dynamics--cross-posted from PKT
Steve.Keen at unsw.edu.au
Steve.Keen at unsw.edu.au
Sun Apr 2 14:58:02 MDT 1995
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Subj: RE: Debreu-Sonnenschein-Mantel
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Date: Sat, 01 Apr 1995 10:49:55 -0700
From: GONZALO FONSECA <FONSECA at newschool.edu>
Subject: Re: Debreu-Sonnenschein-Mantel
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X-Comment: POST-KEYNESIAN THOUGHT
In response to several private requests for the meaning and
implications of the Debreu-Sonnenschein-Mantel Theorem, I have
decided to forward to the list a pretty simple numerical example (from
Kreps) which illustrates essentially what this is all about.
We all know that the WARP essentially preserves (most) of the
axioms of standard Walrasian systems as well as providing the Slutsky
properties which make demand curves downward-sloping. [Note: gross
substitution and WARP are similar but not exactly the same: while gross
substitution implies WARP, the reverse is not true (See Kihlstrom,
Mas-Colell and Sonnenschein (1986)]. The Debreu-Sonnenschein-Mantel
theorem shows that WARP may apply for all agents and yet not apply for
the market, i.e. that individual demand curves are downward-sloping
does not imply that market demand curves are as well.
Numerical Example: take two agents (A and B) and two goods (1
and 2). Let every agent have a budget of $1000. Suppose we are
faced with a set of prices p = ($10, $10) for the two goods. Suppose
the choices of agent A under these are (25, 75) and the choices of
agent B are (75, 25). Obviously, both agents operate within their
budgets (within $1000 each).
Now, suppose prices change to q = ($15, $5). Now, at prices q,
obviously A could choose his old bundle (25, 75) which would imply an
expenditure of only $750. So let A now choose bundle (40, 80).
Obviously, at the original prices, p, this second bundle would have cost
him $1200 and thus was unfeasible. Thus, A has revealed preferred the
second bundle (40, 80) to the first bundle (25, 75). He takes the second
when the first is affordable (at prices q) but when he chose the first, the
second was not affordable (at prices p). Thus, agent A fulfills WARP.
Now, suppose, at prices q, that agent B chooses (64, 8). He could
have afforded this at the original prices (p) since that would have cost
him only $720. But he can no longer afford his first bundle (75, 25) at the
new prices, q, since that would cost him $1250. Therefore, for B, the
first bundle (75, 25) is revealed preferred to the second bundle (64, 8).
Again, it is easy to see that agent B also obeys WARP.
Now, with both agents obeying WARP, does the "market" (or the
"representative agent", RA) also obey WARP? No. The RA has a budget
of $1000 + $1000 = $2000. Under prices p, he chooses (25 + 75, 75 +
25) = (100, 100). Under prices q, he chooses (40 + 64, 80 + 8) = (104,
88). Now, under p = ($10, $10), the cost of the first bundle is $2000
whereas that of the second is $1920. So, the RA takes the first bundle
when the second is affordable (cheaper even). But under q = ($15,
$5), the cost of the first bundle is $2000 and the cost of the second
bundle is $2000. Thus, the RA chooses the second bundle when the
first is affordable. WARP is violated: he chose each bundle when the
other was affordable.
Conclusion: WARP applied for both our individuals. It did not apply for the
RA. This illustrates, then, how demand curves may be
downward-sloping for all individuals but indeterminate at the market (RA)
level.
Implications: DSM claim that the only properties of individual demand
curves which carry over to the market demand curve are three essential
ones: homogeneity, Walras' Law and continuity. Note that this is
sufficient for the proof of the existence of a Walrasian general
equilibrium. Thus, the ADM structure is not affected. Also essentially
unaffected are the local uniqueness properties of the equilibrium set.
What is affected, however, are all the dynamic properties of the
system (and the uniqueness of a single equilibrium). Some authors
(such as Hildenbrand and Grandmont) have accepted this impasse and
moved onto explaining market demand curves from other criteria (e.g.
dispersion and diversity of households) which are not based on
Neo-Walrasian logic but rather on items outside the pure Neoclassical
structure. On this, see, for instance, Hildenbrand's recent book, "Market
Demand" (1994).
Still, much of this is still hand-waving. After all, the "idea" of basing
Neoclassical economics on utility-maximization models rests, essentially,
on the feeling of many Neoclassicals that the simultaneous optimization
of the competing plans of households not only CAN lead to an equilibrium
but also WILL lead to an equilibrium. If, from the utility-maximization
models, all they can say is "CAN", then not much of an argument can be
made for basing demand theory upon it.
[An aside: WARP's creator, Abraham Wald, in his first 1936 paper, did
contend with the issue of whether the WARP could be extended from
individual to market - he concluded simply that "there is a statistical
probability that, from the assumption that [WARP] holds for every
[household], the validity of [WARP for the market] follows" (Wald, 1936,
in Baumol and Goldfeld, 1968). Wald offers no proof. As far as I know,
this issue was ignored until Sonnenschein. Did anyone else ask?]
Gonzalo Fonseca
New School for Social Research
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