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"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)

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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

The "Walras-Cassel" model refers to the general equilibrium
model with production introduced in Léon 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** = **B｢
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=B｢ xconnects output quantities and factor quantities;(iv)

p=Bwconnects 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** = **B｢ 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** = **B｢
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.

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)

x^{h}is a vector of commodities demanded by household h

(ii)x^{f}is a vector of commodities supplied by firm f

(iii)pis the vector of commodity prices.(3) Factors:

(i)

v^{h}is a vector of factors supplied by household h

(ii)v^{f}is a vector of factors demanded by firm f

(iii)wis the vector of factor prices.(4) Technology: (fixed proportions, same technology for all firms)

(i) b

_{ji}= v_{j}^{f}/x_{i}^{f}is a unit output production coefficient

(ii)Bis n ｴ m matrix of unit-input coefficients.(5) Objectives:

(i) Household h: max U

^{h}= U^{h}(x^{h},v^{h}) s.t.px^{h}｣wv^{h}.

(ii) Firm f: no objective;v^{f}=B｢ x^{f}is production function for fth firm

(6) Equilibrium assumptions:

(i) Perfect Competition:

p=Bw(ii) Factor market-clearing:

F(p, w) = v = B｢ 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 b_{ji} i = 1, ..., n, j = 1, .., m. Thus, b_{ji}
= v_{j}^{f}/x_{i}^{f }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:

v^{f}=B｢ x^{f}

where **B｢ **is an m ｴ
n matrix of unit-output coefficients so that factor demands (**v**^{f}) can be
deduced from desired output supplies (**x**^{f}). Thus,
the demand for a particular factor by firm j is:

v

_{j}^{f}= ・/font>_{i}b_{ji}x_{i}^{f}=B_{j｢ }x^{f}

where **B**_{j｢ }is merely the jth
row of **B｢ **. Thus, market demand for factor j is
obtained by summing up over firms:

v

_{j}= ・/font>_{f}v_{j}^{f}= ・/font>_{f}・/font>_{i}b_{ji}x_{i}^{f}=B_{j｢ }x^{f}

or, more generally:

v= ・/font>_{f}v^{f}= ・/font>_{f}B｢ x^{f}=B｢ x

or simply **v** = **B｢ 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 **p｢ x**^{f}
equals total cost **w｢ v**^{f} for every firm f,
or:

p｢ x^{f}=w｢ v^{f}

or, as b_{ji} = v_{j}^{f}/x_{i}^{f},
then the perfect competition assumption implies:

p=Bw

where, note, **B** is the transpose of the earlier matrix of
unit-output coefficients **B｢ **.

Let us now turn to the objectives of households. Each household h has a
utility function U^{h}(**x**^{h}, **v**^{h}) where utility
increases with consumption of produced commodities **x**^{h} and decreases with
supply of factors **v**^{h}. [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 **v**^{h}.
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 U

^{h}= U^{h}(x^{h},v^{h})s.t.

p｢ x^{h}｣w｢ v^{h}

and there are H such programs, one for each household. Household income
comes from the sale of factors (**w｢ v**^{h}) 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 (**p｢ x**^{h}). The result is a set of output demand functions
and factor supply functions of the following general form:

x

_{i}^{h}= D_{i}^{h}(p,w) for each commodity i = 1, ..., n, for each household h = 1, .., Hv

_{j}^{h}= F_{j}^{h}(p,w) for each factor j = 1, .., m, for each household h = 1, ..., H.

and the budget constraint is met, so:

p｢ x^{h}=w｢ v^{h}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:

x

_{i}= ・/font>_{h}x_{i}^{h}= ・/font>_{h}D_{i}^{h}(p,w) = D_{i }(p,w)v

_{j}= ・/font>_{h }v_{j}^{h}= ・/font>_{h }F_{j}^{h}(p,w) = F_{j}(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:

・/font>

_{h }x_{i}^{h}= ・/font>_{f}x_{i}^{f}for each commodity i = 1, ..., n

・/font>

_{h }v_{j}^{h}= ・/font>_{f}v_{j}^{f}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:

D

_{i}(p,w) = x_{i}for each commodity i = 1, .., n.F

_{j}(p,w) = v_{j}=B_{j｢ }xfor 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 |
p |
(n equations) |

output market equilibrium: |
x |
(n equations) |

factor market equilibrium: |
v |
(m equations) |

market factor supplies: |
v |
(m equations) |

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

quantity of produced goods: |
x |
(n unknowns) |

quantity of factors: |
v |
(m unknowns) |

output prices: |
p |
(n unknowns) |

factor prices: |
w |
(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:

・/font>

_{i}p_{i}[・/font>_{h}x_{i}^{h}- ・/font>_{f}x_{i}^{f}] + ・/font>_{j}w_{j}[・/font>_{f}v_{j}^{f}_{ }- ・/font>_{h}v_{j}^{h}] = 0

or, in vector form:

p｢[x^{h}_{ }-x^{f}] +w｢[v^{f }-v^{h}] = 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. p_{1} = 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 v_{j}s
are known and we can omit the market factor supplies equations (the last set of m
equations, v_{j} = F_{j}(**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** = **B｢
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** = **B｢ x **are:

v

_{1}= b_{11}x_{1}+ b_{12}x_{2}v

_{2}= b_{21}x_{1}+ b_{22}x_{2}

To see this graphically, we can depict them in x_{1}, x_{2}
space as in Figure 2. Specifically, notice that the first equation can be rewritten as x_{2}
= v_{1}/b_{12} - (b_{11}/b_{12})x_{1} which yields
us the negatively-sloped line V_{1} in Figure 2. This has vertical intercept v_{1}/b_{12}
> 0, horizontal intercept v_{1}/b_{11} > 0 and slope -(b_{11}/b_{12})
< 0. This curve represents the locus of output level combinations that fulfill
equilibrium in factor market 1 for a given v_{1}. Notice that if factor supply v_{1}
increases, then the V_{1} curve shifts outwards.

Conversely, the second equation can be rewritten as x_{2} = v_{2}/b_{22}
- (b_{21}/b_{22})x_{1} which yields us a second negatively-sloped
line V_{2} in Figure 2. This has vertical intercept v_{2}/b_{22}
> 0, horizontal intercept v_{2}/b_{21} > 0 and slope -(b_{21}/b_{22})
< 0. The curve V_{2} is a locus of output combinations which yield equilibrium
in factor market 2. An increase in v_{2} will also shift the V_{2} curve
out.

Figure 2- Factor Market Clearing

Obviously, equilibrium is obtained when both the equalities hold - in this
case, at the intersection of the V_{1} and V_{2} curves at point E. Thus,
output levels x_{1}* and x_{2}* at point E represent factor market
equilibrium.

It is interesting to to note that a we are assuming here that V_{1}
is steeper than V_{2}. This implies that b_{11}/b_{12} > b_{21}/b_{22}.
Now, let x_{ji} be the amount of factor j used in industry i, then we can easilty
notice that b_{ji} = x_{ji}/x_{i}, the amount of factor j used in
industry i (x_{ji}) divided by the amount of good i. Thus, we can rewrite this
inequality as:

b

_{11}/b_{12 }= (x_{11}/x_{1})/(x_{12}/x_{2}) > (x_{21}/x_{1})/(x_{22}/x_{2}) = b_{12}/b_{22}

or, cross-multiplying and cancelling:

x

_{11}/x_{21}> x_{12}/x_{22}

which implies that industry x_{1} is uses factor v_{1}
relatively more intensively than industry x_{2}, while industry x_{2} uses
factor v_{2} relatively more intensely than industry x_{1}. If we conceive
of factor v_{1} as "capital" and factor v_{2} 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 V_{2} curve to V_{2｢ }. Notice that the equilibrium position moves from E to F. At
F, x_{1}* has fallen and x_{2}* has risen relative to E. As industry 2 is
relatively intensive in factor 2, then the rise in x_{2}* and fall in x_{1}*
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:

p

_{1}= b_{11}w_{1}+ b_{21}w_{2}p

_{2}= b_{12}w_{1}+ b_{22}w_{2}

These price-cost equalities are depicted graphically in w_{1}, w_{2}
space in Figure 3. The first equation can be rewritten as w_{2} = p_{1}/b_{21}
- (b_{11}/b_{21})w_{1} which is the negatively-sloped line P_{1}
in Figure 3 with vertical intercept p_{1}/b_{21} > 0, horizontal
intercept p_{1}/b_{11} > 0 and slope -(b_{11}/b_{21})
< 0. This curve is the locus of factor returns combinations that fulfill the price-cost
equality for industry 1 for a *given* output price, p_{1}. We can note that
in this case if the output price p_{1} increases, then the P_{1} curve
shifts outwards. Similarly, the second equation can be rewritten as w_{2} = p_{2}/b_{22}
- (b_{12}/b_{22})w_{1}, the second negatively-sloped line P_{2}
in Figure 3 with vertical intercept p_{2}/b_{22} > 0, horizontal
intercept p_{2}/b_{12} > 0 and slope -(b_{12}/b_{22})
< 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 p_{2}
will shift the P_{2} curve out.

The first thing to note about Figure 3 is that the intersection of curves
P_{1} and P_{2} at point G yield a particular factor return combination, w_{1}*,
w_{2}*. 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 (p_{1}, p_{2}) and
technology (**B**), we can *immediately* determine the necessary factor returns (w_{1}*,
w_{2}*). 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 P_{1} is steeper than P_{2} which implies that b_{11}/b_{21}
> b_{12}/b_{22}. Following the previous logic, this implies that:

x

_{11}/x_{21}> x_{12}/x_{22}

where x_{ji} is the amount of factor j used in industry i. Thus,
industry x_{1} is relatively intensive in factor v_{1} and industry x_{2}
is relatively intensive in factor v_{2} - 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 p_{2},
the price of good 2, then the P_{2} curve shifts out to P_{2｢
}. The equilibrium position consequently moves from G to H. Notice that at H, w_{1}*
has fallen and w_{2}* 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** = **B**｢**w**
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**

Léon 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.

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 a_{ji} 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 ｴ
n matrix with typical element a_{ji} ｳ 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

x

_{j}=A｢_{j}x+ D_{j}(p,w)

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

x=A｢ 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=B｢ 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 **A**_{i}**p**
+ **B**_{i}**w** where **A**_{i} is the ith row of **A** and **B**_{i}
is the ith row of **B**. Thus, for good i, we have the price-cost equality:

p

_{i}=A_{i}p+B_{i}w

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=A｢ x+D(p,w)(ii)

v=B｢ 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** - **A｢ **)**x** = **D**(**p**, **w**) and (iii) as (**I**-**A**)**p**
= **Bw**. Thus, letting **pｰ **= (**I** - **A**)**p**
denoted "adjusted" or "net" output prices, then this system can be
rewritten:

(i) (

I-A｢)x=D(p,w)(ii)

v=B｢ x(iii)

pｰ=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ｰ
**and **B**. To obtain full equilibrium all we have to guarantee is that the net
market supply (**I** - **A｢ **)**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.

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 Léon 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)^{t}**D**(**p**,
**w**) are consumer demands and (1+g)^{t}**F**(**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, **x _{t} **must meet
not only concurrent consumer demand (1+g)

A｢ x_{t+1}= (1+g)A｢ x_{t}.

Consequently, for goods market equilibrium, we must have:

x_{t}= (1+g)A｢ x_{t}+ (1+g)^{t}D(p,w)

outputs at t (**x**_{t}) 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 **B｢ x**_{t}.
Thus primary factor market equilibrium implies:

v=_{t}B｢ x_{t}

where, as **v**_{t} = (1+g)^{t}**F**(**p**, **w**)
by our assumption of a replicated population, then:

(1+g)

^{t}F(p, w) = B｢ x_{t}

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:

p_{t}= (1+r)Ap_{t}+Bw_{t}

In conclusion, for a Walras-Cassel model with capital, we have the following sets of equations:

(i)

x_{t}= (1+g)A｢ x_{t}+ (1+g)^{t}D(p,w)(ii) (1+g)

^{t}F(p,w) =B｢ x_{t}

(iii)

p_{t}= (1+r)Ap_{t}+Bw_{t}

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:

x_{t}* = (1+g)^{t}B｢^{-1}F(p,w)

provided **B｢ **is invertible and other
conditions are met so that **x**_{t}* ｳ 0. Notice
that iterating for different time periods, we obtain a solution path [**x**_{1}*,
**x**_{2}*, .., **x**_{t}*, ...]. In fact, it can be easily shown
that if **x*** = **B｢ **^{-1}**F**(**p**, **w**),
then:

x_{t}* = (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:

x_{t}* = [I- (1+g)A｢]^{-1}(1+g)^{t}D(p,w)

for which, without detailing, we must appeal to Perron-Frobenius theorems
on non-negative square matrix to assure us that **x**_{t}* exists and is
non-negative. Notice that by iterating t times, we also obtain a path [**x**_{1}*,
**x**_{2}*, .., **x**_{t}*, ...]. Notice, once again, that if **x***
= [**I** - (1+g)**A｢ **]^{-1}**D**(**p**, **w**),
then:

x_{t}* = (1+g)^{t}x*

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:

w_{t}* =B^{-1}[I- (1+r)A]p_{t}

thus we obtain a solution path of factor prices [**w**_{1}*, **w**_{2}*,
.., **w**_{t}*, ...] by inserting a path of output prices [**p**_{1},
**p**_{2}, .., **p**_{t}, ...]. 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).

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 Léon
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・ Nationalkonomie*, 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.

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