LECTURE 15
PROOF OF THE EXISTENCE OF WALRASIAN
EQUILIBRIUM
JACOB T. SCHWARTZ
EDITED BY
KENNETH R. DRIESSEL
Abstract. We prove the existence of Walrasian equilibrium in a
case of single labor sector and in a case of nonhomogeneous labor
force.
1.
Case of a Single Labor Sector
In the previous lecture we derived a set of conditions necessarily satisfied at any price equilibrium. In the present lecture, we shall show
that these conditions are also sufficient, and that they may all be satisfied simultaneously. In this way we shall establish the existence of a
price equilibrium for the model introduced in Lecture 13, i.e., we shall
show the existence of prices pj such that the following conditions are
satisfied
(a) pj ≥ 0, j = 0, · · · , n; moreover, for some k, pk > 0
P P (k)
(b)
ai (δij − πij )pj = max subject to
X X (k)
X (k)
ai φij pj ≤
q j pj .
(See Lecture 13 for a more precise statement.)
(ν)
(c) u(ν) (sj ) = max subject to
X (ν) X X X (k)
pj sj ≤
ai (δij − πij )pj hνk .
j
k
j
i
(See Lecture 13 for a more precise statement.)
2010 Mathematics Subject Classification. Primary 91B50; Secondary 91B24 .
Key words and phrases. Walrasian Equilibrium, Price Theory .
1
2
JACOB T. SCHWARTZ
(d)
P (l)
(k)
ai (δij − πij ) − sj ≥ 0, for all j and
i,k
l
P P (k)
P (k)
ai (δij − πij ) + si = 0 for all j for which pj > 0.
P
i
k
k
We recall that these conditions have the following briefly stated
heuristic significance:
(a) the price of each commodity is nonnegative;
(b) each firm operates so as to maximize its profits subject to its
capital limitations;
(c) each consumption unit optimizes its own utility function subject
to its own budgetary constraints, and
(d) whatever is consumed must be produced, and overproduction
cannot occur for a commodity whose price is positive.
Having established the existence of a set of prices, production schemes,
and consumption schemes satisfying (a)–(d) above, we shall formulate
a more general model, in which not one but several kinds of labor will
be included, and show the existence of a general price equilibrium in
that case also.
2.
A Preliminary Existence Theorem
We shall proceed by setting up an ansatz giving prices, production,
consumption, etc., as functions of a rate-of-profit parameter ρ, and
then choosing ρ so as to obtain a price equilibrium. In determining the
form of our ansatz, we will of course be guided by the analysis carried
out in the preceding lecture. Let us begin by assuming prices pj (ρ) as
solutions of the equation
(15.1)
n
X
j=0
(δij − πij )pj (ρ) = ρ
n
X
φij pj (ρ);
i = 1, · · · , n
j=0
where ρ is a parameter; we know from Lecture 3 that a nonnegative
(and nontrivial) solution of these equations exists for all values of ρ
satisfying 0 ≤ ρ ≤ ρmax and that these prices satisfy condition (a),
◦
◦
provided only that Π is a connected matrix. The matrix Π will be
assumed to be connected throughout the present lecture. Note that
this implies that the prices pj (ρ), j = 1, · · · , n are strictly positive for
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
3
all ρ, 0 ≤ ρ ≤ ρmax . The price p0 (ρ) is positive if and only if ρ 6= ρmax
while p0 (ρmax ) = 0.
(ν)
Next we choose the consumption levels sj so as to maximize each
of the utility functions u(ν) subject to the constraint
X
(ν)
(15.2)
pj (ρ)sj = ∆(ν) (ρ)
j
where, as in the preceding lecture, we put
X (k)
(15.3)
∆(ν) (ρ) = ρ
qj pj (ρ)hνk .
j,k
As we saw in the preceding lecture, this condition determines the quan(ν)
tities sj as functions of the rate of profit ρ. We define the total consumption si (ρ) of the ith commodity as
(15.4)
si (ρ) =
f
X
(ν)
si (ρ).
