Topic 2. Incorporating Financial Frictions in DSGE Models

Transcription

1 Topic 2 Incorporating Financial Frictions in DSGE Models Mark Gertler NYU April

2 Overview Conventional Model with Perfect Capital Markets: 1. Arbitrage between return to capital and riskless rate E t βλ t,t+1 R kt+1 = E t βλ t,t+1 R t+1 where βλ t,t+1 is the household s stochastic discount factor 2. FInancial structure irrelevant. 1

3 Overview (con t) With capital market frictions: 1. External finance premium E t βλ t,t+1 R kt+1 >E t βλ t,t+1 R t+1 2. Premium depends inversely on borrower balance sheets 3. If borrower balance sheets move procyclically, external finance premium move countercyclically: feedback betweeen financial and real sectors ("financial accelerator,") disturbances originating in the financial sector can have real effects. 2

4 Bernanke/Gertler/Gilchrist Financial Accelerator Model Dynamic General Equilibirum Framework with 1. Money 2. Imperfect Competition 3. Nominal Price Rigidities (Calvo staggered price setting.) 4. Financial Accelerator as in Bernanke/Gertler(1989), featuring asset price mechanism in Kiyotaki and Moore (1997) 3

5 Sectors 1. Households 2. Business Sector (a) entrepreneur/firms (b) capital producers (c) retailers 3. Central Bank 4

6 Households Objective subject to max E t X i=0 β i [log (C t+i )+a m log( M t+i P t+i ) a n 1 1+γ n L 1+γ n t+i ] (1) C t = W t P t L t + Π t T t M t M t 1 P t 1 1+i t B t B t 1 P t (2) As in Woodford (2003), we restrict attention to the cashless limit of the economy (the limit as a m 0). 5

7 Decision Rules labor supply W t P t = a n L γ n t+i /( 1 C t ) (3) consumption/saving; ( 1 C = E t (1 + i t ) P t β 1 ) t P t+1 C t+1 (4) 6

8 Entrepreneurs/Firms Produce wholesale output Competitive, risk neutral, face capital market frictions. Ameasureunityinthemarketatanytime. i.i.d survival probability θ : The expected horizon is accordingly to replace exiting entrpreneurs. 1 1 θ. 1 θ enter Exiting entrepreneurs make a small transfer to new entrepreneurs and then consume the rest. 7

9 Production Technology The production technology is given by Y t = ω t A t (K t ) α (L t ) (1 α). (5) where ω t is i.i.d with E{ω t } =1 8

10 Labor Demand F.O.N.C. W t P wt =(1 α) Y t L t 9

11 Capital Demand Gross Return to Capital E t Rkt+1 ª = Et P w+1 P t+1 α Y t+1 K t+1 +(1 δ)q t+1 Q t Opportunity Cost E t ( (1 + i t ) P ) t P t+1 10

12 Capital Demand (con t) Under perfect markets, capital demand given by With imperfect markets: ( ª E t Rkt+1 = Et (1 + i t ) P ) t P t+1 ( ª E t Rkt+1 >Et (1 + i t ) P ) t P t+1 11

13 Capital Demand (con t) The finance of capital is divided between net worth and debt: Q t K t+1 = N t + B t P t. 12

14 Costly State Verification Assume: (i) costly state verification and limited liability (ii) one period contracts (iii) payouts based only on firm-specific contingencies = : 1. Debt with costly default is optimal 2. Agency costs of external finance (expected default costs) 3. Collateral reduces expected default costs 13

15 Optimal Choice of Capital Q t K t+1 = υ( E t Rkt+1 ª E t ½(1 + i t ) P t P t+1 ¾)N t 14

16 Optimal Choice of Capital(con t) Aggregate Demand for Capital (Inverting the previous equation) with ( ª E t Rkt+1 =(1+χt )E t (1 + i t ) P ) t P t+1 and χ t = χ Ã! Qt K t+1 N t χ 0 ( ) > 0, χ(0) = 0, χ( ) = 15

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18 Evolution of Net Worth N t = θv t +(1 θ)d where V t =(1 m t )R kt Q t 1 K t " (1 + i t 1 ) P t 1 P t # Bt P t 1 with R kt = P wt P t α Y tt K tt +(1 δ)q t Q t 1 m t = μg(ω t 1 ) 17

