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

2
Motivation empirical importance of entrepreneurial or capital-income risk
ˆ private businesses account for half of corporate equity, production, and employment
ˆ typical rich household holds more than half his financial wealth in private equity
ˆ extreme variation in entrepreneurial returns
ˆ dramatic lack of diversification yet, little research on uninsurable idiosyncratic investment risk in the neoclassical growth model perhaps more important for business cycles than labor-income risk
( Bewley models , e.g., Aiyagari 1994, Krusell and Smith 1998)

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This paper
ˆ uninsurable idiosyncratic investment risk in a neoclassical growth economy
ˆ standard assumptions for preferences and technology:
CRRA and CEIS, neoclassical technology
ˆ consider both one-sector economy (only private equity) and a two-sector economy (public and private equity)
ˆ despite incomplete markets, closed-form solution
ˆ steady state and transitional dynamics
ˆ novel macroeconomic complementarity

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Methodological Contribution diminishing returns at the aggregate level, but linear returns at the individual level
¶ ¶ with CRRA/CEIS preferences homothetic decision problem (Samuelson-Merton)
¶ ¶ linear individual policy rules
¶ ¶ wealth distribution irrelevant
¶ ¶ closed-form solution for general equilibrium

5
Findings
ˆ very different effects that in Bewley models
ˆ negative effect on capital and output
ˆ in calibrated examples, about 10% loss in output
ˆ non-monotonic effect on interest rates
ˆ pecuniary externality in risk taking
ˆ dynamic macroeconomic complementarity
ˆ amplification and persistence

6
Layout
S The Benchmark Model (only private equity)
S Individual Behavior
S General Equilibrium and Steady State
S The Two-Sector Model (private and public equity)
S Complementarity and Propagation
S Concluding Remarks

7
The Model
S two inputs ( K and L   and a single homogeneous good Ÿ Y
S a continuum of heterogeneous households i ¡ 0, 1 ¢
S competitive product and labor markets
S each household supplies labor in competitive labor market
S each household owns a single firm (family business)
S households can borrow and save in a riskless bond, but can invest capital only in their own firm
S the firm employs labor from the competitive labor market
S production subject to undiversifiable idiosyncratic risk

8
Technology and Risks
Output for firm i in period t : y i t
F Ÿ k i t
, n i t
, A i t
F : R
3 v R is a neoclassical CRS production technology
A i t is an idiosyncratic productivity shock ( F
A
, F
KA
, F
LA
0 and E A 1
Not needed, but useful:
Assumption A1 A is augmented to capital and is lognormally distributed
ˆ @ 2 Var Ÿ ln A   parametrizes incomplete markets

Households
Household capital income firm profits:
= i t y i t
" F t n i t
Budget constraint for household i in period t :
F Ÿ k i t
, n i t
, A i t
" F t n i t c i t k i t 1 b i t 1
F t
= i t
R t b i t
9
Non-negativity constraints: c i t u 0 and k i t u 0
“Natural” borrowing limit:
" b i t t h t q
.
!
j 1 q t j
F t j q t where q t q t 1
/ R t 1
.

Preferences
Kreps-Porteus/Epstein-Zin non-expected utility: u i t
U Ÿ c i t
* U £ CE t
¡ U " 1 Ÿ u i t 1
¢¤ where
CE t
Ÿ u   q & " 1 ¡ E t
& Ÿ u  ¢
U governs intertemporal substitution, & governs risk aversion
CEIS and CRRA :
U Ÿ c   c 1 " 1/ 2
1 " 1/ 2
2 0 elasticity of intertemporal substitution
+ 0 degree of relative risk aversion and & Ÿ c   c 1 " +
1 " +
10

Equilibrium
Definition A competitive equilibrium is a deterministic sequence of prices £ R t
, F t
¤ .
t 0 and a collection of contingent individual plans £ c i t
, n i t
, k i t 1
, b i t 1
¤ .
t 0
, i ¡ 0, 1 ¢ , such that:
(i) The plan £ c i t
, n i t
, k i t 1
, b i t 1
¤ .
t 0 is optimal for all i .
(ii) The labor market clears in every period: ; n i t
1.
(iii) The bond market clears in every period: ; b i t
0.
Remark: in open economy v R exogenous
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Optimal Individual Behavior
By CRS,
= k i t i t F 1, n k i t i t , A i t
" F t n k i t i t
It follows that
Proposition 1 Labor demand and capital income are decreasing in A , decreasing in F , and linear in k n i t n Ÿ A i t
, F t
k i t and = i t r Ÿ A i t
, F t
k i t where r Ÿ A , F   q max
L
¡ F Ÿ 1, L , A   " F L ¢ and n Ÿ A , F   q arg max
L
¡ .
¢
12

Define financial wealth as
By Proposition 1, w i t q F t
= i t
R t b i t w i t
F t r Ÿ A i t
, F t
k i t
R t " 1 b i t
Given £ R t
, F t
¤ .
t 0
, the value function V t
Ÿ w   solves
V t
Ÿ w i t
Ÿ c i t
, k max i t 1
, b i t 1
U Ÿ c i t
* U & " 1 £ E t
¡ & U " 1 V i t 1
Ÿ w i t 1
¢¤ subject to w i t 1
F c i t t 1 k i t 1 b i t 1 w r Ÿ A i t 1
, F t 1
k i t 1 i t
R t 1 b i t 1 c i t u 0 k i t 1 u 0 " b i t 1 t h t 1
13

14
Individual Savings and Investment
Proposition 2 The optimal individual path satisfies w i t c i t k i t 1 b i t 1
F t r Ÿ A i t
, F t
k i t
R t b i t
Ÿ 1 " s t
Ÿ w i t s t
C t
Ÿ w i t h t
h t
s t
Ÿ 1 " C t
Ÿ w i t h t
" h t 1 where
C t
C Ÿ F t 1
, R t 1
arg max
C
> t
> Ÿ F t 1
, R t 1
max
C
;
A
;
A
¡ C r Ÿ A , F t 1
Ÿ 1 " C   R t 1
¢ 1 " +
¡ C r Ÿ A , F t 1
Ÿ 1 " C   R t 1
¢ 1 " +
1
1 " +
" 1 s t
1
.
!
s t s
A t
* 2 >
A
" 1
" 1
1
1 " +

Lemma Under A1,
C t
X ln 6 t 1
+@ 2 and ln > t
X ln R t 1
Ÿ ln 6 t 1
2
+@ 2 where
6 t 1 f U Ÿ K t 1
R t 1 and @ 2 Var ¡ ln A ¢ .
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General Equilibrium
Linear policy rules ® wealth distribution irrelevant
Aggregates satisfy
N t n Ÿ F t
K t
$ t r Ÿ F t
K t
$ t
F t
N t f Ÿ K t
where n Ÿ F   q ;
A n Ÿ A , F   , r Ÿ F   q ;
A r Ÿ A , F   , and f Ÿ K   F Ÿ K , 1, A
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General Equilibrium
Proposition 3 The equilibrium path £ C t
, K t
, H t
, F t
, R t
¤ .
t 0 satisfies
C t
C t
K t 1 f Ÿ K t
Ÿ 1 " s t
¡ f Ÿ K t
H t
¢
1 " s
K t 1
H t t
C t s t
¡ f Ÿ K t
H t
¢
R
1 t 1
¡ F t 1
H t 1
¢ n Ÿ F t
K t
1
1 * 2 > t
2 " 1
1
Ÿ 1 " s t 1
" 1 where C t
C Ÿ F t 1
, R t 1
and > t
> Ÿ F t 1
, R t 1
.
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Steady State
Proposition 4 In steady state, C
.
, K
.
and R
.
solve
* 2
E ¡ C
.
Af
E £¡ C
.
Af U Ÿ K
.
Ÿ 1 " C
.
R
.
¢ " + ¡ f U Ÿ K
.
" R
.
¢¤ 0
U Ÿ K
.
Ÿ 1 " C
.
R
.
¢ 1 " +
2 " 1
1 " + • ¡ C
.
f U Ÿ K
.
Ÿ 1 " C
.
R
.
¢ 1 f Ÿ K
.
Ÿ R
" f
.
U Ÿ K
" 1   K
.
.
K
.
1 "
C
.
C
.
Proposition 5 Under A1, ln f U Ÿ K
.
X ln R
.
@
2 +2
1 2 ln Ÿ * R
.
" 1
® idiosyncratic risk necessarily reduces the capital stock for any given interest rate
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Calibration
+ 2
2 1
* " 1 " 1 5%
) 40%
5%
@ standard deviation of investment return
@ 50% or @ 25%
19

Numerical Simulations
20

σ γ θ β 1 1 α δ
50%
20%
2 1 5% 40% 10%
50%
1
20%
50%
3
20%
1 5% 40% 10%
50%
2
20%
2
50%
.5
20%
5% 40% 10%
50%
8% 40% 10%
20%
50%
2 1
20%
5% 60% 10%
50%
5% 40% 5%
20%
Output
Loss
Capital
Loss
Interest
Rate
Private
Premium
Table 1
Consider the case that private equity accounts for all capital and production. The table reports the impact of idiosyncratic investment risk on income, savings, and interest rates for a series of calibrations. The chosen parameter values are in the first six columns, and the implied effects in the last four columns. “Output loss” and “capital loss” refer to the percentage reduction in the steady-state level of output and capital as compared to complete markets. “Interest rate” is the rate of return in riskless bonds, while “private premium’’ is the excess return earned in private equity.

