A model of working capital with idiosyncratic production risk and rm failure

Transcription

1 A model of oking capital ith idiosyncatic poduction isk and m failue Pof. Mcandless UEMA Novembe, 2 Outline of the talk Intoduction Model Stationay states Dynamic vesion of model onclusions Intoduction Pat of a eseach pogam fo modelling banking systems A simple model ith banks and isky assets Whee a faction of the ms fail Banks need to hold loan loss eseves Need isk avese m manages So less than half the ms fail The model: Fims Fim manages ae like eveyone else except Fim po ts ente thei utility functions X E t= t ln c i t + Bh i t + G( k t )

2 subject to k t = t ' k t kt k h k t t kt k f t t h k t ' k t 2 ' l ; ' u ith a unifom distibution f t is the inteest on oking capital to pay the age bill Given that the utility function is sepaable, ms manages max X E t= t G( k t ) subject to the budget constaint The model: Fims With a unifom distiubtion, the expected utility maximization poblem can be itten max k k t ;hk t ' u ' l Z ' u G( t ' j t k k t h k t t kt k f t t h k t )dj: ' l We use a G() ith constant absolute isk avesion G(x) = ( exp ( x)) is the coe cient of absolute isk avesion Solving the max poblem is ugly The model: Fims FOs give (afte simpli cation) t k k t = f t t h k t ( ) T t = y l t ' u ' l e ('u ' l )y t e ('u ' l )y t + T L t = T t ' l y t 2 2 (' u ' l ) y t D t = ('u y t T t ) 2 2 (' u ' l ) y t The model: Financial intemediaies (banks) 2

3 Take deposits (in money) fom households Lend money to ms fo oking capital ompetitive Zeo po t condition d t N t = f t P t t t P t T L t Equilibium condition fo capital maket N t + ( ) (g t ) M t = P t t t ( ) (g t ) M t is the faction of money goth that goes to the nancial system The model: ouseholds Repesentative household maximizes X t ln (c t ) + h t A ln ( h ) h t= h t =h is the pobability that this family ill be equied to supply h units of labo subject to the budget constaint m t P t + k t+ = t h t + t k t + ( )k t + d t + d t n t P t and the cash-in-advance constaint P t c t = m t + (g t ) M t n t d t is the lump sum dividend payment to the household fom the ms The model: ouseholds FOs ae hee t = E t t+ ( t+ + ( t = Bt d c t P t c t = E t P t+ c t+ d t B A ln ( h ) h )) 3

4 Figue : Stationay state values fo g = The model: Equilibium conditions Since all households ae alike t = h t M t = m t K t = k t D t = d t t = c t N t = n t t = y t Maket cleaing in the oking capital makets N t + ( ) (g t ) M t = P t t t Stationay states, g=, alpha = 4 Stationay states, g=, alpha = 4 Stationay states, g=: Output and alpha (ARA coe cient) Stationay states, g=: Fim failue and alpha (ARA coe cient) Dynamic vesion of the model Log-lineaization of the model (aound stationay state) Use method of undetemined coe cients (a la Uhlig) to solve Find linea policy functions of the fom x t = P x t + Qz t y t = Rx t + Sz t 4

5 Figue 2: Moe stationay state values fo g =.3.2. α = α =.5 α = α = 2 α = 4.6 α = 8.5 α = Figue 3: Stationay state output and isk avesion 5

6 .6.55 α = α =.5 α = α = α = α = 6 α = Figue 4: Faction of ms that fail in the stationay state h hee x t = ekt+ ; M f t ; P e i t, yt = h i z t = et ; eg t using = Ax t + Bx t + y t + Dz t ; h e t ; e t ; e t ; e t ; e t ; N e t ; ; T g i, t ; e t d ; e f t and = E t [F x t+ + Gx t + x t + Jy t+ + Ky t + Lz t+ + Mz t ] ; z t+ = Nz t + " t+ ; = e t E t e t+ + E t e t+ ; = e d t e t + e t ; = e t + P e t E tpt+ e E tt+ e ; = M=P M f h i t + n N=P M=P ept + KK e t+ ( e t + e t ) Ke t ( + )K e K t D e D t d N=P e N t d N=P e d t ; idthheighteqnaay* = ept + e t N=P N e t ; = T T g 'l ' u ' l e u (' 'l ) + e e ('u ' l ) = g T t e t e Kt ; ( + g ) M=P g ( ' u ' l ) (' u ' l ) 2 2 A t e ; e ('u ' l ) 2 fm t M=P eg t + 6

7 3 x 2 5 T d f idthheighteqnaay* = T g t e f t e t t e ; = T LT f T (T ' L l ) t gt (' u ' l ) t + T 2 (' l ) 2 e 2(' u ' l ) t ; = DD e t + T ('u T ) (' gt u ) 2 T 2 (' u ' l ) t e 2(' u ' l ) t = e t t e K e t ( ) e t ; = f e f d N=P t e d t + N e t Pt e T L ft L t + f = N=P N e t e t + e h t ( ) g + ( ) M=P M f t + ( ) M=P eg t ; g = f M t eg t f Mt : Impulse esponse functions (tech shock) ith dif = : Impulse esponse functions (tech shock) ith = :5; dif = :6 Impulse esponse functions (tech shock) ith = 4; dif = :6 Impulse esponse functions (money shock) ith dif = :; = e t + e t ; M=P + N=P i ept 7

8 3 x 2 5 T d f x 2 5 T d f

9 T d f Impulse esponse functions (money shock) ith = :5; dif = :6; = Impulse esponse functions (money shock) ith = 4; dif = :6; = Impulse esponse functions (money shock) ith dif = :; g = :2; = Impulse esponse functions (money shock) ith = :5; dif = :6; g = :2; = Impulse esponse functions (money shock) ith = 4; dif = :6; g = :2; = Impulse esponse functions (money shock) ith = :5; dif = :; g = :2; = Impulse esponse functions (money shock) ith = :5; dif = :6; g = :2; = Impulse esponse functions (money shock) ith = 4; dif = :6; g = :2; = 9

10 T d f T d f

11 T d f T d f

12 T d f x 4 2 T d f

13 3 x 4 2 T d f x 4 2 T d f