Léon Walras
• The son of August Walras, himself an
accomplished economist who was familiar with the writings of Cournot. The younger Walras learned his basic
economic theory from Cournot’s books.
• A mediocre mathematician, Walras
nevertheless fervently believed in mathematical economics. His contributions in the area, given his
relatively poor mathematical abilities, are astounding.
• Walras did much to professionalize
economics by corresponding in five languages to economists throughout
Europe. Mathematics itself helps
overcome the language barrier.
• Walras was dedicated to the method of general equilibrium analysis. He argued that partial equilibrium analysis
was invalid, because it ignored all the effects on other markets of a change in
demand or supply conditions in one market.
Changes in other markets would create feedback effects in the initially
altered market. Everything affects everything else.
• We have seen that Marshall regarded demand
price and supply price as functions of quantities: Dpx
= f(qx) and
Spx = g(qx). Walras viewed the problem the other way
around: quantities demanded and
supplied are functions of price. I.e.,
Qdx = f(px)
and Qsx
= g(px). That is, Marshall regarded quantity as the
independent variable and price the dependent variable, while Walras regarded
their roles as reversed.
• This difference of viewpoint shows up in
their treatments of how markets move to equilibrium. For example, in Marshall’s system, and assuming ordinarily sloped
demand and supply curves, if the quantity traded in the market fell short of
the equilibrium quantity, the demand price for the good would exceed the supply
price. Producers would earn above-normal
profits, inducing them to expand production.
The quantity supplied (and traded) would move toward the equilibrium
level.
• Walras approached the same conditions
differently. If quantity supplied were
less than quantity demanded, and excess demand for the good would exist. The excess demand (qd
- qs) would drive the
market price up, inducing a larger quantity supplied and a smaller quantity
demanded. In the case of ordinary
demand and supply curves, the market equilibrium is stable in either case: a displacement
from equilibrium sets in motion forces that return quantity and price to their
equilibrium levels.
• In the case of negatively sloped supply curves, however, the two approaches yield
different stability results. Suppose
the supply curve cuts the demand curve from above. In this case, at q < qe
the supply price exceeds the demand price.
Firms are taking losses, according to Marshall, and will reduce
production. Output will decline, moving
away from its equilibrium level.
According to Walras, the equilibrium is stable. Quantity supplied exceeds quantity
demanded. Price will fall, increasing
both quantity demanded and supplied until equilibrium is reestablished.
• If a negatively sloped supply curve cuts
the demand curve from below, the Marshallian analysis would say that the
equilibrium is stable, the Walrasian would say it is unstable. (The labor market and the foreign exchange
market may have negatively sloped supply curves, at least over some range of
values.)
• This partial equilibrium analysis
illustrates the concept of market stability.
It is an important – and complex – concept when applied to Walrasian
general equilibrium analysis.
• Walras’s great vision was to develop a
mathematical model of equilibrium of the entire economy. He based his theory on what have come to be
regarded as the basic building blocks of microeconomic theory: utility, input coefficients (on the
production side), and budget constraints.
• Walras set out to answer four questions
concerning general equilibrium:
1) Is there a unique set of prices that results
in equilibrium?
2) Is the unique solution (if it exists)
economically meaningful?
3) Is equilibrium stable or unstable?
4) Is the system determinate?
• To answer these questions, Walras created a
system of mathematical demand and supply equations, based on utility
maximization by consumers and profit maximization by producers. The problem is immensely complex, and the
very fact that Walras was able to state it in a coherent manner was a
tremendous achievement. The fact that
he was unable to answer conclusively any of the four questions does not
diminish his accomplishment. But all he
was able to show is that the number of equations in a fully specified GE model
equals the number of unknowns to be solved for. This is neither a sufficient, nor even a necessary, condition for
general equilibrium to exist.
• In the 1950s, Kenneth Arrow and Gerard
Debreu showed that general equilibrium exists and is meaningful (i.e., equilibrium
prices and quantities are non-negative) if
1) returns to scale are constant or diminishing;
2) there are no joint products or external
effects either in production or in consumption; and
3) all goods are gross substitutes for each other (a rise in the price of one good
produces positive excess demand for the other).
To do this, Arrow and Debreu used
mathematical techniques that were not available to Walras.
• In the course of building his system,
Walras accomplished a number of things.
First, he demonstrated that, in general equilibrium, the marginal rates of substitution of all
goods for the numéraire must equal the prices of the goods. Otherwise, consumers could increase their
total utility by giving up goods with a low marginal rate of substitution ()
in exchange for goods with a higher marginal rate of substitution. This is similar but broader than Gossen’s
second law, in that it applies to all consumers simultaneously.
• In modern terms, the marginal rate of
substitution equals the slope of the indifference curve between two goods. Walras’s student, Vilfredo Pareto, showed the same thing on the production side,
where the marginal rate of technical
substitution is the slope of an isoquant.
• Walras also demonstrated a concept that is
widely used in modern macro theory. It
is known as Walras’ Law, and it
follows directly from the concept of a budget constraint. A budget constraint requires that the total
value of goods demanded be financed by supplying goods of equal value to the
market. Suppose only three goods
exist. Then an individual’s budget
constraint would be
paqa
+ pbqb
+ pcqc
= paq+ pbq+ pcq
where qon the
right-hand side indicates endowments of the goods that can be consumed or
supplied. The left-hand side is demand.
Rearrange by subtracting quantities
supplied from both sides.
pa(qa
- q)+ pb(qb
- q)+ pc(qc
- q)= 0
It is easy to see that you could subtract
off one excess demand term from each side of the equation, and that the
resulting equation would say that the sum of excess demands in all markets but
one equals the negative of the excess demand in the other market.
• Walras’ Law shows that markets are
interrelated. They are not independent
of one another. A change in demand or
supply in one market must affect
excess demand in at least one other market.
Also, if n - 1 markets are in
equilibrium, the nth market must also
be in equilibrium.