Macroeconomic Effects of Financial Shocks Online Appendix


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1 Macroeconomic Effects of Financial Shocks Online Appendix By Urban Jermann and Vincenzo Quadrini Data sources Financial data is from the Flow of Funds Accounts of the Federal Reserve Board. We report the references to the FFA items using the Coded Tables released on September 7, 200. New editions of the Coded Tables may use different coding which should be taken into account by future uses. All series, financial and real, are seasonally adjusted. Data source for Figure and Table Equity Payout is Net dividends of nonfarm, nonfinancial business Table F.02, line 3, plus Net dividends of farm business Table F.7, line 24, minus Net increase in corporate equities of nonfinancial business F.0, line 35, minus Proprietors net investment of nonfinancial business F.0, line 39. Debt Repurchase is the negative of Net increase in credit markets instruments of nonfinancial business Table F.0, line 28. Equity payout and debt repurchase are both divided by business value added from the National Income and Product Accounts Table.3.5. Total GDP used to compute the correlations reported in Table is also from NIPA Table..6. Construction of financial shocks The ξ t series are constructed from the enforcement constraint ξ t k t+ b e t+ = y t, where b e t+ = b t+ / + r t is the end of period liability. The linearized version of this constraint can be written as ˆξt = φ kˆkt+ + φ bˆbe t+ + ŷ t, where φ k = ξ k/ȳ and φ b = ξ b e /ȳ. The hat sign denotes percent deviations from the steady state and the bar sign denotes steady state values. After parameterizing the model using steady state targets see first set of parameters in Table 2 in the paper, we determine the coefficients φ k =.5489 and φ b = We can then use the above equation to construct the ˆξ t series once we have empirical measurements for the end of period capital, ˆk t+, the end of period liabilities, ˆb e t+, and output ŷ t. Following is the description of how we construct these series. The Capital Stock is constructed using the equation k t+ = k t Depreciation + Investment. Jermann: Wharton School of the University of Pennsylvania and NBER, 3620 Locust Walk, Philadelphia, PA 904, Quadrini: Marshall School of Business, University of Southern California, CEPR and NBER, 70 Exposition Blvd, Los Angeles, CA 90089,
2 2 ONLINE APPENDIX Depreciation is measured as Consumption of fixed capital in nonfinancial corporate business Table F.8, line 4 plus Consumption of fixed capital in nonfinancial noncorporate business Table F.8, line 5. Investment is measured as Capital expenditures in nonfinancial business Table F.0, line 4. Both variables are deflated by the price index for business value added from NIPA Table.3.4. We start the recursion in the first quarter of 952. The initial k t is chosen so that the capitaloutput ratio in the business sector does not display any trend during the sample period Since the initial value of k t is important only for the early dates, this is not relevant for our results based on the subperiod The ˆk t+ series used in equation is constructed by linearly detrending the log of k t+ over the period 984.I200.II. The Debt Stock is constructed using the equation b e t+ = b e t + Net New Borrowing. Notice that we use the variable b e t+ = b t+ / + r t instead b t+ because this is the model equivalent of the endofperiod debt reported in the data. Net new borrowing is measured by the Net increase in credit markets instruments of nonfinancial business Table F.0, line 28. Since the constructed stock of debt is measured in nominal terms, we deflate it by the price index for business value added from NIPA Table.3.4. The initial nominal stock of debt is set to 94.2, which is the value reported in the balance sheet data from the Flow of Funds in 952.I for the nonfarm nonfinancial business Table B.02, line 22 and Table B.03, line 24. Since we start the recursion in 952, the initial value is not important for the results of the paper based on the subperiod We do not use directly the series for the stocks of debt because they are not seasonally adjusted. The ˆb e t+ series used in equation is constructed by linearly detrending the log of b e t+ over the period 984.I200.II. For Output we use business value added from NIPA Table.3.5 deflated by the price index for business value added also from NIPA Table.3.4. This gives us y t. The ŷ t series used in equation is constructed by linearly detrending the log of y t over the period 984.I200.II. Construction of productivity shocks To construct the TFP series from the production function y t = z t kt θ n θ t, we need three data series: capital k t, labor n t and output y t. The Capital Stock and Output are constructed through the same procedures as described above. Labor is measured as total private aggregate weekly hours from the Current Employment Statistics, national survey. Data series for the structural estimation GDP is Real gross domestic product from NIPA Table..6. Consumption is Real personal consumption expenditures from NIPA Table..6. Investment is Real gross private domestic investment from NIPA Table..6. Working Hours is total private aggregate weekly hours from the Current Employment Statistics, national survey. Wages are Real hourly compensation in the business sector from the Bureau of Labor Statistics. Nominal Price is the implicit price deflator for GDP from NIPA Table..9. Interest Rate is the Federal fund rate published by the Federal Reserve Board. We construct the quarterly series from the monthly data by averaging over three months. Debt Repurchase is defined in the data source for Figure and Table.
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 46 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 loglinearizing 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 fourdimensional 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 KuhnTucker 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 GaussHermite 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 rearranged 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. Rearranging, the logdeviation 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 rearranging 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 loglinearized 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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