Léon Walras, in Part 5 of his *Elements
of Pure Economics* (1874) presented a theory of capital and interest as the
next-to-last step towards a complete general equilibrium model. Walras's theory of capital
was among the most innovative in his time and rather confusing. The following is a
simplistic elucidation of Walras's theory, and is far from controversial. A more careful
and deeper analysis is contained in Jaff・(1942), van Daal and Jolink (1993), Kompas
(1992) and Walker (1996). See also our discussion of the Walras-Cassel
system.

Walras divided between the stock of capital and the flows of
"services" of capital (i.e. the income flows arising from that stock). Capital,
he argued, was not "used up" in the production process, but rather provided
"services" and it was these services that entered the production process.
Indeed, in this way, Walras considered labor and land to be capital as well because they
provided "services" to production and were themselves (usually) not used up. It
was only in later editions of the *Elements *that he considered the issue of
"circulating" capital explicitly.

In Walras's system, then, given a capital stock, we obtain a given flow amount of capital services. This can be considered the "supply" of capital services. The "demand" for capital services is merely the consumers' demand for an amorphous commodity he called "savings in general" which arise from consumers' desires to transfer some amount of current purchasing power to demand future goods and services. Capital services, then, are the mechanism which makes this transfer possible. The price of capital services is determined by their demand (i.e. savings) and supply (i.e. existing capital services)

However, a few problems soon emerge. Firstly, capital goods have the unique feature that, unlike labor and land, they can themselves be "produced" and thus are not "endowed". As the amount of capital services available depends on the amount of capital stock, then by increasing capital stock, we can increase capital services artificially - thus, there is really no "endowment" of capital services either.

Furthermore, one needs to be able to "price" capital stock. As a produced commodity, it comes under the rule of cost-of-production pricing: "[c]apital goods proper are artificial capital goods: they are products and their prices are subject to the law of cost of production" (Walras, 1874: p.271). However, at the same time, Walras proposed that the price of capital stock is equated to the capitalized net income of capital services, which in turn depends on the price of capital services. This lands Walras's analysis in something of a knot for the price of capital seems to come now under two laws of pricing: by supply and demand (as other factors of production); by cost of production (as other produced goods). The link between these two was never really satisfactorily answered by Walras, even with the introduction of "time" (which enables us to freeze the supply of capital stock temporarily) or the introduction of circulating capital and cost-of-production inequalities in later work.

Walras's theory of capital was developed for a "progressive" or *growing* economy. To get there, it might be
useful to first start from a static state and formulate the general equilibrium equations
for his capital model, and then consider the modifications that have to be made to
incorporate the progressive case.

Let us assume two produced commodities (consumer goods and capital goods)
and two factors (capital and labor). Let X_{c} be the output of consumer goods and
X_{k} be the output of capital goods. Both use capital and labor, let K_{c}
and L_{c} be the amounts of capital and labor used in the production of X_{c}
amount of consumption goods and let K_{k} and L_{k} be the amounts of
capital and labor used in the production of X_{k} amount of labor goods. We can
consider this in two general production functions X_{c} = F_{c}(K_{c},
L_{c}) and X_{k} = F_{k}(K_{k}, L_{k}), but we
shall give it a precise form, as Walras (1874) did, by assuming fixed unit input
coefficients. Define a_{LC} = L_{c}/X_{c}, a_{KC} = K_{c}/X_{c}
and a_{LK} = L_{k}/X_{k}, a_{KK} = K_{k}/X_{k}
as the unit input coefficients (for mnemonic purposes, read the subscript in the order
"input-output" so that the coefficient a_{ij} denotes the amount of
factor i needed to produce a single unit of output j). Thus, the production conditions
become:

X

_{c}= L_{c}+ K_{c}= a_{LC}X_{c}+ a_{KC}X_{c}

X

_{k}= L_{k}+ K_{k}= a_{LK}X_{k}+ a_{KK}X_{k}

as the production requirements for consumer and capital goods respectively. Thus:

L

_{c}+ L_{k}= a_{LC}X_{c}+ a_{LK}X_{k}

K

_{c}+ K_{k}= a_{KC}X_{c}+ a_{KK}X_{k}

are the total demands for labor and capital respectively. Note that factor
demand can be imputed via technology from the amount of outputs produced (X_{c}, X_{k}).
Now, if we *assume* a fixed supply of labor and capital, L and K (thus a static
economy) then it must be that:

L

_{c}+ L_{k}｣ L

K

_{c}+ K_{k}｣ K

i.e. labor and capital demanded must not exceed labor or capital supplied.
If we were to *assume* market clearing in the factor markets, then these inequalities
would become equalities, so that:

L

_{c}+ L_{k}= L

K

_{c}+ K_{k}= K

or, expressing this in terms of unit-input coefficients:

a

_{LC}X_{c}+ a_{LK}X_{k}= L

a

_{KC}X_{c}+ a_{KK}X_{k}= K

If we were to let **x｢ **= [X_{c} X_{k}]｢ and **v｢ **= [L K]｢ and **B** be a 2 ｴ 2 matrix:

a_{LC} a_{LK}

**B** =

a_{KC} a_{KK}

then we can rewrite the entire system as **Bx** = **v**. The implied
generalization to n goods and m factors can be immediately noted by letting **B** be an
(m ｴ n) matrix, **x** be a (n ｴ
1) vector and **v** a (m ｴ 1) vector. However, for our
purposes, let us continue with the simple two-sector, two-factor model.

Let us now *assume* the Walrasian type of perfect competition where
entrepreneurs "break even" in every commodity, i.e. no net profits, so total
revenue must equal total costs. Total revenues are merely the price of the commodity
multiplied by the amount produced, whereas total costs are the wages and rental rates paid
to the labor and capital employed in the production of the goods, i.e.

p

_{c}X_{c}= wL_{c}+ rK_{c}

p

_{k}X_{k}= wL_{k}+ rK_{k}

where (p_{c}, p_{k}) are the prices of consumer and
capital goods respectively while (w, r) are the wage and rental rates per unit of labor
and capital respectively. Substituting for L_{c}, K_{c}, L_{k} and
K_{k}, we obtain:

p

_{c}X_{c}= wa_{LC}X_{c}+ ra_{KC}X_{c}

p

_{k}X_{k}= wa_{LK}X_{k}+ ra_{KK}X_{k}

or simply, dividing through by outputs X_{c} and X_{k}
respectively:

p

_{c}= wa_{LC}+ ra_{KC}

p

_{k}= wa_{LK}+ ra_{KK}

so, letting **p｢ **= [p_{c} p_{k}]｢ , **w｢ **= [w r]｢
and letting **A** be a 2 x 2 matrix:

a_{LC} a_{KC}

**A** =

a_{LK} a_{KK}

we can rewrite the entire system as **p** = **Aw**. Thus, again
generalizing to n goods and m factors, we obtain a (n ｴ 1)
vector **p**, a (n ｴ m) matrix **A** and a (m ｴ 1) vector **w**. Notice that the transpose of matrix **A**
is merely the matrix **B** we derived earlier, **A｢ **= **B**.
Thus, we could write our earlier equation as **v** = **A｢
x**.

We have yet to tackle the issue of the demand for consumer goods and
capital goods. For the sake of argument, let D_{c} be the demand for consumer
goods and D_{k} be the demand for capital goods. Then, in market clearing:

D

_{c}= X_{c}

D

_{k}= X_{k}

However, note that we already have a problem: namely, *who* demands
capital goods? In other models, we can obtain the demand for capital *as factors* by
summing input requirements, K_{c} + K_{k}. Is there any relationship
between D_{k} and (K_{c} + K_{k})? This is where the difficulty
comes in. If we supposed X_{k} was a consumer good, then there would be no problem
as D_{k} would, like D_{c}, be determined by consumer demands and would
have nothing to do with factor demands. However, as capital is a *produced* good,
then there ought to be a relationship between the demand for capital output and the demand
for capital as a factor.

Walras's resolution was to divide the two: D_{k} is the demand for
*new* capital goods, whereas K_{c} + K_{k} is the demand for capital
services on *existing* capital goods. Hence, the output of new capital goods, X_{k},
does not feed into production contemporaneously as it only yields capital services in the
future. Thus, we see immediately the intimate interrelationship between time and capital
in Walras's theory.

But the question remains hanging: what determines the demand for new
capital goods, D_{k}? Walras proposed that this is merely savings. Thus, savings
by households translates into a demand for new capital goods. In this simple two-sector
world, then, households either consume or save - or, equivalently, demand consumer goods D_{c}
and new capital goods, D_{k}. Thus, in aggregate budget constraint faced by
households implies that:

p

_{c}D_{c}+ p_{k}D_{k}｣ Y

where total value of demand for consumer and new capital goods on the left must be less than or equal to total income Y. This is merely total factor income when factor markets clear, i.e. Y = rK + wL, or:

Y = w(a

_{LC}X_{c}+ a_{LK}X_{k}) + r(a_{KC}X_{c}+ a_{KK}X_{k})

then the Walras's Law constraint can be rewritten:

p

_{c}D_{c}+ p_{k}D_{k}= w(a_{LC}X_{c}+ a_{LK}X_{k}) + r(a_{KC}X_{c}+ a_{KK}X_{k})

As this arises from the households' decisions, we must still ask what are
determinants of D_{c} and D_{k}. Essentially, preferences (utility
functions), commodity prices (p_{c}, p_{k}), factor returns (w, r) and
endowments of factors (in total equal to L and K, which we can reduce to Y) will determine
D_{c} and D_{k}. If we aggregated household demands, we would obtain
aggregate demand functions of the form:

D

_{c}= D_{c}(p_{c}, p_{k}, w, r, Y)

D

_{k}= D_{k}(p_{c}, p_{k}, w, r, Y)

for consumption goods and capital goods accordingly. However, households
obtain no utility from capital goods as they cannot be consumed. Thus, D_{k},
rightly, are seen as "savings" and so can be considered as demanded in order to
be able to claim future output.

Walras did not spend much time discussing the desire or demand for future
goods, preferring to just subsume it all under "savings". As a result, a few
more words on this matter are warranted. We have used r to denote the return to capital
services. Walras defined "net income" to a unit of a capital good to be r - (m + v)p_{k} where m is the
depreciation rate and v is the insurance rate and p_{k}, of course, is the price
of the capital good. Ignoring insurance (setting v = 0), then the "net income"
of a unit of capital good would be simply r. The ratio of net income to the price of
existing capital Walras called the "rate of net income" and denoted it by:

i = (r - m p

_{k})/p_{k}= r/p_{k}- m

Thus, we can express the price for the capital good as:

p

_{k}= r/(i - m )

Now, Walras actually went further and claimed that, in equilibrium, ratio
of net income to the price of the capital good would be equal for all capital goods. So,
in general, if we have m capital goods, then i = r_{1}/p_{1} = r_{2}/p_{2}
= ... = r_{m}/p_{m}. Thus, all capital goods, in equilibrium, have the
same rate of net income.

Now, Walras allowed for two types of savings: savings "in kind",
in which households save entire vectors of capital goods and receive a stream of various
net returns, or savings in a *numeraire* commodity. Let us consider the latter case
where agents save "purchasing power" in the form of a homogeneous substance he
called "perpetual net income" and denoted E. We can think of it, heuristically,
as an asset which pays a unit of the numeraire good every period forever. Thus, the price
of a unit of E is p_{e} = 1/i, i.e. the value of a unit of E is the perpetually
discounted unit of account. But i, as we noted earlier, was merely r/p_{k}. Thus,
the households' consumption and savings decisions are reduced to considering merely i
instead of r and p_{k} (or, when we have multiple capital goods, a whole
array of r_{i} and p_{i}).

As noted earlier, households save and thus purchase E in order to be able
to claim future output. In turn, the assets E are sold by firms to households in order to
enable them to purchase the new capital goods. Thus, firms sell asset E to households and
purchase the newly produced capital goods X_{k}. Thus, p_{k}D_{k}
= E/i and the Walras's Law constraint becomes:

p

_{c}D_{c}+ E/i = w(a_{LC}X_{c}+ a_{LK}X_{k}) + r(a_{KC}X_{c}+ a_{KK}X_{k})

Households, thus, maximize utility subject to a constraint similar to this
one (just add the subscript for each individual household). Note that, now, having
subsumed r and p_{k} into i, then the aggregate demand functions become:

D

_{c}= D_{c}(p_{c}, w, i, Y)

E = E(p

_{c}, w, i, Y)

where D_{c} is the demand for consumption goods and E is merely
"savings". As noted earlier, Walras, himself, did not explicitly consider future
"utility" from future consumption - thus he allows E to represent it.

One last condition is necessary. To tie up our entire model in a simple
fashion, let us eliminate time by assuming a stationary state condition so that new
capital goods are demanded merely to replace depreciated capital, i.e. D_{k} = m K. In this manner, we can eliminate the issue of
"expectations" of different future capital requirements and thus determine a
single level of K instead of a moving path of K. Consequently, let us recapitulate the
equations and unknowns in Walras's capital model.

Equation Type |
Number of such equations |
Description |

D_{c} = D_{c}(p_{c}, w, i, Y) |
(n) | Demand for consumer goods |

E = E(p_{c}, w, i, Y) |
(1) | Savings-in-general |

Y = wL + rK | (1) | Total income |

a_{LC}X_{c} + a_{LK}X_{k} = L |
(m) | Non-produced factor market equilibrium |

a_{KC}X_{c} + a_{KK}X_{k} = K |
(k) | Produced factor market equilibrium |

D_{c} = X_{c} |
(n) | Consumer goods equilibrium |

D_{k} = X_{k} |
(k) | Capital goods equilibrium |

p_{c} = wa_{LC} + ra_{KC} |
(n) | Consumer sector price-cost equality |

p_{k} = wa_{LK} + ra_{KK} |
(k) | Capital sector price-cost equality |

E/i = p_{k}D_{k} |
(1) | Investment-savings equilibrium |

i = r/p_{k} - m |
(k) | Perpetual net income equality |

D_{k} = m K |
(k) | Demand for new capital goods (investment)/stationary state condition |

where we thus have twelve types of equations. The types of unknowns are D_{c},
E, Y, X_{c}, X_{k}, p_{c}, p_{k}, w, r, i, D_{k}
and K, which are also twelve. By Walras's Law, one of the equations is redundant and,
setting a numeraire good, p_{c} = 1, one of the prices can be eliminated, thus we
have a system of eleven types of equations and eleven types of unknowns. It is important
to note that K is one of the unknowns, thus we do not assume it is an exogenously endowed
factor but a producible means of production, i.e. "capital" properly speaking.

