Asset Pricing in a Production Economy with Heterogeneous Investors

Asset Pricing in a Production Economy with
Heterogeneous Investors
Jin E. Zhang1
Faculty of Business and Economics
The University of Hong Kong
Pokfulam Road, Hong Kong
Email: jinzhang@hku.hk
Tiecheng Li
Department of Mathematical Science
Tsinghua University
Beijing 100084, P. R. China
Email: tli@math.tsinghua.edu.cn
First Version: January 2006
This paper is a theoretical examination of the stochastic behavior of equilibrium asset
prices in an economy consisting of a production process controlled by a state variable
representing the state of technology. The investors with different degrees of risk aversion
and time preferences trade and lend among themselves in order to maximize their individual
utilities of life time consumption. The allocation of wealth fluctuates randomly among
them and acts as a state variable against which each investor wants to hedge. This hedging
motive complicates the investor’s portfolio choice and the equilibrium in the production
economy. A general method of constructing equilibrium asset prices is developed and the
wealth effect in the general equilibrium is discussed.
Keywords: Asset pricing; Heterogeneous preferences; Market price of risk; Interest rate
JEL Classification Code: G12; E43; D51; D91
1Corresponding author. Tel: (852) 2859 1033, Fax: (852) 2548 1152. The authors acknowledge helpful
comments and suggestions from Andrew Carverhill, Y. Stephen Chiu and Tao Lin, and seminar participants
at the University of Hong Kong (HKU). This paper is supported by HKU under Small Project Funding
Scheme (Project No. 200507176196).

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Asset Pricing
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Asset Pricing in a Production Economy with
Heterogeneous Investors
Abstract
This paper is a theoretical examination of the stochastic behavior of equilibrium asset
prices in an economy consisting of a production process controlled by a state variable
representing the state of technology. The investors with different degrees of risk aversion
and time preferences trade and lend among themselves in order to maximize their individual
utilities of life time consumption. The allocation of wealth fluctuates randomly among
them and acts as a state variable against which each investor wants to hedge. This hedging
motive complicates the investor’s portfolio choice and the equilibrium in the production
economy. A general method of constructing equilibrium asset prices is developed and the
wealth effect in the general equilibrium is discussed.

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Asset Pricing
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1
Introduction
This paper is a theoretical examination of the stochastic behavior of equilibrium asset prices
in an economy consisting of a production process controlled by a state variable representing
the state of technology. The investors in the economy maximize their individual utilities
of life time consumption. Heterogeneity in preferences introduces trading and lending
among investors. Market clearing condition determines the general equilibrium in the
production economy. The allocation of wealth fluctuates randomly among the investors and
acts as a state variable against which each investor wants to hedge. This hedging motive
complicates the investor’s portfolio choice and the equilibrium in the production economy.
As a result, the equilibrium market prices of risk and interest rate, and the investor’s optimal
investment and consumption strategies will fluctuate according to the wealth fluctuation
among investors. Our objective will be to understand the relation between the dynamic
equilibrium and the wealth distribution, and the interaction of the optimal indirect utilities
between the heterogeneous investors.
In three related papers, Cox, Ingersoll and Ross (1985), Dumas (1989) and Vasicek
(2005) consider equilibrium asset prices in production economy. Cox, Ingersoll and Ross
develop an equilibrium model in a production economy with multiple production processes
controlled by several state variables. They only deal with one representative agent2, there-
fore their market clearing condition is superficial. Dumas uses a production economy with
no technology change to model the capital market and investigates equilibrium conditions
in the economy with two investors, say A and B. Investor A is myopic, i.e., has logarithmic
utility function; investor B has non-logarithmic isoelastic utility function. He assumes that
the myopic investor A’s optimal strategy is not affected by that of investor B. He then
studies the influence of investor A upon investor B. Vasicek derives equilibrium conditions
2Both terms agent and investor are used in the literature. They have the same meaning in this paper.

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Asset Pricing
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in a production economy with technology change and with several investors. Our paper is
more general than Dumas (1989). It includes Dumas’ result as one of the special cases. It
improves Vasicek (2005) in the ways of finding joint optimal conditions between investors
and handling the wealth effect of each investor. With a proper procedure of applying
market clearing condition, we find that the wealth distribution among investors plays an
important role in the equilibrium.
Other related work includes equilibrium models in pure exchange economy developed
by Lucas (1978), Wang (1996) and Chan and Kogan (2002). Lucas considers equilibrium
in a one-good pure exchange economy with one representative investor. Wang looks at an
exchange economy with two heterogeneous investors. He observes that in a pure exchange
economy, there is no intertemporal transformation of resources. The intertemporal resource
constraint is simply the collection of resource constraints for each date and each state. Max-
imizing the expected intertemporal welfare function is equivalent to maximizing the welfare
function period by period and state by state subject to the corresponding resource con-
straint. With this clever observation, Wang obtains a closed-form market equilibrium in the
economy of two heterogeneous investors. Chan and Kogan analyze an exchange economy
with heterogeneous investors, where each individual’s utility is a function of consumption
measured in units of an average aggregate endowment. They obtain the equilibrium for
heterogeneous investors with a continuous weight distribution.
In a production economy, the wealth of each participant can be invested in a production
process for a possible growth. The intertemporal transformation of resources plays a crucial
role in the dynamic equilibrium. The joint optimization problem of investors’ expected
utility functions is much more difficult to solve. Market clearing between the supply and
demand created by the investors plays a critical role in determining the general equilibrium.
In general, one is unable to find a closed-form market equilibrium. Our target in this
paper is to establish a proper economic model for the equilibrium conditions, so that the

