Chapter Nine
Buying and Selling
购买和销售
Structure
?Endowments （ 禀赋）
?Budget constraints with endowments
?Net demands
–Price offer curve
?Example,Labor supply
–Comparative statics
?Slutsky equation revisited
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?
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?
Endowments
?The list of resource units with which
a consumer starts is his endowment
（ 禀赋）,
?A consumer’s endowment will be
denoted by the vector (omega).?
Endowments
?E.g.
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?
? ? ?? ?(,) (,)1 2 10 2
Endowments
?p1=2 and p2=3 so the value of the
endowment is
?Q,For which consumption bundles
may the endowment be exchanged?
?A,For any bundle costing no more
than the endowment’s value.
(,) (,)? ?1 2 10 2?
p p1 1 2 2 2 10 3 2 26? ?? ? ? ? ? ?
Budget Constraints Revisited
?So,given p1 and p2,the budget
constraint for a consumer with an
endowment is
?The budget set is
(,)? ?1 2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?,
?
?
(,),
,.
x x p x p x p p
x x
1 2 1 1 2 2 1 1 2 2
1 20 0
? ? ?
? ?
? ?
Budget Constraints Revisited
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
Budget Constraints Revisited
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
Budget set?
?
(,),
,
x x p x p x p p
x x
1 2 1 1 2 2 1 1 2 2
1 20 0
? ? ?
? ?
? ?
Changing Endowments
x2
x1
(?1 ’,?2 ’)
(?1,,?2,)
(?1,?2 )
Changing the
endowment
Changing Prices
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
p x p x p p1 1 2 2 1 1 2 2' ' ' '? ? ?? ?
Changing prices of the
endowment
Changing Prices
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
p x p x p p1 1 2 2 1 1 2 2' ' ' '? ? ?? ?
Budget set
Changing Prices
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
The endowment point is always on
the budget constraint.
p x p x p p1 1 2 2 1 1 2 2' ' ' '? ? ?? ?
So price changes pivot the
constraint about the
endowment point.
Net Demand
?The constraint
is
?That is,the sum of the values of a
consumer’s net demands （ 净需求）
is zero.
p x p x1 1 1 2 2 2 0( ) ( ),? ? ? ?? ?
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
Net Demands
?Suppose and
p1=2,p2=3,Then the constraint is
?If the consumer gross demands (毛需
求） (x1*,x2*) = (7,4),then 3 good 1
units exchange for 2 good 2 units,
Net demands are
x1*- ?1 = 7-10 = -3 and
x2*- ?2 = 4 - 2 = +2.
(,) (,)? ?1 2 10 2?
p x p x p p1 1 2 2 1 1 2 2 26? ? ? ?? ?,
Net Demands
p1=2,p2=3,x1*-?1 = -3 and x2*-?2 = +2 sop x p x1 1 1 2 2 2
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,
Net Demands at Different Prices
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
x2*
x1*
At prices (p1,p2) the consumer
sells units of good 1 to acquire
more units of good 2.
Net seller of good 1
x2
x1?1
?2 p x p x p p
1 1 2 2 1 1 2 2' ' ' '? ? ?? ?x2*
x1*
At prices (p1’,p2’) the consumer
sells units of good 2 to acquire
more of good 1.
Net buyer of good 1
Net Demands at Different Prices
x2
x1
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
x2*=?2
x1*=?1
At prices (p1”,p2”) the consumer
consumes her endowment; net
demands are all zero.
p x p x p p1 1 2 2 1 1 2 2" " " "? ? ?? ?
Autarky
Net Demands at Different Prices
Offer Curve
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
Price-offer curve contains all the
utility-maximizing gross demands
for which the endowment
can be exchanged.
Net Demands
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
Price-offer curve
Sell good 1,buy good 2
Net Demands
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
Price-offer curve
Buy good 1,sell good 2
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.
Labor Supply
?A worker is endowed with $m of
nonlabor income and R hours of time
which can be used for labor or
leisure,? = (R,m).
?Consumption good’s price is pc.
?w is the wage rate.
?
?
Labor Supply
?The worker’s budget constraint is
where C,R denote gross demands
for the consumption good and for
leisure,That is
p C w R R mc ? ? ?( )?
p C wR wR mc ? ? ?? ??
endowment
value
expenditure
Labor Supply
p C w R R mc ? ? ?( )?
rearranges to
C w
p
R m wR
pc c
? ? ? ?,?
Labor Supply
C
RR?
endowmentm
($)
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ??
m
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ?m wR
p c
? ??
m
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ?m wR
p c
? ??
m
slope =,the ‘real wage rate’?
w
pc
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ?m wR
p c
? ??
m
C*
R*
leisure
demanded
labor
supplied
Labor Supply Curve
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
does change with price,How does
this modify Slutsky’s equation?
y p p? ?1 1 2 2? ?
Slutsky’s Equation Revisited
?A change in p1 or p2 changes
so there will be
an additional income effect,called the
endowment income effect （ 禀赋收入效应）,
?Slutsky’s decomposition will thus have
three components
–a pure substitution effect
–an (ordinary) income effect,and
–an endowment income effect.
y p p? ?1 1 2 2? ?
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1’
Initial prices are (p1’,p2’).
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1”
x2”
Initial prices are (p1’,p2’).
Final prices are (p1”,p2”).
x1’
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1”
x2”
Initial prices are (p1’,p2’).
Final prices are (p1”,p2”).
How is the change in demand
from (x1’,x2’) to (x1”,x2”) explained?
x1’
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1’
Initial prices are (p1’,p2’).
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1”
x2”
Initial prices are (p1’,p2’).
Final prices are (p1”,p2”).
x1’
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Endowment income effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Endowment income effect?
Slutsky’s Equation Revisited
Overall change in demand caused by a
change in price is the sum of:
(i) a pure substitution effect
(ii) an ordinary income effect
(iii) an endowment income effect
Slutsky Equation with
Endowments
1 1 1 1 1 1 1 1 1
11
11
(,( ) ) ( ) (,) (,)sx p m p x p x p m x p mx
p p m m
?? ? ? ?? ? ?
? ? ? ?
1 1 1 1 1 1 1
11
11
(,( ) ) ( ) (,)()sx p m p x p x p mx
p p m
?? ? ?? ? ?
? ? ?
Or
(-) (+) (+)
Buying and Selling
购买和销售
Structure
?Endowments （ 禀赋）
?Budget constraints with endowments
?Net demands
–Price offer curve
?Example,Labor supply
–Comparative statics
?Slutsky equation revisited
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?
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?
Endowments
?The list of resource units with which
a consumer starts is his endowment
（ 禀赋）,
?A consumer’s endowment will be
denoted by the vector (omega).?
Endowments
?E.g.
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?
? ? ?? ?(,) (,)1 2 10 2
Endowments
?p1=2 and p2=3 so the value of the
endowment is
?Q,For which consumption bundles
may the endowment be exchanged?
?A,For any bundle costing no more
than the endowment’s value.
(,) (,)? ?1 2 10 2?
p p1 1 2 2 2 10 3 2 26? ?? ? ? ? ? ?
Budget Constraints Revisited
?So,given p1 and p2,the budget
constraint for a consumer with an
endowment is
?The budget set is
(,)? ?1 2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?,
?
?
(,),
,.
x x p x p x p p
x x
1 2 1 1 2 2 1 1 2 2
1 20 0
? ? ?
? ?
? ?
Budget Constraints Revisited
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
Budget Constraints Revisited
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
Budget set?
?
(,),
,
x x p x p x p p
x x
1 2 1 1 2 2 1 1 2 2
1 20 0
? ? ?
? ?
? ?
Changing Endowments
x2
x1
(?1 ’,?2 ’)
(?1,,?2,)
(?1,?2 )
Changing the
endowment
Changing Prices
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
p x p x p p1 1 2 2 1 1 2 2' ' ' '? ? ?? ?
Changing prices of the
endowment
Changing Prices
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
p x p x p p1 1 2 2 1 1 2 2' ' ' '? ? ?? ?
Budget set
Changing Prices
x2
x1?1
?2
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
The endowment point is always on
the budget constraint.
p x p x p p1 1 2 2 1 1 2 2' ' ' '? ? ?? ?
So price changes pivot the
constraint about the
endowment point.
Net Demand
?The constraint
is
?That is,the sum of the values of a
consumer’s net demands （ 净需求）
is zero.
p x p x1 1 1 2 2 2 0( ) ( ),? ? ? ?? ?
p x p x p p1 1 2 2 1 1 2 2? ? ?? ?
