L08
Buying and Selling
Review


Model of choice
We know preferences U  x1 x2 and
p1 , p2 , m
we find demands
*
1
x ,x



*
2
Q: Where does the mysterious income m
come from?
From selling goods (e.g. labor)!
Today: Model of choice with endowments
Endowments
 Instead
of nominal income: goods
 The list of commodities with which a
consumer starts is his endowment.
 A consumer’s endowment will be
denoted by the vector (omega).
  1 ,  2 
 Example
  3,6 
Budget constraint
 Suppose
p1=2 and p2=3 and
(1 ,  2 )  (10,2)
what is the value of endowment?
m
 What
is a collection of all affordable
bundles (budget set)?
Budget Constraints Revisited
 Given
p1 and p2, the budget constraint
for a consumer with an endowment
  (1 ,  2 )
is
p1 x1  p2 x2  m  p11  p2 2 .
 Example:
p1  1, p2  1
Budget Constraints Revisited
  (5,5)
p1  12
3,, pp22 11
x2
x1
More generally
x2
  1 ,  2 
p1 ' p1

1
p 2 ' p2
2
1
x1
Net Demands
 Net
demands: actual trades of a
consumer
x  1
*
1
x  2
*
2
 Example
x*  (10,4)   (1,10)
 Net
demands (buying, selling)?
Budget Constraints Revisited
 The
is
 The
constraint
p1 x1  p2 x2  p11  p2 2
p1 ( x1  1 )  p2 ( x2   2 )  0.
sum of the values of a
consumer’s net demands is zero.
 Buying, selling?
Buying, Selling?
  (5,5)
x2
p1  1, p2  1
x1  x2  10.
2=5
1=5
x1
Optimal Choice
 Almost
the same as before
 We only need to find m first
m  p11  p2 2
 When
are we net buyers of good 1?
 We first answer it graphically
 Price offer curve   (5,5) p 2  1
p1  2, p1  1, p1  1 / 2
Magic Formulas
 Cobb
Douglass
 Perfect
Complements
x2
Optimal Choice
  (5,5)
p1  2, p2  1
(5,5)
|MRS( )| =1
p1 / p 2  2
x1
Optimal Choice
x2
  (5,5)
(5,5)
p1  1 / 2, p2  1
|MRS( )| =1
p1 / p2  1 / 2
x1
Optimal Choice
x2
  (5,5)
(5,5)
p1  1, p2  1
MRS(  ) =1
p1 / p2  1
x1
Gains-to-Trade
 Consumer
 In
engages in trade if:
particular:
Price Offer Curve
x2
  (5,5)
p1  1, p1  2, p1  1 / 2
(5,5)
x1
Price offer curve
 With
initial endowments price offer
curve is ``enveloped’’ by the
indifference curve that passes
though endowment
 Intuition:
Agents engage in trade
only trade gives higher utility
Cobb-Douglass
U ( x1 , x2 )  x1 x2
p1 m
2
1
0.5
x1* , x2* nd
p2  1
  (5,5)