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quantitative stochastic behavior of equilibrium market prices of risks and interest rate can
be studied numerically if necessary.
In the next section, we describe the model of production economy and heterogeneous
investors. Our main results are presented in Section 3. For completeness, we also include
the results of one representative investor discussed by Cox, Ingersoll and Ross (1985) in
this section. The paper is concluded in Section 4.
2
The model
To model the production economy, we use Vasicek’s (2005) setup for the economy with
one production process controlled by one state variable, which is a simplified version of
Cox, Ingersoll and Ross’ (1985) economy with multiple production processes controlled by
several state variables. We also inherit Vasicek’s (2005) notations for the convenience of
the reader.
Consider a production process whose rate of return dA/A on an investment in production
variable A is
dA = µdt + σdy,
(1)
A
where y(t) is a Wiener process that models the production risk. The development of the
production process is affected by a state variable, X(t), representing the state of tech-
nology. Both the expected return function, µ = µ(X(t), t), and the volatility function,
σ = σ(X(t), t), are exogenously given.
The dynamics of the state variable is also exogenously given
dX = ζdt + ψdy + φdx,
(2)
where x(t) is another Wiener process independent of y(t). The new Wiener process is used
to model the state risk that is independent of the production risk. The parameters ζ, ψ
and φ are exogenously given functions of X(t) and t. The function ψ here is designed to

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Asset Pricing
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allow a freedom to model the correlation between the production process and the progress
of the state of technology.
The economy allows unrestricted borrowing and lending at any maturity. The risk-free
rate is denoted by r. Then the money market account, M(t), follows
dM = rdt.
(3)
M
The asset price, P (t) of any asset in the economy must satisfy the equation
dP = (r + βλ + δη)dt + βdy + δdx,
(4)
P
where β and δ are the risk exposures of the asset to the production risk y and the state
risk x. And λ is the market price of production risk, η is the market price of state risk.
Since there are only two risk sources, we only need one more asset with δ = 0 to
complete3 the economy. The asset with δ = 0 can be a derivative contract, such as an
option written on the production variable A with certain strike price and maturity date.
The price of the derivative depends on both the production risk y and the state risk x.
The production variable itself can be understood as a traded asset with β = σ and δ = 0,
then the expected return of an investment in the production process satisfies following
relationship
µ = r + σλ,
=⇒
r = µ − σλ.
(5)
The equality will be used to derive interest rate r once we know the market price of
production risk λ from the equilibrium.
In particular, there exists a numeraire asset Z(t) (Long Jr 1990) with the dynamics
dZ = (r + λ2 + η2)dt + λdy + ηdx,
(6)
Z
3By complete we mean that any asset in the economy can be replicated dynamically by the production
variable and this additional derivative contract.

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such that the price P of any asset satisfies
P (t)
P (s)
= E
,
(7)
Z(t)
t
Z(s)
which means that P (s)/Z(s) is a martingale. If we write π(t) = 1/Z(t), then
1
P (t) =
E
π(t) t [π(s)P (s)] .
(8)
The pricing kernel or stochastic discount factor, π(t), follows
dπ = −rdt − λdy − ηdx.
(9)
π
In integral form, the pricing kernel is written analytically as
π(s)
s
1
s
s
s
= exp −
rdτ −
(λ2 + η2)dτ −
λdy −
ηdx .
(10)
π(t)
t
2 t
t
t
We now describe the investors with heterogeneous preferences. Suppose that the econ-
omy has n participants, k = 1, 2, , · · · , n and let Wk(0) be the initial wealth of kth
investor. Each investor maximizes the expected utility of his life time consumption,
T
max E0
pk(t)Uk(ck(t))dt,
(11)
0
where ck(t) is the rate of consumption at time t, Uk(c) is a utility function with U > 0,
k
U < 0, and p
k
k(t) ≥ 0, 0 ≤ t ≤ T is a time preference function.
The time preference
function can be very general. If it is concentrated at time T , then the investor maximizes
the expected utility of terminal wealth, i.e, max E0Uk(Wk(T )). The problem becomes a
pure investment problem without consumption.
We consider the class of isoelastic utility functions

 c(γk−1)/γk
U
γk > 0, γk = 1,
k(c) =
(12)

γk − 1
ln c
γk = 1.

