Idiosyncratic Investment (or Entrepreneurial) Risk in a Neoclassical Growth Model. George-Marios Angeletos. MIT and NBER

Transcription

1 Idosyncratc Investment (or Entrepreneural) Rsk n a Neoclasscal Growth Model George-Maros Angeletos MIT and NBER

2 Motvaton emprcal mportance of entrepreneural or captal-ncome rsk ˆ prvate busnesses account for half of corporate equty, producton, and employment ˆ typcal rch household holds more than half hs fnancal wealth n prvate equty ˆ extreme varaton n entrepreneural returns ˆ dramatc lack of dversfcaton yet, lttle research on unnsurable dosyncratc nvestment rsk n the neoclasscal growth model perhaps more mportant for busness cycles than labor-ncome rsk (Bewley models, e.g., Ayagar 1994, Krusell and Smth 1998)

3 Ths paper ˆ unnsurable dosyncratc nvestment rsk n a neoclasscal growth economy ˆ standard assumptons for preferences and technology: CRRA and CEIS, neoclasscal technology ˆ consder both one-sector economy (only prvate equty) and a two-sector economy (publc and prvate equty) ˆ despte ncomplete markets, closed-form soluton ˆ steady state and transtonal dynamcs ˆ novel macroeconomc complementarty

4 Methodologcal Contrbuton dmnshng returns at the aggregate level, but lnear returns at the ndvdual level wth CRRA/CEIS preferences homothetc decson problem (Samuelson-Merton) lnear ndvdual polcy rules wealth dstrbuton rrelevant closed-form soluton for general equlbrum

5 Fndngs ˆ very dfferent effects that n Bewley models ˆ negatve effect on captal and output ˆ n calbrated examples, about 10% loss n output ˆ non-monotonc effect on nterest rates ˆ pecunary externalty n rsk takng ˆ dynamc macroeconomc complementarty ˆ amplfcaton and persstence

6 Layout The Benchmark Model (only prvate equty) Indvdual Behavor General Equlbrum and Steady State The Two-Sector Model (prvate and publc equty) Complementarty and Propagaton Concludng Remarks

7 The Model two nputs (K and L and a sngle homogeneous good Y a contnuum of heterogeneous households 0,1 compettve product and labor markets each household supples labor n compettve labor market each household owns a sngle frm (famly busness) households can borrow and save n a rskless bond, but can nvest captal only n ther own frm the frm employs labor from the compettve labor market producton subject to undversfable dosyncratc rsk

8 Technology and Rsks Output for frm n perod t: y t Fk t,n t,a t F : R 3 R s a neoclasscal CRS producton technology A t s an dosyncratc productvty shock (F A,F KA,F LA 0andEA 1 Not needed, but useful: Assumpton A1 A s augmented to captal and s lognormally dstrbuted ˆ 2 VarlnA parametrzes ncomplete markets

9 Households Household captal ncome frm profts: t y t t n t Fk t,n t,a t t n t Budget constrant for household n perod t: c t k t1 b t1 t t R t b t Non-negatvty constrants: c t 0 and k t 0 Natural borrowng lmt: b t h t j1 qtj tj q t where q t q t1 /R t1.

10 Preferences Kreps-Porteus/Epsten-Zn non-expected utlty: where u t Uc t UCE t U 1 u t1 CE t u 1 E t u U governs ntertemporal substtuton, governs rsk averson CEIS and CRRA: Uc c11/ 1 1/ and c c1 1 0 elastcty of ntertemporal substtuton 0 degree of relatve rsk averson

11 Equlbrum Defnton A compettve equlbrum s a determnstc sequence of prces R t, t t0 and a collecton of contngent ndvdual plans c t,n t,k t1 () The plan c t,n t,k t1,b t1 t0 s optmal for all. () The labor market clears n every perod: n t 1. () The bond market clears n every perod: b t 0.,b t1 t0, 0,1, such that: Remark: n open economy R exogenous

12 Optmal Indvdual Behavor By CRS, t k t F 1, n t k,a t t n t t k t It follows that Proposton 1 and lnear n k Labor demand and captal ncome are decreasng n A, decreasng n, n t na t, t k t and t ra t, t k t where ra, max L F1,L,A L and na, arg max L.

13 Defne fnancal wealth as w t t t R t b t By Proposton 1, w t t ra t, t k t R t1 b t Gven R t, t t0, the value functon V t w solves V t w t max c t,k t1,b t1 Uc t U 1 E t U 1 V t1 w t1 subject to c t k t1 b t1 w t w t1 t1 ra t1, t1 k t1 R t1 b t1 c t 0 k t1 0 b t1 h t1

14 Indvdual Savngs and Investment Proposton 2 The optmal ndvdual path satsfes w t t ra t, t k t R t b t c t 1 s t w t h t k t1 b t1 s t t w t h t s t 1 t w t h t h t1 where t t1,r t1 arg max t t1,r t1 max s t 1 st s t 1 A ra, t1 1 R t1 1 A ra, t1 1 R t

15 Lemma Under A1, t ln t1 and ln 2 t lnr t1 ln t1 2 2 where t1 f K t1 R t1 and 2 VarlnA.

16 General Equlbrum Lnear polcy rules wealth dstrbuton rrelevant Aggregates satsfy N t n t K t t r t K t t t N t fk t where n A na,, r A ra,, and fk FK,1,A

17 General Equlbrum Proposton 3 The equlbrum path C t,k t,h t, t,r t t0 satsfes C t K t1 fk t C t 1 s t fk t H t K t1 t s t fk t H t H t 1 R t1 t1 H t1 n t K t 1 1 s t 1 1 t 1 1 s t1 1 where t t1,r t1 and t t1,r t1.