19 EvolutionofNetWorth(con t) Main Sources of Net Worth Fluctutions Unexpected movements in Q t and P t Irving Fisher s debt-deflation hypothesis: unanticipated declines in price level raies real debt burdens. 18

20 TheRoleofLeverage Given Q t 1 K t = N t 1 + B t 1 P t 1 t V t = {[(1 m t )R kt R t ]φ t 1 + R t }N t 1 with φ t 1 = Q t 1K t N t R t =(1+i t 1 ) P t 1 P t The sensitivity of net worth to unanticipated returns is increasing in the leverage ratio φ t

21 Capital Producers Capital Producers are competitive. They produce new capital and sell at the price Q t. Evolution of capital K t+1 = Φ( I t K t )K t +(1 δ)k t Φ 0 > 0, Φ 00 < 0, Φ( I K )= I K 20

22 Optimal Choice of Investment E t 1 {Q t [Φ 0 ( I t K t )] 1 } =0 i.e.,q is increasing I t K t as in Tobin s Q theory Note: Marginal product of capital used in producing new capital goods is zero within a local region of the steady state. See BGG. 21

23 Retailers Buy wholesale output and sell as differentiated product Set prices on a staggered basis as in Calvo (1983) P t (μ P t w ) λ E t ( P t+1 P t 1 P t P t ) β in loglinear form π t = λ(p wt p t )+βe t π t+1 Note: p t p wt is the log price markup. 22

24 Resource Constraint Let C e t entrepreneurial consumption and M t total monitoring costs: Y t = C t + C e t + I t + G t + M t with C e t =(1 φ)(v t D) M t = m t R t Q t 1 K t 23

25 Monetary and Fiscal Policy Monetary Rule: i t = ρi t 1 +(1 ρ)[γ π π t + γ y (y t y n t )] + ε rn t i t = r t+1 E t π t+1 Fiscal Policy: Gov t spending exoxgenous and finance by lum sum taxes. 24

26 Investment, Finance and Monetary Policy in BGG I t /K t = φ(q t ) (6) where Ã! ( E t Rt+1 k Qt =(1+χ K t+1 ) (1 + i t ) P ) t N t+1 P t+1 E t R k t+1 = E t P w+1 P t+1 α Y t+1 K t+1 +(1 δ)q t+1 Q t (7) (8) 25

27 Investment, Finance and Monetary Policy in BGG (con t) Note: N t = θ{(1 m t )R kt Q t 1 K t (1 + i t 1 ) P t 1 P t B t P t 1 } +(1 θ)d Thus: i. Positive feedback between asset prices and investment (financial accelerator) ii. Strength depends positively on leverage ratio ratio Q t K t+1 /N t. iii. Monetary Policy has additional impact via balance sheets 26

28 LOG-LINEARIZED BGG MODEL Aggregate demand y t = C Y c t + I Y inv t + G Y g t + Ce Y ce t +... c t = σr t+1 + E t c t+1 c e t = 1 φ φ n t+1 27

29 (inv t k t )=ϕq t E t r kt+1 =(1 ϑ)e t (p wt+1 p t+1 + y t+1 k t+1 )+ϑe t q t+1 q t E t r kt+1 r t+1 = v(n t q t k t+1 ) 28

30 LOG-LINEARIZED BGG MODEL (con t) Aggregate supply y t = a t + αk t +(1 α)l t y t l t = μ t + γ l l t + c t π t = κ(p wt p t )+βe t π t+1 29

31 LOG-LINEARIZED BGG MODEL (con t) Evolution of state variables k t+1 = δinv t +(1 δ)k t n t = θrk N [rk t r t ]+θr(r t + n t 1 ) with r r = i t 1 π t 1 30

32 LOG-LINEARIZED BGG MODEL (con t) Monetary Policy Rule i t = ρi t 1 +(1 ρ)[γ π π t + γ y (y t y n t )] + ε rn t i t = r t+1 E t π t+1 31

33 Calibrating Financial Sector Parameters Choose (i) survival probability θ, (ii) monitoring costs μ, and (iii) the moments of the idiosyncratic shock to match evidence on: 1. Steady state external finance premium: R k /R.. 2. Steady state leverage ration QK/N 3. Annual business failure rate. 32

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