Financial Frictions in DSGE Models

Transcription

1 Financial Frictions in DSGE Models Noah Williams University of Wisconsin-Madison Noah Williams (UW Madison) New Keynesian model 1 / 1

2 Overview Conventional Model with Perfect Capital Markets: 1. Arbitrage between return to capital and riskless rate E t t;t+1 R kt+1 = E t t;t+1 R t+1 where 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 nance premium ) E t t;t+1 R kt+1 > E t t;t+1 R t+1 2. Premium depends inversely on borrower balance sheets ) 3. If borrower balance sheets move procyclically, external nance premium move countercyclically: ) feedback betweeen nancial and real sectors (" nancial accelerator,") ) disturbances originating in the nancial sector can have real e ects. 2

4 Objective Illustrate the following key concepts: 1. Asymmetric information and/or costly contract enforcement as foundations of nancial market imperfections 2. Premium for external nance 3. Rationing vs. non-rationing equilibria 4. Balance sheets and the external nance premium 5. Idiosyncratic risk and the external nance premium. 1

5 Objective (con t) Illustrate with two simple models: 1. Costly State Veri cation Model (CSV) (Townsend, 1979) 2. Costly Enforcement Model 2

6 Basic Environment Two Periods: 0 and 1: Risk Neutral Entrepreneur: Has project that requires funding in 0 and pays o in 1: Competive Risk Neutral Lender: Has opportunity cost of funds R: 3

7 Basic Environment (con t) Project Finance: QK = N + B K Capital Input N Entrepreneurs s Net Worth (Equity Finance) B Debt Finance 4

8 Basic Environment (con t) Period 1 Payo e!r k QK R k Average Gross Return on Capital e! Idiosyncratic Shock Entrepreneur takes e!r k as given, but K is a choice variable. 5

9 Basic Environment (con t) Idiosyncratic Shock Distribution: Efe!g = 1 e! [!;!] H(!) = prob(e!!) h(!) = dh d! 6

10 Perfect Information and Perfect Contract Enforcement Given E e!r k = R k ; entrepreneurs operates if R k R where R is the opportunity cost. If R k > R; entrepreneur s demand for funds is in nite Competitive market forces ) R k = R in equilibrium. Miller-Modigliani theorem applies: Real Investment Decision is independent of nancial structure Financial Structure is indeterminate 7

11 Private Information and Limited Liability Private Information: Only entrepreneurs can costlessly observe returns. Lenders must pay a cost equal to a xed fraction of the realized return!r k K: Interpretable as a bankruptcy cost. Limited Liability: Entrepreneurs minimum payo bounded at zero. 8

12 Private Information and Limited Liability (con t) Implications: Entrepreneur has incentive to misreport returns. Financial structure matters to real investment decisions, due to expected bankruptcy costs. Financial structure determinate: Designed to reduce expected bankruptcy costs. 9

13 Entrepreneur s Optimization Problem: 1. Investment Decision (choice of K) 2. Financial contract: payment schedule baced on! and decision to monitor 3. Constraint: Lender must receive opportunity cost in expectation. 10

14 Optimal Contract 1. Induce Truth-Telling (revelation principle) ) 2. Minimize Expected Monitoring Costs Optimal Contract is Standard Debt: i.e, Debt with bankruptcy 11

15 Optimal Contract (con t) Let D face value of debt and! the cuto value of! D =! R k QK The contract then works as follows: If!! : Lender s payo is D =! R k QK; Borrower s payo is (!! )R k QK If! <! ; The borrower announces default and then the lender monitors. Lender s payo is (1 )!R k K; Borrower s payo is 0: Observe that the deadweight bankruptcy cost is!r k QK: 12

16 Intuition for Optimal Contract Optimal Contract (con t) 1. There is no incentive for the entrepreneur to lie: In non-default states the payment to lenders is xed In default states there is monitoring. 2. Expected bankruptcy costs are minimized. By giving the lender everything in the default state, the non-default payment D is minimized. Given D =! R k K, the bankruptcy probability H(! ) is which is increasing in D. H(! D ) = H( R k QK ) 13