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The wealth process of kth investor is written as
dWk = [Wk(r + βkλ + δkη) − ck]dt + Wkβkdy + Wkδkdx,
(13)
where the risk exposures βk and δk can be achieved by a portfolio of investing in the
production A, a basic derivative asset P and the money market account M. These risk
exposures fully describe the particular investment strategy of investor k.
By summing up equation (13) according to the index k for all investors, we obtain that
the total wealth of the n investors in the economy,
n
W (t) =
Wk(t),
(14)
k=1
follows the process
n
n
n
n
n
dW = W
r + λ
ωkβk + η
ωkδk −
ck dt + W
ωkβkdy + W
ωkδkdx,
k=1
k=1
k=1
k=1
k=1
(15)
where ωk = Wk/W is the wealth ratio of investor k’s wealth Wk to the total wealth W . By
definition, the wealth ratios satisfy
n
ωk = 1.
(16)
k=1
The market clearing condition is that the total wealth must be invested in the production
process, i.e.,
n
dW =
µW −
ck dt + σW dy.
(17)
k=1
Comparing (15) and (17) gives two restrictions on the investment strategies
n
n
ωkβk = σ,
ωkδk = 0.
(18)
k=1
k=1
The problem is to determine the equilibrium risk-free rate r, the market prices of pro-
duction risk λ, and the market price of state risk η. Once we know their dynamics, the
stochastic discount factor can be determined by equation (10). Any asset with a known
payoff, P (s), at future time s, can be priced by equation (8) accordingly.

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Asset Pricing
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3
Main Results
The new results in this paper are the equilibrium conditions for the production economy
with heterogeneous investors. For completeness, we also present the results of one repre-
sentative investor, which has been studied by Cox, Ingersoll and Ross (1985).
3.1
One representative investor
Proposition 1 In a production economy of one representative investor with a general util-
ity function U(c), the equilibrium market prices of risk and risk-free rate are
V
V
V
λ = − WW W σ − WX ψ,
η = − WX φ,
r = µ − σλ.
(19)
VW
VW
VW
The indirect utility function, V = V (W, X, t), is determined by a nonlinear partial differ-
ential equation (PDE)4
1
Vt + VW (µW − U −1(VW /p)) + VXζ + V
2 WW W 2σ2
1
+ VWXW σψ + V
2 XX(ψ2 + φ2) + pU(U −1(VW /p)) = 0,
(20)
subject to a final condition V (W, X, T ) = 0. The notation U −1 stands for the inverse
function of the marginal utility function of consumption, U (c).
The investor’s optimal investment strategy is β = σ, δ = 0. His optimal consumption
strategy is c = U −1(VW /p). His wealth process is described by
dW = [µW − U −1(VW /p)]dt + σW dy.
(21)
Proof. See appendix A.
Since there is only one representative investor in the economy, there is no counter-party
to trade or borrow. The market clearing condition requires that the investor must invest all
4By nonlienar PDE we mean that the dependent variable V appears in the PDE in a nonlinear way.
For example, U −1(VW /p)VW is a nonlinear term in equation (20).

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Asset Pricing
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his wealth into the production process, i.e., β = σ, δ = 0. The strategy is the only choice
to the investor, but in order to make it optimal, the market prices of risk have to satisfy
certain relations given by equation (19), which are associated with the marginal utility of
wealth, VW .
With Ito’s Lemma, the process of marginal utility of wealth can be written as
dVW
V
V
V
= µ
dt +
W W W σ + WX ψ dy + WX φdx = µ dt − λdy − ηdx,
V
VW
VW
W
VW
VW
VW
where µV represents the drift of the process. It depends on the partial derivatives of V
W
W
w.r.t. to W , X and t. One may observe that the market price of risk for each risk source
is the negative of the volatility on the corresponding risk source of the marginal utility of
wealth. The interest rate is written as
V
V
1
dW dV
r = µ + WW W σ2 + WX σψ = µ +
cov
,
W
.
VW
VW
dt
W
VW
It says that the equilibrium interest rate r is the sum of expected rate of return on wealth
µ and covariance of the rate of return on wealth with the rate of change in the marginal
utility of wealth. The result is presented by Cox, Ingersoll and Ross (1985) in a more
general setting of multiple production processes controlled by several state variables.
For a given utility function U(c), both market prices of risk and risk-free rate are
functions of wealth level W , state of technology X and time t. A full description on them
relies on the detailed structure of the indirect utility function of wealth V (W, X, t), which
has to be determined by solving the nonlinear PDE (20). In general, the problem cannot
be solved analytically, but it can be simplified for some specified utility functions such as a
logarithmic utility function and isoelastic utility function. The equilibrium conditions for
the case of these two kinds of utility functions will be discussed in the next two propositions.
Proposition 2 In a production economy of one representative investor with a logarithmic

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