General Equilibrium Theory Lecture Notes, Bisin PDF
General Equilibrium Theory Lecture Notes, Bisin PDF
General Equilibrium Theory Lecture Notes, Bisin PDF
Lecture notes
Alberto Bisin
Dept. of Economics
NYU1
November 4, 2014
1 These
notes constitute the material for the second section of the first year
graduate Micro course at NYU. The first section on Decision theory, is taught by
Ariel Rubinstein. The notes owe much to the brilliant TAs Ariel and I had, Sevgi
Yuksel, Bernard Herskovic, Andrew McClellan.
ii
Contents
1 Introduction
1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Abstract exchange economies
2.1 Pareto efficiency . . . . . . . . . . . . . . . . . . . . .
2.1.1 Characterization of Pareto efficient allocation
2.2 Competitive market equilibrium . . . . . . . . . . . .
2.2.1 Welfare . . . . . . . . . . . . . . . . . . . . .
2.2.2 Existence . . . . . . . . . . . . . . . . . . . .
2.2.3 Uniqueness . . . . . . . . . . . . . . . . . . .
2.2.4 Local uniqueness . . . . . . . . . . . . . . . .
2.2.5 Characterization of the structure of equilibria
2.3 Some useful math . . . . . . . . . . . . . . . . . . . .
2.3.1 Convexity and separation . . . . . . . . . . .
2.3.2 Fixed point theorems . . . . . . . . . . . . . .
2.3.3 A primer on differential topology . . . . . . .
2.3.4 References . . . . . . . . . . . . . . . . . . . .
3 Two-period economies
3.1 Arrow-Debreu economies . . . . . . . . . . . . . .
3.2 Financial market economies . . . . . . . . . . . .
3.2.1 The stochastic discount factor . . . . . . .
3.2.2 Arrow theorem . . . . . . . . . . . . . . .
3.2.3 Existence . . . . . . . . . . . . . . . . . .
3.2.4 Constrained Pareto optimality . . . . . . .
3.2.5 Asset pricing . . . . . . . . . . . . . . . .
3.2.6 Some classic representation of asset pricing
3.2.7 Production . . . . . . . . . . . . . . . . .
iii
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iv
CONTENTS
Chapter 1
Introduction
The standard approach to graduate teaching of general equilibrium theory
involves introducing a series of theorems on existence, characterization, and
welfare properties of competitive equilibria under weaker and weaker assumptions in larger and larger (and sometimes weirder) commodity spaces. Such
an approach introduces the students to precise rigorous mathematical analysis and invariably impresses them with the elegance of the theory. Various
textbooks take this approach, in some form or another:
A. Mas-Colell, M. Whinston, and J. Green (1995): Microeconomic Theory,
Oxford University Press, Part 4 - the main modern reference; it also
contains a short introduction to two-period economies.
L. McKenzie (2002), Classical General Equilibrium Theory, MIT Press - a
beautiful modern treatment of the classical theory.
K. Arrow and F. Hahn (1971): General Competitive Analysis, North Holland
- the classical treatment of the classical theory.1
B. Ellickson (1994): Competitive Equilibrium: Theory and Applications,
Cambridge University Press - somewhat heterodox in the choice of the
main themes; it contains a useful chapter on non-convex economies.
1
CHAPTER 1. INTRODUCTION
1.1
Preliminaries
Chapter 2
Abstract exchange economies
Consider an economy populated by agents with exogenously given preferences
over and endowments of commodities. There is no production. Nonetheless
agents do not necessarily consume their own endowments but rather participate in an allocation mechanism. We now formalize this structure.
The economy is populated by an infinity of agents. Agents are categorized
in a finite set I = {1, .., I} of types, with generic element i.1 We also assume
an infinite number of agents is of type i, for any i I. The consumption set
X is the set of admissible levels of consumption of L existing commodities.
We shall assume
X = RL+ , with generic element x.
X is then a convex set, bounded below.
Each agent of type i I has a utility function
Ui : X R
which represents his preferences.2 Each agent of type i I also has an
1
endowment i X.
Definition 1 An allocation for the economy is an array x =(x1 , .., xI )
X I := RLI
+ . An allocation is feasible if it satisfies
I
X
xi
i=1
I
X
i.
i=1
2.1
Pareto efficiency
U i (x + (1 )y) > U i (x) + (1 )U i (y) min{U i (x), U i (y)} for all (0, 1).
Pareto Efficient allocations are solutions of the following problem (convince yourself this is just a formal translation of the definition):
max U i (xi )
(PE pb)
xX
s.t.
X
i
j
xi
U (x ) U j for all j =
6 i
for some given vector U j j6=i RI1 .
Varying the values of U j for j 6= i we obtain the set of Pareto efficient
allocations.
Let the Utility possibility set be defined as
(
)
X
X
I
i i I
LI
i
i
U = U R U U (x ) i=1 , for some x R+ such that
x
.4
j
Problem 1 Show that U is bounded above and closed. Under which conditions is U also convex?
The Pareto utility frontier is defined as
UP = {U U |@U 0 U such that U 0 > U } .
It contains the image of all Pareto optimal allocations in the space of utility
levels.
Problem 2 Show that UP ( bdry(U).
There exist another formal characterization of Pareto efficiency.
4
the image of the feasible allocations in the space of utility levels; in particular U is
unbounded below, which is not necessarily the case for U0 .
i U i (xi )
(Negishi pb)
iI
s.t.
xi
iI
i.
iI
(Only if ) Furthermore, any solution of the Negishi pb, for some RI+ ,
6= 0, is a Pareto efficient allocation.
Proof. Consider a Pareto efficient allocation x X I , so that U (x) =
{U i (xi )}iI UP. U RI is convex by assumption (in Problem 1 you
are asked for conditions under which this is the case). Furthermore, U (x)
bdry(U). The Supporting hyperplane theorem (see Theorem 7) then implies
that there exists a RI , 6= 0, such that
U (x) U, for any U U;
that is,
U (x) arg max
i U i
iI
s.t. U U.
By the definition of U:
x arg max
i U i (xi )
iI
s.t.
X
iI
iI
2.1.1
2.2
An allocation mechanism is a rule which maps the preferences and endowments of each of the agents in the economy into an allocation. The allocation
mechanism standing at the core of most of economics is that of competitive
markets. But this is not the only possible mechanism. We now study in
detail the competitive equilibrium concept, but we shall mention another allocation mechanism, the jungle equilibrium, introduced formally and studied
in detail by Piccione and Rubinstein (2007).
At a competitive equilibrium agents trade in perfectly competitive markets, where:
prices are linear : the unit price pl of each commodity l is fixed independently
of level of individual trades and is the same for all agents;
prices are non-negative: this is typically justified under free disposal, that
is, by the assumption that agents can freely dispose of any amount of
any commodity;5
markets are complete: for each commodity l in X there is a market where
the commodity can be traded.
Definition 3 A competitive equilibrium is an allocation x X I and a price
p RL+ such that:
i) each agent i I solves:
max
U i (xi )
i
(Consumer pb)
x X
s.t. pxi p i
for given price p RL+ ; and
ii) markets clear (the allocation is feasible):
X
X
xi
i.
i
We assumed utilities are strictly monotonic increasing and hence, as we shall later see,
prices will be strictly positive at any equilibrium.
x +
I
X
i=1
I
X
i;
i=1
10
smoothness: z (p, ) is C 1 ;
homogeneity of degree 0: z (p, ) = z (p, ) ,for any > 0;
Walras Law: pz (p, ) = 0, p >> 0;
lower boundedness: s such that zl (p, ) > s, l L;
boundary property:
pn p 6= 0, with pl = 0 for some l, max {z1 (pn , ), ..., zL (pn , )} .
The simple proof of this Proposition is left to the reader. [Hint: lower
boundedness follows from demand being non-negative. The boundary property
is written as confusingly as it is it is because if the prices of more than one
commodity converge to 0, it is possible that only a subset of the corresponding
excess demands explodes.]
2.2.1
Welfare
11
By construction
B(x) is convex. Furthermore, x being Pareto efficient implies
P
/ B(x).6 The supporting hyperplane theorem implies that there
that iI xi
exist a p RL such that py px, for any y B(x).
It remains to show that i) indeed p RL++ and that ii) at such prices any
agent i I, with endowment xi X will not trade, that is, we need to show
that py i > pxi , for any y i B i (xi ).
We show i) first, proceeding by contradiction. Suppose first pl < 0 for
some l = 1, ..., L. We can then construct for any arbitrary i I a vector
y i such that yli0 = xil0 , for any l0 6= l, and yli > xil so that y i B i (xi ) and
py i < pxi . Since i I is arbitrary, this implies the desired contradiction
with the implication of the supporting hyperplane theorem obtained above.
Suppose now that pl = 0 for some l = 1, ..., L. We can then construct for
any arbitrary i I a vector y i such that yli0 = xil0 , for any l0 6= l, and
yli > xil . By continuity of U i we can in fact choose to be small enough and
yli xil large enough so that y i B i (xi ). But by construction then py i < pxi .
Since i I is arbitrary, this again implies the desired contradiction with the
implication of the supporting hyperplane theorem.
As for ii), note that for any i I, pxi > 0, since xi RL++ by assumption.
As consequence, there exist a cheaper bundle xi , such that p
xi < pxi . Coni
i i
i
sider now any y B (x ) and assume by contradiction py pxi . Construct
6
12
2.2.2
Existence
the L-simplex, a compact ad convex set. The equilibrium equations can thus
be always reduced to L 1 equations in L 1 unknowns. Since equations are
typically nonlinear, having number of unknowns less or equal than number
of independent equations does not ensure a solution exists.
13
j=1
, l = 1, ..L
, l = 1, ..L.
Then
zl (p)pl
L
X
j=1
L
R . Consider an arbitrary continuous extension of the excess demand to the
whole simplex; call it z : L1 RL . It is straightforward to construct
this extension so that Walras law, pz (p) = 0 is satisfied. Construct the
corresponding map : L1 L1 by substituting z for z. Consider
now a sequence of trimmed simplexes, as 0 and the corresponding
sequences of maps {z } and { }. It is straightforward to show that, for any
14
if p int L1
if p bdry L1
Fixed points are not necessarily unique. In this case the sequence is of sets of fixed
points.
8
The proof is taken, with minor changes, from Mas Colell et al. (1995), Proposition
17.C.1, p. 585-7.
L1
9
With the notation 2
it is meant the power set (the set of all subsets) of L1 .
The map f can equivalently be said to be a correspondence from L1 into L1 .
15
2. ql = 0 if pl > 0 and furthermore, any p bdry L1 cannot satisfy
p f (p) as pp > 0 while pq = 0 for all q f (p).
But then 2) implies that any fixed point p f (p) must satisfy p int L1 .
In turn 1) implies that any p int L1 cannot satisfy p f (p) if z(p) 6= 0.
As a consequence, any p f (p) must satisfy p int L1 and z(p) = 0. In
other words, any fixed point of the map f is a competitive equilibrium price.
It remains to show that a fixed point p f (p) exist. This is a consequence
of Kakutani fixed point theorem, Theorem 9, if we can prove that i) L1
is a non-empty, compact, and convex set and ii) f is upper-hemi-continuous,
non-empty and convex valued. i) is straightforward and hence we concentrate
on ii).
Non-empty and convex valuedness. For any p L1 , f (p) is a face of the
simplex L1 ,10 hence non-empty and convex.
Upper-hemi-continuity. Consider sequences {pn , q n } L1 L1 such
that pn p, q n q, and q n f (pn ) for all n. We need to show that
q f (p). Considerthe following three distinct
cases: ii1) p int L1 ,
ii2) p bdry L1 and pn bdry L1 , indeed in the same face of the
simplex as p, for n large enough, ii3) p bdry L1 and pn int L1
for n large enough. Note that as pn p, pn cannot be on a different face of
the simplex as p for n large enough and as a consequence we can disregard
this case. Case ii1) is straightforward: pn >> 0 for n large enough and
z(p) is continuous and so the arg maxqL1 qz(pn ) is upper-hemi-continuous.
Consider case ii2) and ii3). Let pl > 0, without loss of generality. Case ii2)
implies that for n large enough pn is on the same face of the simplex as p. As
a consequence, ql = 0 by construction of the map f . This is enough to show
that q f (p)in this case. Finally consider case ii3). Note that, as pl > 0
and pnl > 0 for n large enough, zl pn is bounded above for n large enough, as
the budget set is bounded.
On the contrary, for some l0 L, l0 6= l, pl0 = 0,
as p bdry L1 in this case. Then zl0 (pn ) , for some l0 L, l0 6= l,
by the boundary property of z(p) in Proposition 1. Therefore, for n large
enough, zl0 (pn ) > zl (pn ) and hence qln = 0. This concludes the proof of
upper-hemi-continuity of f .
Problem 5 Which steps of the proof of a) Negishis theorem, b) First and
Second Welfare theorem, c) Existence, relies crucially on i) strict monotonic10
16
2.2.3
Uniqueness
Existence of a competitive equilibrium can be proved under quite general conditions.11 Equilibria are however unique only under very strong restrictions.
Several examples of such restrictions are listed in the following, without any
detail.
Pareto efficiency. If endowments are Pareto efficient, there exists a unique
equilibrium which is autarchic: xi = i for all i I.
Aggregation. If preferences are identical and homothetic, then an aggregation result implies that the economy is equivalent to one with a single
representative agent and hence the exists a unique equilibrium which
is effectively autarchic.
Gross substitution. If the aggregate demand satisfies the gross substitution property,
p0l > pl and p0j = pj for all j 6= l = zj (p0 ) > zj (p),
the law of demand holds at any equilibrium price and there exists a
unique equilibrium. Gross substitution holds for instance for Cobb
Douglas and CES utility functions under restrictions on the elasticity
of substitutions across goods.
Problem 6 Prove that indeed uniqueness holds in each of the above three
environments, Pareto efficiency, aggregation, and gross substitution.
2.2.4
Local uniqueness
17
18
2.2.5
i(p, ).
p:z(p,)=0
19
20
21
2.3
2.3.1
22
23
2.3.2
24
function correspondence14 mapping A into itself and such that the set f (x)
A is non-empty and convex for any x A . Then f has a fixed point in A,
that is,
x A such that x f (x)
2.3.3
1 (x1 ) =
2 (x1 ) =
3 (x2 ) =
4 (x2 ) =
14
q
2
if x2 > 0
x1 , 1 (x1 )
q
2
x1 , 1 (x1 )
if x2 < 0
q
2
1 (x2 ) , x2 if x1 > 0
q
2
1 (x2 ) , x2 if x1 < 0
See Mas Colell et al. (1995), Definition M.H.3, p. 950 for a definition of upper-hemicontinuous correspondence.
25
26
27
2.3.4
References
28
Chapter 3
Two-period economies
In a two-period pure exchange economy we study financial market equilibria. In particular, we study the welfare properties of equilibria and their
implications in terms of asset pricing.
In this context, as a foundation for macroeconomics and financial economics, we study sufficient conditions for aggregation, so that the standard
analysis of one-good economies is without loss of generality, sufficient conditions for the representative agent theorem, so that the standard analysis of
single agent economies is without loss of generality.
The No-arbitrage theorem and the Arrow theorem on the decentralization of equilibria of state and time contingent good economies via financial
markets are introduced as useful means to characterize financial market equilibria.
3.1
Arrow-Debreu economies
30
will assume:
n
Assumption 1 i R++
for all i. Furthermore, ui is continuous, strongly
monotonic, strictly quasi-concave and smooth, for all i. Finally, ui has a
Von Neumann-Morgernstern representation:
ui (xi ) = ui (xi0 ) +
S
X
probs ui (xis )
s=1
0i )
S
X
s (xis si ) 0
(3.1)
s=0
S
X
s (xs si ) 0, and
s=0
I
X
2.
i=1
Observe that the dynamic and uncertain nature of the economy (consumption occurs at different times t = 0, 1 and states s S) does not
manifests itself in the analysis: a consumption good l at a time t and state s
is treated simply as a different commodity than the same consumption good
1
We write the budget constraint with equality. This is without loss of generality under
monotonicity of preferences, an assumption we shall maintain.
31
3.2
It will be convenient to define as to be the s-th row of the matrix. Note that
it contains the payoff of each of the assets in state s.
Let p = (p0 , p1 , ..., pS ), where ps RL++ for each s, denote the spot price
vector for goods. That is, for a price pls agents trade one unit of good l
in state s. Recall the definition of prices for state contingent commodities
in Arrow-Debreu economies, denoted and note the difference (different
commodity spaces are everything in the world of general equilibrium theory)!
Let good l = 1 at each date and state represent the numeraire; that is,
p1s = 1, for all s = 0, ..., S.
Let xisl denote the amount of good l that agent i consumes in good s. Let
2
32
q = (q1 , ..., qJ ) RJ+ , denote the prices for the assets.4 Note that the prices
of assets are non-negative, as we normalized asset payoff to be non-negative.
Given prices (p, q) Rn++ Rj+ and the asset structure A RSJ
+ , any agent
i picks a consumption vector xi X and a portfolio z i RJ to maximize
presend discounted utility. s.t.
p0 (xi0 0i ) qz i
ps (xis si ) As z i , for s = 1, ...S.
Definition 13 A Financial markets equilibrium is a (x, z, p, q) X I Rj
Rn++ Rj+ such that
1. xi arg max ui (x) s.t.
I
X
2.
i=1
si
p0 (xi0 0i ) qz i
;
ps (xis si ) as z i , for s = 1, ...S
I
X
z i = 0.
i=1
Quantities will be column vectors and prices will be row vectors, to avoid the annoying
use of transposes.
33
No Arbitrage
Before deriving the properties of asset prices in equilibrium, we shall invest
some time in understanding the implications that can be derived from the
milder condition of no-arbitrage. This is because the characterization of noarbitrage prices will also be useful to characterize financial markets equilbria.
For notational convenience, define the (S + 1) J matrix
W =
q
A
.
RS+1
= {0}.
+
= {0}
RS+1
W = 0.
RS+1
+
++ such that
Recall that W z > 0 requires that all components of W z are 0 and at least one of
them > 0.
34
...
...
W =
S ajS z = 0 ;
0 q j +
1 aj1 + ... +
... Jx1
...
a condition which must hold for any z RJ , hence implying, after rearranging:
q j = 1 aj1 + ... + S ajS , for s =
s
and any asset j J.
(3.2)
is strictly positive,
> 0, the desired contradiction.
RS+1
+ \{0}. Since
A few final remarks to this section.
35
Remark 3 An asset which pays one unit of numeraire in state s and nothing in all other states (Arrow security), has price s ; this is an immediate
consequence of (3.2). Such asset is called Arrow security.
Remark 4 Is the vector
obtained by the No-arbitrage theorem unique? Notice how (3.2) defines a system of J equations and S unknowns, represented
by . Define the set of solutions to that system as
R(q) = { RS++ : q = A}.
Suppose, the matrix A has rank J 0 J (that it, A has J 0 linearly independent
column vectors and J 0 is the effective dimension of the asset space). In general, then R(q) will have dimension S J 0 . It follows then that, in this case,
the No-arbitrage theorem restricts to lie in a S J 0 dimensional set (or,
equivalently,
in a S J 0 + 1 dimensional set). If we had S linearly independent assets, the solution set has dimension zero, and there is a unique
vector that solves (3.2). The case of S linearly independent assets is referred
to as complete markets.
Remark 5 Recall we assumed preferences are Von Neumann-Morgernstern:
X
ui (xi ) = ui (xi0 ) +
probs ui (xis ),
s=1,...,S
s=1,...,S
s
.
probs
Then
(3.4)
3.2.1
In the previous section we showed the existence of a vector that provides the
basis for pricing assets in a way that is compatible with equilibrium, albeit
milder than that. In this section, we will strengthen our assumptions and
study asset prices in a full-fledged economy. Among other things, this will
allow us to provide some economic content to the vector
36
ui (xis )
xi1s
ui (xi0 )
xi10
Let IM RS i (xi ) = (. . . IM RSsi (xi ) . . .) RS+ denote the vector of intertemporal marginal rates of substitution for agent i, an S dimentional vector. Note that, under the assumption of strong monotonicity of preferences,
IM RS i (xi ) RS++ .
By taking the First Order Conditions (necessary and sufficient for a maximum under the assumption of strict quasi-concavity of preferences) with
respect to zji of the individual problem for an arbitrary price vector q, we
obtain that
j
q =
S
X
probs IM RSsi (xi )ajs = E IM RS i (xi ) aj ,
(3.5)
s=1
for all j = 1, ..., J and all i = 1, ..., I, where of course the allocation xi is the
equilibrium allocation. At equilibrium, therefore, the marginal cost of one
j
more unit of asset j, qP
, is equalized to the marginal valuation of that agent
for the assets payoff, Ss=1 probs IM RSsi (xi )ajs .
Compare equation (3.5) to the previous equation (3.4). Clearly, at any
equilibrium, condition (3.5) has to hold for each agent i. Therefore, in equilibrium, the vector of marginal rates of substitution of any arbitrary agent i
can be used to price assets; that is any of the agents vector of marginal rates
of substitution (normalized by probabilities) is a viable stochastic discount
factor m.
