Appendix: Financial Intermediation and Capital Reallocation

Transcription

1 Appendix: Financial Intermediation and Capital Reallocation Hengie Ai, Kai Li, and Fang Yang A Data Construction A. Measurement of the efficiency of capital reallocation We first derive an aggregation result that is similar to Hsieh and Klenow 2009 and Hopenhayn and Neumeyer In fact, the product market of our model is a special case of the above papers without labor market distortions. Consider the maximization problem in 2, first order conditions with respect to k and l imply: αp y = MPL l, αp y = MPK k A. Together, the above imply: k = MPL α l MPK α A.2 To save notation, we denote A = Aa in this section. Note also, total output of firm can be written as: y = A k α l α = A l k ] α k = A l A.3 l ] α k. A.4 We can use A.2 and A.3 to write l as a function of y, and use A.2 and A.4 to write k as a function of y : l = y A αmpl αmpk ] α, k = y αmpl A αmpk ] α. A.5 Note that the final goods producer s optimality condition implies that y = p η Y. Using this equation to replace y in the above equation and integrate across all, we have: K = L = p η A p η A ] α ] αmpl α d Y MPK α A.6 ] α ] αmpl α d Y, MPK α A.7 where K and L stands for the total capital and total labor employed for production, respectively.

2 Together, A.6 and A.7 imply Y = K α L α ] ] αd α p η ] ]. αd α A MPK A MPK p η A.8 We now express p in A.8 as a function of productivity and prices. Note that A. implies MPK k +MPL l = p y. A.9 Also, A.5 implies Combining A.9 and A.0, we have: MPK k +MPL l = y ] MPL α MPK A α α p = ] MPL α MPK A α α Note that because of the price of the final goods is normalized to one, we have Integrating A. over, we have: Together, A. and A.2 imply ] { MPL α = α A MPK α ] α. A.0 ] α. A. ] p η η d =. ] α ] η d} η. A.2 p = ] MPK α A α { MPK A α ] α ] ηd } η. A.3 Replacing p in equation A.8 with A.3, and using A = Aa, we can write Y = TFP K α L α, where { a η d } η η +α TFP = A F, where F = MPK α { a MPK α } η α. A.4 MPK d Under the assumption 3, it is straightforward to show that TFP = A if MPK = MPK for all. The F defined above is our measure of the efficiency measure of capital reallocation. Under 2

3 the assumption lna and lnmpk are ointly normally distributed, we can show that lnf = 2 αη +]ασ2, A.5 where σ 2 is the cross-sectional variance of marginal product of capital. A.2 Misallocation and TFP In Figure, we plot the measure of capital misallocation and total factor productivity. We measure the cross-sectional dispersion of TFPR following Hsieh and Klenow In the context of our model, equation A. implies MPK = α p y k. Following Chen and Song 203, we measure MPK by the ratio of Operating Income before Depreciation OIBDP to one-year-lag net Plant, Property and quipment PPNT. As in Hsieh and Klenow 2009, we focus on the manufacturing sector and compute the cross-sectional dispersion measure within narrowly defined industries as classified by the 4-digit standard industry classification code. Specifically, for firm in industry i, we compute MPK i, MPK i = α p i, y i, k i, α p y k = p i, y i, k i, p y k, where p y k is measured at the industry level. We then compute the variance of MPK i, MPK i within each industry and take average across all the industries for each year. This is our empirical measure of σ 2 in equation A.5. We use the first order approximation in A.5 to construct the time series of the efficiency measure of capital reallocation, which is the solid line in Figure. The measure of total factor productivity is directly taken from the published TFP series on the U.S. Bureau of Labor Statistics website. Both series are HP filtered. A.3 Capital reallocation We follow isfeldt and Rampini 2006 to construct the capital reallocation measure from the annual firm level data in Compustat. Reallocation is defined as the sum of acquisitions and sales of property, plant and equipment. When we calculate the its ratio to investment, investment is defined as capital expenditures plus acquisitions. We deflate all the series using CPI from Federal Reserve Bank of St Louis website to remove any effects from variation in nominal prices. A.4 Total volume of bank loans We measure the total volume of bank loans of non-financial corporate sector through the aggregate balance sheet of nonfinancial corporate business Table B.03 as reported in the U.S. Flow of 3