How is this generalized to multiple commodities and multiple factors? Let
there be n produced consumption goods, k produced factors (i.e. capital goods) and m
non-produced factors (i.e. labor, land, etc.). In this case, the letters on the right hand
side above denote the number of equations of each type. Thus, reading from the top, we
have n of the D_{c} equations, one E equation, one Y equation, m non-produced
factor market clearing conditions, k produced factor market clearing conditions, n
consumption goods market equilibrium conditions, k capital goods market equilibrium
conditions, n price-cost equalities for consumption goods, k price-cost equalities for
capital goods, one definition of savings E, k net income equations and k stationary state
conditions. Thus, we have (3 + 3n + m + 5k) equations.

The generalization is straightforward enough, only the following might
merit noticing. Firstly, we only have one equation for the earlier E/i = p_{k}D_{k}.
This is because, with multiple goods, p_{k}D_{k} now becomes the sum of
such terms over all capital goods, i.e. E/i = p_{1}D_{1} + p_{2}D_{2}
+ .... p_{k}D_{k}, so that we still only have one definition of E. In
contrast, we have k conditions i = r/p_{k} - m ,
because, as noted earlier, it is supposed to hold for every capital good and thus every p_{k}.

The number of unknowns are as follows: we have n consumption demands (n
times D_{c}), one savings (E), one total income (Y), n outputs of consumption
goods (n times X_{c}), k outputs of capital goods (k times X_{k}), n
consumption good prices (p_{c}), k capital goods prices (p_{k}), m factor
returns on non-producible factors (m times w), k factor returns on producible factors (k
times r), one rate of net income (i), k capital demands (k times D_{k}) and k
capital stocks (k times K). Thus the total number of unknowns is (3 + 3n + m + 5k).

Thus, even with multiple goods and factors, the number of equations is equal to the number of unknowns. We can eliminate one of the equations by Walras's Law and one of the unknowns by the numeraire, and thus end up with (2 + 3n + m + 5k) equations and unknowns.

One bizarre notion must be underlined: the stationary state condition.
Knut Wicksell (1893) pointed out that the
interest rate was indeterminate in Walras's model. However, as Enrico Barone (1895) noted and we can see above clearly, the
stationary condition allows for determination. However, by effectively reducing the demand
for new capital goods to replacement capital, we have thus reduced the determination of
the interest rate to the market for depreciation. But suppose capital did *not*
depreciate. Then Wicksell's critique is legimitate: there is no determinate rate of
interest.

To reincorporate the determinateness of interest in an economy without depreciation, we must have a "progressive" or "growing" economy with net savings and thus replace the stationary state condition above with some other, more dynamic, condition. This would require capital demands to be dependent on future demands, etc., which brings in expectations and thus might imply the concept of a temporary sequential equilibrium in a growing economy. This seems pretty much what Walras himself had in mind.

This point is interesting because it brings forth a notion that was quite unique to Walras. After Walras, the Neoclassical theories of capital (e.g. J.B. Clark, 1899) seemed to be able to determine interest rates in static economies. But in Walras, the rate of interest is only determinate in a "progressive" economy. If one thinks of the work of Schumpeter (1911) or von Neumann (1937), where stationary economies are indeed associated with zero interest rates, one can immediately realize their affinity to Walras's notion (see our discussion of steady-state growth and also the Walras-Cassel system.)

E. Barone (1895) "Sopra un Libro del Wicksell", *Giornale
degli Economisti*, Vol. 11, p.524-39.

J.v. Daal and A. Jolink (1993) *The Equilibrium Economics of Léon
Walras*. London: Routledge.

W. Jaff・(1942) "Léon Walras' Theory of Capital Accumulation",
in O. Lange, F. McIntyre and O. Yntema, editors, *Studies in Mathematical Economics and
Econometrics*. Chicago: University of Chicago Press.

T. Kompas (1992) *Studies in the History of Long-Run Equilibrium Theory*.
Manchester, UK: Manchester University Press.

D.A. Walker (1996) *Walras's Market Models*. Cambridge, UK: Cambridge
University Press.

L. Walras (1874) *Elements of Pure Economics: Or the theory of social
wealth*. 1954 translation of 1926 edition, Homewood, Ill.: Richard Irwin.

K. Wicksell (1893) *Value, Capital and Rent*. 1970 reprint of 1954
edition, New York: Augustus M. Kelley.

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