 0 0 Nominal price: ϕd,t G 2,t + E m t+ G,t+ χ t ϕ d,t = 0 ϕ d,t+ fγ t,, c t,, d t, c t, Y t, χ t, γ t+, +, d t+, c t+, Y t+ = 0 Firm s value: d t + Em t+ V t+ V t = 0 fγ t, c t, d t, c t, V t, γ t+, c t+, V t+ = 0 2 Enforcement constraint: ξ t k t+ b t+ F t = 0 + r t

9 ONLINE APPENDIX 9 fz t, ξ t, k t, R t,, n t, u t, k t+, b t+ = 0 3 Firm s budget: 4 Household s budget: F t Ψ t k t + b t+ R t b t W t n t G t ϕ t i t = 0 fz t,, k t, b t, W t, R t,, n t, u t, d t, i t, Y t, b t+ = 0 W t n t + d t b t+ + r t + b t c t T t = 0 fb t, W t, R t,, n t, d t, c t, T t, b t+ = 0 5 Government budget: G t + B t+ R T t = 0 + r t 6 Monetary policy linearized: fg t, R t,, T t, B t+ = 0 a r t + a 2 + a 3 Y t Y t + a 4 Y t Y t + ς t r t = 0 where a = ρ R, a 2 = ρ R ν, a 3 = ρ R ν 2 + ν 3, a 4 = ν 3. fς t, R t,, Y t, R t,, Y t = 0 7 Output: 8 Debt repurchase: F t Y t = 0 fz t, k t, n t, u t, Y t = 0 b t / + r t b t+ / + r t Y t x t = 0 fr t, b t, R t, Y t, x t, b t+ = 0 Taking into account that + r = R τ/ τ, we can use the linearized version of these equations to solve for 8 variables R t,, c t, n t, u t, d t, µ t, χ t, x t, Q t, i t, w t, W t, Y t, V t, T t, k t+, b t+ as a function of 6 states z t, ζ t, γ t, η t, υ t, G t, ς t, ξ t, p t, i t, c t, W t, R t, Y t, k t, b t.

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