17 Optimal Contract (con t) Given the form of the optimal contract) Lender s expected payment: Z! Z!!! R k QKdH + (1 )!R kqkdh [ (! ) G(! )]R k QK! with (! ) = Z! Z!!! dh +!dh! =! [1 H(!)] + Z!!!dH G(! ) = Z!!!dH 14

18 Optimal Contract (con t) (! ) is increasing and concave 0 (! ) = 1 H(! ) > 0 00 (! ) = h(! ) < 0 G(! ) is increasing and convex, assuming! h(! ) is increasing G 0 (! ) =! h(! ) > 0 G 00 (! ) > 0 (! ) G(! ) is increasing so long as the default prob H(! ) is not too large 0 (! ) G 0 (! ) = 1 H(! )! h(! ) which is positive under reasonable values for H(! ); and! h(! ) 15

19 Entrepreneur s Decision Problem Objective: subject to max! ;K fmaxf[1 (! )]R k QK RN; 0gg [ (! ) G(! )]R k QK = R(QK N) constraint multiplier = shadow value of N 16

20 Entrepreneur s Decision Problem Combining equations: subject to max! ;K fmaxf[r k R G(! )R k ]QK; 0gg [ (! ) G(! )]R k QK = R(QK N) where G(! )R k expected default costs (related to premium for external nance) 17

21 Entrepreneur s Decision Problem (con t) F.O.N.C:! = 1 + G 0 (! ) 0 (! ) G 0 (! ) K R k f[1 (! )] + [ (! ) G(! )]g R = 0 [ (! ) G(! )]R k = R(1 N K ) 18

22 Entrepreneur s Decision Problem (con t) Given 0 (! ) 0 G(! ) > 0 ) three observations: 1. is increasing in! (from FONC for! ) 2.! increasing in R k =R (from FONC for K) 3. f[1 (! )]+[ (! ) G(! )]g > 1 is the premium for external nance. Note that the premium is increasing in! : 19

23 Optimal Choices of! and K The following two equations determine! and QK : Lender s voluntary participation constraint: Optimal Choice of Captial [ (! ) G(! )]R k = R(1 N QK ) R k (! )R = 0 with (! ) = (! ) f[1 (! )] + (! )[ (! ) G(! )]g > 1; 0 (! ) > 0 20

24 The Demand for Capital and Net Worth Inverting the lender s voluntary participation constraint: QK N = 1 1 [ (! ) G(! )]R k =R! is increasing in R k =R from FONCs for! and K: ) with QK N = (R k R ) 0 ( R k R ) > 0 21

25 Aggregate Demand for Capital and Financial Crises Capital demand QK = ( R k R )N where ( R k R ) is the optimal leverage ratio. ( R k R ) does not depend on rm speci c factors ) Can aggregate capital demand across entrepreneurs: QK = ( R k R )N where N is aggregate net worth and K is aggregate capital demand. Financial Crisis: Sharp drop in N or in ( R k R ) that reduces QK: 22

26 Inverting yields Balance Sheet Strength and the Spread with R k R = (QK N ) 0 ( QK N ) > 0 where is the gross spread. Thus, in the market equilibrium, the spread is inversely related to aggregate balance sheet strength ) during a crisis the balance sheet weakens and the spread increases. 23

27 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

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

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

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

31 Produce wholesale output Entrepreneurs/Firms Competitive, risk neutral, face capital market frictions. A measure unity in the market at any time. i.i.d survival probability : The expected horizon is accordingly 1 1 : 1 enter to replace exiting entrepreneurs. Ensures borrowers do not save their way out of the nancial constraint. (A way of modeling dividend payouts). optimal to retain earnings until exit. consume wealth upon exit. Exiting entrepreneurs make a small transfer to new entrepreneurs and then consume the rest. 7