In other words, any vector (. . . probs IM RSsi (xi ) . . .) belongs to R(q) and
is hence a viable for the asset pricing equation (??). But recall that
R(q) is of dimension S J 0 , where J 0 is the effective dimension of the asset space. The higher the the effective dimension of the asset space (intuitively said, the larger the set of financial markets) the more aligned are
agents marginal rates of substitution at equilibrium (intuitively said, the
smaller are unexploited gains from trade at equilibrium). In the extreme
case, when markets are complete (that is, when the rank of A is S) and the
37
set R(q) is a singleton, IM RS i (xi ) are equalized across agents i at equilibrium: IM RS i (xi ) = IM RS, for any i = 1, ..., I.
Let M RSlsi (xi ) denote agent is marginal rate of substitution between
consumption the good l and consumption of the numeraire good 1 in state
s = 0, 1, ..., S:
M RSlsi (xi ) =
ui (xis )
xils
ui (xis )
xi1s
let also M RSsi (xi ) = (. . . M RSlsi (xi ) . . .) RL+ and M RS i (xi ) = (. . . M RSsi (xi ) . . .)
RLS
+ .
Problem 8 Write the Pareto problem for the economy and show that, at
any Pareto optimal allocation, x, it is the case that
IM RS i (xi ) = IM RS
M RS i (xi ) = M RS
for any i = 1, ..., I. Furthermore, show that an allocation x which satisfies
the feasibility conditions (market clearing) for goods and is such that
IM RS i (xi ) = IM RS
M RS i (xi ) = M RS
for any i = 1, ..., I, is a Pareto optimal allocation.
We conclude that, when markets are Complete, equilibrium allocations
are Pareto optimal. That is, the First Welfare theorem holds for Financial
market equilibria when markets are Complete.
Problem 9 (Economies with bid-ask spreads) Extend our basic two-period
incomplete market economy by assuming that, given an exogenous vector
J
R++
:
the buying price of asset j is qj + j
while
the selling price of asset j is qj
for any j = 1, ..., J. Write the budget constraint and the First Order Conditions for an agent is problem. Derive an asset pricing equation for qj in
terms of intertemporal marginal rates of substitution at equilibrium.
38
3.2.2
Arrow theorem
j
j
n
Proposition 5 Let (x, z, p, q) RnI
++ R R++ R+ represent a Financial
market equilibrium of an economy with rank(A) = S. Then (x, ) RnI
++
Rn++ represents an Arrow-Debreu equilibrium if s = s ps , for any s = 1, ..., S
S
n
and some R++
. The converse also holds. Let (x, ) RnI
++ R++
nI
j
n
represent an Arrow-Debreu equilibrium. Then (x, z, p, q) R++ R R++
Rj+ represents a Financial market equilibrium of a complete market economy
(that is, whose asset structure satisfies rank(A) = S) if
S
X
s=1
Proof. = Financial market equilibrium prices of assets q satisfy Noarbitrage. There exists then a vector
RS+1
W = 0, or
++ such that
q = A. The budget constraints in the financial market economy are
p0 xi0 0i + qz i = 0
ps xis si = as z i , for s = 1, ...S.
39
Substituting q = A, expanding the first equation, and writing the constraints at time 1 in vector form, we obtain:
p0 xi0
0i
S
X
s as z
p0 xi0
s=1
0i
S
X
s ps xis si = 0(3.6)
s=1
.
.
i
ps (xs si )
.
.
<A>
(3.7)
i
S
But if rank(A) = S, it follows that < A >= R , and the constraint ps (xs si )
.
.
A > is never binding. Each agent is problem is then subject only to
p0 xi0
0i
S
X
s ps xis si = 0,
s=1
S
X
s=1
and using s = probs IM RSsi (xi ), for any s = 1, ..., S, proves the result.
(Recall that, with Complete markets IM RS i (xi ) = IM RS, for any i =
1, ..., I.)
3.2.3
Existence
<
40
xi (, p) i = z F M (, p) = 0.
iI
i
i
)
p
(x
restricted by
s
s
s
< A > . This is where the Cass trick comes
.
.
in handy. It is in fact an important Lemma.
Cass trick. For any Financial
market economy,
consider a modified econ.
i
i
< A > is imposed on all
p
(x
)
omy where the constraint
s
s
s
.
.
agents i = 2, ..., I but not on agent i = 1. Any equilibrium of the
Financial Market economy is an equilibrium of the modified economy,
and any equilibrium of the modified economy is a Financial market
equilibrium.
41
Proof. Consider an P
equilibrium of the modified economy
statePI in the
I
i
i
i
i
ment. At equilibrium,
i=1 ps (xs
s ) = 0. Therefore, i=2 ps (xs s ) =
i
i
1
1
ps (xs s ) . But ps (xs s ) < A >, for any i = 2, ..., I, and hence
.
.
PI
PI
i
i
i
i
1
1
i=2 ps (xs s ) < A > . Since
i=2 ps (xs s ) = ps (xs s ) , it follows that ps (x1s s1 ) < A >, and hence that ps (x1s s1 ) < A > .
Therefore, the constraint ps (x1s s1 ) < A > must necessarily hold at
an equilibrium of the modified economy. In other words, the constraint
ps (x1s s1 ) < A > is not binding at a Financial market equilibrium. The
equivalence between the modified economy and the Financial Market economy is now straightforward.
In the modified economy, now, agent 1 faces complete markets without
loss of generality. His excess demand, therefore, will satisfy the boundary
conditions; these properties will transfer than to the aggregate excess demand
and the existence proof will proceed exactly as in the standard Arrow-Debreu
economy.
3.2.4
Under Complete markets, the First Welfare Theorem holds for Financial
market equilibrium. This is a direct implication of Arrow theorem.
j
j
n
Proposition 6 Let (x, z, p, q) RnI
++ R R++ R+ be a Financial market
equilibrium of an economy with Complete markets (with rank(A) = S). Then
x RnI
++ is a Pareto optimal allocation.
However, under Incomplete markets (with rank(A) < S), Financial market equilibria are generically inefficient in a Pareto sense. That is, a planner
could find an allocation that improves some agents without making any other
agent worse off. Note that of course a Pareto optimal allocation is a Financial
Market equilibrium (with no trade), independently of the asset matrix A in
the economy. As a consequence, it is immediate that, even with Incomplete
markets, equilibria are a most generically (not always) Pareto inefficient.
j
j
n
Theorem 15 Let (x, z, p, q) RnI
++ R R++ R+ be a Financial market
42
(3.8)
s=1
.
.
i
ps (xs si )
.
.
<A>
(3.9)
S
. Pareto optimality of xrequires that there does not exist
for some R++
an allocation y such that
1. u(y i ) u(xi ) for any i = 1, ..., I (strictly for at least one i), and
I
X
2.
y i si = 0, for any s = 0, 1, ..., S
i=1
i
i
/ A >, for some i = 1, ..., I;
a y exists, it must be that
ps (ys s ) <
.
.
otherwise the allocation y would be budget feasible for all agent i at the
equilibrium prices. Generic Pareto sub-optimality of x follows then directly
from the following Lemma.
j
j
n
Lemma 2 Let (x, z, p, q) RnI
++ R R++ R+ be a Financial market
equilibrium of an
economy withrank(A) < S. For a generic set of economies,
.
i
i
the constraints ps (xs s ) < A > are binding for some i = 1, ..., I.
.
.
