Net Demands
p1=2, p2=3, x1*-ω1 = -3 and x2*-ω2 = +2 so p1(x1 − ω1) + p2(x2 − ω2 ) =
2 × ( −3) +
3× 2
= 0.
The purchase of 2 extra good 2 units at $3 each is funded by giving up 3 good 1 units at $2 each.
Budget Constraints Revisited
So, given p1 and p2, the budget constraint for a consumer with an endowment (ω1,ω2 ) is
p1x1 + p2x2 = p1ω1 + p2ω2.
{(x1, x2 ) p1x1 + p2x2 ≤ p1ω1 + p2ω2,
ω
Endowments
E.g. ω = (ω1,ω2 ) = (10,2) states that the consumer is endowed with 10 units of good 1 and 2 units of good 2. What is the endowment’s value? For which consumption bundles may it be exchanged?
y = p1ω1 + p2ω2
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
ω2
ω1
x1
Some Results about Price Change
If a consumer is a net seller of a good, when the price of the good decreases, if she remains a buyer then the consumer is worse off. If a consumer if a net buyer of a good, when the price decreases, the consumer will remain a buyer and be better off.
ω1
x1
Net Demand
The constraint
p1x1 + p2x2 = p1ω1 + p2ω2
is
p1(x1 − ω1) + p2(x2 − ω2 ) = 0.
That is, the sum of the values of a 净需求） consumer’s net demands （净需求） is zero.
{
pcC + wR = wR + m
endowment value

Labor Supply
pcC = w(R − R) + m
rearranges to

w m+ wR C= − R+ . pc pc
($) C
Labor Supply
m

endowment
R
R
Labor Supply
C
w m+ wR C= − R+ pc pc

m

endowment
R
R
Labor Supply
m+ wR pc
C
w m+ wR C= − R+ pc pc

m

endowment
R
R
Labor Supply
m+ wR pc
C
w m+ wR C= − R+ pc pc w slope = − , the ‘real wage rate’ pc
Chapter Nine
Buying and Selling 购买和销售
中级微观经济学9
Structure
Endowments （禀赋） 禀赋） Budget constraints with endowments Net demands – Price offer curve Example: Labor supply – Comparative statics Slutsky equation revisited
Endowments
The list of resource units with which a consumer starts is his endowment 禀赋） （禀赋）. A consumer’s endowment will be denoted by the vector (omega).
x1 ≥ 0, x2 ≥ 0}.
The budget set is
Budget Constraints Revisited
x2
p1x1 + p2x2 = p1ω1 + p2ω2
ω2
ω1
x1
Budget Constraints Revisited
x2
p1x1 + p2x2 = p1ω1 + p2ω2
endowment


m
R
R
Labor Supply
m+ wR pc
C* C
w m+ wR C= − R+ pc pc

m

endowment
R* leisure demanded labor supplied
R
R
Labor Supply Curve
Slutsky’s Equation Revisited
p' x1 + p' x2 = p' ω1 + p' ω2 1 2 1 2
ω1
x1
Changing Prices
x2 The endowment point is always on the budget constraint.
p1x1 + p2x2 = p1ω1 + p2ω2
ω2
So price changes pivot the constraint about the endowment point. p' x1 + p' x2 = p' ω1 + p' ω2 1 2 1 2
x2
p1x1 + p2x2 = p1ω1 + p2ω2
ω2
Changing prices of the endowment
p' x1 + p' x2 = p' ω1 + p' ω2 1 2 1 2
ω1
x1
Changing Prices
x2
p1x1 + p2x2 = p1ω1 + p2ω2
ω2
Budget set
x 2*Biblioteka ω2ω1x 1*
x1
Net buyer of good 1
Net Demands at Different Prices
x2
p1(x1 − ω1) + p2(x2 − ω2 ) = 0
At prices (p1”,p2”) the consumer consumes her endowment; net demands are all zero.
ω2
x1* ω1 Net seller of good 1
x1
Net Demands at Different Prices
x2 At prices (p1’,p2’) the consumer sells units of good 2 to acquire more of good 1.
' ' ' ' p1x1 + p2x2 = p1ω1 + p2ω2
x2*=ω2
p"x1 + p"x2 = p"ω1 + p"ω2 1 2 1 2
x1*=ω1 Autarky x1
Offer Curve
x2
p1(x1 − ω1) + p2(x2 − ω2 ) = 0
Price-offer curve contains all the utility-maximizing gross demands for which the endowment can be exchanged.
ω2
Budget set {(x1, x2 ) p1x1 + p2x2 ≤ p1ω1 + p2ω2,
x1 ≥ 0, x2 ≥ 0}
x1
ω1
Changing Endowments
x2 Changing the endowment
(ω1,ω2 )
(ω1 ’,ω2 ’)
(ω1 ”,ω2 ”)
x1
Changing Prices
Net Demands Suppose (ω1,ω2 ) = (10,2)
p1x1 + p2x2 = p1ω1 + p2ω2 = 26.
and p1=2, p2=3. Then the constraint is If the consumer gross demands (毛需 毛需 求）(x1*,x2*) = (7,4), then 3 good 1 units exchange for 2 good 2 units. Net demands are x1*- ω1 = 7-10 = -3 and x2*- ω2 = 4 - 2 = +2.