Transcript Chapter Eight

Endowments
1
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?

2
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?

3
Endowments
The list of resource units with which a
consumer starts is his endowment.
 A consumer’s endowment will be
denoted by the vector ω (omega).
ω1 = endowment in good 1
ω2 = endowment in good 2

4
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 p1=2 and p2=3
What is the endowment’s value?
Endowment value is
p1w1 + p2 w2 =
This value can be exchanged for any
consumption bundle costing no more than the
endowment’s value.
5
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

6
Budget Constraints Revisited

So, given p1 and p2, the budget constraint for a
consumer with an endowment (ω1, ω2 ) is
p1x1 + p2x2 = p1w1 + p2w2

The budget set is
{(x1,x2) | p1x1 + p2x2 ≤ p1w1 + p2w2}
where x1 ≥ 0 and x2 ≥ 0
7
Budget Constraints Revisited
x2
p1ω1 + p2ω2
p2
Budget constraint
p1x1 + p2x2 = p1w1 + p2w2
slope =
w2
-
p1
p2
Budget set
{(x1,x2)|p1x1+ p2x2 ≤ p1w1+ p2w2}
w1
x1
8
Prices change from p1 to p′
1 and p2 to p′
2
x2
p1x1 + p2x2 = p1w1 + p2w2
p′
1ω1 + p′
2 ω2
p′
2
w2
New slope =
- pp
1
2
New constraint
p′
1x1 + p′
2 x 2 = p′
1ω1 + p′
2 ω2
w1
New budget set
x1
9
Prices change from p1 to p′
1 and p2 to p′
2
The endowment point is always on the
budget constraint.
 So price changes pivot the constraint
about the endowment point.

10
Net Demands

Definition: net demand is x – w
for example
If x1 – w1 > 0
the consumer is a buyer of good 1
If x1 – w1 < 0
the consumer is a seller of good 1
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Net Demands

The constraint
p1x1 + p2x2 = p1w1 + p2w2

can be written as
p1(x1 – w1) + p2(x2 – w2) = 0

That is, the sum of the values of a
consumer’s net demands is zero.
12
Net Demands

Suppose (w 1 , w 2 ) = (10, 2) and p1=2, p2=3.
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.
13
Net Demands
x2
p1(x1 – w1) + p2(x2 – w2) = 0
At prices (p1,p2), the consumer
x 2*
w2
x1* w1
x1
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Net Demands
x2
At prices (p1',p2'), the consumer
w2
p1x1  p2x2 = p1w1  p2w2
x 2*
w1
x 1*
x1
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Net Demands
x2
p1(x1 – w1) + p2(x2 – w2) = 0
At prices (p1'' p2''), the consumer
x2*=w2
p1x1  p2 x2 = p1w1  p2w2
x1*=w1
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
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Effect of a Price Decrease for a Seller
x2
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
x 2'
w2
U
U'
x1 x1' w1
x1
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Effect of a Price Decrease for a Buyer
At prices (p1,p2), the consumer
is a seller of good 1.
The consumer MUST
remain a buyer of good 1,
He is
U' U
x2
w2
x2
x 2'
U'
w1 x1
x 1'
U
x1
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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
20
Net Demands and Price-Offer Curve
x2
p1(x1 – w1) + p2(x2 – w2) = 0
Price-offer curve
good 1,
good 2
w2
w1
x1
21
Net Demands and Price-Offer Curve
x2
p1(x1 – w1) + p2(x2 – w2) = 0
Price-offer curve
good 1,
good 2
w2
w1
x1
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Net Demands and Price-Offer Curve
x2
Price-offer curve contains all the
utility-maximizing gross demands
for which the endowment can be
exchanged.
w2
w1
x1
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Slutsky’s 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 Slutsky’s equation?
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Slutsky’s Equation Revisited
A change in p1 or p2 changes
y = p1w1 + p2w2, so there will be
an additional income effect, called the
endowment income effect.
 Slutsky’s decomposition will thus have
three components

a
pure substitution effect
 an (ordinary) income effect, and
 an endowment income effect.
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Slutsky’s Equation Revisited

Slutsky’s equation is now
Total effect = Substitution Effect
w
+ Ordinary Income Effect
+ Endowment Income Effect
i

Suppose pi changes by ∆pi
The change in money income
∆m = ∆pi wi or ∆m = wi
∆pi
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Slutsky’s Equation Revisited

We can write Slutsky’s Identity as
∆xi
∆pi
∆xis
∆pi
∆xim xi(pi, m)
∆m
∆xim ∆m
∆m ∆pi
Endowment income effect ∆xim ∆m = ∆xim ωi
∆m ∆pi ∆m

Alternatively Slutsky’s equation is
∆xi
∆xis (ωi – x i) ∆xim
+
∆pi
∆pi
∆m
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Slutsky’s Equation Revisited
x2
Initial prices are (p1, p2)
Final prices are (p1', p2')
How is the change in demand
from (x1, x2) to (x1', x2') explained?
x2
w2
x 2'
x1
w1
x 1'
x1
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Slutsky’s Equation Revisited
x2

Pure substitution effect
w2
w1
x1
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Slutsky’s Equation Revisited
x2


Pure substitution effect
Ordinary income effect
w2
w1
x1
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Slutsky’s Equation Revisited
x2



Pure substitution effect
Ordinary income effect
Endowment income effect
w2
w1
x1
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Slutsky’s Equation Revisited
Overall change in demand caused by a
change in price is the sum of:
(i) A pure substitution effect
Change in demand at constant real income
(ii) 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
32
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 good’s price is pc.
 w is the wage rate.

33
Labor Supply
The worker’s budget constraint is

pcC = w (R  R )  m
where C, R denote gross demands for the
consumption good and for leisure. That is

expenditure
{
pcC  wR = wR  m
{

endowment
value
34
Labor Supply

pcC = w (R  R )  m
rearranges to

w
m  wR
C=  R
.
pc
pc
35
Labor Supply
C

m  wR
pc

w
m  wR
C=  R
pc
pc
w
slope = 
, the ‘real wage rate’
pc
endowment
m

R
R
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Labor Supply

w
m  wR
C=  R
pc
pc
C
C*
endowment
m

R*
leisure
demanded
R
labor
supplied
R
37