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Definitions of Economics Terms
These are taken from the book [#!MWG95!#].
- Preference Relations:
- A relation that captures the decision makers taste! Its a usual
binary relation, denoted by
(greater than equal to) on the
set X, where X denotes the set of alternatives. A rational
preference relation is the one which is complete and
transitive. By complete we mean that any two elements in the set
X are comparable.
- Utility Functions:
- A utility function assigns a numerical value to each element of
the set X (i.e. the set of choices/alternatives). If ,
then
,
where
.
Here it is assumed
that there is a rational preference relation on the set X, and
the utility function preserves that preference relation.
- Commodities:
- The objects of choice by consumers in a market economy.
Commodities are finite in number. The commodity vector represents
amounts of different commodities, usually referred to as a
consumption bundle/vector. Let us assume that there are in all
L different commodities. Consumption set is a subset of the
commodity space RL, where each element of the subspace is a
consumption bundle.
- Who is Walras?
- Here is a great web-site about Who is
Walras
and on his Contributions to
Economics.
- Walrasian Set/ Competitive Budget Set:
- Assume that each commodity has a given price (per unit). The
consumption bundle
is affordable for a consumer if its
total cost does not exceed the wealth of the consumer w (i.e.,
). The Walrasian set or the competitive budget set
refers to the set of affordable consumption bundles (i.e.,
).
- Walrasian demand function:
- For each price wealth pair, (p,w), x(p,w) refers to the set of
consumption bundles satisfying (p,w). In general x(p,w) is a
relation, but in the case it is single valued, it is called a
demand function, otherwise demand correspondence. Walrasian
demand correspondence is homogeneous of degree zero if for any
constant
r >0, x(rp, rw)=x(p,w) (i.e. scaling of price and
wealth by the same proportion does not alter the choices for the
consumers).
- Walras's Law:
- Walrasian demand correspondence satisfies Walras's Law if for
every p and w >0, p.x = w, for all
.
Essentially it says that consumers fully expends their wealth!
- Monotone preference relation:
- A preference relation on X is monotone, if ,
and ,
then
(consumption of larger amounts of goods is
always feasible).
- Non-satiated preference relation:
- For every ,
and every
,
there is a ,
such that
and .
(Always in the
neighborhood of x, we can find another consumption bundle y,
that is preferred over x, and y is not same as x.)
- Convex preference relation:
- A preference relation is convex, if for every ,
the set
is convex (upper contour set is convex).
- Continuous preference relation:
- It is continuous if it is preserved under limits. For any sequence
of pairs
with
,
and
and
,
when
,
then .
Alternatively, the upper contour set
,
and the lower contour set
are
closed. Consumer preferences do not exhibit jumps!! It turns out
that for every continuous preference relation, one can find a
continuous utility function!
- Utility maximization problem:
- Assume that consumers have rational, continuous, and locally
non-satiated preference relation over the set X=R+L and the
corresponding continuous utility function u(x). Then the
maximization problem is to max u(x), subject to
and
.
Why does utility maximization problem has a solution?
Note that continuous functions always have an optimum value on a
compact set. Utility function is continuous. The set of feasible
solutions, namely the budget set,
is
compact, since it is bounded and closed.
Next: Competitive Equilibrium
Up: Models, Algorithms and Economics
Previous: Brouwer's Fix Point Theorem
Anil Maheshwari
2003-03-05