The Walras-Cassel System









The
Walras-Cassel System

________________________________________________________

"To some people (including no doubt Walras himself) the system
of simultaneous equations determining a whole price-system seems to have vast
significance. They derive intense satisfaction from the contemplation of such
a system of subtly interrelated prices; and the further the analysis can be
carried (in fact it can be carried a good way)...the better they are pleased,
and the profounder the insight into the working of a competitive economic
system they feel they get."


(John Hicks,
Value and Capital, 1939: p.60)
"The fundamental Anglo-Saxon quality is satisfaction with the
accumulation of facts. The need for clarity, for logical coherence and for
synthesis is, for an Anglo-Saxon, only a minor need, if it is a need at all.
For a Latin, and particularly a Frenchman, it is exactly the opposite."


(Maurice Allais,
Trait� d'Economie Pure, 1952:
p.58)
________________________________________________________
Contents
(1) Introduction(2)
TheWalras-Cassel
Model(3) The Linear
Production Conditions: A simple illustration(4) Incorporating
Capital and Growth        (A)
Circulating Capital        (B) Steady-State
Growth
Selected
References
Back
(1) Introduction
The "Walras-Cassel" model refers to the general equilibrium model
with production introduced in L�on Walras's
Elements of Pure Economics (1874). The Walrasian model fell into disuse
soon after 1874 as general equilibrium theorists, particularly in the 1930s in
the English-speaking world, opted for the Paretian
system. The Walrasian model was resurrected in Gustav Cassel's Theory
of Social Economy (1918), but even after that, its analysis was confined to
the German-speaking world, notably in the Vienna Colloquium in
the 1930s, where it was corrected and
expanded by Abraham Wald (1936). It only
really broke through the English-speaking barrier in the 1950s, when there was a
resurgence of interest in general equilibrium with linear production technology
and existence of equilibrium questions. However, in the dextrous hands of Arrow, Debreu, Koopmans and the
Cowles
Commission, the Walras-Cassel model was quickly replaced by the more nimble
"Neo-Walrasian"
model, which fused aspects of Walrasian and Paretian traditions.
As outlined by Walras, the basics of the model are the following:
individuals are endowed with factors and demand produced goods; firms demand
factors and produce goods with a fixed coefficients production technology.
General equilibrium is defined as a set of factor prices and output prices such
that the relevant quantities demanded and supplied in each market are equal to
each other, i.e. both output and factor markets clear. Competition ensures that
price equal cost of production for every production process in operation.
Despite its superficial resemblance to some elements of Classical
Leontief-Sraffa models (e.g. fixed production coefficients, price-cost
equalites, steady-state growth, etc.), the Walras-Cassel model is inherently and
completely Neoclassical.
Equilibrium is still identified where market demand is equal to market
supply in all markets rather than being conditional on replication and
cost-of-production conditions. The Walras-Cassel model yields a completely
Neoclassical subjective theory of value based on scarcity, rather than a
Classical objective theory of value based on cost. Furthermore, in the
Walras-Cassel system equilibrium prices and quantites are only obtained jointly
by solving the system simultaneously, whereas the Classicals would solve
for prices and quantities separately.
It might be worthwhile to run down a quick preliminary description
of the Walras-Cassel model in order to get up the intuition for what is to
follow. [Those wishing to jump ahead, can go
here.] Let v denote factors, x denote produced outputs,
w be factor prices and p denote output prices. Individuals are
endowed with factors and desire produced outputs. They decide upon their supply
of factors (which we call F(p, w)) and their demand for
outputs (which we call D(p, w)) by solving their
utility-maximizing problem. Firms have no independent objective function: they
mechanically take the factors supplied to them by consumers and convert them to
the produced goods the consumers desire via a fixed set of production
coefficients, which we denote B.
We face two further sets of equations which form the heart of the
Walras-Cassel system: one set makes factor supply equal to factor demand by
firms ("factor market clearing") and is written as v = Bó x; a second set says that the output price equals cost
of production for each production process ("perfect competition") and is written
p = Bw. We shall refer to both of these as the linear
production conditions of the Walras-Cassel model. It is important to note
that these are not functions, but rather equilibrium conditions.

Notice then what is given: consumer's preferences (utility),
endowments of factors and production technology. From these components we should
be able to derive in equilibrium: (1) factor prices, w*; (2) output
prices, p*; (3) quantity of factors, v* and (4) quantity of
produced outputs, x*. An equilibrium is defined when these components are
such that (1) households maximize utility; (2) firms do not violate perfect
competition; (3) factor and output markets clear.
The four sets of equations we have outlined connect the entire
system together in equilibrium. Their functions can be outlined as follows:

(i) D(p, w) connects output prices and
output quantities;

(ii) F(p, w) connects factor prices and
factor quantities;
(iii) v = Bó x connects
output quantities and factor quantities;
(iv) p = Bw connects output prices and factor
prices.
To ground our intuition more clearly, we can appeal to Figure 1,
where we schematically depict the logic of the Walras-Cassel equations.
Heuristically speaking, suppose we have two markets, one for factors (on the
left) and one for outputs (on the right). Note that supply of factors
F(p, w) on the left is upward-sloping with respect to
factor prices w, while demand for outputs D(p,
w) on the right is downward-sloping with respect to output prices
p. The elasticities of factor supply and output demand curves reflect the
impact of prices and wages on household utility-maximizing decisions.
[Two caveats: firstly, yes, these are all supposed to be vectors
and, yes, Figure 1 makes no sense in that context; but the diagram is merely a
heuristic device, not a graphical depiction of the true model; secondly, the
output demand function is also a function of w and the factor supply
function is also a function of p, so there is interaction between the
diagrams which will cause the curves to shift around; for simplicity, we shall
suppress these cross-effects by assuming that factor supplies do not respond to
p and output demands do not respond to w.]



Figure 1 - Schematic Depiction of the Walras-Cassel
Model
It is important to note how the factor supply and output demand
decisions of households sandwich this entire problem, with the linear production
conditions sitting passively in the middle. Fixing any one of the four items
(w, p, x or v) at its equilibrium value, we can
determine the rest [although to do so, we must assume that output demand and
factor supply functions are invertible: e.g. given v, we can determine
what w is by the factor supply function F(p, w) and
given x, we can determine what p by the output demand function
D(p, w); naturally, this is a very strong assumption and
not a very clear one in the manner it is stated].
It might be worthwhile to go through it "algorithmically" from
some starting point (trace this with the arrows in Figure 1). Suppose
equilibrium output prices, p*, are given. From p*, we get
x* by the output demand function D(p, w) and we
obtain w* by the competition condition p = Bw. In their
turn, x* gives us v* via the factor market clearing condition
v = Bó x while w* gives us
v* via the factor supply function, F(p, w). If this
is truly equilibrium, then it had better be that the v*s computed via the
two different channels are identical to each other.
Equivalently, suppose we start from equlibrium output demands,
x*. Thus, given x*, we get p* by the output demand function
D(p, w) and v* by the factor market clearing
condition v = Bó x. In their turn,
p* gives us w* by the competition condition p = Bw
and v* gives us w* by the factor supply function
F(p, w). For equilibrium, we need it that both of the
w* are the same. We go through analogous stories when we start with
equilibrium factor quantities, v*, or equilibrium factor prices,
w*.
The main lesson is this: in the Walras-Cassel system, there is no
necessary direction of determination from one thing to another. The
Walras-Cassel system is a completely simultaneous system where
equilibrium prices (w*, p*) and equilibrium quantities (v*,
x*) are determined jointly. It does not matter whether we say "prices
determine cost of production" or "cost of production determines prices", etc. In
equilibrium, price equals cost of production, but this is obtained as a solution
to a simultaneous system, not by causal direction. The only exogenous
data are preferences of households, endowments and technology.
(2) The Walras-Cassel Model
Let us then set out the Walras-Cassel economy. The basic
environment can be laid out as following summarizing form:

(1) Economy: H households, F firms, n produced commodities, m
primary factors.
(2) Produced Commodities:

(i) xh is a vector of commodities demanded
by household h(ii) xf is a vector of commodities
supplied by firm f(iii) p is the vector of commodity
prices.
(3) Factors:

(i) vh is a vector of factors supplied by
household h(ii) vf is a vector of factors demanded by
firm f(iii) w is the vector of factor prices.
(4) Technology: (fixed proportions, same technology for all
firms)

(i) bji =
vjf/xif is a unit output
production coefficient(ii) B is n � m
matrix of unit-input coefficients.
(5) Objectives:

(i) Household h: max Uh =
Uh(xh, vh) s.t.
pxh Ł
wvh.(ii) Firm f: no objective; vf =
Bó xf is production function for
fth firm

(6) Equilibrium assumptions:


(i) Perfect Competition: p = Bw(ii) Factor
market-clearing: F(p, w) = v = Bó x(iii)
Output market-clearing: x = D(p, w)
It might be worthwhile detailing the components (4) to (6) a bit
further. Let us then begin with technology. Firms face fixed-proportions
technology with unit-output coefficients bji i = 1, ..., n, j = 1,
.., m. Thus, bji = vjf/xif
represents the amount of factor j necessary to produce a unit of input i.
We assume all firms face the same technology. The fth firm has a production
function of the form:

vf = Bó
xf
where Bó is an m � n matrix of unit-output coefficients so that factor demands
(vf) can be deduced from desired output supplies
(xf). Thus, the demand for a
particular factor by firm j is:

vjf = �ibjixif =
Bjó
xf
where Bjó is merely the
jth row of Bó . Thus, market demand for factor j
is obtained by summing up over firms:

vj = � f
vjf = � f�ibjixif =
Bjó
xf
or, more generally:

v = �f
vf = �f Bó xf = Bó
x
or simply v = Bó x where
vector x is the supply of produced goods and v is the demand for
factors.
Let us now turn to the issue of competition. Walras assumed
"perfect competition" by which he meant entrepreneurs make no positive profits
and no losses (Walras, 1874:
p.225). This implies that, for a viable production process, total revenue
pó xf equals total cost wó vf for every firm f, or:

pó xf = wó vf
or, as bji =
vjf/xif, then the perfect
competition assumption implies:

p = Bw
where, note, B is the transpose of the earlier matrix of
unit-output coefficients Bó .
Let us now turn to the objectives of households. Each household h
has a utility function Uh(xh, vh)
where utility increases with consumption of produced commodities
xh and decreases with supply of factors vh.
[we should note that Gustav Cassel (1918) did
not have utility functions but worked directly with demand]. Each household is
endowed with a set of factors vh. Note that we have not
allowed here for produced means of production (i.e. capital) - thus all
factors are endowed. Facing an announced set of prices, (p, v),
the hth household maximizes the following:

max Uh = Uh(xh,
vh)

s.t. pó xh Ł wó vh

and there are H such programs, one for each household. Household
income comes from the sale of factors (wó
vh) and, possibly, profits distributed by firms - but these are
set to zero by the perfect competition assumption. Household expenditure is the
purchase of produced commodities (pó
xh). The result is a set of output demand functions and factor
supply functions of the following general form:

xih = Dih(p,
w) for each commodity i = 1, ..., n, for each household h = 1, ..,
H
vjh = Fjh(p,
w) for each factor j = 1, .., m, for each household h = 1, ...,
H.
and the budget constraint is met, so:

pó xh = wó vh for each household h = 1, ...,
H.
Thus, household output demands and factor supplies are functions
of commodity prices (p), factor prices (w). Market demand for
goods and supply of factors is thus obtained by simply summing these over H,
so:

xi = �hxih = �h Dih(p, w)
= Di (p, w)
vj = �h
vjh = �h
Fjh(p, w) = Fj(p,
w)
Notice that we are using the same notation for market commodity
demand, x, and market factor supply, v, as we did for market
commodity supply and market factor demand before. This implies we are
already imposing market-clearing in the output market. More explicitly
the market clearing conditions are:

�h xih = �f xif for each commodity
i = 1, ..., n

�h vjh = �f vjf for each factor j =
1, ..., m.
so that market commodity demand is equal to market commodity
supply for each produced good and market factor supply equal to market factor
demand for each factor. Given our earlier notation, this can be rewritten:

Di (p, w) = xi for each
commodity i = 1, .., n.
Fj(p, w) = vj =
Bjó x for each factor j = 1,
.., m.
Let us now turn to the existence of equilibrium. Walras attempted
to prove existence by counting equations and unknowns in his system and, when he
found they were equal, he assumed this was sufficient for existence. From
the terms set out above, the following are the set of relevant equations:




price-cost equalities

pi = �jbjiwj =
Biw

(n equations)


output market equilibrium:

xi = Di(p, w)

(n equations)


factor market equilibrium:

vj = Bjó
x

(m equations)


market factor supplies:

vj = Fj(p, w)

(m equations)
so we have (2n + 2m) equations. The unkowns are:




quantity of produced goods:

xi

(n unknowns)


quantity of factors:

vj

(m unknowns)


output prices:

pi

(n unknowns)


factor prices:

wj

(m unknowns)
so we have (2n + 2m) unknowns. We can remove one equation by
Walras's Law: summing up budget constraints over households yields, after some
rearrangement:

� i pi [� h xih - � f xif] + � j wj [�
f vjf - �
h vjh] = 0
or, in vector form:

pó [xh
- xf] + wó
[vf - vh] = 0
which is the familiar statement of Walras's Law. Note that if all
markets clear but one, then that last one will necessarily clear too. Thus, we
can exclude one of the market-clearing conditions from our list. Thus, now, the
number of equations becomes (2n + 2m - 1). This seems to make unkowns exceed
equations, but we forgot the numeraire good. We can thus set, say, the
price of the first commodity to 1 (i.e. p1 = 1) and so one of the
unknowns drops out. Thus total unknowns are now (2n + 2m - 1), thus, the total
number of equations equal the total number of unknowns.  Walras (1874)
thought this was enough to prove existence of equilibrium.
We should note that, Cassel (1918)
originally assumed that factors were supplied inelastically. In this case, the
vjs are known and we can omit the market factor supplies equations
(the last set of m equations, vj = Fj(p,
w)). Thus, Cassel only had (2n + m - 1) equations and (2n + m -1)
unknowns.
(3) The Linear Production Conditions: A
simple illustration
As noted, the Walras-Cassel model has four sets of equations:
output demands D(p, w) and factor supplies
F(p, w) derived from the household's problem and, in
addition, two sets of linear production conditions: the factor market-clearing
equalities v = Bó x and price-cost
equalities p = Bw, both generated by the linear production
technology. It might be useful to examine these linear production conditions
further by employing a simple two-sector version (two outputs, two factors). In
this case, the factor market equations v = Bó
x are:

v1 = b11x1 +
b12x2
v2 = b21x1 +
b22x2
To see this graphically, we can depict them in x1,
x2 space as in Figure 2. Specifically, notice that the first equation
can be rewritten as x2 = v1/b12 -
(b11/b12)x1 which yields us the
negatively-sloped line V1 in Figure 2. This has vertical intercept
v1/b12 > 0, horizontal intercept
v1/b11 > 0 and slope -(b11/b12)
< 0. This curve represents the locus of output level combinations that
fulfill equilibrium in factor market 1 for a given v1. Notice that if
factor supply v1 increases, then the V1 curve shifts
outwards.
Conversely, the second equation can be rewritten as x2
= v2/b22 - (b21/b22)x1
which yields us a second negatively-sloped line V2 in Figure 2. This
has vertical intercept v2/b22 > 0, horizontal intercept
v2/b21 > 0 and slope -(b21/b22)
< 0. The curve V2 is a locus of output combinations which yield
equilibrium in factor market 2. An increase in v2 will also shift the
V2 curve out.



Figure 2 - Factor Market
Clearing
Obviously, equilibrium is obtained when both the equalities hold -
in this case, at the intersection of the V1 and V2 curves
at point E. Thus, output levels x1* and x2* at point E
represent factor market equilibrium.
It is interesting to to note that a we are assuming here that
V1 is steeper than V2. This implies that
b11/b12 > b21/b22. Now, let
xji be the amount of factor j used in industry i, then we can easilty
notice that bji = xji/xi, the amount of factor
j used in industry i (xji) divided by the amount of good i. Thus, we
can rewrite this inequality as:

b11/b12 =
(x11/x1)/(x12/x2) >
(x21/x1)/(x22/x2) =
b12/b22
or, cross-multiplying and cancelling:

x11/x21 >
x12/x22
which implies that industry x1 is uses factor
v1 relatively more intensively than industry x2, while
industry x2 uses factor v2 relatively more intensely than
industry x1. If we conceive of factor v1 as "capital" and
factor v2 as "labor", we would say that the inequality implies that
industry 1 is more "capital-intensive" and industry 2 is more "labor-intensive".

Interestingly, we can obtain from this the famous Rybszynski
Theorem from international trade theory (Rybczynski, 1955). The Rybsczynski
Theorem can be succinctly stated as the following:

Theorem: (Rybczynski) in a simple two-sector model,
if product prices are held constant, an increase in the supply of a particular
factor will lead to an increase in the output of the good intensive in that
factor and a fall in the output of the other good.
We can see this result immediately in Figure 2. Suppose we incease
the supply of factor 2. We consequently shift the V2 curve to
V2ó . Notice that the equilibrium position
moves from E to F. At F, x1* has fallen and x2* has risen
relative to E. As industry 2 is relatively intensive in factor 2, then the rise
in x2* and fall in x1* effectively shows that the
Rybczynski Theorem holds here.
Let us now turn to the price side. In this two-output, two-factor
case, our p = Bw becomes:

p1 = b11w1 +
b21w2
p2 = b12w1 +
b22w2
These price-cost equalities are depicted graphically in
w1, w2 space in Figure 3. The first equation can be
rewritten as w2 = p1/b21 -
(b11/b21)w1 which is the negatively-sloped line
P1 in Figure 3 with vertical intercept p1/b21
> 0, horizontal intercept p1/b11 > 0 and slope
-(b11/b21) < 0. This curve is the locus of factor
returns combinations that fulfill the price-cost equality for industry 1 for a
given output price, p1. We can note that in this case if the
output price p1 increases, then the P1 curve shifts
outwards. Similarly, the second equation can be rewritten as w2 =
p2/b22 - (b12/b22)w1, the
second negatively-sloped line P2 in Figure 3 with vertical intercept
p2/b22 > 0, horizontal intercept
p2/b12 > 0 and slope -(b12/b22)
< 0. This curve is the locus of factor return combinations that yield
price-cost equalities in the second industry. Obviously, an increase in output
price p2 will shift the P2 curve out.
The first thing to note about Figure 3 is that the intersection of
curves P1 and P2 at point G yield a particular factor
return combination, w1*, w2*. This is the only set
of factor returns which fulfill the price-cost equalities.
 


Figure 3 - Price-Cost Equalities
Notice how this price-cost equality is different from the Classical system:
it is not that cost of production determines prices, but rather output
prices that determine cost of production. This, of course, is merely the Austrian
principle of imputation, as initially outlined by Carl Menger (1871) and
Friedrich von Wieser (1889):

"The value of goods of lower order [i.e. commodities] cannot,
therefore, be determined by the value of goods of higher order [i.e. factors]
that were employed in their production. On the contrary, it is evident that
the value of goods of higher order is always and without exception determined
by the prospective value of the goods of lower order in whose production they
serve." (C. Menger, 1871: p.149-50).
In short, given output prices (p1,
p2) and technology (B), we can immediately determine
the necessary factor returns (w1*, w2*). Thus, factor
returns can be "imputed" from product prices. But where do Menger and Wieser
suppose the output prices come from? Presumably, these come from the
utility-maximization problem: an output price is high if that output is very
much demanded by consumers. Thus, the imputation principle captures the idea
that it is the demand for goods bearing down on a fixed supply of factors
that gives value to those factors. Of course, this statement must be qualified
in a general equilibrium system: as we shall see, in the end, prices and
cost of production are determined simultaneously, with no necessary direction of
causality assumed.
The second result we obtain is the Stolper-Samuelson Theorem. To
see this, notice that P1 is steeper than P2 which implies
that b11/b21 > b12/b22. Following
the previous logic, this implies that:


x11/x21 >
x12/x22
where xji is the amount of factor j used in industry i.
Thus, industry x1 is relatively intensive in factor v1 and
industry x2 is relatively intensive in factor v2 - as in
our earlier case. As a result, we can now state the Stolper-Samuelson Theorem
(from Stolper and Samuelson
(1941)):

Theorem: (Stolper-Samuelson) in a simple two-sector
model, if outputs are held constant, a rise in the relative price of a good
will raise the return to the factor in which it is relatively more intensive
and reduce the return to the factor in which it is relatively less
intensive.
This is again immediately obvious in Figure 3. If we increase
p2, the price of good 2, then the P2 curve shifts out to
P2ó . The equilibrium position consequently
moves from G to H. Notice that at H, w1* has fallen and
w2* has risen relative to G. Yet recall that good 2 was relatively
intensive in factor 2. Thus, the Stolper-Samuelson Theorem holds here.
Finally, we should note that the linear production conditions by
themselves seem to betray a resemblance to the Classical
Sraffa-Leontief system in that it seems as we have a dichotomy
between prices and quantities. In other words, as p = Bów and as B is given, then if we know
p, then w is known uniquely and vice-versa, so that we are talking
about prices being determined without referring to demand or supply quantities.
Similarly, as v = Bx, then if we know v, then
x is determined uniquely, and vice-versa, which seems as if we are
talking about quantities being determined without referred to prices of any
sort.
However, it is erroneous to deduce from this that the
Walras-Cassel system exhibits a Classical price-quantity dichotomy. We should
reiterate here that the Walras-Cassel system is not these linear
production conditions in isolation but it is these equations plus the
output demand functions D(p, w) and the factor supply
functions F(p, w) which tie everything together and make it
non-dichotmous.
(4) Incorporating Capital and
Growth
L�on Walras (1874)
included capital in his model of general equilibrium. This is, in fact, not
difficult to incorporate - provided we try to confine ourselves to circulating
capital. An examination of Walras's
original theory of capital is contained elsewhere.
(A) Circulating Capital
Capital, or intermediate goods, are produced goods which
also enter into the process of production. We can incorporate these via unit
production coefficients as well. Let aji denote the input of
intermediate good j necessary to produce a unit of good i. Assuming all produced
goods are potentially intermediate goods, then the matrix of unit input
coefficients A is an n � n matrix with typical
element aji ł 0 (with strict equality if j
does not enter into the production of good i). As a result, both firms and
consumers can demand a produced output. Thus, for a particular produced good j,
the market for good j is in equilibrium if

xj = Aó
jx + Dj(p, w)
where Aó j is the jth
row of Aó . The term Aó jx represents the total demand for
good j by firms, i.e. the demand for good j as an intermediate
good. Dj(p, w) represents market consumer demand for
good j as a final good. Thus, in equilibrium, the supply of good j,
xj, must equal total demand by both firms and consumers,
Aó jx +
Dj(p, w). As we have n produced goods, then we have n
such market-clearing conditions which we can summarize as follows:

x = Aó x +
D(p, w)
i.e. output supplies (x) must equal input demands
(Ax) and consumer demands (D(p,w)).
We should not forget non-produced or primary factors as we
had before. Thus, letting B a matrix of input demands for primary
factors, then the factor market equilibrium conditions are:

v = Bó x
i.e. supply of primary factors is equal to the demand for primary
factors.
We now need to turn our attention to price-cost equalities where
we must now add the costs of purchasing intermediate goods at their market
prices, p. Notice that we are purchasing capital and not renting
it. This follows from our assumption that all capital is circulating as opposed
to fixed. Thus, by "capital" we mean things like wool or iron which are
completely used up in the production process, and not things like weaving looms
or hammers which remain standing after the production process is carried
through. [Note: if we wish to incorporate fixed capital, we might follow John von Neumann (1937)
by reducing it to dated, circulating capital. The only substantial modifications
required, in that case would be then to incorporate "joint production".]
Assuming zero profits, then the cost of production of a unit of
good i is Aip + Biw where
Ai is the ith row of A and Bi is the
ith row of B. Thus, for good i, we have the price-cost equality:

pi = Aip +
Biw
As we have n such equations, then we can summarize the price-cost
equalities as:

p = Ap + Bw.
We thus have now three sets of equations:

(i) x = Aó x +
D(p, w)
(ii) v = Bó x
(iii) p = Ap + Bw.
which are similar to the ones we had before, only now with the
addition of intermediate goods. Note that we are ignoring/suppressing the factor
market supply functions, F(p, w), for simplicity (i.e. we
are assuming endowed factors are supplied inelastically), but they could be
included with no loss of generality.
How do we solve this? The magic of this system is that we can
easily adjust these equations and reduce them effectively to the older
Walras-Cassel model. To see this, notice that (i) can be rewritten as (I
- Aó )x = D(p, w)
and (iii) as (I-A)p = Bw. Thus, letting p� = (I - A)p denoted "adjusted" or
"net" output prices, then this system can be rewritten:

(i) (I - Aó )x =
D(p, w)
(ii) v = Bó x
(iii) p� =
Bw.
and we return to the structure of our old Walras-Cassel system!
From equation (ii), we solve for x* given v and B, whereas
for equation (iii), we can solve for w* given (adjusted) prices,
p� and B. To obtain full equilibrium all
we have to guarantee is that the net market supply (I - Aó )x, i.e. the supply of goods not allocated to
firms as inputs, equals the consumer demand for those goods, D(p,
w). Thus, incorporating circulating capital into the Walras-Cassel model
implies no substantial change in the structure of the model.
(B) Steady-State Growth
The circulating capital model we outlined above resurrects Knut Wicksell's (1893)
accusation that there is no rate of profit in this model. To incorporate it,
however, we must assume a "progressive" or "growing" economy, so that our
equations would take the form for a steady-state equilibrium. In a steady-state
equilibrium we do not want prices to change over time but rather to
remain constant. In this case, factor markets and goods markets may "expand" in
size over time, but proportions of factors employed and goods produced cannot
change (otherwise prices would change).
Steady-state
growth or a "uniformly progressing" economy was intimated by L�on Walras (1874), but
was really the brainchild of Gustav Cassel (1918:
pp.33-41, 152-64). Let g be the uniform rate of growth that is going to rule our
steady-state growing economy. Thus, primary factor supplies every period are
growing at a constant, uniform rate g, consumer demands are
growing at the rate g and thus the outputs produced will need to grow at
rate g. If this holds true, then neither factor prices nor output prices will
change over time as we are merely "scaling" everything up (demands and supplies)
at the same rate.
The first thing that needs to be handled is consumer demands and
factor supplies. Suppose, for the sake of argument, population is growing at the
rate g and people are being merely "replicated" with the same preferences and
endowments. Let the initial consumers' market demand for output be
D(p, w) and supply of factors be F(p,
w). Thus, the next period, because of pure replication, consumers' output
demand is (1+g)D(p, w) and factor supply is
(1+g)F(p, w) and so on. Thus, at any time t,
(1+g)tD(p, w) are consumer demands and
(1+g)tF(p, w) are primary factor supplies.
For circulating capital, let us assume a sequential structure so
that inputs that will be used in production in time t+1 must have already been
produced in the previous time period, t. This means that output at time t,
xt must meet not only concurrent consumer demand
(1+g)tD(p, w) but also industry demand for
production next period, Aó
xt+1. As we want a steady-state, then we impose the condition
that xt+1 = (1+g)xt so that outputs in
period t+1 are a proportional amount g greater than outputs at t, thus:

Aó xt+1 = (1+g)Aó xt.
Consequently, for goods market equilibrium, we must have:

xt = (1+g)Aó
xt + (1+g)tD(p,
w)
outputs at t (xt) must meet both demand for
inputs by firms and consumer demands. Primary factors, however, are supplied
concurrently to production. Thus, factor demands at t are Bó xt. Thus primary factor market equilibrium
implies:

vt = Bó
xt
where, as vt =
(1+g)tF(p, w) by our assumption of a replicated
population, then:

(1+g)tF(p, w) = Bó
xt
We now must turn to price-cost equalities. In this case, we no
longer want pure equality as a surplus must be produced in order for
investment to happen. Assuming "perfect competition" implies that there must be
a uniform rate of profit, r, on circulating capital (otherwise, firms would move
from low profit to high profit industries over time, and thus the proportions
woulc change). In this case:

pt = (1+r)Apt +
Bwt
In conclusion, for a Walras-Cassel model with capital, we have the
following sets of equations:

(i) xt = (1+g)Aó xt + (1+g)t D(p,
w)
(ii) (1+g)tF(p, w) = Bó xt

(iii) pt = (1+r)Apt +
Bwt
We shall not solve this system explicitly here, but only give an
idea of what the solution would be. For (ii), note that we can obtain
directly:

xt* = (1+g)tBó -1F(p,
w)
provided Bó is invertible and
other conditions are met so that xt* ł 0. Notice that iterating for different time periods, we
obtain a solution path [x1*, x2*, ..,
xt*, ...]. In fact, it can be easily shown that if x* =
Bó -1F(p, w),
then:

xt* = (1+g)t-x*
thus the solution path is generated simply by expanding the
solution x* to the static problem by the uniform growth rate over
time.
Now, let us turn to (i), note that by inverting:

xt* = [I - (1+g)Aó ]-1(1+g)tD(p,
w)
for which, without detailing, we must appeal to Perron-Frobenius
theorems on non-negative square matrix to assure us that xt*
exists and is non-negative. Notice that by iterating t times, we also obtain a
path [x1*, x2*, .., xt*,
...]. Notice, once again, that if x* = [I - (1+g)Aó ]-1D(p, w), then:

xt* = (1+g)tx*
thus, again, we obtain the path by expanding the solution to the
problem, x*, by the uniform growth factor. Obviously, this x*
depends on D(p, w), A and g in this case and in the
previous case, x* depended on F(p, w) and B.
Thus for steady-state equilibrium in the end, we must guarantee that the paths
are the same.
Now, (iii) remains. Notice that this can be converted to:

wt* = B-1[I -
(1+r)A]pt
thus we obtain a solution path of factor prices
[w1*, w2*, .., wt*, ...]
by inserting a path of output prices [p1,
p2, .., pt, ...]. Notice also that if
p is constant, then w* is constant. This is what we would like to
obtain in steady-state.
To guarantee this steady-state equilibrium, we must appeal to
Perron-Frobenius, fixed point theorems, etc. to ensure that there is a uniform
rate of profit, r, a uniform rate of growth, g, and a constant p* and
w* that yields a D(p*, w*) and F(p*,
w*) such that the solution x* generated by (i) is the same as the
x* generated by (ii). This is by no means easy, but the important point
to note here is that the solution to a uniformly progressive Walras-Cassel model
is effectively achieved by the same means as in the Walras-Cassel model, except
that we now have to additionally determine r and g. For more details on dynamic
Walras-Cassel models, consult the discussions in Dorfman, Samuelson and Solow (1958), Morishima (1964,
1969, 1977), and Hicks
(1965).
Selected References
G. Cassel (1918) The Theory of Social Economy. 1932
edition, New York: Harcourt, Brace and Company.
J.v. Daal and A. Jolink (1993) The Equilibrium Economics
of L�on Walras. London: Routledge.
R. Dorfman, P.A. Samuelson and R.M. Solow (1958) Linear
Programming and Economic Analysis. New York: McGraw-Hill.
J. Hicks (1965) Capital and Growth. Oxford: Clarendon.
C. Menger (1871) Principles of Economics. 1981 edition of
1971 translation, New York: New York University Press.
M. Morishima (1964) Equilibrium, Stability and Growth: A
multi-sectoral analysis. Oxford: Clarendon Press.
M. Morishima (1969) Theory of Economic Growth. Oxford:
Clarendon Press.
M. Morishima (1977) Walras' Economics: A pure theory of capital
and money. Cambridge: Cambridge University Press.
T.M. Rybczynski (1955) "Factor Endowment and Relative Commodity
Prices", Econometrica, Vol. 22, p.336-41.
W.F. Stolper and P.A. Samuelson (1941) "Protection and Real
Wages", Review of Economic Studies, Vol. 9, p.58-73.
A. Wald. (1936) "On Some Systems of Equations of Mathematical
Economics", Zeitschrift f�r National�konomie, Vol.7. Translated, 1951,
Econometrica, Vol.19 (4), p.368-403.
L. Walras (1874) Elements of Pure Economics: Or the theory of
social wealth. 1954 translation of 1926 edition, Homewood, Ill.: Richard
Irwin.
F. von Wieser (1889) Natural Value. 1971 reprint of 1893
translation, New York: Augustus M. Kelley.
D.A. Walker (1996) Walras's Market Models. Cambridge, UK:
Cambridge University Press.


________________________________________________________
Top

Back
 







Home
Alphabetical Index
Schools of Thought 
Surveys and Essays

Web
Links
References
Contact
Frames