4 Funds Table. In particular, the bank loan is calculated as the difference between total credit market liability, the sum of debt securities Line 26 and loans Line 30, and corporate bond Line 26. Under this construction, bank loans consist of the following credit market liability items: commercial paper Line 27, municipal securities Line 28, depository institution loans Line 3, other loans and advances Line 32 and mortgages Line 33. The bank loan measure is deflated using CPI from Federal Reserve Bank of St Louis website. B Misallocation and Aggregation on the Product Market B. Proof of Proposition Using the fact that y = p η Y and equation A.2, we can write p y = y η Y η = A k α l α ] η Y η. B. Combining A. and B., we can write the marginal product of capital as MPK = αa η and write the marginal product of labor as k α η l α η Y η, B.2 α A k a ] η Y η = MPLl α η. B.3 have: ] η quation B.3 implies l A k a α η. Using the resource constraint, l d = L, we ] A k a l = ] A k a η α η η α η d L B.4 Together, B.2 and B.4 imply that MPK can be written as a function of A,k : { MPK = αa η ξ k ξ } A η ξ k ξ d η ξ, B.5 where we normalize total labor supply L = and denote ξ = αη α +αη α as in the proposition. Using 4

5 B.5 to replace MPK in the expression of F to get F = ] A k a η d { k d } α B.6 η α η α η Under our assumptions, A = Āa and Ā = AK α, we have A H = AK α A L = AK α πˆφ+ π ˆφ πˆφ+ π η η. Replacing A in equations B.5 and B.6, and using the fact that the total amount of capital utilized in production is k d = uk, we can derive the expression for total output and the marginal product of capital in the proposition. and C Proof of Proposition 2 and 3 Note thepolicy functionsfor φand uaredetermined by conditions 23, 24, 26 and27. Clearly, they depend only on the state variables A and s, which we will denote as φa,s and ua,s. We first compute the difference between the left-hand side of inequality 24 and that of 23 as = We define { } u θq L A,φ,u ωqu θq L A,φ,u] πφ+ π { } µφ θq H A,φ,u ωqu θq H A,φ,u] πφ+ π u πϕ+ π { θq LA,φ,u θφq H A,φ,u+ ωφ Qu}. A,φ,u = θq L A,φ,u θφq H A,φ,u+ ωφ Qu. Clearly, A,φ,u > 0 is equivalent to only one constraint, 23 binds, and A,φ,u = 0 is equivalent to both constraints, 23 and 24 bind simultaneously. It is convenient to define MPKu = Qu δ. Using the no arbitrage equation 5, we can simplify the above equation as A,φ,u = θmpk L A,φ,u θφmpk H A,φ,u + ωφ MPK u+θ ωφ δ. Throughout, we maintain the assumption θ > ω, which means that bankers are better than households in enforcing contracts. We also define the left-hand side of inequality 23 as a function 5

6 of A,φ,u: ΨA,φ,u = θmpk H A,φ,u+ δ] C. uφ { ωmpku θmpk H A,φ,u+θ ω δ} πφ+ π C. First best case, no constraint binds In the first best case, Q H A,φ,u = Q L A,φ,u = QA,φ,u = αu α A+ δ. The optimal capital utilization condition 9 implies αu α A+ δ = b 0 u ν. C.2 quation C.2 defines capital utilization rate as a function of productivity, ûa. We define ˆQA as the price of capital in the first best case as a function of A: ˆQA = αûa α A+ δ, where ûa is implicitly defined by equation C.2. Also, define ŝa to be the highest level of s such that there is no capital misallocation: One simplify the expression C. and show Ψ A,ˆφ,ûA = ˆQA πˆφ+ π ŝa = Ψ A,ˆφ,ûA. { } ω θ ωûa]ˆφ ω π ˆφ. It is straightforward to prove the following claim, which establishes the first part of Proposition 2. Claim A.. If s ŝa, then the optimal policy is given by: φa,s = φ, ua,s = ûa. C.3 Proof. We need to show that 23, 24, 26 and 27 are satisfied with appropriate choices of the 6