Net Demands
?Suppose and
p1=2,p2=3,Then the constraint is
?If the consumer gross demands (毛需
求） (x1*,x2*) = (7,4),then 3 good 1
units exchange for 2 good 2 units,
Net demands are
x1*- ?1 = 7-10 = -3 and
x2*- ?2 = 4 - 2 = +2.
(,) (,)? ?1 2 10 2?
p x p x p p1 1 2 2 1 1 2 2 26? ? ? ?? ?,
Net Demands
p1=2,p2=3,x1*-?1 = -3 and x2*-?2 = +2 sop x p x1 1 1 2 2 2
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,
Net Demands at Different Prices
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
x2*
x1*
At prices (p1,p2) the consumer
sells units of good 1 to acquire
more units of good 2.
Net seller of good 1
x2
x1?1
?2 p x p x p p
1 1 2 2 1 1 2 2' ' ' '? ? ?? ?x2*
x1*
At prices (p1’,p2’) the consumer
sells units of good 2 to acquire
more of good 1.
Net buyer of good 1
Net Demands at Different Prices
x2
x1
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
x2*=?2
x1*=?1
At prices (p1”,p2”) the consumer
consumes her endowment; net
demands are all zero.
p x p x p p1 1 2 2 1 1 2 2" " " "? ? ?? ?
Autarky
Net Demands at Different Prices
Offer Curve
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
Price-offer curve contains all the
utility-maximizing gross demands
for which the endowment
can be exchanged.
Net Demands
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
Price-offer curve
Sell good 1,buy good 2
Net Demands
x2
x1?1
?2
p x p x1 1 1 2 2 2 0( ) ( )? ? ? ?? ?
Price-offer curve
Buy good 1,sell good 2
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.
Labor Supply
?A worker is endowed with $m of
nonlabor income and R hours of time
which can be used for labor or
leisure,? = (R,m).
?Consumption good’s price is pc.
?w is the wage rate.
?
?
Labor Supply
?The worker’s budget constraint is
where C,R denote gross demands
for the consumption good and for
leisure,That is
p C w R R mc ? ? ?( )?
p C wR wR mc ? ? ?? ??
endowment
value
expenditure
Labor Supply
p C w R R mc ? ? ?( )?
rearranges to
C w
p
R m wR
pc c
? ? ? ?,?
Labor Supply
C
RR?
endowmentm
($)
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ??
m
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ?m wR
p c
? ??
m
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ?m wR
p c
? ??
m
slope =,the ‘real wage rate’?
w
pc
Labor Supply
C
RR?
endowment
C wp R m wRp
c c
? ? ? ?m wR
p c
? ??
m
C*
R*
leisure
demanded
labor
supplied
Labor Supply Curve
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
does change with price,How does
this modify Slutsky’s equation?
y p p? ?1 1 2 2? ?
Slutsky’s Equation Revisited
?A change in p1 or p2 changes
so there will be
an additional income effect,called the
endowment income effect （ 禀赋收入效应）,
?Slutsky’s decomposition will thus have
three components
–a pure substitution effect
–an (ordinary) income effect,and
–an endowment income effect.
y p p? ?1 1 2 2? ?
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1’
Initial prices are (p1’,p2’).
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1”
x2”
Initial prices are (p1’,p2’).
Final prices are (p1”,p2”).
x1’
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1”
x2”
Initial prices are (p1’,p2’).
Final prices are (p1”,p2”).
How is the change in demand
from (x1’,x2’) to (x1”,x2”) explained?
x1’
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1’
Initial prices are (p1’,p2’).
Slutsky’s Equation Revisited
x1
?2
?1
x2
x2’
x1”
x2”
Initial prices are (p1’,p2’).
Final prices are (p1”,p2”).
x1’
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Endowment income effect?
Slutsky’s Equation Revisited
x1
?2
?1
x2 Pure substitution effect?
Ordinary income effect?
Endowment income effect?
Slutsky’s Equation Revisited
Overall change in demand caused by a
change in price is the sum of:
(i) a pure substitution effect
(ii) an ordinary income effect
(iii) an endowment income effect
Slutsky Equation with
Endowments
1 1 1 1 1 1 1 1 1
11
11
(,( ) ) ( ) (,) (,)sx p m p x p x p m x p mx
p p m m
?? ? ? ?? ? ?
? ? ? ?
1 1 1 1 1 1 1
11
11
(,( ) ) ( ) (,)()sx p m p x p x p mx
p p m
?? ? ?? ? ?
? ? ?
Or
(-) (+) (+)
