finalexam

History of Economic Thought

Notes for Week 9

Leon Walras (1834–1910) on General Competitive Equilibrium

As mentioned in the previous note, Walras saw himself as resolving, once and for all, the

question of whether or not competitive market systems are capable of arriving at some kind of

beneficial or optimal outcome. He was correct in claiming that the classicals had never managed

to prove that laissez-faire policies bring about the benefits they claimed it would provide. But in

his attempt to resolve this theoretical issue, he inadvertently introduced a conception of

competition, and a vision of economic behavior, that’s at odds with the way Smith, Ricardo, or

Marx would have thought of these issues. Indeed, it’s not unreasonable to suggest that Walras’

criticism of the classicals had more to do with the writings of Bentham, J. B. Say, and Nassau

Senior, than with the writings of Smith, Ricardo, or Marx.

The classical economists in the tradition of Smith, Ricardo, and Marx, were intent on

explaining how capitalist economies grow over time. Their focus was on figuring out the logic of

economic growth as well as the distribution of functional income in a context of economic

freedom. Competition among capitalists would generate a set of prices, called normal or natural

prices (Marx called them prices of production), consistent with the effective demand for

commodities. And the effective demand for commodities was determined by the spending

patterns typical of workers, capitalists, and landlords out of their respective flows of normal or

natural income. The normal prices, normal spending patterns, and normal income flows were

conceived of as “equilibrium” magnitudes in the sense that they were consistent with the on-

going reproduction of the system. What brought about this equilibrium was the achievement of a

system-wide common rate of profit and a system-wide common wage rate, as a result of the

competitive search for income on the part of capitalists and workers. Landlords were the

recipients of a passive form of income, rent, in that they did not engage in productive economic

activity like capitalists and workers.

None of this is present in Walras’ conception of equilibrium. His conception of

equilibrium has little to do with an economy growing over time. Instead, he’s imagining a

stationary state in which the amount and composition of goods (resources in the sense of land,

labor, and capital, as well as commodities in the sense of final output) is given. Those goods are

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distributed randomly among the members of society and the question then becomes, does the

free market distribute those goods in such a fashion that demand equals supply in every market

and every individual is achieving a maximum of utility?

With that proviso out of the way, let’s jump into his theory. Here we’ll rely on the

equilibrium he thought would emerge in a capitalist economy in a stationary state. The way to

think of this is to imagine that there’s a random distribution of resources among the people

making up the society. The resources consist of land, labor, and capital, and there are firms

(industries) that transform the services of those resources into finished goods that are demanded

by the resource owners.

Since he’s thinking in very broad terms, Walras notes that in this hypothetical economy

there are n productive services (resources) and m products. But, to make this way of thinking a

bit more concrete we’ll imagine that n is 3 (though it could be a million) and that m is also 3

(though, again, it could also be in the millions). The three types of productive services

(resources) are land (T), labor (L) and capital (K); while the three types of products are A, B, and

C. He imagines that all the exchanges that could take place are conducted in terms of a

commonly accepted numéraire, a good that’s used by everyone to measure the worth of a service

or a good. But it’s important to note that this numéraire is not money in the contemporary sense

of that term. It’s instead, literally, one of the goods produced in the system, it’s not fiat money

nor is it a debt issued by banks.

The individuals in this system own various amounts of land, labor and capital, and make

utility maximizing decisions regarding the amounts of land, labor and capital to offer at various

possible prices. At the same time, those same individuals make utility maximizing decisions

regarding the amounts of A, B and C to purchase at various possible prices. The products (A, B

and C) are produced through industries that use the services (T, L and K) in combinations

determined by the existing technology of production. The individuals are thus the ultimate

providers of services (T, L, and K) as well as the ultimate consumers of products (A, B, and C).

At the same time, the industries that produce A, B, and C operate under the principles of

competition ensuring that, in the long-run, the prices of A, B, and C are equal to their unit cost.

All of the individuals and firms in this system are price takers.

The following n (3) equations represent the sum of all the utility maximizing offer of

services on the part of all of the individuals in the system. In this example we’re not showing the

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mathematics of the utility maximizing choice of offers on the part of the individual, since the

underlying logic has already been explained. Walras imagines that individuals derive utility from

owning productive services (i.e. resources), so that the utility maximizing choice is over the

amount of net services retained (the difference between the original endowment of the service

and the amount offered for sale). Summing up all these decisions for all of the individuals in the

system provides us with the following n (3) equations showing the utility maximizing offer of

services for all possible prices on the part of all the individuals.

𝑂𝑡 = 𝑓𝑡 (𝑝𝑡 , 𝑝𝑙 , 𝑝𝑘 , 𝑝𝑏, 𝑝𝑐 )

𝑂𝑙 = 𝑓𝑙 (𝑝𝑡 , 𝑝𝑙 , 𝑝𝑘 , 𝑝𝑏 , 𝑝𝑐 )

𝑂𝑘 = 𝑓𝑘 (𝑝𝑡 , 𝑝𝑙 , 𝑝𝑘 , 𝑝𝑏, 𝑝𝑐 )

Where Ot, Ol, Ok represent the offer of T, L, and K respectively, and pt, pl, pk, pb, pc represent the

prices of T, L, K, B, and C respectively. Note that the price of A, pa, is not shown since it’s the

numéraire and its price must be equal to one, i.e. pa = 1.

At the same time, the individual makes utility maximizing choices over the amount of

products to purchase. The choice in this case follows the same principles of utility maximization;

i.e., each individual observes his/her budget constraint and purchases an amount such that the

marginal utility of the desired good just matches the marginal utility of A, since it’s the

numéraire. Summing these decisions for all of the individuals in the system provides us with the

following m (3) equations showing the utility maximizing demand for products at all possible

prices on the part of all the individuals.

𝐷𝑏 = 𝑓𝑏 (𝑝𝑡 , 𝑝𝑙 , 𝑝𝑘 , 𝑝𝑏 , 𝑝𝑐 )

𝐷𝑐 = 𝑓𝑐 (𝑝𝑡 , 𝑝𝑙 , 𝑝𝑘 , 𝑝𝑏 , 𝑝𝑐 )

𝐷𝑎 = 𝑂𝑡 ∙ 𝑝𝑡 + 𝑂𝑙 ∙ 𝑝𝑙 + 𝑂𝑘 ∙ 𝑝𝑘 − (𝐷𝑏 ∙ 𝑝𝑏 + 𝐷𝑐 ∙ 𝑝𝑐 )

where Da, Db, and Dc represent the demand for A, B, and C. The other variables have already

been defined. Note that, while there are m (3) equations, only m-1 (2) of them are independent

since Da depends on the other equations in the system.

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So far, we have outlined the offer of services (the above n equations) and the demand for

products (the above m equations) on the part of all the individuals in the system. We can think of

the n offer equations as representing the offer side of the input market and the m demand

equations as representing the demand side of the output market. Now we need to consider the

transformation of the services into products through the various industries in the system.

Walras imagines that, in a system of extreme competition and efficiency, it must be the

case that, in equilibrium, the volume of services (T, L, and K) demanded by each industry (used

in production) is equal to the amount of services offered by all the individuals. The following n

(3) equations capture this equilibrium condition for the entire economy, showing that the amount

of services used in each industry is equal to the utility maximizing amount of services offered by

all of the individuals.

𝑎𝑡 ∙ 𝐷𝑎 + 𝑏𝑡 ∙ 𝐷𝑏 + 𝑐𝑡 ∙ 𝐷𝑐 = 𝑂𝑡

𝑎𝑙 ∙ 𝐷𝑎 + 𝑏𝑙 ∙ 𝐷𝑏 + 𝑐𝑙 ∙ 𝐷𝑐 = 𝑂𝑙

𝑎𝑘 ∙ 𝐷𝑎 + 𝑏𝑘 ∙ 𝐷𝑏 + 𝑐𝑘 ∙ 𝐷𝑐 = 𝑂𝑘

where at represents the amount of T that is used in the production of one unit of A, bt represents

the amount of T that is used in the production of one unit of B, and ct represents the amount of T

that is used in the production of one unit of C. Similar interpretations apply for al, bl, cl, ak, bk, ck.

These are the technical coefficients of production that Walras assumed were fixed by technology.

Note that at∙Da must represent the total amount of T that is used in the production of A, since it is

the product of the unit usage of T in the production of A times the total amount of A demanded.

Similar interpretations apply to the other products: bt∙Db, ct∙Db, al∙Da, … etc. Thus, the left hand

side of the above system of equations shows the amount of the various services (T, L, and K)

demanded by all the industries, while the right hand side shows the amount of the various

services offered by all the individuals.

At the same time that the above is taking place it must also true that, in a system of

extreme competition and efficiency, the price of each good must be equal its unit cost of

production. Walras is here giving expression to the notion that, in very competitive markets, the

long run price of a good must be equal to the minimum unit cost of production. The

contemporary neoclassical version of this idea is that, in the long run under conditions of

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extreme competition, the firm will be earning zero economic profits. Thus, for the system as a

whole, the following system of m (3) equations must exist showing that the unit cost of

production just matches the price of the product.

𝑎𝑡 ∙ 𝑝𝑡 + 𝑎𝑙 ∙ 𝑝𝑙 + 𝑎𝑘 ∙ 𝑝𝑘 = 1

𝑏𝑡 ∙ 𝑝𝑡 + 𝑏𝑙 ∙ 𝑝𝑙 + 𝑏𝑘 ∙ 𝑝𝑘 = 𝑝𝑏

𝑐𝑡 ∙ 𝑝𝑡 + 𝑐𝑙 ∙ 𝑝𝑙 + 𝑐𝑘 ∙ 𝑝𝑘 = 𝑝𝑐

where at∙pt represents the value of T used in the production of one unit of A and al∙pl represents

the value of L used in the production of one unit of A and so on. Thus, the left-hand side of the

above system of equations shows the unit cost of production while the right-hand side represents

the price of the product. Note that the price of the first product, A, is equal to 1. The reason for

this is that Walras designates product A as the numéraire good. As a result, its price must be

equal to one.

Now notice that there are n+m (i.e. 3+3 = 6) equations capturing the above two sets of

conditions (namely, that the services used in each industry are equal to the services offered, and

that the price of each product is equal to its unit cost of production). When this last set of

equations is added to the previous set of equations representing the utility maximizing offer of

services and the utility maximizing demand for products (i.e. n+m-1 or 3+2=5), then for the

system as a whole, the total number of independent equations must be 2n+2m-1 (i.e. 6+5=11).

At the same time, in this system, the total numbers of unknowns are: n (3) services, m (3)

products, n (3) prices of services, and m-1 (2) prices of goods. Note that there are m-1 (2) prices

of goods because all goods (including services) are being priced in terms of the numéraire, A.

When the unknowns are added up, we have a total of 2n+2m-1 (i.e. 6+5=11) unknowns.

Since the total number of independent equations just matches the total number or

unknowns, it must be the case, so Walras reasoned, that a general equilibrium exists. He saw this

system as proving the proposition that utility maximizing behavior, in the context of extremely

competitive markets, will bring about a state of affairs in which the following three things will be

occurring simultaneously:

1. Every individual is maximizing utility

2. The price of every good is equal to its unit cost

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3. Demand is equal to supply in every market.

Interpretation

Walras believed that the above mathematical model was sufficient to prove that a GCE

exists at least as a theoretical possibility. He believed that competitive market systems are

forever moving toward such an equilibrium and are always operating fairly close to such a state

of affairs. Yet, even though he did his best to demonstrate that a system of markets will move

toward that outcome, his demonstration fell far short of the mark and, to this day, there does not

exist a satisfactory demonstration of a competitive market system’s ability to move toward a

GCE. So, it’s more appropriate to think of the above model as a characterization of what Walras

believed would be occurring in a system of markets that’s already in equilibrium, without going

through the contortions of trying to demonstrate (since it can’t be done anyway) how market

systems converge on a GCE.

So, the above model depicts an economy in a state of equilibrium where the volume and

composition of the produced goods (the A, B, C’s), as well as the volume and composition of the

productive services (the T, L, Ks), are forever being demanded and offered at the same rates.

Every year, month, or day, the exact same amount and composition of goods is being produced.

And, at the same time, the exact same amount and composition of productive services are being

used up by the industries. Productive services are offered at the rate that’s just enough to

generate the produced goods which in turn are produced at a rate that’s just enough to maximize

the utility of each and every individual. And finally, all of the produced goods are selling at

prices that equal their unit cost of production.

If we lived in such a world, every day would be exactly like the day before and the day

that’s yet to come. We would be offering the same amount of services and purchasing the same

amount of produced goods each and every day. And each of the produced goods would be

produced using the same technology and resources unique to each good. There would be no

haggling over price since the prices remain the same each and every day.

It’s important to keep in mind that the above equilibrium is presumed to be the outcome

of a system of competitive exchanges for a given volume and composition of resources. That is,

Walras’s GCE depicts an equilibrium that’s consistent with the existing set of resources. In that

sense, and assuming his conception of a GCE is correct, the equilibrium is unique to the existing

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configuration of resources. It is not the kind of equilibrium which the Classical political

economists had in mind. The latter imagined that a growing economy could be depicted as being

in an equilibrium where there’s an ongoing circulation of commodities, with the output of

today’s production serving as the input for tomorrow’s production. So, instead of there being a

fixed amount of resources, as Walras imagines, there’s instead a changing (and generally

growing) volume of resources. The Classical’s conception of an equilibrium took for granted a

growing economy. But Walras’s conception of an equilibrium has nothing to do with economic

growth, it’s instead a depiction of the equilibrium that would be achieved with the existing set of

resources. If there were to be an increase in resources between today and tomorrow, then today’s

equilibrium would differ from tomorrow’s equilibrium. It’s an equilibrium for a moment in time,

assuming we could hold time fixed while everyone figured out their utility maximizing offers

and demands, and industries behaved with extreme efficiency. It’s for this reason that the most

charitable way of interpreting this equilibrium is that imagines an economy in a stationary state.

Another feature of this model that must be kept in mind is that Walras imagines that

resource owners are offering not their resources but rather the services of those resources. So, for

example, those who only have their labor to offer don’t offer themselves, their bodies, they

instead offer the use of their laboring activity per time period. Likewise, those who own capital

goods, don’t sell the capital goods they own, they instead offer the services of those capital

goods per time period. Similarly, those who own land don’t offer the land as such, they instead

offer the use of a specific amount of land per time period. What this means is that there’s a fixed

amount of land, labor, and capital, but the services of those resources can be used for a variety of

different purposes. What’s more, those who own capital goods aren’t depicted as capitalists in

the way the Classicals thought of these agents. That is, the capitalist, in Walras’s scheme of

things, is nothing more than an owner of capital goods who generates a revenue from selling the

services of the capital goods she/he owns to entrepreneurs.

Walras starts the tradition of thinking of the person that organizes production and sales as

an entrepreneur. The entrepreneur is an individual that takes advantage of market conditions to

purchase resources (or the services of those resources) to produce and sell a commodity at a price

that exceeds its unit cost. The entrepreneur, in short, lives in the context of disequilibrium. But,

once the economy attains a GCE, there is no need for an entrepreneur. And, indeed, the income

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of the entrepreneur (profits) ceases to exist in a GCE, since all goods are now selling at their unit

cost.

This brings up another closely related point. In the system developed by the Classical

political economists the growth equilibrium they imagined was brought about as a result of

capitalists figuring out where to invest their capital. This would have the effect of bringing about

a common rate of profit and a set of natural or normal prices (long run equilibrium prices) that

reflected the going rate of profit. But in Walras’ system, the rate of profit vanishes, since there is

no economic profit. The owners of the capital goods earn, at best an amount that covers the

depreciation and interest of having their capital goods employed in some industry.

The Limits of Walrasian Theory

In a state of general equilibrium, the system is said to be efficient in the sense that there’s

no way that more of any one good could be produced without having to give up on the

production of something else. But it’s also efficient in that there’s no way in which any one

individual can experience a higher level of utility without having to reduce the utility of at least

one other individual in society. This idea has come to be known as a Pareto Optimum, in honor

of Vilfredo Pareto who extended the Walrasian notion of a general equilibrium to the broader

notion of efficiency. While Walras tended to view efficiency as the position toward which

competitive markets are forever moving, Pareto – while not denying this possibility – viewed the

conditions of a GCE as a state of efficiency that can exist independently of markets.

