Lecture Notes, January 21, 2010, and following
Convexity
A set of points S in
is said to be convex if the line segment between any two points
R
N
of the set is completely included in the set.
S is convex
if
x, y
S, implies
.
z
z
x
1
y
,0
1
S
S is said to be strictly convex
if x,
, implies
y
S
,
x
y
1
.
x
1
y
interior
S
The notion of convexity is that a set is convex if it is connected, has no holes on the inside and
no indentations on the boundary.
A set is strictly convex if it is convex and has a continuous
strict curvature (no flat segments) on the boundary.
Economically, this notion corresponds to "diminishing marginal utility"
"diminishing marginal
rate of substitution" "diminishing marginal product" .
Properties of Convex Sets
Let
be convex subsets of R
N
.
Then:
C
1
,
C
2
is convex,
C
1
C
2
is convex,
C
1
C
2
is convex
C
1
The Market, Commodities and Prices
N commodities
x = (x
1
, x
2
, x
3
, .
.., x
N
)
R
N
,
a commodity bundle
The market takes place at a single instant, prior to the rest of economic activity.
commodity
= good or service completely specified
description
location
date (of delivery)
Price system
:
.
p
i
0 for all i = 1, .
.., N.
p
p
1
,
p
2
,...,
p
N
0
Value of a bundle x
R
N
at prices p is p
x.