This paper considers a two period general equilibrium production model with incomplete markets (GEI). The novelty of this model is the endogenous smooth asset structure introduced in Stiefenhofer (2010). It is shown that incomplete markets is a consequence of idiosyncratic risk.

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. 10/13 
 
Endogenous Incomplete Markets: A Production Model 
with Idiosyncratic Risk 
 
By 
 
Pascal Stiefenhofer, University of York  Endogenous Incomplete Markets: A Production Model
with Idiosyncratic Risk
Pascal Christian Stiefenhofer
University of York
June 7, 2010
Abstract
This paper considers a two period general equilibrium production model with incom-
plete markets (GEI). The novelty of this model is the endogenous smooth asset structure
introduced in Stiefenhofer (2010). It is shown that incomplete markets is a consequence of
idiosyncratic risk.
JEL: D20,D52
Keywords: Incomplete Markets, Production
1 Introduction
Most of the general equilibrium literature on production with incomplete markets models
incomplete markets by hypothesis, and derives equilibrium properties for exogenous asset
structures. These models consider aggregate risk as the only source of uncertainty. For a
summary of this class of models see i.e. [1] or [2].
This paper considers an endogenous asset formation model introduced in [3],[4]. The
paper shows that incomplete markets is a consequence of rm specic risk. This follows
from long run market entry barriers, where rms cannot issue stocks in each state of the
world.
2 The Endogenous Asset Formation 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 consumers
which are characterized by assumptions of smooth economies. There are k 2 f1;:::;lg
physical commodities and j 2 f1;:::;ng nancial assets, referred to as stocks. Physical
goods are traded on each of the (S + 1) spot markets. Producers 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.
Contact address: Department of Economics and Related Studies University of York.
ps515@york.ac.uk.
1There are in total l(S+1) physical goods available for consumption. The consumption
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 consumption space for each
consumer i 2 f1;:::;mg is Xi = R
l(S+1)
++ , the strictly positive 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 resources !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.
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 in 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. 1
The objective function of the rm 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)
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
1Assumptions of smooth production sets and utility functions apply. These are introduced in detail
in Stiefenhofer (2009).
2maximize utility subject to a sequence of (S + 1) budget constraints.
Denote 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 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.  is the full payo matrix of the economy of dimension (S  n).
Algebraically, each i 2 f1;:::;mg
(xi;zi)argmax
n
ui(xi;zi) : zi 2 R
n
+;xi 2 Bzi
o
: (3)
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)n  Rn
  of consumption, production and nancial quantities.
(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
(4)
Stiefenhofer [3] shows that equilibria for this stock market model always exist.
3 Result
We formally introduce the assumption of idiosyncratic risk.
Assumption 1 (T) For each producer j 2 f1;:::;ng, technological uncertainty is repre-
sented by a technology correspondence jZ(s)j, a period t = 1 map from Rm
   R++ !
Rn
   R++ for all s 2 f1;:::;Sg such that for each j 2 f1;:::;ng, S  2.
Theorem 1 Incomplete markets, n < S (=
P
j Yj j z is a consequence of assumption 1
(T).
Proof 1 (Theorem 1) Let Sj = 1 for every j 2 f1;:::;ng; and
P
j Sj = S: Then long
run prot prospects (p) > 0 imply long run capacity adjustment and market entrance until
n = S. Similar for negative long run prospects, the number of rms decreases until n = S;
and (p) = 0 satised. This violates assumption (T). Let S > 1 for every j 2 f1;:::;ng,
and
P
j Sj = S: Then (p) > 0 implies market entrance and the issue of new securities
such that in the limit as (p) ! 0 the number of rms increases until j ! n < S by
assumption (T): Similar for (p) < 0; rms exit the market and as (p) ! 0 the number
of rms decreases until j ! n < S by assumption (T); and (p) = 0 satised.
4 Conclusion
The paper considers idiosyncratic risk in an endogenized asset model of production. It
shows that incomplete markets is a consequence of technological uncertainty. This im-
proves on the literature in GEI with production which considers aggregate risk, and models
incomplete markets by assumption.
3References
[1] J. Geanokopolos. "An introduction to general equilibrium with incomplete asset mar-
kets". Journal of Mathematical Economics, 19:1{38, 1990.
[2] M. Magill and W. Shafer. Incomplete markets. In W. Hildenbrand and H. Son-
nenschein, editors, Handbook of Mathematical Economics, volume 4 of Handbook of
Mathematical Economics, chapter 30, pages 1523{1614. Elsevier, 1991.
[3] P. Stiefenhofer. "Production in incomplete markets". Disucssion Papers in Economics,
Univeristy of York, 06, 2009.
[4] P. Stiefenhofer. "A theorem on the decentralization of the objective function of the
rm in GEI". Discussion Papers in Economics, University of York, 08, 2010.
4

Endogenous Incomplete Markets: A Production Model with Idiosyncratic Risk

http://www.york.ac.uk/media/economics/documents/discussionpapers/2010/1013.pdf

Endogenous Incomplete Markets: A Production Model with Idiosyncratic Risk

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics