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Endowments
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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?
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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?
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Endowments
The list of resource units with which a consumer starts is his endowment.
A consumer’s endowment will be denoted by the vector (omega).
= endowment in good 1
= endowment in good 2
ω
1ω2ω
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Endowments Example:
Let (1,2 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
p11 + p2 2 = This value can be exchanged for any consumption bundle costing no more than the endowment’s value.
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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
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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
),( 21 ωω
p1x1 + p2x2 = p11 + p22
{(x1,x2) | p1x1 + p2x2 ≤ p11 + p22}
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Budget Constraints Revisited
2
2211
pp+p ωω
x2
x1
Budget set
Budget constraint
2
1
pp
=slope -
{(x1,x2)|p1x1+ p2x2 ≤ p11+ p22}
p1x1 + p2x2 = p11 + p22
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x2
x1
New constraint
Prices change from to and to
22112211 p′+p′=xp′+xp′ ωω
2
1
p
pslopeNew
-
1p 1p′ 2p′2p
New budget set
p1x1 + p2x2 = p11 + p22
2
2211
p′p′+p′ ωω
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Prices change from to and to 1p 1p′ 2p′2p
The endowment point is always on the budget constraint.
So price changes pivot the constraint about the endowment point.
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Net Demands
Definition: net demand is x –
for example
If x1 – 1 > 0
the consumer is a buyer of good 1
If x1 – 1 < 0
the consumer is a seller of good 1
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The constraint
p1x1 + p2x2 = p11 + p22
can be written as
p1(x1 – 1) + p2(x2 – 2) = 0
That is, the sum of the values of a consumer’s net demands is zero.
Net Demands
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Net Demands Suppose and p1=2, p2=3.
Then the constraint is
p1x1 + p2x2 = p11 + p22 = 26
If the consumer demands (x1*,x2*) = (7,4), Net demands are x1*- 1 = 7-10 = -3 and x2*- 2 = 4 - 2 = +2.
p1(x1 – 1) + p2(x2 – 2) =The purchase of 2 extra good 2 units at $3 each is funded by giving up 3 good 1 units at $2 each.
( , ) ( , ) 1 2 10 2
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Net Demandsx2
x1
x2*
x1*
At prices (p1,p2), the consumer
p1(x1 – 1) + p2(x2 – 2) = 0
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Net Demandsx2
x1
x2*
x1*
At prices (p1',p2'), the consumer
22112211 ppxpxp
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Net Demandsx2
x1
x2*=
x1*=
At prices (p1'' p2''), the consumer
22112211 ppxpxp
p1(x1 – 1) + p2(x2 – 2) = 0
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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
Effect of a Price Decrease
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Effect of a Price Decrease for a Seller
x2
x1
x2
x1
At prices (p1,p2), the consumer is a seller of good 1.
x1'
x2'
If after p1 decreases, the consumer remains the seller of good 1, he must be
U' U
UU'
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Effect of a Price Decrease for a Buyer
x2
x1
x2
x1
At prices (p1,p2), the consumer is a seller of good 1.
x1'
x2'
The consumer MUST remain a buyer of good 1, He is
U' U
UU'
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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 – 1) + p2(x2 – 2) = 0
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Price-offer curve
good 1, good 2
Net Demands and Price-Offer Curve
x2
x1
p1(x1 – 1) + p2(x2 – 2) = 0
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Price-offer curve
good 1, good 2
Net Demands and Price-Offer Curve
x2
x1
p1(x1 – 1) + p2(x2 – 2) = 0
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Net Demands and Price-Offer Curve
Price-offer curve contains all theutility-maximizing gross demands
for which the endowment can be exchanged.
x2
x1
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Slutsky’s Equation Revisited Slutsky: changes to demands caused by a
price change are the sum ofa pure substitution effect, andan income effect.
This assumed that income y did not change as prices changed. But
y = p11 + p22
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 = p11 + p22, so there will be
an additional income effect, called the endowment income effect.
Slutsky’s decomposition will thus have three componentsa pure substitution effectan (ordinary) income effect, andan endowment income effect.
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Slutsky’s Equation Revisited
Slutsky’s 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 =
i
i i∆pi
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We can write Slutsky’s Identity as
∆xi ∆xis
∆xim xi(pi, m) ∆xi
m ∆m
∆pi ∆pi ∆m ∆m ∆pi
Endowment income effect ∆xim ∆m = ∆xi
m ωi
∆m ∆pi ∆m
Slutsky’s Equation Revisited
Alternatively Slutsky’s equation is∆xi ∆xi
s (ωi – x i) ∆xi
m ