APPENDIX TO RESERVE REQUIREMENTS AND OPTIMAL CHINESE STABILIZATION POLICY

APPENDIX TO RESERVE REQUIREMENTS AND OPTIMAL CHINESE STABILIZATION POLICY CHUN CHANG ZHENG LIU MARK M. SPIEGEL JINGYI ZHANG Abstract. This appendix shows some additional details of the model and equilibrium conditions in Chang Liu Spiegel and Zhang (217). Date: March 19 218. Key words and phrases. Reserve requirements China off-balance sheet loans shadow banking financial accelerator reallocation aggregate productivity welfare. JEL classification: E44 E52 G28. Chang: Shanghai Advanced Institute of Finance Shanghai Jiao Tong University; Email: cchang@saif.sjtu.edu.cn. Liu: Federal Reserve Bank of San Francisco; Email: Zheng.Liu@sf.frb.org. Spiegel: Federal Reserve Bank of San Francisco; Email: Mark.Spiegel@sf.frb.org. Zhang: Shanghai Advanced Institute of Finance Shanghai Jiao Tong University; Email: jyzhang.11@saif.sjtu.edu.cn. The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of San Francisco or the Federal Reserve System. 1

APPENDIX: RESERVE REQUIREMENTS 2 I. Optimizing conditions I.1. Household decisions. The household optimizing conditions are summarized by the following equations: Λ t = 1 C t w t = ΨHη t Λ t 1 = E t βr t Λ t+1 Λ t π t+1 (1) (2) (3) Ψ m 1 = + E t β Λ t+1 Λ t M t /P t Λ t π t+1 [ 1 = qt k 1 Ω ( ) 2 ( ) ] k It It It g I Ω k g I 2 I t 1 I t 1 I t 1 + βe t q k t+1 ( Λ t+1 It+1 Ω k g I Λ t I t ) ( It+1 q k t = βe t Λ t+1 Λ t [q k t+1(1 δ) + r k t+1]. (6) where Λ t denotes the Lagrangian multiplier for the budget constraint π t = Pt P t 1 denotes the inflation rate from period t 1 to period t and qt k Λk t Λ t is Tobin s q with Λ k t being the Lagrangian multiplier for the capital accumulation equation. I t (4) ) 2 (5). I.2. Optimal debt contracts. Under the loan contract characterized by ω jt and B jt the expected nominal income for a firm in sector j is given by ω jt à jt ω jt (N jt 1 + B jt )df (ω) (1 F (ω jt ))Z jt B jt = Ãjt(N jt 1 + B jt )[ Ãjt(N jt 1 + B jt )f(ω jt ) ω jt ωdf (ω) (1 F (ω jt ))ω jt ] where f(ω jt ) is the share of production revenue going to the firm under the loan contract. The expected nominal income for the lender is given by (1 F (ω jt ))Z jt B jt + ωjt {(1 m j )Ãjtω(N jt 1 + B jt ) +l j [Z jt B jt (1 m j )Ãjtω(N jt 1 + B jt )]}df (ω) = Ãjt(N jt 1 + B jt ){[1 (1 l j )F (ω jt )] ω jt + (1 m j )(1 l j ) ωt ωdf (ω)} Ãjt(N jt 1 + B jt )g jt (ω jt ) (7)

APPENDIX: RESERVE REQUIREMENTS 3 where g j (ω jt ) is the share of production revenue going to the lender. Note that ωjt ωjt f(ω jt ) + g jt (ω jt ) = 1 m j ωdf (ω) + l j [ω jt (1 m j )ω]df (ω). (8) The optimal contract is a pair ( ω jt B jt ) chosen at the beginning of period t to maximize the borrower s expected period t income max Ãjt(N jt 1 + B jt )f(ω jt ) (9) subject to the lender s participation constraint à jt (N jt 1 + B jt )g j (ω jt ) R jt B jt. (1) The optimizing conditions for the contract characterize the relation between the leverage ratio and the productivity cut-off N jt 1 = g j(ω jt ) B jt + N jt 1 f (ω jt ) à jt f(ω jt ) R jt. (11) II. Summary of equilibrium conditions On a balanced growth path output consumption investment real bank loans and real wage rates all grow at a constant rate g. To obtain balanced growth we make the stationary transformations y f t = Y f t g γ t mt = Γ t g c t t = C t g i t t = I t g k t t = K t g gdp t t = GDP t λ g t t = Λ t g t y st = Y st g t y pt = Y pt g t k st = K st g t k pt = K pt g t w t = w t g w t set = w set g t n st = N st P t g t n pt = N pt P t g t b st = B st P t g t b pt = B pt P t g t m t = M t P t g t ms t = M s t P t g t. w pet = w pet g t On the balanced growth path the transformed variables the interest rate and the inflation rate are all constants. The balanced growth equilibrium is summarized by the following equations:

APPENDIX: RESERVE REQUIREMENTS 4 1) Households. k t = 1 δ g k t 1 + i t [1 Ω k 2 ( i tg i t 1 g) 2 ] λ t = 1 c t (A1) (A2) w t = Ψ Hη t λ t 1 = E t β λ t+1 λ t R t π t+1 g 1 = Ψ m λ t m t + E t β λ t+1 λ t 1 π t+1 g 1 = qt k [1 Ω k 2 ( i tg i t 1 g) 2 Ω k ( gi t i t 1 g) i tg i t 1 ] + βe t qt+1 k λ t+1 λ t g Ω k( i t+1g i t 2) Firms and banks. q k t = βe t λ t+1 λ t g [(1 δ)qk t+1 + r k t+1]. y st = A m t (A3) (A4) (A5) g)( i t+1g i t (A6) ) 2 (A7) Ā s k 1 α st (H θ st) α (A8) w t H st = αθ( n st 1 + b st ) (A9) w set = ( n st 1 + b st )α(1 θ) (A1) k st r k t = (1 α)( n st 1 + b st ) (A11) à st = p sty st n st 1 +b st (A12) à st ( n st 1 + b st )g s (ω st ) = b st R st (A13) n st 1 n st 1 +b st = g s(ω st )f(ω jt ) à st f (ω st ) R st (A14) n st = w set + ξ s à st ( n st 1 + b st )f(ω st ) (A15) y pt = A m t Ā p k 1 α pt (H θ pt) α (A16) w t H pt = αθ( n pt 1 + b pt ) (A17) w pet = α(1 θ)( n pt 1 + b pt ) (A18) r k t k pt = (1 α)( n pt 1 + b pt ) (A19) à pt = p pty pt n pt 1 +b pt (A2) à pt ( n pt 1 + b pt )g p (ω pt ) = b pt R pt. (A21)

APPENDIX: RESERVE REQUIREMENTS 5 n pt 1 n pt 1 +b pt = g p(ω pt )f(ω pt ) à pt f (ω pt ) R pt (A22) n pt = w pet + ξ p à pt ( n pt 1 + b pt )f(ω pt ) (A23) (R st 1)(1 τ t ) = R t 1 (A24) R pt = R t. (A25) 3) Pricing market clearing and monetary policy. p wt = ɛ 1 ɛ + Ω p ɛ 1 ln ( µt µ ln ( τ t τ y t [( π t π 1)π t π c λ t βe t+1 t λ t ( π t+1 π 1) π t+1 π c t+1] (A26) ) = ψ mp ln ( π t ) ( ) π + ψmy ln γt γ (A27) ) = ψτp ln ( π t ) ( ) π + ψτy ln γt γ (A28) µ t = ms t m s t 1 (A29) γ t = GDP t GDP t 1 m s t = m t + d t b pt (A3) (A31) g t = gdp t g c (A32) y f Ω t = i t + c t + g t + c p t 2 (π t π 1)2 + Ãst( n st 1 +Ãpt( n pt 1 ( γ mt = φy σm 1 σm st + b st )m s ωst ωdf (ω) + b pt )m p ωpt ωdf (ω) (A33) y f t = γ t + (1 φ)y σm 1 σm pt ) σm σm 1 (A34) (A35) ( ) y st = φ σ σm m pst p yt wt (A36) y pt = (1 φ) σ m ( ppt p wt ) σm yt (A37) k t 1 g = k st + k pt (A38) H t = H st + H pt (A39) gdp t = g t + i t + c t. (A4)

APPENDIX: RESERVE REQUIREMENTS 6 where f(ω st ) = 1 k 1 ωk mω 1 k st (A41) f (ω st ) = ω k mω k st (A42) k g s (ω st ) = ω m k 1 (1 l s)(1 m s ) + l s ω st + (1 l s )[1 (1 m s)k k 1 ]ωmω k 1 k st (A43) g s(ω st ) = l s + (1 l s )(1 m s k)ω k mω k st (A44) f(ω pt ) = 1 k 1 ωk mω 1 k pt (A45) f (ω pt ) = ω k mω k pt (A46) k g p (ω pt ) = ω m k 1 (1 l p)(1 m p ) + l p ω pt + (1 l p )[1 (1 m p)k k 1 ]ωmω k 1 k pt (A47) g p(ω pt ) = l p + (1 l p )(1 m p k)ω k mω k ωst pt (A48) ωdf (ω) = k k 1 (ω m ωmω k 1 k st ) (A49) ωpt ωdf (ω) = k k 1 (ω m ωmω k 1 k pt ). (A5) The system of 4 equations from (A1) to (A4) determine the equilibrium solution for the 4 endogenous variables summarized in the vector [y f t γ mt c t i t g t gdp t k t λ t q k t H t H st H pt y st y pt k st k pt n st n pt b st b pt Ãst Ãpt ω st ω pt m t m s t µ t γ t w t w set w pet r k t R t R st R pt π t p wt p st p pt τ t ] III. Macro effects of SOE-specific TFP shocks In the text we discuss the macro implications of an aggregate TFP shock and a POEspecific TFP shock. Here we summarize our findings about the macro effects an SOE-specific TFP shock. Table 1 shows the volatilities of several macroeconomic variables and welfare under the benchmark policy regime as well as under the three alternative optimal policy rules. The table shows that in response to a positive SOE TFP shock the individually optimal τ rule and µ rule both call for a tightening of monetary policy. Since the shock raises GDP and lowers inflation under the benchmark policy the central bank raises τ in response to the