7 Lagrangian multipliers. Under the proposed policies and prices, the LHS of 23 is Also, θ MPK H A,ˆφ,ûA ] + δ ωmpk H A,ˆφ,ûA θmpk H A,ˆφ,ûA +θ ω δ = ŝa s. = Ψ A,ˆφ,ûA = θmpk L A,ˆφ,ûA θφmpk H A,ˆφ,ûA = θ ωφ MPK H A,ˆφ,ûA > 0 ûaˆφ πˆφ+ π + ωφ MPKûA Therefore, both23, 24 aresatisfied. Finally, notethatc.3 impliesthatmpk H A,ˆφ,ûA = MPK L A,ˆφ,ûA = MPKûA = αûa α A, and therefore ζ H A,φ,u = ζ L A,φ,u = 0. As a result, the Kuhn-Tucker conditions 26 and 27 for optimality are satisfied. C.2 Only the constraint on high productivity islands binds In this case, Q H A,φ,u > Q L A,φ,u = QA,φ,u, where ξ Q H A,φ,u = αau α πφ+ π ˆφ f φ πˆφ ξ + δ, φ ξ + π φ Q L A,φ,u = αau α πφ+ π f φ πˆφ ξ + δ. φ ξ + π The optimality condition for capital utilization implies αau α πφ+ π f φ πˆφ ξ + δ = b 0 u ν. φ ξ + π C.4 The above equation define the capital utilization rate as a function of A,φ, which we will define as u L A,φ. Let φa be the unique solution to A,φ,u L A,φ = 0, and define sa to be the highest level of s such that only the constraint on high productivity islands are binding sa = Ψ A, φa,u L A, φa. 7

8 Given the definition of u L A,φ, we can show that ΨA,φ,u L A,φ = MPK LA,φ,u L A,φ πφ+ π { θu L A,φˆφ ξ φ ξ ωφ π φ u L A,φ] + δ πφ+ π { θu LA,φφ ωφ π φ u L A,φ]}, } and ] A,φ,u L A,φ = MPK L A,φ,u L A,φ ωφ θˆφ ξ φ ξ + δθ ωφ. Using the above expressions, we can prove that ΨA,φ,u L A,φ is strictly decreasing in φ and A,φ,u L A,φ is a strictly increasing function of φ. As a result, i φ φa if and only if A,φ,u L A,φ 0; ii φ φa if and only if ΨA,φ,u L A,φ Ψ A, φa,u L A, φa. We can now prove the second part of Proposition 2 by verifying the following claim. Claim A.2. If ŝa s sa then the optimal policy φa,s is implicitly defined by the unique solution to ΨA,φ,u L A,φ = s. C.5 Given φa,s, the optimal policy ua,s is given by ua,s = u L A,φA,s. C.6 Proof. First, by construction, ΨA,φ,u L A,φ = s and 23 holds with equality. Also, the assumption that s sa implies φ φa and A,φ,u L A,φ 0; therefore, 24 is satisfied. Finally, condition C.6 implies MPKu L A,φ = MPK L A,φA,s,u L A,φ and ζ L A,φ,u L A,φ = 0; therefore, the Kuhn-Tucker conditions 26 and 27 are satisfied. C.3 Both constraints bind In the case where both constraints are binding, from = 0, we can express MPK as a function of A,φ,u: { MPK = θ ˆφ ξ } φ ξ MPK L A,φ,u θ ω δ. ω φ The optimality condition for capital utilization, 9 implies { θ ˆφ ξ } φ ξ MPK L A,φ,u θ ω δ ω φ + δ = b 0 u ν. C.7 8