32 Production Technology Gross rm output GY t ( rm output Y t plus leftover rm capital): with GY t =! t [A t (K t ) (L t ) (1 ) + (1 )K t ]: (5) Y t =! t A t (K t ) (1 ) (L t ) where! t is i.i.d across rms and across time, with Ef! t g = 1 8

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

34 Capital Demand Gross Return to Capital E t Rkt+1 = E t 8 >< >: P w+1 P t+1 Y t+1 K t+1 + (1 )Q t+1 Q t 9 >= >; Opportunity Cost E t ( (1 + i t ) P ) t P t+1 10

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

36 Capital Demand (con t) The nance of capital is divided between net worth and debt: N t is accumulated via retained earnings. Q t K t+1 = N t + B t P t : 12

37 Costly State Veri cation Assume: costly state veri cation and limited liability one period loan contracts between bank and rm entrepreneurs absorb aggregate risk: banks diversify idiosyncratic risk! households receive sure nominal return from banks no need for households to monitor banks =) 1. (Aggregate state-contingent) debt with costly default is optimal 2. Agency costs of external nance (expected default costs) 3. Net worth reduces expected default costs 13

38 Costly State Veri cation with Aggregate Risk R kt+1 eu t+1 R kt+1 eu t+1 aggregate risk R kt+1 E t fr kt+1 g Ex post return on capital Expected real return on deposits R t+1 Deposits o er sure nominal return R kt+1 =! t+1 eu t+1 R kt+1 Q t K t R t+1 = (1 + i t )E t ( P t P t+1 ) 14

39 Costly State Veri cation with Aggregate Risk (con t) Given eu, bank s expected debt payo = opportunity cost [ (! ) G(! )]euqk = R(QK N) (! ) = [1 H(!)]! + G(! ) = Entrepreneur s objective: Z!!!dH Z!!!dH max K where! is chosen ex post and K ex ante. Efmax! f[1 (! )]eur k QK RN; 0gg 15

40 Entrepreneur s Decision Problemg Objective: subject to max K Efmax! f[ eur k R G(! )eur k ]QK; 0gg [ (! ) G(! )]eur k QK = R(QK N) (eu) (state-contingent) constraint multiplier = state-contingent shadow value of net worth 16

41 Optimal Choices of! and K Optimality conditions: [ (! ju ) G(! ju )] eur k = R(1 N QK ) R k R = 0 = Ef(! ju )g (! ju ) = (! ju ) f[1 (! ju )] + (! ju )[ (! ju ) G(! ju )]g > 1; 0 (! ) > 0 (eu) = 1 + G 0 (! ju ) 0 (! ju ) G0 (! ju ) 17

42 Optimal Choice of Capital Take the expectation of the lenders vpc and combine equations (see lecture 6) Q t K t+1 = ( R kt+1 R t+1 )N t ; 0 () > 0 Aggregate Demand for Capital (Inverting the previous equation) with R kt+1 = t R t+1 t = Q! tk t+1 N t 0 () > 0; (0) = 1; (1) = 1 18

43 Evolution of Net Worth N t = V t + (1 )X where V t = (1 m t )R kt Q t 1 K t " (1 + i t 1 ) P # t 1 Bt P t P t 1 and X = total transfers to new entrepreneurs, with R kt = P wt P t Y tt K tt + (1 Q t 1 )Q t m t = G(! t 1 ) where G(! t 1 ) is total defaults, so that.m tr kt Q t 1 K t is total default costs. 19

44 Evolution of Net Worth (con t) Main Sources of Net Worth Fluctutions Unexpected movements in Q t and P t Irving Fisher s debt-de ation hypothesis: unanticipated declines in price level raises real debt burdens. 20

45 The Role of Leverage Given Q t 1 K t = N t 1 + B t 1 P t 1 t V t = f[(1 m t )R kt R t ] t 1 + R t gn t 1 with t 1 = Q t 1K t N t 1 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 t 1 :. 21

46 Capital Producers Capital Producers are competitive. They produce new capital and sell at the price Q t. Evolution of capital K t+1 = ( I t K t )K t + (1 )K t 0 > 0; 00 < 0; ( I K ) = I K 22

47 Optimal Choice of Investment no lags Q t 0 ( I t K t ) = 0 Q is increasing I t K t as in Tobin s Q theory planning lags E t 1 fq t [ 0 ( I t K t )] 1 g = 0 I t picked at t-1 based on expected Q t : (to get investment delays to shocks) Note: Marginal product of capital used in producing new capital goods is zero within a local region of the steady state. See BGG.) 23

48 Retailers Buy wholesale output and sell as di erentiated product Set prices on a staggered basis as in Calvo (1983) P t P t ( P t w ) E t ( P t+1 1 P t P t ) in loglinear form t = (p wt p t ) + E t t+1 Note: p t p wt is the log price markup. 24

49 Resource Constraint and Asset Markets Ct e entrepreneurial consumption ; M t total monitoring costs: Y t = C t + Ct e + I t + G t + M t with Ct e = (1 )(V t X) M t = m t R t Q t 1 K t Bank balance sheet B t = D t 25

50 Monetary and Fiscal Policy Monetary Rule: i t = i t 1 + (1 )[ t + y (y t y n t )] + " rn t i t = r t+1 E t t+1 Fiscal Policy: Gov t spending exoxgenous and nance by lum sum taxes. 26

51 Investment, Finance and Monetary Policy in BGG I t =K t = (Q t ) (6) E t Rt+1 k = Q! ( tk t+1 (1 + i t ) P ) t N t+1 P t+1 (7) where E t R k t+1 = E t 8 >< >: P w+1 P t+1 Y t+1 K t+1 + (1 )Q t+1 Q t 9 >= >; (8) 27

52 Investment, Finance and Monetary Policy in BGG (con t) Note: N t = f(1 m t )R kt Q t 1 K t (1 + i t 1 ) P t 1 P t B t P t 1 g + (1 )D Thus: i. Positive feedback between asset prices and investment ( nancial 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 28

53 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 = r t+1 + E t c t+1 c e t = 1 n t+1 29

54 (inv t k t ) = 'q t E t r kt+1 = (1 #)E t (p wt+1 p t+1 + y t+1 k t+1 ) + #E t q t+1 q t E t r kt+1 r t+1 = v(n t q t k t+1 ) 30

55 LOG-LINEARIZED BGG MODEL (con t) Aggregate supply y t = a t + k t + (1 )l t y t l t = t + l l t + c t t = (p wt p t ) + E t t+1 31

56 LOG-LINEARIZED BGG MODEL (con t) Evolution of state variables k t+1 = inv t + (1 )k t n t = RK N [rk t r t ] + R(r t + n t 1 ) with r r = i t 1 t 1 32

57 LOG-LINEARIZED BGG MODEL (con t) Monetary Policy Rule i t = i t 1 + (1 )[ t + y (y t y n t )] + " rn t i t = r t+1 E t t+1 33

58 Calibrating Financial Sector Parameters Choose (i) survival probability ; (ii) monitoring costs ; and (iii) the moments of the idiosyncratic shock to match evidence on: 1. Steady state external nance premium: R k =R:. 2. Steady state leverage ration QK=N 3. Annual business failure rate. 34

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63 Figure 1: DSGE forecasts of the Great Recession SWπ SWFF Output Growth Inflation Notes: The figure is taken from Del Negro and Schorfheide (2013). The panels show for each model/vintage the available real GDP growth (upper panel) and inflation (GDP deflator, lower panel) data (black line), the DSGE model s multi-step (fixed origin) mean forecasts (red line) and bands of its forecast distribution (shaded blue areas; these are the 50, 60, 70, 80, and 90 percent bands, in decreasing shade), the Blue Chip forecasts (blue diamonds), and finally the actual realizations according to the May 2011 vintage (black dashed line). All the data are in percent, Q-o-Q.shows the filtered mean of λ t (solid black line) and the 50%, 68% and 90% bands in shades of blue.