This paper introduces a separation theorem for a two period general equilibrium model with endogenized asset structures and incomplete income transfer space. This theorem separates the activities of the consumers from the activities of the firms. The result improves on the objective function introduced in Drèze (1974) and Grossman and Hart (1979), which is not independent of an exogenously assigned present value derived from the utilities of the owners of the firm.Objective Function, Incomplete Markets, Prot Maximization

P Stiefenhofer

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Discussion Papers in Economics 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Department of Economics and Related Studies 
University of York 
Heslington 
York, YO10 5DD 
 
No. 10/08 
 
 
A Theorem on the Decentralization of the Objective Function of the Firm in 
GEI 
 
By 
 
Pascal Stiefenhofer, University of York  A Theorem on the Decentralization of the
Objective Function of the Firm in GEI
Pascal Christian Stiefenhofer
University of York
June 2, 2010
Abstract
This paper introduces a separation theorem for a two period general equilibrium
model with endogenized asset structures and incomplete income transfer space. This
theorem separates the activities of the consumers from the activities of the rms. The
result improves on the objective function introduced in Dr eze (1974) and Grossman
and Hart (1979), which is not independent of an exogenously assigned present value
derived from the utilities of the owners of the rm.
Keywords: Objective Function, Incomplete Markets, Prot Maximization
JEL Classication: D20, D21, D52
1 Introduction
The seminal research program on the objective function of the rm in general equi-
librium with incomplete markets is concentrated around the question: Do spot and
equity prices provide rms with enough information to deduce what the appropriate
objective function of the rm should be? The literature on the classical GEI model
with production replies no to this question. It then adds the sux, not without
further extra information ([3],[5],[6],[2],[4], and others). This additional information
comes from the group of owners of the rm in form of an average present value
derived from the utilities of the share holders. This implies the centralization of the
objective functions of the rms.
Stiefenhofer [7] initiates another research program on the objective function of
the rm in general equilibrium with incomplete markets by asking, what role do
intermediate goods and nancial assets play in determining the production set of
the rm. He considers, at variance with the two period xed production sets of
classical GEI models, one period endogenized productions sets. In this more realistic
economic scenario, production sets are not independent of the nancial activities of
the rm. This allows to consider a new class of objective functions.
This paper shows that the extensive form of the two argument, sequential objec-
tive function of the rm introduced in Stiefenhofer [7] has a nice property, namely,
Contact address: Department of Economics and Related Studies University of York.
ps515@york.ac.uk.
I would like to thank Peter Simmons for commenting on an early draft of the model presented
here.
1the independence of any average utility of the owners of the rm. This property
separates the activities of the agents as consumers and producers. The result is
interesting because it rehabilitates the prot maximization criterion of the Arrow-
Debreu model for the more general economic scenario when markets are incomplete.
It essentially follows from the way capital and nancial assets enter the model.
The paper is organized in four parts. Section two introduces the model. Section
three states the separation theorem. Section four is a conclusion.
2 The Model
We consider a two period t 2 T = f0;1g model with technological uncertainty in
period 1 represented by states of nature. An element in the set of mutually exclusive
and exhaustive uncertain events is denoted s 2 f1;:::;Sg, where by convention s = 0
represents the certain event in period 0. We count in total (S +1) states of nature.
The economic agents are the j 2 f1;:::;ng producers and i 2 f1;:::;mg con-
sumers which are characterized by sets of assumptions of smooth economies in-
troduced below. There are k 2 f1;:::;lg physical commodities and j 2 f1;:::;ng
nancial assets, referred to as stocks. In fact, stocks are the only nancial assets
considered here. Physical goods are traded on each of the (S+1) spot markets. Pro-
ducers issue stocks which are traded at s = 0, yielding a payo in the next period
at uncertain state s 2 f1;:::;Sg. The quantity of stocks issued by rm j 2 f1;:::;ng
is denoted zj 2 R , where
Pn
j=1 zj = ^ z.
There are in total l(S + 1) physical goods available for consumption. The con-
sumption bundle of agent i 2 f1;:::;mg is denoted xi = (xi(0);xi(s);:::;xi(S)) 2
R
l(S+1)
++ , with xi(s) = (x1
i(s);:::;xl
i(s)) 2 Rl
++, and
Pm
i=1 xi = x. The consump-
tion space for each consumer i 2 f1;:::;mg is Xi = R
l(S+1)
++ , the strictly posi-
tive orthant. The associated price system is a collection of vectors represented
by p = (p(0);p(s);:::;p(S)) 2 R
l(S+1)
++ , with p(s) = (p1(s);:::;pl(s)) 2 Rl
++, the
strictly positive orthant. Each consumer i 2 f1;:::;mg is endowed with initial re-
sources !i 2 
, where 
 = RlT
++, and !i = (!i(0);!i(1)) a collection of strictly
positive vectors. Denote an initial resource vector at time period t 2 T = f0;1g,
!i(t) = (!1
i(t);:::;!l
i(t)) 2 Rl
++, and the sum of total initial resources,
Pm
i=1 !i = !.
In total, there are n nancial assets traded in period t = 0. Denote the quantity
vector of stocks purchased by consumer i 2 f1;:::;mg, zi = (zi(1);:::;zi(n)) 2 Rn
+,
a collection of quantities of stocks purchased from producers j 2 f1;:::;ng, and
denote
Pm
i=1 zi = z, with associated stock price system q = (q(1);:::;q(n)) 2 Rn
++.
Denote producer j's period one vector of capital purchase yj(0) 2 Rl
 , and denote
his period two state dependent net activity vector yj(s) = (y1
j(s);:::;yl
j(s)) 2 Rl. Let
yj(t = 1) = (yj(s);:::;yj(S)) 2 RlS denote the collection of state dependent period
t=1 net activity vectors. A period two input of production for every s 2 f1;:::;Sg
is by convention denoted yk
j(s) < 0, and a production output in state s 2 f1;:::;Sg
satises yk
j(s)  0.
2.1 The model of the rm
Each rm j 2 f1;:::;ng issues stocks zj at stock price qj in period one in order
to build up production capacity. A rm's total cash acquired via stock market
determines the upper bound of the total value of production capacity it can install
2in the same period. Denote this liquidity constraint qjzj = Mj, where Mj 2 R+
is a non-negative real number and zj 2 R  a feasible nancial policy of the rm
j 2 f1;:::;ng. Mj constraints the quantity of capital y(0) 2 Rl
  a producer j can
purchase at spot price system p(0) 2 Rl
++. The quantity of intermediate goods yj(0)
purchased in period one determines a correspondence jjZ. This correspondence
denes the technology of the rm at feasible nancial policy Z. Let the production
set available to each producer j 2 f1;:::;ng in period two be described by this
technology, jjZ : Rm
  ! Rn
 , a correspondence dened on the set of period two
inputs, and denote it Yjjz  Rl. Let S denote the set of all exogenously given states
of nature. Then for each producer j 2 f1;:::;ng let the t = 1 one period production
set be dened by a map jjZ with domain Rm
   R++ and range Rn
   R++, and
denote it Yjjz(s)  Rl(S+1), where m + n = l.
Assumption 1 (Technological Uncertainty) For the production set of each rm
j 2 f1;:::;ng, denoted Yjjz(s)  Rl(S+1), the number of uncertain states of the world
satises S  2:
Assumption 2 (Smooth Endogenous Production Structures) Each produc-
tion set Yjjz(s)  Rl(S+1) for j 2 f1;:::;ng, satises the dierentiability properties
for smooth economies.1
Using assumption 1 (Technological Uncertainty), the implicit assumption of long
run prots maximization, and assumption 2 (Smooth Production Sets), a closed form
objective function can be assigned to each producer j 2 f1;:::;ng. The algebraic
form of the long run prot maximization objective function is shown in equation
(1).
argmax
(  yj^ z(s);(^ z; y(0)))j
8
> <
> :
 qzj +
XS
s=1  p(s) yjj^ z (s) :
 q
m X
i=1
 zi(j)   q^ zj =  p(0)yj(0)
yjj^ z (s) 2 Yjj^ z (s)
s 2 S
9
> =
> ;
;
(1)
2.2 The consumers
Consumers play the same role in this production model as in the classical GEI
model with production. They invest into rms because they want to transfer wealth
between future uncertain states of nature, and to smooth out consumption across
states of nature. Each consumer i 2 f1;:::;mg purchases stocks zi at stock price q
in period one in return for a dividend stream in the next period. The consumer's
optimization problem is to maximize utility subject to a sequence of (S +1) budget
constraints. Each consumer i 2 f1;:::;mg is characterized by the set of standard
assumptions for smooth economies introduced in Debreu [1].
Assumption 3 (Smooth Utilities) For each consumer i 2 f1;:::;mg, prefer-
ences are represented with smooth utilities dened on the strictly positive orthant,
Xi = R
l(S+1)
++ .
1All assumptions and their implications are introduced in Stiefenhofer (2009) in great detail.
3Denote consumer i's sequence of (S + 1) budget constraints
Bzi =
(
(xi;zi) 2 R
l(S+1)
++  R
n
+
 