The way in which this idea is usually introduced to beginning econ students is with the

notion of a production possibilities frontier. Without any sense of irony or suggesting that what’s

about to be discussed never happens in the real world, the instructor will introduce a production

possibilities curve as a way of claiming that market economies operate so efficiently that there’s

no way that more of any one good can be produced without giving up on some other good. And,

of course, in a state of Pareto Optimality (i.e. in a state of perfect efficiency) that’s indeed true.

But, Walras and Pareto never claimed that real economies are always in a state of efficiency,

operating on the frontier. Instead, they viewed real capitalist economies as being in a perpetual

process of moving toward the frontier, operating at any moment in time in various degrees away

from it. The General Equilibrium was viewed as the position toward which the economy would

move if all else could somehow be held constant. As a result, in the real world, economic

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systems are generally able to produce a little bit more of a lot of different things without having

to give up on the production of other things; that is, the economy is not operating on the frontier,

it’s instead operating within the frontier. So, from Walras perspective the real world was messy

and subject to inefficiencies but not so much that the theory of general equilibrium invalided the

direction toward which the system is forever moving toward. He was convinced that a

competitive market system was always within a stone’s throw of Pareto Optimality.

It’s important to note, however, that Leon Walras did not interpret this theory to mean

that all of economic society should be organized along the lines of laissez faire. He was

convinced that his theory proved that markets bring about a state of efficiency but only in those

areas of economic activity where goods are private (as opposed to public goods) and markets are

truly competitive (what we’d now-a-days describe as purely or perfectly competitive). In those

sectors of the economy where these conditions apply, it makes sense to organize economic

activity on the basis of economic freedom. But in those sectors of the economy were there are

public goods (or where there should be) then the government must be used to offer them to

society. In other words, he did not see his theory as invalidating the need for a wide arena of

public services, goods that would normally not be provided in sufficient amounts by the private

market system.

At the same time, he was quite comfortable with the idea that market structures which are

not competitive, such as natural monopolies, should be regulated or nationalized to ensure that

their outcomes mimic the efficiencies that competitive markets are presumed to bring about. In

short, he did not think that his theory meant that all markets should be unregulated. If there are

markets that are monopolies or oligopolies, then they should be controlled by government so as

to ensure outcomes that are consistent with the welfare of society as a whole.

Lastly, Walras was aware of the fact that his theory said absolutely nothing about the

fairness of the allocation of goods brought about by competitive market systems. That is, while

his theory may “prove” that competitive markets are capable of moving the system toward a

Pareto Optimum, there’s nothing in that outcome which claims that the goods and services will

be equitably distributed. A Pareto Optimum can emerge in a context where 1% owns 99% of all

the resources and the remaining 99% must fight over the remaining 1% of all resources. In short,

extreme inequality and poverty is consistent with a Pareto Optimum. As a result, Walras had no

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problem with suggesting that government should be used to redistribute wealth and/or income so

as to bring about the equity that competitive markets may not.

The History of General Equilibrium Theory since Walras.

Despite the advances that Walras made toward our understanding of the meaning of

market equilibrium and efficiency he did not, in fact, prove the existence of a general

competitive equilibrium. The proof of the existence of such a state of affairs has eluded

economists to this very day. It’s true that a “proof” was eventually developed (by Kenneth Arrow

and Gerard Debreu) and is used to this very day to claim that market systems are forever moving

toward a Pareto Optimum, but no one would claim that such proofs can be used to say anything

concrete about the real world. The contemporary “proofs” of the existence of a general

equilibrium are too abstract (unrealistic) to make such a claim. For example, among the various

conditions that must exist for contemporary mathematical proofs of the existence of a general

equilibrium to hold are: instantaneous production, all consumers have perfect knowledge of all

goods (even goods yet to be produced), no economies of scale, a futures market for every good

in the system, and all consumers must own a little bit of everything. Clearly such proofs say very

little about real economies that do not abide by these conditions. What’s more, and more

importantly, there are well-established alternative theories (namely those of Keynes and the Post

Keynesians) which argue that a competitive market system may indeed move toward an

equilibrium, but one in which there’s an excess supply of labor, i.e. unemployment. And this

latter set of theories fit more closely the facts of the real world than Walras’s GCE theory.