Endowments

1
Endowments
2
Buying and Selling

• Trade involves exchange -- when something is
  bought something else must be sold.
• What will be bought? What will be sold?
• Who will be a buyer? Who will be a seller?

3
Buying and Selling

• And how are incomes generated?
• How does the value of income depend upon
  commodity prices?
• How can we put all this together to explain
  better how price changes affect demands?

4
Endowments

• The list of resource units with which a consumer
  starts is his endowment.
• A consumers endowment will be denoted by the
  vector (omega).
• endowment in good 1
• endowment in good 2

5
Endowments

• Example
• Let w (w1, w2) (10, 2)
• This states that the consumer is endowed with 10
  units of good 1 and 2 units of good 2.
• If p12 and p23
• What is the endowments value?
• Endowment value is
• p1w1 p2 w2
• This value can be exchanged for any consumption
  bundle costing no more than the endowments value.

6
Consumption bundles

• The amount of goods that a consumer can choose to
  consume
• A consumer consumption of good will be denoted by
  x
• x1 consumption of good 1
• x2 consumption of good 2

7
Budget Constraints Revisited

• So, given p1 and p2, the budget constraint for a
  consumer with an endowment
  is
• The budget set is
• where x1 0 and x2 0

p1x1 p2x2 p1w1 p2w2
(x1,x2) p1x1 p2x2 p1w1 p2w2
8
Budget Constraints Revisited
x2
Budget constraint
p1x1 p2x2 p1w1 p2w2
w2
Budget set
(x1,x2)p1x1 p2x2 p1w1 p2w2
w1
x1
9
Prices change from to and to
x2
p1x1 p2x2 p1w1 p2w2
w2
New constraint
w1
x1
New budget set
10
Prices change from to and to

• The endowment point is always on the budget
  constraint.
• So price changes pivot the constraint about the
  endowment point.

11
Net Demands

• Definition net demand is x w
• for example
• If x1 w1 gt 0
• the consumer is a buyer of good 1
• If x1 w1 lt 0
• the consumer is a seller of good 1

12
Net Demands

• The constraint p1x1 p2x2 p1w1 p2w2
• can be written asp1(x1 w1) p2(x2 w2) 0
• That is, the sum of the values of a consumers
  net demands is zero.

13
Net Demands

• Suppose and
  p12, p23. Then the constraint is
• p1x1 p2x2 p1w1 p2w2 26
• If the consumer demands (x1,x2) (7,4),
  Net demands are x1- w1 7-10 -3 and
  x2- w2 4 - 2 2.
• p1(x1 w1) p2(x2 w2)
• The purchase of 2 extra good 2 units at 3 each
  is funded by giving up 3 good 1 units at 2
  each.

14
Net Demands
x2
p1(x1 w1) p2(x2 w2) 0
At prices (p1,p2), the consumer
x2
w2
x1
w1
x1
15
Net Demands
x2
At prices (p1',p2'), the consumer
w2
x2
w1
x1
x1
16
Net Demands
p1(x1 w1) p2(x2 w2) 0
x2
At prices (p1'' p2''), the consumer
x2w2
x1w1
x1
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Effect of a Price Decrease

• From Revealed Preference
• A seller of good i who remains a seller of i
  after price of i has decreased must be worse off
• A buyer of good i must remain a buyer of i after
  price of i has decreased

18
Effect of a Price Decrease for a Seller
At prices (p1,p2), the consumer is a seller of
good 1.
x2
If after p1 decreases, the consumer remains the
seller of good 1, he must be U' U
x2
x2'
U
w2
U'
x1'
x1
w1
x1
19
Effect of a Price Decrease for a Buyer
At prices (p1,p2), the consumer is a seller of
good 1.
x2
The consumer MUST remain a buyer of good 1, He
is U' U
w2
x2
x2'
U'
U
w1
x1
x1'
x1
20
Net Demands and Price-Offer Curve

• Price-offer curve represents bundles of goods
  that would be demanded at different prices.
• It contains all the utility-maximizing gross
  demands for which the endowment can be exchanged
  such that the budget constraint is not violated
  i.e.

p1(x1 w1) p2(x2 w2) 0
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Net Demands and Price-Offer Curve
p1(x1 w1) p2(x2 w2) 0
x2
Price-offer curve
good 1, good 2
w2
w1
x1
22
Net Demands and Price-Offer Curve
p1(x1 w1) p2(x2 w2) 0
x2
Price-offer curve
good 1, good 2
w2
w1
x1
23
Net Demands and Price-Offer Curve
Price-offer curve contains all theutility-maximiz
ing gross demands for which the endowment can be
exchanged.
24
Slutskys Equation Revisited

• Slutsky changes to demands caused by a price
  change are the sum of
• a pure substitution effect, and
• an income effect.
• This assumed that income y did not change as
  prices changed. But y p1w1 p2w2 does
  change with price. How does this modify
  Slutskys equation?

25
Slutskys Equation Revisited

• A change in p1 or p2 changesy p1w1 p2w2, so
  there will bean additional income effect, called
  the endowment income effect.
• Slutskys decomposition will thus have three
  components
• a pure substitution effect
• an (ordinary) income effect, and
• an endowment income effect.

26
Slutskys Equation Revisited

• Slutskys equation is now
• Total effect Substitution Effect
• Ordinary Income Effect
• Endowment Income Effect
• Suppose pi changes by ?pi
• The change in money income
• ?m ?pi or ?m

?pi
27
Slutskys Equation Revisited

• We can write Slutskys Identity as
• ?xi ?xis ?xim xi(pi, m)
  ?xim ?m
• ?pi ?pi ?m ?m ?pi
• Endowment income effect ?xim ?m ?xim ?i
• ?m ?pi ?m

• Alternatively Slutskys equation is
• ?xi ?xis (?i x i) ?xim
  ?pi ?pi ?m

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Slutskys Equation Revisited
x2
Initial prices are (p1, p2)
x2
w2
x1
w1
x1
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Slutskys Equation Revisited
x2
Þ
Pure substitution effect
w2
w1
x1
30
Slutskys Equation Revisited
x2
Þ
Pure substitution effect
w2
w1
x1
31
Slutskys Equation Revisited
x2
Þ
Pure substitution effect
w2
w1
x1
32
Slutskys Equation Revisited

• Overall change in demand caused by achange in
  price is the sum of
• A pure substitution effect
• Change in demand at constant real income
• An ordinary income effect
• Change in demand holding money income fixed
• (iii) An endowment income effect
• Change in demand due to a change in endowment
  value

33
Labor Supply

• A worker is endowed with m of nonlabor income
  and R hours of time which can be used for labor
  or leisure. w (R,m).
• Consumption goods price is pc.
• w is the wage rate.

¾
¾
34
Labor Supply

• The workers budget constraint iswhere C, R
  denote gross demands for the consumption good and
  for leisure. That is

¾
¾


endowment value
expenditure
35
Labor Supply
¾
rearranges to
¾
36
Labor Supply
C
¾
slope , the real wage rate
endowment
m
¾
R
R
37
Labor Supply
¾
C
C
endowment
m
¾
R
R
R
leisuredemanded
laborsupplied