p(0)  (xi(0)   !i(0)) =  qzi
p(s)2(xi(s)   !i(1)) = (p1;jZ)i(zi)
)
; (2)
where i in (2) denotes the endogenously determined ownership structure of con-
sumer i 2 f1;:::;mg, a (n  1) vector dened by the mappings
ij : R+ ! R+ for all j 2 f1;:::;ng; (3)
where zi(j) 2 R+ is a non-negative real number for every j 2 f1;:::;ng. ij =
zi(j)[
P
i zi(j)] 1 is the proportion of total payo of nancial asset j 2 f1;:::;ng
hold by consumer i 2 f1;:::;mg after trade at the stock market took place in period
one.  represents the payo matrix of the economy of order (S  n).
The sequential optimization problem of the consumer i 2 f1;:::;mg is to invest
into rms in period one in order to smooth out future uncertain consumption and to
optimize consumption of goods in every (S+1) spot market. For a given price system
p = (p(0);p(1);:::;p(S)) 2 R
l(S+1)
++ of consumption goods and price system q 2 Rn
++
of nancial assets (stocks), a consumer chooses bundles of consumption goods and
quantities of stocks (x;z)i 2 Xi  Rn
+ such that ui(xi;zi) is maximized subject to
the sequence of (S + 1) constraints in Bzi. Algebraically, each i 2 f1;:::;mg
(xi;zi)argmax
n
ui(xi;zi) : zi 2 R
n
+;xi 2 Bzi
o
: (4)
2.3 Equilibrium equations
We introduce following prize normalization S = fp 2 R
l(S+1)
++ : kpk = g such
that the Euclidean norm vector of the spot price system p is a strictly positive
real number R++. A competitive equilibrium of the production economy dened
by the initial resource vector ! 2 
 is a price pair (p;q) 2 S  Rn
++ if equality
between demand and supply of physical goods and nancial assets is satised in all
states of nature, s = 0;1;:::;S. Its associated competitive equilibrium allocation is
a collection of vectors [(x;z);(y;e z)] 2 R
l(S+1)m
++ Rn
+Rl(S+1)nRn
  of consumption,
production and nancial quantities. Market clearance conditions are determined by
the aggregate excess demands for physical goods and for nancial assets as expressed
by the equilibrium equations:
(i)
Pm
i=1(xi(0)   !i(0)) =
Pn
j=1 yj(0)
(ii)
Pm
i=1(xi   !i(1)) =
PS
s=1
Pn
j=1 yj(s)
(iii)
Pn
j=1
Pm
i=1(zi)j = 0,
Pm
i=1 (zi)j = 1 for all j 2 f1;:::;ng
(5)
Stiefenhofer [7] shows that equilibria for this stock market model always exist.
3 Result
Theorem 1 For every producer j 2 f1;:::;ng, the period two net activity vector
yj(s) in available production set YjjZ(s) is independent of any present value vector
i for all i 2 f1;:::;mg, and s 2 f1;:::;Sg.
4Proof 1 Using assumption (1) and assumption (2), let qj(i) =
PS
s=1 i(s)[p(s)  yj(s)],
where i denotes i's marginal evaluation of one additional unit of future income for
i(s) 6= i0(s) ( i^  = 0 at S > n; for all i 2 f1;:::;mg; where
^ (p1;q;y) =
2
6 6 6
4
 q1   qn
p(1)  y1(1)  p(1)  yn(1)
. . .
. . .
p(S)  y1(S)  p1(S)  yn(S)
3
7 7 7
5
represents the full payo matrix
of order ((S + 1)  n):
Denote zi(j) the consumer's i 2 f1;:::;mg demand of quantity of stocks j 2 f1;:::;ng
evaluated at qj(i) and i:
Now, let
Xm
i=1  zi(j) =  zj for all j 2 f1;:::;ng )  q
Xm
i=1 ( xi(0)   !i(0)) =
Xn
j=1  yj(0) )  p(0)
Xm
i=1 ( xi(s)   !i(1)) =
Xn
j=1  yj(s); 8s 2 S )  p(s)
Consider t = 0 optimization problem for i 2 f1;:::;mg and j 2 f1;:::;ng: For given
equity prices system  q and spot prices system  p(0), let each consumer i 2 f1;:::;mg
max
x(0)2Bzi;zi
ui(xi;zi) )  xi(0);  zi
and each producer j 2 f1;:::;ng
max
z(j)
 qz(j) :  qz(j) =  q
Xm
i=1  zi(j) )  z(j):
Given maximum quantity of stocks producer j 2 f1;:::;ng can sell on the stockmar-
ket,  z(j); the problem of the producer j 2 f1;:::;ng is then to purchase capital yj(0):
The problem of produce j 2 f1;:::;ng is then to maximize the level of capital at given
spot price system  p(0), and given cash acquired by issuing stocks,  q z(j) =  Mj. Let
max
y(0)j
 p(0)yj(0) :  Mj =  p(0)yj(0) )  yj(0):
Let the level of capital  yj(0), at nancial policy  z(j) determine a correspondence
jz : This correspondence maps Rm
  into Rn
+ for every state of nature s 2 f1;:::;Sg:
This correspondence describes the production set available to the rm in period two,
denoted Yj j z(s) for all s 2 f1;:::;Sg. Consider t = 1 optimization problem for
i 2 f1;:::;mg and j 2 f1;:::;ng: For given  p(s); each consumer i 2 f1;:::;mg
max
x(s)2Bzi
ui(xi;  zi) )  xi(s);
and each producer j 2 f1;:::;ng, given his endogenized production set Yj j z(s) for all
s 2 f1;:::;Sg
max
yj(s)
f p(s)  yj(s) : yj(s) 2 Yj j z (s)g )  yj(s):
We have constructed a sequential two argument linear objective function with en-
dogenous asset structure
argmax
( z; y(s); y(0))j

 qzj +
XS
s=1  p(s)  yj(s)
   
yj(s) 2 Yjjz (s)
 q
Pm
i=1  zi(j)   qzj =  p(0)yj(0) 8s 2 S

;
(6)
5which is independent of  j, and consequently the choice of yj is independent of all
i 2 f1;:::;mg:
The result shows, by construction, the independence of the objective function
of the rm from the utility of the stock holders. This decentralizes the objective
function of the rm, and introduces a new class of objective functions, where rms
maximize long run prots.
4 Conclusion
The paper considers a property of the objective function of the model of the rm
introduced in Stiefenhofer [7]. It shows the separation of economic decisions of
the agents in an incomplete markets environment. This follows from endogenizing
the asset structure. The result generalizes the prot maximization criterion of the
Arrow-Debreu model to an economic scenario, where incomplete markets are present,
and the asset structure endogenized.
References
[1] G. Debreu. Smooth preferences. Econometrica, 40:603{615, 1972.
[2] J. Dr eze. (uncertainty) the rm in general equilibrium theory. The Economic
Journal, 95:1{20, 1985.
[3] J. H. Dr eze. Investment under private ownership: Optimality, equilibrium and
stability. Wiley, 1974.
[4] J. Geanokopolos. An introduction to general equilibrium with incomplete asset
markets. Journal of Mathematical Economics, 19:1{38, 1990.
[5] S. Grossman and O. Hart. A theory of competitive equilibrium in stock market
economies. Econometrica, 47:293{330, 1979.
[6] M. Magill and W. Shafer. Characterization of generically complete real asset
structures. Journal of Mathematical Economics, 19:167{194, 1990.
[7] P. Stiefenhofer. Production in incomplete markets. Disucssion Papers in Eco-
nomics, Univeristy of York, 06, 2009.
6

A Theorem on the Decentralization of the Objective Function of the Firm in GEI

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