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. 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 v_js 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:

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 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:

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 V_2｢ . 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 x_ji 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 P_2｢ . 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 = 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.

(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 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｢ _jx + D_j(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. 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｢ _jx + 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_ip + B_iw 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_ip + B_iw

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.

(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, x_t must meet not only concurrent consumer demand (1+g)^tD(p, w) but also industry demand for production next period, A｢ x_t+1. As we want a steady-state, then we impose the condition that x_t+1 = (1+g)x_t so that outputs in period t+1 are a proportional amount g greater than outputs at t, thus:

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)^tD(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)^tF(p, w) by our assumption of a replicated population, then:

(1+g)^tF(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)^tF(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)^tB｢ ^-1F(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 [x₁*, x₂*, .., x_t*, ...]. In fact, it can be easily shown that if x* = B｢ ^-1F(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)^tD(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 [x₁*, x₂*, .., x_t*, ...]. Notice, once again, that if x* = [I - (1+g)A｢ ]^-1D(p, w), then:

x_t* = (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:

w_t* = B^-1[I - (1+r)A]p_t

thus we obtain a solution path of factor prices [w₁*, w₂*, .., w_t*, ...] by inserting a path of output prices [p₁, p₂, .., 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).

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・ 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.

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