APPENDIX: RESERVE REQUIREMENTS 7 increase in GDP growth and lowers τ to accommodate the decline in inflation. Similarly when the central bank can optimally choose the reaction coefficients in the µ rule it reduces money growth in response to the acceleration in GDP growth and raises money growth to mitigate declines in inflation. Under the jointly optimal rule the optimal reaction coefficients in both policy rules indicate more aggressive monetary policy responses. Acting alone each of the two alternative optimal rules (i.e. the τ rule and the µ rule) can better stabilize real GDP and inflation fluctuations and also improve welfare relative to the benchmark policy regime. The jointly optimal rule leads to further welfare improvement relative to the individually optimal rules. In this sense adjusting reserve requirements is a complementary policy instrument to the conventional policy instrument (money supply growth) in the economy buffeted by SOE-specific TFP shocks. This finding is qualitatively similar to that under POE-specific TFP shocks as we discuss in the text. In both cases the sector-specific shocks call for resource reallocation across sectors and the use of reserve requirements in combination with money supply adjustments helps enhance welfare and macro stability. Figure 1 shows the impulse responses to a positive SOE TFP shock. In the benchmark economy with a constatnt τ and calibrated parameters in the µ rule (the black solid lines) the shock raises real GDP and lowers inflation. It also raises SOE output and reduces POE output relative to their steady-state levels. Under the optimal τ rule (the red dashed lines) the central bank responds to the increase in real GDP by tightening reserve requirements (as shown in Table 1). On the other hand the central bank also responds to the decline in inflation by easing reserve requirements which stimulates aggregate demand. Overall the leads to a small decline in τ. These policy reactions help stabilize fluctuations in both real GDP and inflation. The decline in τ also reallocates resources from the POE sector to the SOE sector which experiences a positive TFP shock. Although the lower τ reduces the funding costs for the SOE sector and thus reduces the incidence of SOE failures it also stimulates aggregate demand. Through the financial accelerator that operates in the POE sector (but not for SOEs) the expansion in aggregate demand raises POE output relative to SOE output despite the improvement in SOE productivity and the reallocation effects. Under the optimal µ rule (the blue dashed lines) the central bank reduces money supply to dampen the increase in real GDP and raises money supply to stimulate aggregate demand and therefore mitigate the declines in inflation. The policy is very effective for stabilizing both real GDP and inflation as shown in Table 1. Similar to the case with the optimal τ rule POE output is more responsive to the aggregate demand expansion than SOE output because the financial accelerator is operating in the POE sector but not in the SOE sector.

APPENDIX: RESERVE REQUIREMENTS 8 Finally under the jointly optimal rule (the magenta dashed lines) the central bank reacts to the SOE TFP shock by aggressively cutting both mony growth and reserve requirements. These policy reactions further improves welfare relative to the individually optimal policies. References Chang C. Z. Liu M. M. Spiegel and J. Zhang (217): Reserve Requirements and Optimal Chinese Stabilization Policy Federal Reserve Bank of San Francisco Working Paper 216-1.

APPENDIX: RESERVE REQUIREMENTS 9 Table 1. Volatilities and welfare under alternative policy rules: SOE-specific TFP shock Variables Benchmark Optimal τ rule Optimal money rule Jointly optimal rule Policy rule coefficients ψ mp.65.65 54.63 128.1 ψ my.3.3 3.11 62.29 ψ τp. 49.58. 1. ψ τy. 7.56. 69.25 Volatility and welfare GDP 1.46% 1.425% 1.413% 1.43% π.56%.86%.32%.45% C 1.139% 1.13% 1.86% 1.93% H.289%.283%.262%.32% R.139%.157%.136%.153% Y s 7.34% 7.329% 7.257% 7.64% Y p 1.423% 1.352% 1.337% 1.577% Welfare gains.26%.265%.272%

APPENDIX: RESERVE REQUIREMENTS 1 Impulse responses to SOE TFP shock.35 Real GDP.2 Inflation.3.25 -.2 -.4.2 5 1 15 2 -.6 5 1 15 2 5 Money growth rate 5 Required reserve ratio -5-1 -5-15 5 1 15 2-1 5 1 15 2 2.5 SOE output.2 POE output 2 1.5 -.2 1 -.4.5 5 1 15 2 -.6 5 1 15 2 1 SOE bankruptcy ratio 1 POE bankruptcy ratio -1-2 -3 5 1 15 2 Benchmark -1 Optimal money supply rule Optimal rule Jointly optimal rules -2 5 1 15 2 Figure 1. Impulse responses to a positive SOE TFP shock under alternative policy rules. Benchmark rule: black solid lines; optimal money growth rule: blue dashed lines; optimal reserve requirement rule: red dashed lines; jointly optimal rule: magenta dashed lines. The horizontal axes show the quarters after the impact period of the shock. The units on the vertical axes are percent deviations from the steady state levels.