ν=1
Finally, we determine the total production ai =
commodity by the equation
(15.5)
n
X
ai (ρ)(δij − πij ) = −sj (ρ),
Pm
k=1
(k)
ai
of the ith
j = 1, · · · , n.
i=1
Pm
(k)
We put qj = k=1 qj , j = 1, · · · , n for convenience.
The following lemma then reduces the problem of establishing the
existence of a price equilibrium to a more tractable form.
Lemma 15.1. Let the prices, consumption, and total production be
determined from ρ as above. Then there will exist a determination of
(k)
the production levels aj for the various firms satisfying
(15.6)
m
X
(k)
aj = aj (ρ)
k=1
which, together with the prices and consumption schemes determined
above, will satisfy the conditions (a)–(d), i.e., will define a price equilibrium, provided that
4
JACOB T. SCHWARTZ
(1 ) either
or
P
P
ai (ρ)φij pj (ρ) = qj pj (ρ)
i,j
j
P
P
ρ = 0 and
ai (0)φij pj (0) ≤ qj pj (0)
i,j
and provided also that
m
P
(2 ) either
or
ρ = ρmax
and
ai (ρ)πi0 = −s0 (ρ)
i=1
n
P
ai (ρ)πi0 ≤ −s0 (ρ).
i=1
Proof: If the prices satisfy (15.1), they are nonnegative and not
all zero so that (a) is satisfied. If in condition (1 ) we have ρ > 0,
it is plain that we may divide the total production levels aj (ρ) into
(k)
individual-firm contributions aj as in (15.5), in such a way that
X (k)
X (k)
qj pj (ρ)
aj φji pi (ρ) =
i,j
j
for each k, k = 1, · · · , m. Similarly, if in condition (1 ) we have ρ = 0,
(k)
we may divide the total levels into individual-firm contributions aj in
such a way that
X (k)
X (k)
qj pj (0)
aj φji pi (0) ≤
i,j
j
for each k, k = 1, · · · , m. Thus, if condition (1 ) is satisfied, we may
in any case divide the total production levels aj (ρ) into individual-firm
contributions in such a way that no firm’s production plans will require
the outlay of more capital than it possesses; moreover, unless ρ = 0,
each firm will make use of the whole of its capital. As we have seen in
the previous lecture, this, coupled with (15.1), is just the condition that
each firm’s profits be maximized. Thus (b) of our conditions, defining
a price equilibrium, is satisfied. We have determined the consumption
(ν)
schemes si (ρ) in such a way that (c) is automatically satisfied. Finally,
since all the relations in (d) except the first, i.e., the j = 0 relations, are
(k)
satisfied in consequence of the way the ai have been defined, and since
condition (2 ) is the condition that the j = 0 relation be satisfied also
(recall that ρ = ρmax means that the price of labor is zero), condition
(d) is also satisfied. Q.E.D.
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
5
In the trivial case ρmax = 0, we have p0 = 0 and ρ = 0, so that
= 0 for j ≥ 1 by (15.2) and (15.3); thus sj = 0 for j ≥ 1 by
(15.4) and aj = 0 by (15.5), so that the above lemma tells us that a
price equilibrium exists and is to be determined by our formulas (15.1)–
(15.5). We consequently may, and shall unless the contrary is explicitly
indicated, assume in what follows that ρmax > 0.
Let us define the demand for labor d0 (p) by
X
(15.7)
d0 (ρ) ≡
ai (ρ)πi0
(ν)
sj
i
and recall that we have already defined the supply of labor −s0 (ρ) by
(15.4). The following useful identity reduces the conditions of Lemma
1 to a more manageable form.
Lemma 15.2. Let ai (ρ), pi (ρ), s0 (ρ), and d0 (ρ) be as above. Then
!
n
n
X
X
(−s0 (ρ) − d0 (ρ))p0 (ρ) = ρ
ai (ρ)φij pj (ρ) −
qj pj (ρ) .
j=1
j=1
Proof: Consider the expression
(15.8)
n
n
X
X
j=0
!
ai (ρ)(δij − πij )pj (ρ) − sj (ρ)pj (ρ) .
i=1
All terms in this expression, except possibly the j = 0 term, are zero
by (15.5). Therefore (15.8) is just
−
n
X
ai (ρ)πi0 p0 (ρ) − s0 (ρ)p0 (ρ)
i=1
or, equivalently,
(−s0 (ρ) − d0 (ρ))p0 (ρ).
But (15.8) can be written (using (15.1), (15.2), and (15.3)) as
(15.9)
ρ
n
n X
X
ai (ρ)φij pj (ρ) − ρ
j=0 i=1
n
X
qj pj (ρ).
j=0
Therefore
(15.10) (−s0 (ρ)−d0 (ρ))p0 (ρ) = ρ
hX X
ai (ρ)φij pj (ρ)−
X
i
qj pj (ρ) .
6
JACOB T. SCHWARTZ
Q.E.D.
Putting together Lemmas 1 and 2 yields the following corollary:
Corollary 15.3. Let prices and total production be determined
from ρ as above. Then there will exist a determination of the production levels for the various firms satisfying (15.6), which, together
with the prices and consumption schemes determined above, will define
a price equilibrium, provided that
either (a) 0 < ρ < ρmax and d0 (ρ) + s0 (ρ) = 0
or
(b) ρ = ρmax
and d0 (ρ) + s0 (ρ) ≤ 0
P
P
or
(c) ρ = 0
and
ai (0)φij pj (0) ≤ qj pj (0).
i,j
j
Lemma 2 also yields trivially the following corollary, which states a
fact used in the construction of the diagrams of the previous lecture
and also gives an equivalent alternate version of the condition under
(c) of the preceding corollary.
Corollary 15.4.
(i) d0 (0) + s0 (0) = 0
P
P
(ii) (d/dρ)(d0 (ρ) + s0 (ρ))|ρ=0 =
qj pj (0) − ai (0)φij (0)pj (0).
j
i,j
Proof: Statement (i) follows from Lemma 2, and statement (ii)
follows from statement (i) and Lemma 2. Q.E.D.
We have supposed that ρmax > 0. Using what we have proved we
may distinguish three possibilities for s0 (ρ) + d0 (ρ). The first is that
for some ρ, 0 < ρ ≤ ρmax , supply and demand for labor are equal,
i.e., s0 (ρ) + d0 (ρ) = 0. The second is that for all ρ, 0 < ρ ≤ ρmax ,
−s0 (ρ) < d0 (ρ). The third possibility is that for all ρ, 0 < ρ ≤ ρmax , it
is the case that −s0 (ρ) > d0 (ρ). Note that in any case, by Corollary 4,
s0 (0) + d0 (0) = 0.
In the first case, condition (a) of Corollary 3 is satisfied. In the
second case, since s0 (ρ) + d0 (ρ) is positive for ρ > 0 and zero for ρ = 0,
the derivative of s0 (ρ) + d0 (ρ) is nonnegative at ρ = 0, so that by
Corollary 4, condition (c) of Corollary 3 is satisfied. If s0 (ρ) + d0 (ρ) is
negative for ρ > 0, the condition (b) of Corollary 3 is satisfied. This
establishes the existence of a price equilibrium in every case.
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
7
The following theorem summarizes the necessary conditions for equilibrium derived in the preceding lecture and the sufficient conditions
derived in the present section.
Theorem 15.5. Let φ0 (ρ) = s0 (ρ) + d0 (ρ), where s0 (ρ), d0 (ρ) are
defined as above. Then price equilibria correspond to those values 0 <
ρ < ρmax for which φ0 (ρ) = 0; and also to the value ρ = ρmax if and
only if (d/dρ)[φ0 (ρ)]ρ=0 ≥ 0; and also to the value ρ = ρmax if and
only if φ0 (ρmax ) ≤ 0. At each such equilibrium the prices are unique;
the total activity levels are also unique, and these totals may be divided
among the firms in any way consistent with the requirement
X (k)
X (k)
qj pj (ρ), k = 1, · · · , m.
aj πji pi (ρ) ≤
j
i,j
3.
Price Equilibrium for a Nonhomogeneous Labor Force
We now extend our price-equilibrium model to include several labor
sectors, considered as originating labor commodities C0 , C−1 , · · · , C−L .
If ρmax = 0, no wages can be paid; hence the distinction between
labor sectors is entirely irrelevant, and we return to the model of the
preceding section. Hence we may and shall assume without loss of
generality that ρmax > 0.
In the present more general context, we shall reestablish results much
like those of the two preceding lectures. We will find, in particular,
that price equilibria exist whenever supply and demand for labor are
simultaneously in balance in all of the labor sectors.
As we have seen in Lecture 3, prices may be determined by the basic
relation (15.1) (the range of summation of j being extended from −L
to n) as functions of ρ and of the ratios θj (j = 0, · · · , L) of the wages,
P
where we normalize these ratios by demanding that Lj=0 θj = 1. Thus,
letting pj (ρ, θ) be the prices so determined, we can write
(15.11)
n
X
(δij − πij )pj (ρ, θ) = ρ
j=−L
n
X
j=−L
φij pj (ρ, θ).
8
JACOB T. SCHWARTZ
We shall assume throughout the present section that each form of
labor is required as input to the production of at least one commodity,
◦
and shall also assume that the n × n matrix Π, obtained by deleting
all elements of πij but those with both i and j between 1 and n, is
connected.
To construct a price equilibrium, we imitate the procedure of the
(ν)
preceding sections, as follows. The consumption levels sj are again
chosen so as to maximize
u(ν) (sj )
(15.12)
over all sj such that
(15.13)
n
X
pj (ρ, θ)sj = ρ
X
j=−L
(k)
qi pi (ρ, θ)hνk .
i,k
This determines total consumption, defined by
X (ν)
(15.14)
si (ρ, θ) ≡
si (ρ, θ).
ν
Total activity levels are then defined by the equations
(15.15)
n
X
ai (ρ, θ)(δij − πij ) = sj (ρ, θ),
j = 1, · · · , n,
i=1
and we define the demand for labor in the various sectors by
(15.16)
d−j (ρ, θ) =
n
X
ai (ρ, θ)πi,−j ,
j = 0, · · · , L.
i=0
The above formulae define individual-family consumption schemes,
prices, and total production as functions of ρ and θ, but not the division
of total production among the individual firms. It will be convenient
in what follows to speak of those values of ρ and θ for which the total
production levels aj may be divided into the sum of individual-firm
(k)
contributions aj to yield a set of individual-firm production levels,
family consumption and labor schemes, and prices satisfying the conditions (a)–(d) defining a price equilibrium as values ρ, θ corresponding
to a price equilibrium. This terminology will be employed in stating a
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
9
number of the theorems and lemmas which are to follow. Put
φ−j (ρ, θ) = d−j (ρ, θ) + s−j (ρ, θ).
We aim to obtain the following result.
Theorem 15.6. A. A price equilibrium will correspond to each
of those and only those points 0 < ρ ≤ ρmax , θ0 , · · · , θL for which
φ−j (ρ, θ) ≤ 0 for all j = 0, · · · , L, with equality whenever p−j (ρ, θ) 6= 0;
and also to those and only those points ρ = 0, θ0 , · · · , θL for which
φ−j (0, θ) is nonnegative for j = 0, · · · , L and
" L
#
X
(15.17)
d/dρ
p−j (ρ, θ)φ−j (ρ, θ)
≥ 0.
j=0
ρ=0
B. Furthermore, there always exists at least one equilibrium point.
The remainder of this lecture will be devoted to proving the above
theorem. The proof of part A follows the development in Section 2 of
this lecture quite closely, but to prove part B we must employ some
deeper topological theorems, since we cannot make use of the basically
one-dimensional division into three collectively exhaustive cases φ > 0,
φ = 0, φ < 0 as in the last part of Section 2.
Since equation (15.11) is homogeneous in the prices pj , it only determines these prices up to a scalar factor. In the arguments which
are to follow, it is important to make this factor definite and convenient to choose it in an appropriate way. For this reason, we preface
our proof by a lemma characterizing a manner of normalizing the solutions pj (ρ, θ) of (15.11), to which normalization we shall adhere in the
remainder of the present section.
Lemma 15.7. The solution pj (ρ, θ) of equations (15.11) may be made
definite by imposing the additional requirement
(15.18)
ρ+
L
X
j=0
p−j (ρ, θ) ≡ ρmax .
10
JACOB T. SCHWARTZ
The solutions determined in this way depend continuously on ρ and θ;
moreover, they satisfy the condition
(15.19)
pj (ρ, θ) > 0,
j = 1, · · · , n.
P
Proof: The factors θj have been chosen to satisfy Lj=0 θj = 1. Thus,
a solution of (15.11) will satisfy (15.18) if and only if p−j (ρ, θ) = (ρmax −
ρ)θj ; hence what we must do is prove the continuity and positivity of
the solutions pj (ρ, θ) of the equation
(15.20)
L
X
(δij − (1 + ρ)πij )pj (ρ, θ) = (1 + ρ)(ρmax − ρ)
j=1
L
X
πi,−j θj ;
j=0
i = 1, · · · , n.
If we let v(ρ, θ) be the vector whose ith. component is
(1 + ρ)
L
X
πi,−j θj
j=0
and let p1 (ρ, θ) be the vector of length n whose components are pj (ρ, θ),
j = 1, · · · , n, we may write the solution of (15.20) in vectorial form as
(15.21)
◦
p1 (ρ, θ) = (ρmax − ρ)(I − (1 + ρ)Π)−1 v(ρ, θ).
This formula makes it evident that the proof of the present lemma will
result immediately once we establish the following lemma.
Lemma 15.8. Let A be a connected positive matrix with dom(A) < 1,
and let α be determined by the equation dom((1 + α)A) = 1, i.e.,
α = (dom(A))−1 − 1. Then the matrix
(15.22)
(α − ρ)(I − (1 + ρ)A)−1 = J(ρ)
is continuous and strictly positive in the whole closed interval 0 ≤ ρ ≤
α.
Proof: The correctness of our statement in the interval 0 ≤ ρ < α
follows at once from Lemmas 3.3, 3.5, and 3.6. Thus we have only to
show that the matrix (15.22) has a limit as ρ approaches α from below,
and that this limiting matrix is positive.
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
11
Let v be the dominant eigenvector of A, so that (1 + α)Av = v.
Then
(15.23) (α − ρ)(I − (1 + ρ)A)−1 v = {(α + ρ)/1−[(1 + ρ)/(1 + α)]}v
= (1 + α)v,
so that J(ρ)v is bounded. Since every component of v is positive,
and since all the entries of J(ρ) are positive for ρ < α, it follows at
once that J(ρ) remains bounded as ρ → α. Thus we may extract
a convergent subsequence from the family of matrices J(ρ); i.e., we
may find a sequence ρn → α such that the sequence of matrices has a
limit: limn→∞ J(ρn ) = J. Since the limit of a sequence of nonnegative
matrices is nonnegative, it follows from (15.23) that J is nonnegative
and Jv = (1 + α)v.
Since (I − (1 + ρ)A)J(ρ) = (α − ρ), it follows on letting ρ approach
α that (1 + α)AJ = J; thus every column of the matrix J must be
proportional to the column vector v; similarly, we may show that every row of the matrix J must be proportional to the uniquely defined
adjoint vector u0 which satisfies u(1 + α)A = u0 . Hence the elements
Jik of J must have the form Jik = cvi uk . Since Jv = (1 + α)v, we must
have cu0 v = 1 + α, so that c = (u0 v)−1 (1 + α) is positive, and thus J
is positive and uniquely defined.
Let ρ0n be any other sequence of numbers approaching α from below for which limn→∞ J(ρ0n ) = J 0 exists. Then, by the argument of
the preceding paragraph, we must have J 0 = J. This shows that
limρ→α J(ρ) = J exists, establishing the continuity and positivity of
J(ρ) in the closed interval 0 ≤ ρ ≤ α and completing the proof of the
present lemma and of Lemma 7. Q.E.D.
Next we give a lemma generalizing the key Lemma 2 of the preceding
section.
Lemma 15.9. Let φ−j (ρ, θ) = −d−j (ρ, θ) − s−j (ρ, θ), j = 0, · · · , L as
above, and put
X
X
(15.24)
φ−L−1 (ρ, θ) =
qj pj (ρ, θ) −
ai (ρ, θ)πij pj (ρ, θ),
j
i,j
12
where qj =
JACOB T. SCHWARTZ
P
(k)
qj . Moreover, put
θj∗ = [1 − (ρ/ρmax )]θj ,
(15.25)
i = 0, · · · , L
and
∗
θL+1
= ρ/ρmax .
(15.26)
Then
L+1
X
(15.27)
φ−j (ρ, θ)θj∗ = 0.
j=0
Moreover,
θj∗ ≥ 0 and
(15.28)
L+1
X
θj∗ = 1.
j=0
Proof: By (15.11), (15.13), (15.14), and (15.15) we have
"
#
X
X
(15.29) ρ
ai (ρ, θ)φij pj (ρ, θ) −
qj pj (ρ, θ)
i,j
=
X
ai (ρ, θ)(δij − πij )pj (ρ, θ) −
i,j
=
n
X
X
sj (ρ, θ)pj (ρ, θ)
j
sj (ρ, θ)pj (ρ, θ) −
X
0
X
i
j=−L
j=1
−
n
X
sj (ρ, θ)pj (ρ, θ) −
j=1
ai (ρ, θ)πij pj (ρ, θ)
0
X
sj (ρ, θ)pj (ρ, θ)
j=−L
L
X
=
(−s−j (ρ, θ) − d−j (ρ, θ))p−j (ρ, θ)
j=0
=
L
X
φ−j (ρ, θ)p−j (ρ, θ).
j=0
Our lemma follows immediately on using the definitions (15.25) and
(15.26). Q.E.D.
We may now conveniently make use of a result from elementary
topology.
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
13
To state this result, we have first to make a number of simple geometric definitions.
Definition. The k-dimensional simplex is the subset σ of Euclidean
k + 1-dimensional space defined by the formula
( )
k+1
X
σ = x x ≥ 0,
xi = 1 .
i=1
Definition. The vertices of the k-dimensional simplex σ are the
k + 1 points p(j) defined by
(15.30)
(j)
pi = δji ,
i, j = 1, · · · , k + 1.
Definition. Let p(j1 ) , · · · , p(jr ) be a set of r distinct vertices of the
simplex σ. Then the side determined by these vertices is the subset σ0
of σ defined by the formula
(15.31) σ0 = {x ∈ σ | xj = 0 unless
j = j1
or
j = j2
or · · · or
j = jr }.
In terms of these definitions we may state the following useful topological theorem.
Theorem 15.10. Suppose an n − 1-dimensional simplex is covered
by n closed sets Σ1 , · · · , Σn , in such a way that for each r the r − 1dimensional side determined by the vertices p(i1 ) , · · · , p(ir ) is contained
in the union of the r sets Σi1 , · · · , Σir . Then the sets Σ have a common
point, i.e.,
Σ1 ∩ Σ2 ∩ · · · ∩ Σn 6= ∅.
(For an elementary proof, see L. Graves, Theory of Functions of Real
Variables (1946 Edn., p. 148).)
We can use this theorem to establish the following.
Lemma 15.11. Let σ be the simplex defined by
( k
)
X
σ= x
xj = 1; xi ≥ 0 .
j=1
14
JACOB T. SCHWARTZ
Let fj (x1 , · · · , xk ), j = 1, · · · , k be continuous functions defined on σ
such that
k
X
xj fj (x1 , · · · , xn ) = 0, x ∈ σ.
j=1
Then there is a point x01 , · · · , x0n such that for every j
fj (x01 , · · · , x0n ) ≥ 0,
with equality wherever x0j 6= 0.
Proof: For each j = 1, · · · , k define the closed set Σj by writing
(15.32)
Σj = {x | fj (x) ≥ 0}.
We first show that the r − 1-dimensional side σ 0 of σ with vertices
p(i1 ) , · · · , p(ir ) is contained in the union of the sets Σi1 , · · · , Σir . After
permuting the variables x, we may assume without loss of generality
that i1 , · · · , ir = 1, · · · , r. Our side is then the subset of (15.32) defined
by the equations
xr+1 = xr+2 = · · · = xk = 0
so that
k
X
xj fj (x) = 0
j=1
0
everywhere along the side σ . Therefore for each x in σ 0 , at least one
function fj must be nonnegative. Thus the side σ 0 is contained in
Σ1 ∪ · · · ∪ Σr .
By Theorem 10, then, the intersection of all the sets Σj is not void.
Any point in the intersection is a point (x01 , · · · , x0n ) satisfying the conditions of the present lemma. Q.E.D.
We may use Lemma 9 and Theorem 10 to derive the following lemma,
which practically contains the main Theorem 6 which we are in the
course of proving.
Lemma 15.12. There exists a point (ρ, θ) such that
(15.33)
φ−j (ρ, θ) ≥ 0,
j = 0, · · · , L + 1.
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
15
At this point we have φ−j (ρ, θ) = 0 if j 6= L + 1 and p−j (ρ, θ) 6= 0 while
φ−L−1 (ρ, θ) = 0 if ρ 6= 0.
Proof: Let θi∗ , i = 0, · · · , L + 1 be the quantities defined in the
statement of Lemma 9. Since ρ may be varied arbitrarily between 0
and ρmax , and since the quantities θj may be varied arbitrarily subject
P
only to the restrictions θj ≥ 0 and Lj=0 θj = 1, it is plain that by
choosing ρ and θ, the quantities θj∗ may be varied arbitrarily subject
only to the restrictions (15.28). It is clear from (15.25) and (15.26)
∗
that as long as θL+1
6= 1, i.e., as long as ρ 6= ρmax , we may write ρ and
θ as continuous functions of θ∗ :
ρ = ρ(θ∗ ),
(15.34)
θ = θ(θ∗ ).
Thus the functions φ−j (ρ(θ∗ ), θ(θ∗ )) depend continuously on θ∗ except
∗
at the vertex θL+1
= 1. However, since (cf. Lemma 15.7) p−j (ρ, θ) = θj∗ ,
even in the vicinity of ρ = ρmax , it follows from (15.11), (15.12)–(15.14),
(15.15), (15.16) and the formula following (15.16), and (15.24)–(15.26)
∗
that φ−j (ρ(θ∗ ), θ(θ∗ )) depends continuously on θ∗ even near θL+1
= 1.
Thus, by Lemma 11, there exists a ρ and θ such that φ−j (ρ, θ) ≥ 0,
j = 0, · · · , L + 1, with equality if θj∗ 6= 0.
Using (15.25) and (15.26), the present lemma follows at once. Q.E.D.
Lemma 15.13. A point ρ = 0 satisfies
φ−L−1 (0, θ) ≥ 0
(15.35)
if and only if
(15.36)
(d/dρ)
L
X
j=0
φ−j (ρ, θ)p−j (ρ, θ)
≤ 0.
ρ=0
Proof: Use the identity
(15.37)
L
X
φ−j (ρ, θ)p−j (ρ, θ) + φ−L−1 (ρ, θ) = 0
j=0
given by Lemma 9; differentiate and put ρ = 0. Q.E.D.
Now we shall prove part A of Theorem 6.
16
JACOB T. SCHWARTZ
Proof of Part A of Theorem 6. Suppose first that ρ > 0. Let ρ, θ
be a rate of profit and a set of wage ratios such that φ−j (ρ, θ) ≥ 0,
0 ≤ j ≤ L+1 with equality whenever θj 6= 0. It is clear from Lemma 10
that not all the prices pj (ρ, θ) are zero; thus condition (a) is satisfied.
(ν)
Since the consumption schemes sj (ρ, θ) are determined by (15.12)–
(15.13), they satisfy condition (c). Since the total activity levels ai (ρ, θ)
are defined by (15.15) and since by hypothesis we have φ−j (ρ, θ) ≥ 0
with equality whenever p−j (ρ, θ) 6= 0, we see that condition (d) is
also satisfied. It only remains to show that condition (b) is satisfied;
since with prices giving a uniform positive rate of profit a firm makes
maximum profits if and only if it employs all its capital it is sufficient
to show that the total activity levels ai (ρ, θ) can be divided into the
(k)
sum of individual-firm contributions ai in such a way that
X (k)
ai = ai (ρ, θ),
(15.38)
k
and
(15.39)
X
(k)
ai φij pj (ρ, θ) =
X
(k)
qj pj (ρ, θ).
k,i,j
It is plain that (15.38) and (15.39) can be satisfied simultaneously if
and only if
X
X
(15.40)
ai (ρ, θ)φij pj (ρ, θ) =
qj pj (ρ, θ),
i,j
P (k)
where qj = k qj . Formula (15.24) shows that (15.40) holds, since,
by hypothesis, φ−j (ρ, θ) = 0 unless p−j (ρ, θ) = 0.
Conversely, suppose that we have a price equilibrium corresponding
to the rate of profit ρ > 0 and the wage ratios θ. We know that
condition (b) implies that the equilibrium prices are (proportional to)
pj (ρ, θ), j = −L, · · · , n. Since we have seen that all the prices pj (ρ, θ),
j = 1, · · · , n are then strictly positive, it follows from condition (d)
P (k)
that the total activity levels ai = k ai , i = 1, · · · , n must satisfy
(15.15). Moreover, by condition (b), each firm must employ its total
capital; thus, the dividend income of the νth family is necessarily given
15. PROOF OF EXISTENCE OF WALRASIAN EQUILIBRIUM
17
by
(15.41)
ρ
X
(k)
qi pi hνk ,
i,k
which by condition (c) implies that the family’s labor-consumption
scheme must satisfy (15.12)–(15.13). Next we observe that condition
(d) implies that at equilibrium we have φ−j (ρ, θ) ≥ 0, with equality if
p−j (ρ, θ) 6= 0. Thus part A of Theorem 6 is proved in case ρ > 0.
If ρ = 0, we may argue in almost the same way, making use however of the information given by Lemma 13. We have only to amend
the sixth sentence of the first paragraph of the present proof to state
that if ρ = 0, a firm makes no profit in any case, so that condition
(b) degenerates by the further argument of the present proof to the
condition
X
X
(15.42)
ai (0, θ)φij pj (0, θ) ≤
qj pj (0, θ),
i,j
i.e., to the condition φ−L−1 (0, θ) ≥ 0. The converse argument in case
ρ = 0 is again much like the converse argument in case ρ > 0; we have
only to amend the preceding paragraph with the remark that if profits
are all zero, dividend income for each family is zero also. Thus part A
of Theorem 6 is fully proved.
Proof of Part B of Theorem 6: This follows at once from Part A and
from Lemmas 12 and 13. Q.E.D.
4.
A Summary Remark
The analysis presented in the present lecture shows the Walrasian
theory to be quite satisfactory from the purely mathematical point of
view. On the other hand, we have seen above that its mathematical
formulations are based upon unrealistically drastic assumptions, which
artificially assume away all the Keynesian phenomena studied in Lectures 5–12. In order to incorporate the neoclassical ideas of the three
preceding lectures into a less hopelessly unrealistic model, it is necessary for us to construct a generalized equilibrium model combining
neoclassical and Keynesian features. We now turn to this task.