18 Steady State Proposton 4 In steady state,, K and R solve E Af K 1 R f K R 0 E Af K 1 R f K 1 R 1 fk f K K R 1K 1 Proposton 5 Under A1, lnf K lnr 2 1 lnr 1 dosyncratc rsk necessarly reduces the captal stock for any gven nterest rate

19 Calbraton % 40% 5% standard devaton of nvestment return 50% or 25%

20 Numercal Smulatons

21 σ γ θ β -1-1 α δ Output Loss Captal Loss Interest Rate Prvate Premum 50% 20% 2 1 5% 40% 10% 50% 20% 50% 20% % 40% 10% 50% 20% 50% 20% % 40% 10% 50% 20% 8% 40% 10% 50% 20% 2 1 5% 60% 10% 50% 20% 5% 40% 5% Table 1 Consder the case that prvate equty accounts for all captal and producton. The table reports the mpact of dosyncratc nvestment rsk on ncome, savngs, and nterest rates for a seres of calbratons. The chosen parameter values are n the frst sx columns, and the mpled effects n the last four columns. Output loss and captal loss refer to the percentage reducton n the steady-state level of output and captal as compared to complete markets. Interest rate s the rate of return n rskless bonds, whle prvate premum s the excess return earned n prvate equty.

22 Two Sectors: Prvate and Publc Equty Publc equty no dosyncratc rsk Let X t and L t denote captal and labor n publc equty; output s GX t,l t where G s a neoclasscal producton functon Assumpton A2 For 1, GX,L FX,L, A/ ˆ pns down the prvate equty premum when both sectors are open

23 Indvdual Behavor The household budget: c t k t1 x t1 b t1 ra t, t k t R t x t R t b t t N. where x t1 denotes nvestment n publc equty The optmal plan satsfes w t ra t, t k t R t x t R t b t t N c t 1 s t w t h t k t1 s t t w t h t x t1 b t1 s t 1 t w t h t h t where t, t,ands t are defned as before.

24 General Equlbrum By proft maxmzaton, L t l t X t and R t R t where R max L G1,L L and l arg max L. Lemma Under A1 and A2, n l and r R It follows that t ln 2 and t R t1 exp ln 2 2

25 General Equlbrum Proposton 6 In any equlbrum n whch both sectors are actve, the equlbrum dynamcs satsfy C t K t1 X t1 W t FK t, n t K t, A GX t,l t X t C t 1 s t W t H t K t1 t s t W t H t H t 1 R t1 H t1 t1 n t K t l t X t 1 R t R t 1 s t 1 1 t 1 1 s t1 1 where t ln 2, t R t1, exp ln 2 2.

26 Steady State Proposton 7 A steady state n whch both sectors are actve s unque whenever t exsts, and t exsts f and only f s suffcently hgh. The steady state satsfes R R R K 1/l /R 1 1/ 1 X 1/l K Proposton 7 There exsts 1 such that, whenever, an ncrease n rases R, has an ambguous effect on K X, but necessarly reduces K, Y, C, Y/L and Y/K.

27 Numercal Smulatons

28 σ γ θ β -1-1 α δ Output Loss Captal Loss Interest Rate Prvate Premum 50% 20% 2 1 5% 40% 10% 50% 20% 50% 20% % 40% 10% 50% 20% 50% 20% % 40% 10% 50% 20% 8% 40% 10% 50% 20% 2 1 5% 60% 10% 50% 20% 5% 40% 5% Table 2 Consder the case that rsk-free publc equty coexsts wth rsky prvate equty. The table reports the mpact of dosyncratc nvestment rsk on ncome, savngs, and nterest rates for a seres of calbratons. Output loss and captal loss now refer to the combned output and captal n prvate and publc equty. The nterest rate s the rate of return n ether rskless bonds or publc equty. The prvate premum s pnned down by the technologcal parameter µ and s calbrated so that prvate and publc equty each account for half of the aggregate captal stock.

29 Propagaton Mechansm Along the transton, K t1 t s t fk t H t H t j1 qtj q t K tj. Hence K t1,k t2,... ncreases wth H t,h t1,... H t,h t1,... ncreases wth K t1,k t2,... dynamc macroeconomc complementarty amplfcaton and persstence

30 Remarks on Propagaton Mechansm a general-equlbrum phenomenon derves from a pecunary externalty reles on two premses: (1) nvestment subject to undversfable dosyncratc rsk (2) rsk takng senstve to antcpated future economc actvty (1) absent from Ayagar (1994), Krusell and Smth (1998); (2) absent from Bernanke and Gertler (1989, 1990), Kyotak and Moore (1997), as well as Bencvenga and Smth (1991), Obstfeld (1994), Krebs (2003).

31 Numercal Smulatons

32 Concludng Remarks/Future Research lower captal and output n the steady state amplfcaton and persstence n transtonal dynamcs pecunary externalty ˆ neffcency? coordnaton falure? stablzaton polcy? optmal taxaton? welfare cost of busness cycles? wealth dstrbuton? quanttatve analyss (back to Krusell and Smth)