9 quation C.7 defines u as a function A,φ, which we will denote as u HL A,φ. The fact the the constraint for H binds implies ΨA,φ,u HL A,φ = s. Part three of Proposition 2 can therefore be proved as the result of the following claim. Claim A.3. For s < sa, the optimal policy {φa,s,ua,s} are ointly determined by: ΨA,φ,u HL A,φ = s, ua,s = u HL A,φA,s. C.8 Proof. Clearly, by construction, both 23, 24 holds with equality. Also, we can show that ua,s < u L A,φA,s; therefore, MPKu < MPK A,φA,s,u, with u = u HL A,φA,s for = H,L. As a result, the Kuhn-Tucker conditions 26 and 27 are satisfied with ζ A,φ,u L A,φ > 0 for = H,L. To prove Proposition 3, note that the optimization problem of both households and banks are standard convex programing problems. Therefore, first order conditions are equivalent to optimality. D Recursive policy function iteration In this section, we describe an operator that maps the space of equilibrium functionals into itself such that if a fixed point for the operator exists, it constitutes a Markov equilibrium described in Section of the paper. Although the construction of the operator may not be unique, our procedure is aimed toward numerical efficiency, because it leads naturally to a recursive approach to compute the equilibrium functionals. D. The iteration procedure First, we observe that Proposition 2 allows us to determine the policy functions φz and uz without any iteration. Second, given an initial guess of next period consumption, cz and the value of bank net worth, µz, we can use the intertemporal uler equation 30 to determine the current period consumption and investment policies and use the envelop condition 33 to determine the current period value of bank net worth. At the same time, we need to verify that the policy functions and the law of motion of the state variable, 37 are consistent with each other. Because both 30 and 33 are discounting relationships, it is reasonable to expect that if we iterate this procedure, the policy functions, cz and µz will converge. Below are the details.. Using Proposition 2 to construct the policy functions φa,s and ua,s. Note that the discussion in Section C implies that the policy functions for φ and u are only functions of A, s and can be computed independently of the iterations. 9

10 2. Starting from an initial guess of the equilibrium functionals { c 0 z,µ 0 z }. 3. Given a set of equilibrium functionals, {c n z,µ n z}, we use the equilibrium conditions 28, 30, 33, and 34 to solve the optimal investment policy ix,s and the endogenous law of motion of the state variable s = Γs,x. Update the equilibrium functional using the above policy function to compute { c n+ z,µ n+ z }. We provide the details of this step of calculation in the next section. 4. Iterate on step 3 and 4 until the error is smaller than a preset convergence criteria, ε, i.e., sup z c n+ z c n z +supz µ n+ z µ n z < ε. The advantage of our approach is that it makes full use of the first order optimality conditions to improve numerical efficiency. In addition, thanks to the simplification of Proposition 2, the dependence of policy functions on the occasionally binding limited enforcement constraints is fully determined before any iteration. D.2 Updating policy functions The first order and envelope conditions Given the policy functions φa,s and ua,s, we can represent the prices for capital and the Lagrangian multipliers as functions of state variables A,s, by using equations 7, 8, 9, 5, 26, and 27. With a slight abuse of notation, we denote these pricing functionals as {MPK A,s} =H,L, MPKA,s, {Q A,s} =H,L, QA,s, and { ζ A,s }. Using the above pricing functionals, we can combine the first order condition =H,L 30 and the envelope condition 33 as M { +ζ H A,s +ζ L A,s }] R f = M { + ω ζ H A,s +ζ L A,s ]} Q A,s ]. D. We start with an initial guess of policy functions of normalized consumption and marginal value of net worth, c n x,s and µ n x,s and construct the SDF using c n x,s as the next period consumption policy: where we denote M, = β C C = βcx,s c n x,s K K = ma,s = Au α A,sf φa,s, βma,s i] c n x,s δa,s+i], δa,s = h ua,s+ δua,s. 0