43
Proof. We shall only sketch the proof here. Consider Financial market equilibria as the zeroes of the excess demand system for this economy, as defined
earlier in this section (but making explicit the dependence on endowments
FM
(, p, ) = 0. Take any two distinct agents i and j and
RnI
++ ): z
note that Pareto optimality requires that IM RS i (xi , ) = IM RS j (xj , ),
where once again we make explicit the dependence on endowments RnI
++ .
Consider now the system
z F M (, p, )
h(, p, ) =
= 0.
IM RS i (xi , ) IM RS j (xj , )
n1
n
Because of the normalizations, the system maps R++
RnI
++ into R++ (recall that n = L(S + 1)). Suppose we could show that, at any (, p, )
nI
Rn1
++ R++ such that h(, p, ) = 0, D h(, p, ) has rank n. Then, the
Transversality Theorem would immediately imply that h(, p, ) = 0 has
generically no solutions in RnI
++ . The proof that D h(, p, ) has rank
n at equilibrium can be found in Magill-Shafer, ch. 30 in W. Hildenbrand
and H. Sonnenschein (eds.), Handbook of Mathematical Economics, Vol. IV,
Elsevier, 1991.
Pareto optimality might however represent too strict a definition of social
welfare of an economy with frictions which restrict the consumption set,
as in the case of incomplete markets. In this case, markets are assumed
incomplete exogenously. There is no reason in the fundamentals of the model
why they should be, but they are. Under Pareto optimality, however, the
social welfare notion does not face the same contraints. For this reason,
we typically define a weaker notion of social welfare, Constrained Pareto
optimality, by restricting the set of feasible allocations to satisfy the same
set of constraints on the consumption set imposed on agents at equilibrium.
In the case of incomplete markets, for instance, the feasible wealth vectors
across states are restricted to lie in the span of the payoff matrix. That can be
interpreted as the economys financial technology and it seems reasonable
to impose the same technological restrictions on the planners reallocations.
The formalization of an efficiency notion capturing this idea follows.
S
i
i S
SL
Let xit=1 = (xis )s=1 RSL
++ ; and similarly t=1 = (s )s=1 R++ , pt=1 =
S
SL
J
SL
(ps )s=1 RSL
++ . Let gt=1 (t=1 , ), mapping R++ R into R++ , denote the
equilibrium map for t = 1 spot markets at when each agent i = 1, ..., I has
i
i
i
endowment (s1
+ as i , s2
, ..., sL
), for any s S.
44
2.
1.
I
X
i=1
and
3.
i
yt=1
B(gt=1 (t=1 , )), for any i = 1, ..., I.
The constraint on the consumption set restricts only time 1 consumption allocations. More general constraints are possible but these formulation
is consistent with the typical frictions we encounter in economics, e.g., on
financial markets. It is important that the constraint on the consumption
set depends in general on gt=1 (t=1 , ), that is on equilibrium prices for spot
markets opened at t = 1 after income transfers to agents. It implicit identifies income transfers (besides consumption allocations at time t = 0) as the
instrument available for Constrained Pareto optimality; that is, it implicitly
constrains the planner implementing Constraint Pareto optimal allocations
to interact with markets, specifically to open spot markets after transfers.
On the other hand, the planner is able to anticipate the spot price equilibrium map, gt=1 (t=1 , ); that is, to internalize the effects of different transfers
on spot prices at equilibrium. Consider first a degenerate case:
j
j
n
Proposition 7 Let (x, z, p, q) RnI
++ R R++ R+ be a Financial market
equilibrium of an Arrow-Debreu economy whose consumption set at time t =
1 is restricted by
SL
xit=1 B R++
, for any i = 1, ..., I
45
46
.
.
.
.
i
i
ps (ysi si ) <
/ A >, for some i = 1, ..., I; while gs (s , ) (ys s ) =
.
.
.
.
i
A , for all i = 1, ..., I. Generic Constrained Pareto sub-optimality of x follows then directly from the following Lemma, which we leave without proof.8
j
j
n
Lemma 3 Let (x, z, p, q) RnI
++ R R++ R+ be a Financial market equilibrium of an economy with Incomplete markets (with rank(A) <
2I
S). For a generic set of economies (, ) RnI
++ R , the constraints
.
P i .
that
dz = 0, are weakly relaxed for all i = 1, ..., I, strictly for at least
iI
one.9
There is a fundamental difference between incomplete market economies,
which have typically not Constrained Optimal equilibrium allocations, and
economies with constraints on the consumption set, which have, on the contrary, Constrained Optimal equilibrium allocations. It stands out by comparing the respective trading constraints
gs (s , )(xis si ) = As i , for all i and s,
vs.
47
The constrained inefficiency due the dependence of constraints on equilibrium prices is sometimes called a pecuniary externality.10 Several examples of
such form of externality/inefficiency have been developed recently in macroeconomics. Some examples are:
- Thomas, Charles (1995): The role of fiscal policy in an incomplete markets
framework, Review of Economic Studies, 62, 449468.
- Krishnamurthy, Arvind (2003): Collateral Constraints and the Amplification Mechanism,
Journal of Economic Theory, 111(2), 277-292.
- Caballero, Ricardo J. and Arvind Krishnamurthy (2003): Excessive Dollar Debt: Financial Development and Underinsurance, Journal of Finance, 58(2), 867-94.
- Lorenzoni, Guido (2008): Inefficient Credit Booms, Review of Economic
Studies, 75 (3), 809-833.
- Kocherlakota, Narayana (2009): Bursting Bubbles: Consequences and
Causes,
http://www.econ.umn.edu/nkocher/km bubble.pdf.
- Davila, Julio, Jay Hong, Per Krusell, and Victor Rios Rull (2005): Constrained Efficiency in the Neoclassical Growth Model with Uninsurable
Idiosyncractic Shocks, mimeo, University of Pennsylvania.
Remark 6 Consider an economy whose constraints on the consumption set
depend on the equilibrium allocation:
xit=1 B(xt=1 , z ), for any i = 1, ..., I
This is essentially an externality in the consumption set. It is not hard to
extend the analysis of this section to show that this formulation introduces
inefficiencies and equilibrium allocations are Constraint Pareto sub-optimal.
10
48
i
0
i
i
3.
ps (xs s ) = Az , for any i = 1, ..., I
.
.
where p is the spot market Financial market equilibrium vector of prices. That
is, the planner takes the equilibrium prices as given. It is immediate to prove
that, with this definition of Constrained Pareto optimality, any Financial
market equilibrium allocation x of an economy with Incomplete markets is
in fact Constrained Pareto optimal, independently of the financial markets
available (rank(A) S).
Problem 10 Consider a Complete market economy (rank(A) = S) whose
feasible set of asset portfolios is restricted by:
z i Z ( RJ , for any i = 1, ..., I
A typical example is borrowing limits:
z i b, for any i = 1, ..., I
Are equilibrium allocations of such an economy Constrained Pareto optimal
(also if L > 1)?
Problem 11 Consider a 1-good (L = 1) Incomplete market economy (rank(A) <
S) which lasts 3 periods. Define an Financial market equilibrium for this
economy as well as Constrained Pareto optimality. Are Financial market
equilibrium allocations of such an economy Constrained Pareto optimal?
49
50
and assets are all determined simultaneously. The financial and the real
sectors of the economy cannot be isolated. Under some special conditions,
however, the consumption and portfolio decisions of agents can be separated.