Two Sectors: Private and Public Equity
Public equity no idiosyncratic risk
Let X t and L t denote capital and labor in public equity; output is
G Ÿ X t
, L t
where G is a neoclassical production function
Assumption A2 For 6 1,
G Ÿ X , L   F Ÿ X , L , A / 6
ˆ 6 pins down the private equity premium when both sectors are open
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Individual Behavior
The household budget: c i t k i t 1 x i t 1 b i t 1 t r Ÿ A i t
, F t
k i t
R t x i t
R t b i t
F t
N i .
where x i t 1 denotes investment in public equity
The optimal plan satisfies w i t x i t 1 r Ÿ A i t
, F t
k i t
R t x i t
R t b i t
F t
N i c i t k i t 1
Ÿ 1 " s t
Ÿ w i t s t
C t
Ÿ w i t h h i t
i t
b i t 1 s t
Ÿ 1 " C t
Ÿ w i t h i t
" h i t where C t
, > t
, and s t are defined as before.
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General Equilibrium
By profit maximization,
L t l Ÿ F t
X t and R t
R Ÿ F t
where R Ÿ F   q max
L
¡ G Ÿ 1, L   " F L ¢ and l Ÿ F   q arg max
L
¡ ¢ .
Lemma Under A1 and A2, n Ÿ F   6 l Ÿ F   and r Ÿ F   6 R Ÿ F
It follows that
C t
C X ln 6
+@ 2 and
R
> t t 1
N X exp ln 6
2 +@ 2
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General Equilibrium
Proposition 6 In any equilibrium in which both sectors are active, the equilibrium dynamics satisfy
C t
K t 1
X t 1
W
C t t
F Ÿ K t
, n Ÿ F t
K t
, A   G Ÿ X t
, l Ÿ F t
X t
Ÿ 1 " s t
¡ W t
H t
¢
Ÿ 1 " s t
H
K t t 1
C t s t
¡ W t
H t
¢
1
R t 1
¡ F t 1
H t 1
¢ n Ÿ F t
K t l Ÿ F t
X t
1
R t
R Ÿ F t
1 * 2 Ÿ > t
1
2 " 1 Ÿ 1 " s t 1
" 1 where C t
C X ln 6
+@ 2
, > t
N R t 1
, N X exp ln 6
2 +@ 2
.
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Steady State
Proposition 7 A steady state in which both sectors are active is unique whenever it exists, and it exists if and only if @ is sufficiently high. The steady state satisfies
Ÿ * R
.
2 N 2 " 1 Ÿ C6 1 " C   1
K
.
R Ÿ F
.
R
.
1/ l Ÿ F
.
F
.
/ Ÿ R
6 1/ C " 1
.
" 1
X
.
1/ l Ÿ F
.
" 6 K
.
Proposition 7 There exists 2 1 such that, whenever 2 2 , an increase in @ raises
R , has an ambiguous effect on K X , but necessarily reduces K , Y , C , Y / L and Y / K .
25

Numerical Simulations
26

σ
50%
20%
γ θ β 1 1 α δ
2 1 5% 40% 10%
50%
1
20%
50%
3
20%
1 5% 40% 10%
50%
2
20%
2
50%
.5
20%
5% 40% 10%
50%
8% 40% 10%
20%
50%
2 1
20%
5% 60% 10%
50%
5% 40% 5%
20%
Output
Loss
Capital
Loss
Interest
Rate
Private
Premium
Table 2
Consider the case that risk-free public equity coexists with risky private equity. The table reports the impact of idiosyncratic investment risk on income, savings, and interest rates for a series of calibrations. “Output loss” and “capital loss” now refer to the combined output and capital in private and public equity. The “interest rate” is the rate of return in either riskless bonds or public equity. The “private premium” is pinned down by the technological parameter µ and is calibrated so that private and public equity each account for half of the aggregate capital stock.

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Propagation Mechanism
Along the transition,
K t 1
H t
C t s t f Ÿ K t
H t
.
!
j 1 q q t j t
F Ÿ K t j
.
Hence
£ K t 1
, K t 2
, . . .
¤ increases with £ H t
, H t 1
, . . .
¤
£ H t
, H t 1
, . . .
¤ increases with £ K t 1
, K t 2
, . . .
¤
Ž dynamic macroeconomic complementarity
Ž amplification and persistence

Remarks on Propagation Mechanism
 a general-equilibrium phenomenon
 derives from a pecuniary externality
 relies on two premises:
(1) investment subject to undiversifiable idiosyncratic risk
(2) risk taking sensitive to anticipated future economic activity
(1) absent from Aiyagari (1994), Krusell and Smith (1998);
(2) absent from Bernanke and Gertler (1989, 1990), Kiyotaki and Moore (1997), as well as Bencivenga and Smith (1991), Obstfeld (1994), Krebs (2003).
28

Numerical Simulations
29

Concluding Remarks/Future Research
S lower capital and output in the steady state
S amplification and persistence in transitional dynamics
S pecuniary externality ˆ inefficiency? coordination failure?
S stabilization policy? optimal taxation?
S welfare cost of business cycles?
S wealth distribution?
S quantitative analysis (back to Krusell and Smith)
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