11 Because the risk-free interest rate R f satisfies 28, we have: R f = βma,s i] c n x,s δa,s+i] ]. D.2 Assuming that µ n x,s is the marginal value of net worth in the next period, the left-hand side of D. can now be written as: = ] βma,s i] c n x,s δa,s+i] { Λ +Λ µ n x,s }{+ζ H A,s +ζ L A,s } βma,s i] c n x,s δa,s+i] { Λ +Λ µ n x,s } c n x,s {+ζ H A,s +ζ L A,s } Similarly, the RHS of of D. is c n x,s ]. βma,s i]{ Λ +Λ µ n x,s } { + ω ζh A c n x,s,s +ζ δa,s+i] L A,s ]} Q A,s ] Therefore, equation D. can be written as: ] ] = βma,s i]{ Λ +Λ µ n x,s } { + ω ζh A c n x,s,s +ζ δa,s+i] L A,s ]} Q A,s ] ] { Λ +Λ µ n x,s } c n x,s {+ζ H A,s +ζ L A,s } ], D.3 c n x,s or ma,s i δa,s+i] = β ] { Λ +Λ µ n x,s } c n x,s {+ζ H A,s +ζ L A,s } ] { Λ +Λ µ n x,s } c n x,s {+ ωζ H A,s +ζ L A,s ]}QA,s D.4 c n x,s ]. The law of motion of state variables Let N denote the aggregate net worth of the bankingsector in the current period. Because only a fraction Λ of the banks survive to the next period, total bank net worth in the period, N is N = Λ {πn H + πn L }. Using equations 2, πn H + πn L = π { Q H K +RA H ] Q RA H R f B f } + π { Q L K +RA L ] Q RA L R f B f } = πq H K +RA H ] + πq L K +RA L ] Q πra H + πra L ] R f B f D.5

12 Using K H = K +RA H and K L = K +RA L, we write the first two terms of the above equation as πq H K H + πq L K L = πmpk H K H + πmpk L K L + δ πk H + πk L]. D.6 Using the fact that capital share is α, πmpk H K H + πmpk L K L = αy. Together with πk H + πk L = u K, the equation D.6 can be written as: πq H K H + πq L K L = αy + δu K. The resource constraint 7 implies πra H + πra L = u K. We can combine the first three terms in D.5 as: πq H K H + πq L K L Q πra H + πra L ] = αy + δu K Q u K D.7 Because Q u K = MPK + δ ] u K = MPK u K + δ u K, equation D.6, or equivalently, D.7 can now be simplified to πq H K H + πq L K L Q πra H + πra L ] = αy + u MPK K + δk. Therefore, D.5 can be simplified as: πn H + πn L = αy + u MPK K + δk R f B f. We have: N = Λ αy + u MPK K + δk R f B f ]. D.8 Because B I = 0 in equilibrium, banks budget constraint, implies B f = K N. Therefore, by the definition of s, s = R f B f K = R f NK ] = R f KK K K N ]. D.9 K 2

13 Using equation D.8, N = ΛαY + umpkk + δk R f B f, ], we can express all the terms on the right-hand side of the above equation as functions of the state variables: Y = ma,sk, u = ua,s, R f B f, = sk. Therefore, n = N K can be written as a function of the state variable A,s: N K = ΛαmA,s+ ua,smpka,s+ δ s] D.0 Now we can combine D.9 and D.0, and use D.2 to derive the law of motion of the state variable s: or s = { δa,s+i] ΛαmA,s+ ua,smpka,s+ δ s]} ], δa,s+i] βma,s i] c n x,s δa,s+i] s = { δa,s+i] ΛαmA,s+ ua,smpka,s+ δ s]} ] D. βma,s i] c n x,s Given the policy functions {c n x,s,µ n x,s}, equations D.4 and D. are two nonlinear equations with two unknowns, i and s. For each point in the state space, x,s, we solve these two equations for the policy function ix,s and the law of motion of next period s as a function of s,x. We update the policy functions for the next iteration by setting c n+ x,s = ma,s ix,s, { Λ µ n+ +Λ µ n x,s } { x,s = +ζh A c n x,s,s +ζ L A,s }] / c n x,s ]. References Chen, K., and. Song Financial frictions on capital allocation: A transmission mechanism of tfp fluctuations. Journal of Monetary conomics 60: isfeldt, A. L., and A. A. Rampini Capital reallocation and liquidity. Journal of Monetary conomics 53: Hopenhayn, H., and Neumeyer Productivity and distortions. Hsieh, C.-T., and P. J. Klenow Misallocation and manufacturing tfp in china and india. The Quaterly Journal of conomics CXXIV4: