The Walras-Cassel System

Tricolor above the Paris Bourse


"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)



(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


(1) Introduction

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 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.]

walcass1.gif (3891 bytes)

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.

(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 wvh.
(ii) Firm f: no objective; vf = B xf 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 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 = B xf

where B 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 = ・/font>ibjixif = Bj xf

where Bj is merely the jth row of B . Thus, market demand for factor j is obtained by summing up over firms:

vj = ・/font> f vjf = ・/font> f・/font>ibjixif = Bj xf

or, more generally:

v = ・/font>f vf = ・/font>f B xf = 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 xf equals total cost w vf for every firm f, or:

p xf = w 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 B .

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. p xh w vh

and there are H such programs, one for each household. Household income comes from the sale of factors (w 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 (p 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:

p xh = w 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 = ・/font>hxih = ・/font>h Dih(p, w) = Di (p, w)

vj = ・/font>h vjh = ・/font>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:

・/font>h xih = ・/font>f xif for each commodity i = 1, ..., n

・/font>h vjh = ・/font>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 = Bj 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 = ・/font>jbjiwj = Biw

(n equations)

output market equilibrium:

xi = Di(p, w)

(n equations)

factor market equilibrium:

vj = Bj 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:


(n unknowns)

quantity of factors:


(m unknowns)

output prices:


(n unknowns)

factor prices:


(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 pi [・/font> h xih - ・/font> f xif] + ・/font> j wj [・/font> f vjf - ・/font> h vjh] = 0

or, in vector form:

p [xh - xf] + w [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 = 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:

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.

walcass2.gif (2718 bytes)

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 V2 . 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.

walcass3.gif (2761 bytes)

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 P2 . 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 = Bw 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.

(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 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 = A jx + Dj(p, w)

where A j is the jth row of A . The term A 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, A 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 = 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 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 = 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 = (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 = 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 - 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.

(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 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)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, A 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:

A xt+1 = (1+g)A xt.

Consequently, for goods market equilibrium, we must have:

xt = (1+g)A 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 B xt. Thus primary factor market equilibrium implies:

vt = B xt

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

(1+g)tF(p, w) = B 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)A xt + (1+g)t D(p, w)

(ii) (1+g)tF(p, w) = B 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)tB -1F(p, w)

provided B is invertible and other conditions are met so that xt* 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* = B -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)A ]-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)A ]-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 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.