This is typically very useful when the analysis is centered on financial issue.
In order to concentrate on asset pricing issues, most finance models deal in
fact with 1-good economies, implicitly assuming that the individual financial
decisions and the market clearing conditions in the assets markets determine
the financial equilibrium, independently of the individual consumption decisions and market clearing in the goods markets; that is independently of the
real equilibrium prices and allocations. In this section we shall identify the
conditions under which this can be done without loss of generality. This is
sometimes called the problem of aggregation.
The idea is the following. If we want equilibrium prices on the spot
markets to be independent of equilibrium on the financial markets, then
the aggregate spot market demand for the L goods in each state s should
must depend only on the incomes of the agents in this state (and not in
other states) and should be independent of the distribution of income among
agents in this state.
Theorem 16 Budget Separation. Suppose that each agent is preferences are separable across states, identical, homothetic within states, and
von Neumann-Morgenstern; i.e. suppose that there exists an homothetic
u : RL R such that
i
u (x ) =
u(xi0 )
S
X
s=1
consumption vector xi R+
51
to
max ui (xi )
s.t.
p0 xi0 = y0i
ps xis = ysi , f or s = 1, ...S.
Let the L(S + 1) demand functions be given by xils (p, y i ), for l = 1, ..., L,
s = 0, 1, ...S. Define now the indirect utility function for income by
v i (y i ; p) = ui (xi (p, y i )).
The Income allocation problem. Given prices (p, q), endowments i , and the
asset structure A, agent i has to pick a portfolio z i RJ and an income
S+1
stream y i R++
to
max v i (y i ; p)
s.t.
p0 0i qz i = y0i
ps si + as z i = ysi , f or s = 1, ...S.
By additive separability across states of the utility, we can break the consumption allocation problem into S +1 spot market problems, each of which
yields the demands xis (ps , ysi ) for each state. By homotheticity, for each
s = 0, 1, ...S, and by identical preferences across all agents,
xis (ps , ysi ) = ysi xis (ps , 1);
and since preferences are identical across agents,
ysi xis (ps , 1) = ysi xs (ps , 1)
Adding over all agents and using the market clearing condition in spot markets s, we obtain, at spot markets equilibrium,
xs (ps , 1)
I
X
ysi
i=1
I
X
si = 0.
i=1
I
X
i=1
ysi )
I
X
i=1
si = 0.
(3.10)
52
Recall from the consumption allocation problem that ps xis = ysi , for s =
0, 1, ...S. By adding over all agents, and using market clearing in the spot
markets in state s,
I
X
ysi
= ps
i=1
I
X
(3.11)
i=1
= ps
I
X
si , for s = 0, 1, ...S.
i=1
I
X
si )
i=1
I
X
si .
(3.12)
i=1
Note how we have passed from the aggregate demand of all agents in the
economy to the demand of an agent owning the aggregate endowments. Observe also how equation (3.12) is a system of L equations with L unknowns
that determines spot prices ps for each state s independently of asset prices
i
q. Note also
PI thati equilibrium spot prices ps defined by (3.12) only depend
through i=1 s .
The Budget separation theorem can be interpreted as identifying conditions under which studying a single good economy is without loss of generality. To this end, consider the income allocation problem of agent i, given
equilibrium spot prices p :
max
i
J
y i RS+1
++ ,z R
v i (y i ; p)
s.t. y0i = p0 0i qz i
ysi = ps si + as z i , for s = 1, ...S
If preferences ui (xi ) are identical, homothetic within states, and von NeumannMorgenstern, that is, if they satisfy
i
u (x ) =
u(xi0 )
S
X
s=1
v (y ; p) =
v(y0i ; p)
S
X
s=1
53
Note that homotheticity in (y0i , y1i , ..., ySi ) is guaranteed by the von NeumannMorgenstern property. Let w0i = p0 0i , wsi = ps si , for any s = 1, ..., S; and
disregard for notational simplicity the dependence of v(y; p) on p. The income
allocation problem can be written as:
max
i
J
y i RS+1
++ ,z R
s.t.
y0i
ysi
w0i
wsi
v(y0i ) +
S
X
probs v(ysi )
s=1
= qz
= As z i , for s = 1, ...S
which is homeomorphic to any agent is optimization problem in the definition of Financial market equilibrium with l = 1. Note that ysi gains the
interpretation of agent is consumption expenditure in state s, while wsi is
interpreted as agent is income endowment in state s.
The representative agent theorem
A representative agent is the following theoretical construct.
Definition 16 Consider a Financial market equilibrium (x, z, p, q) of an
economy populated by i = 1, ..., I agents with preferences ui : X R and
endowments i . A Representative agent for this economy is an agent with
preferences U R : X R and endowment R such that the Financial market
equilibrium of an associated economy with the Representative agent as the
only agent has prices (p, q).
In this section we shall identify assumptions which guarantee that the
Representative agent construct can be invoked without loss of generality.
This assumptions are behind much of the empirical macro/finance literature.
Theorem 17 Representative agent. Suppose preferences satisfy:
ui (xi ) = u(xi0 ) +
S
X
s=1
.
.
ps si
.
.
54
S+1
a map uR : R+
R such that:
I
X
si ,
i=1
R
U (x) = u (y0 ) +
S
X
s=1
I
X
xis , s = 0, 1, ..., S
i=1
I
X
i=1
= 0
If the Representative agents preferences can be constructed independently of the equilibrium of the original economy with I agents, then equilibrium prices can be read out
agents marginal rates of
P
P of the Representative
substitution evaluated at Ii=1 i . Since Ii=1 i is exogenously given, equilibrium prices are obtained without computing the consumption allocation
and portfolio for all agents at equilibrium, (x , z ).
Proof. The proof is constructive. Under the assumptions on preferences in
the statement, we need to shown that, for all agents
i = 1, ...,
oI, equilibrium
L(S+1) PI
i
i
asset prices q are constant in R++
i=1 given .If preferences
P
satisfy ui (xi ) = u(xi0 ) + Ss=1 probs u(xis ), for all i = 1, .., I, with an homothetic u(x), then by the Budget separation
spot
n theorem, equilibrium
o prices p
L(S+1) PI
i
i
are independent of q and constant in R++
i=1 given . There
.
.
i
fore,
ps s < A > can be written as an assumption on fundamentals,
.
.
in particular on i . Furthermore, we can restrict our analysis to the single
55
max
i
J
y i RS+1
++ ,z R
s.t.
y0i
ysi
v(y0i )
S
X
probs v(ysi )
s=1
w0 = qz
ws = As z i , for s = 1, ...S
y0i
w0i
S
X
s ysi wsi
= 0
s=1
But,
.
.
wsi
.
.
.
.
ysi wsi
.
.
<A>
i
< A > implies that there exist a zw such that
.
.
wsi
.
.
= Azwi .
.
.
i
i
i
i
Therefore,
ws < A > implies that ys = As (z + zw ), for any s S. We
.
.
can then write each agent is optimizationPproblem in terms P
of (y0i , z i ), and
S
0
i
i
i
the value of agent i s endowment is w0 + s=1 s ws = w0 + Ss=1 s as zwi =
w0i + qzwi .
56
max
y i RS+1
+
v(y0i )
S
X
probs v(ysi ),
s=1
s.t.
y0i +
S
X
s ysi = W i
s=1
.
.
ysi
.
.
<A>
w0i + qzwi y0 (q, 1)
w0i + qzwi ys (q, 1) , for any s S
At equilibrium then
!
y0 (q, 1)
i
w0i + qzw
= y0
q,
i
w0i + qzw
w0i
iI
iI
iI
!
ys (q, 1)
w0i + qzwi
= ys
q,
= As
PI
i=1
w0i and
X
iI
iI
iI
w0i + qzwi
PI
i
i=1 zw .
57
.
.
psi
.
.
.
.
i
ws < A > . Convince yourself that the assumption is necessary in the
.
.
proof.
The Representative agent theorem, as noted, allows us to obtain equilibrium prices without computing the consumption
allocation and portfolio for
PI
i
the assumptions
of
all agents at equilibrium, (x, z).Let w = i=1
P
Pw . Under
i
i
the Representative agent theorem, let w0 = iI w0 , and ws = iI ws , for
any s S. Then
q=
S
X
s=1
v(ws )
ws
v(w0 )
w0
.
.
i
p
restriction on endowments,
s < A >, for all agents i, is trivially sat
.
.
isfied. Does this assumption imply Pareto optimal allocations in equilibrium?
Problem 15 Assume all agents have identical quadratic
preferences. Derive
.
.
i
individual demands for assets (without assuming
ps < A >) and show
.
.
that the Representative agent theorem is obtained.
58
I
X
i,
i=1
R
U (x) =
max
(xi )Ii=1
I
X
i i
I
X
u (x ) s.t.
i=1
xi = x,
i=1
where i = (i )1 and i =
u (x )
xi10
S
X
s=1
U R (ws )
ws
.
U R (w0 )
w0
ui (xi )
xi10
1
.
This is left to the reader to check; its part of the celebrated Negishi theorem.
This result is certainly very general, as it does not impose identical homothetic preferences, however, it is not as useful as the real Representative
agent theorem to find equilibrium asset prices. The reason is that to define
the specific weights for the planners objective function, (i )Ii=1 , we need to
know what the equilibrium allocation, x, which in turn depends on the whole
distribution of endowments over the agents in the economy.
3.2.5
59
Asset pricing
3.2.6
Often in finance, especially in empirical finance, we study asset pricing representation which express asset returns in terms of risk factors. Factors are
to be interpreted as those component of the risks that agents do require a
higher return to hold.
How do we go from our basic asset pricing equation
q = E(mA)
to factors?
Single factor beta representation
Consider the basic asset pricing equation for asset j,
qj = E(maj )
Let the return on asset j, Rj , be defined as Rj =
equation becomes
1 = E(mRj )
Aj
.
qj
1
This equation applied to the risk free rate, Rf , becomes Rf = Em
. Using
the fact that for two random variables x and y, E(xy) = ExEy + cov(x, y),
we can rewrite the asset pricing equation as:
ERj =
1
cov(m, Rj )
cov(m, Rj )
= Rf
Em
Em
Em
cov(m, Rj )
Em
60
Finally, letting
j =
cov(m, Rj )
var(m)
and
var(m)
Em
we have the beta representation of asset prices:
=
ERj = Rf + j m
(3.13)
ER = R +
F
X
jf mf
(3.14)
f =1
where (mf )Ff=1 are orthogonal random variables which take the interpretation
of risk factors and
cov(mf , Rj )
jf =
var(mf )
is the beta of factor f , the loading of the return on the factor f .
Proposition 9 A single factor beta representation
ERj = Rf + j m
is equivalent to a multi-factor beta representation
f
ERj = R +
F
X
f =1
jf mf with m =
F
X
f =1
bf mf
61
var(mf )bf
Em
The CAPM
The CAPM is nothing else than a single factor beta representation of the
following form:
ERj = Rf + jf mf
where
mf = a + bRw
the return on the market portfolio, the aggregate portfolio held by the investors in the economy.
It can be easily derived from an equilibrium model under special assumptions.
For example, assume preferences are quadratic:
S
1
1 X
probs (xis x# )2
u(xio , xi1 ) = (xi x# )2
2
2 s=1
P
Moreover, assume agents have no endowments at time t = 1. Let Ii=1 xis =
P
xs , s = 0, 1, ..., S; and Ii=1 w0i = w0 . Then budget constraints include
xs = Rsw (w0 x0 )
Then,
ms =
xs x#
(w0 x0 ) w
x#
=
R
x0 x#
(x0 x# ) s
x0 x#
#
(w0 x0 )
.
(x0 x# )
62
0 x0 )
Note however that a = x0x
and b = (w
are not constant, as they
x#
(x0 x# )
do depend on equilibrium allocations. This will be important when we study
conditional asset market representations, as it implies that the CAPM is
intrinsically a conditional model of asset prices.
ERj Rf
(m)
|
(Rj )
Em
pT = A(AT A)1 q.
63
Clearly, q T = AT pT , that is, pT satisfies the asset pricing equation. Furthermore, such pT belongs to < A >, since pT = Azp for zp = (AT A)1 q. Prove
uniqueness.
We can now exploit this uniqueness result to yield a characterization of
the multiplicity of stochastic discount factors when markets are incomplete,
and consequently a bound on (m). In particular, we show that, for a given
(q, A) pair a vector m is a stochastic discount factor if and only if it can
be decomposed as a projection on < A > and a vector-specific component
orthogonal to < A >. Moreover, the previous corollary states that such a
projection is unique.
S
Let m R++
be any stochastic discount factor, that is, for any s =
s
1, . . . , S, ms = probs and qj = E(mAj ), for j = 1, ..., J. Consider the orthogonal projection of m onto < A >, and denote it by mp . We can then write any
stochastic discount factors m as m = mp + , where is orthogonal to any
vector in < A >, in particular to any Aj . Observe in fact that mp + is also
a stochastic discount factors since qj = E((mp + )aj ) = E(mp aj ) + E(aj ) =
E(mp aj ), by definition of . Now, observe that qj = E(mp aj ) and that we
just proved the uniqueness of the stochastic discount factors lying in < A > .
In words, even though there is a multiplicity of stochastic discount factors,
they all share the same projection on < A >. Moreover, if we make the economic interpretation that the components of the stochastic discount factors
vector are marginal rates of substitution of agents in the economy, we can
interpret mp to be the economys aggregate risk and each agents to be the
individuals unhedgeable risk.
It is clear then that
(m) (mp )
the bound on (m) we set out to find.
3.2.7
Production
Assume for simplicity that L = 1, and that there is a single type of firm in the
economy which produces the good at date 1 using as only input the amount
k of the commodity invested in capital at time 0.11 The output depends on
11
It should be clear from the analysis which follows that our results hold unaltered
if the firms technology were described, more generally, by a production possibility set
Y RS+1 .
64
k according to the function f (k; s), defined for k K, where s is the state
realized at t = 1. We assume that
- f (k; s) is continuously differentiable, increasing and concave in k,
- , K are closed, compact subsets of R+ and 0 K.
In addition to firms, there are I types of consumers. The demand side of
the economy is as in the previous section, except that each agent i I is also
endowed with 0i units of stock of the representative firm. Consumer i has
von Neumann-Morgernstern preferences over consumption in the two dates,
represented by ui (xi0 ) + Eui (xi ), where ui () is continuously differentiable,
strictly increasing and strictly concave.
Competitive equilibrium
Let the outstanding amount of equity be normalized
to 1: the initial distriP i
bution of equity among consumers satisfies i 0 = 1. The problem of the
firm consists in the choice of its production plan k..
Firms are perfectly competitive and hence take prices as given. The
firms cash flow, f (k; s), varies with k. Thus equity is a different product
for different choices of the firm. What should be its price when all this
continuum of different products are not actually traded in the market? In
this case the price is only a conjecture. It can be described by a map Q(k)
specifying the market valuation of the firms cash flow for any possible value
of its choice k.12 The firm chooses its production plan k so as to maximize
its value. The firms problem is then:
max k + Q(k)
k
(3.15)
65
the price of assets, q and the price of equity Q. In the present environment a
consumers long position in equity identifies a firms equity holder, who may
have a voice in the firms decisions. It should then be treated as conceptually
different from a short position in equity, which is not simply a negative
holding of equity. To begin with, we rule out altogether the possibility of
short sales and assume that agents can not short-sell the firm equity:
i 0, i
(3.16)
x0 ,xi ,z i ,i
(3.17)
(3.18)
(3.19)
x (s)
zi = 0
(3.20)
= 1
(3.21)
In addition, the equity price map faced by firms must satisfy the following
consistency condition:
13
We state here the conditions for the case of symmetric equilibria, where all firms take
the same production and financing decision, so that only one type of equity is available
for trade to consumers. They can however be easily extended to the case of asymmetric
equilibria.
66
i) Q(k ) = Q;
This condition requires that, at equilibrium, the price of equity conjectured by firms coincides with the price of equity, faced by consumers in the
market: firms conjectures are correct in equilibrium.
We also restrict out of equilibrium conjectures by firms, requiring that
they satisfy:
ii) Q(k) = maxi E [M RS i f (k)], k, where M RS i denotes the marginal rate
of substitution between consumption at date 0 and at date 1 in state
s for consumer i, evaluated at his equilibrium consumption allocation
(xi0 , xi ).
Condition ii) says that for any k (not just at equilibrium!) the value of
the equity price map Q(k) equals the highest marginal valuation - across all
consumers in the economy - of the cash flow associated to k. The consumers
i
marginal rates of substitutions M RS (s) used to determine the market valuation of the future cash flow of a firm are taken as given, unaffected by
the firms choice of k. This is the sense in which, in our economy, firms are
competitive: each firm is small relative to the mass of consumers and each
consumers holds a negligible amount of shares of the firm.
To better understand the meaning of condition ii), note that the consumers with the highest marginal valuation for the firms cash flow when
the firm chooses k are those willing to pay the most for the firms equity in
that case and the only ones willing to buy equity - at the margin - when
its price satisfies ii). Given i) such property is clearly satisfied for the firms
equilibrium choice k. Condition ii) requires that the same is true for any
other possible choice k: the value attributed to equity equals the maximum
any consumer is willing to pay for it. Note that this would be the equilibrium
price of equity of a firm who were to deviate from the equilibrium choice
and choose k instead: the supply of equity with cash flow corresponding to
k is negligible and, at such price, so is its demand.
In this sense, we can say that condition ii) imposes a consistency condition on the out of equilibrium values of the equity price map; that is, it
corresponds to a refinement of the equilibrium map, somewhat analogous
to bacward induction. Equivalently, when price conjectures satisfy this condition, the model is equivalent to one where markets for all the possible types
67
of equity (that is, equity of firms with all possible values of k) are open, available for trade to consumers and, in equilibrium all such markets - except the
one corresponding to the equilibrium k - clear at zero trade.14
It readily follows from the consumers first order conditions that in equilibrium the price of equity and of the financial assets satisfy:
Q = max E M RS i f (k)
(3.22)
i
q = E M RS i A
The definition of competitive equilibrium is stated for simplicity for the
case of symmetric equilibria, where all firms choose the same production plan.
When the equity price map satisfies the consistency conditions i) and ii) the
firms choice problem is not convex. Asymmetric equilibria might therefore
exist, in which different firms choose different production plans. The proof
of existence of equilibria indeed requires that we allow for such asymmetric
equilibria, so as to exploit the presence of a continuum of firms of the same
type to convexify firms choice problem. A standard argument allows then
to show that firms aggregate supply is convex valued and hence that the
existence of (possibly asymmetric) competitive equilibria holds.
Proposition 10 A competitive equilibrium always exist.
Objective function of the firm
Starting with the initial contributions of Diamond (1967), Dreze (1974),
Grossman-Hart (1979), and Duffie-Shafer (1986), a large literature has dealt
with the question of what is the appropriate objective function of the firm
when markets are incomplete.The issue arises because, as mentioned above,
firms production decisions may affect the set of insurance possibilities available to consumers by trading in the asset markets.
If agents are allowed infinite short sales of the equity of firms, as in the
standard incomplete market model, a small firm will possibly have a large
effect on the economy by choosing a production plan with cash flows which,
when traded as equity, change the asset span. It is clear that the price
14
68
Independence of the kernel is guarantee by the fact that M RS i (s), for any i, is
evaluated at equilibrium.
16
A minimal consistency condition on Q (k) is clearly given by i) in the previous section,
which only requires the conjecture to be correct in correspondence to the firms equilibrium
69
Such condition requires the price conjecture for any plan k to equal the
pro rata marginal valuation of the agents who at equilibrium are the firms
equity holders (that is, the agents who value the most the plan chosen by
firms in equilibrium). It does not however require that the firms equity
holders are those who value the most any possible plan of the firm, without
contemplating the possibility of selling the firm in the market, to allow the
new equity buyers to operate the production plan they prefer. Equivalently,
the value of equity for out of equilibrium production plans is determined using
the - possibly incorrect - conjecture that the firms equilibrium shareholders
will still own the firm out of equilibrium.
Grossman-Hart (1979) propose another consistency condition and hence
a different equilibrium notion. In their case
"
#
X
QGH (k) = E
0i M RS i f (k) , k
i
We can interpret such notion as describing a situation where the firms plan
is chosen by the initial equity holders (i.e., those with some predetermined
stock holdings at time 0) so as to maximize their welfare, again without
contemplating the possibility of selling the equity to other consumers who
value it more. Equivalently, the value of equity for out of equilibrium production plans is again derived using the conjecture belief that firms initial
shareholders stay in control of the firm out of equilibrium.
Unanimity
Under the definition of equilibrium proposed in these notes, equity holders
unanimously support the firms choice of the production and financial decisions which maximize its value (or profits), as in (3.15). This follows from
the fact that, when the equity price map satisfies the consistency conditions
i) and ii), the model is equivalent to one where a continuum of types of equity
choice. Duffie-Shafer (1986) indeed only impose such condition and find a rather large
indeterminacy of the set of competitive equilibria.
70
X
i
(3.24)
xi (s)
(3.25)
(3.26)
To keep the notation simple, we state both the definition of competitive equilibria and
admissible allocations for the case of symmetric allocations. The analysis, including the
efficiency result ,extends however to the case where asymmetric allocations are allowed
are admissible; see also the next section.
71
(3.27)
72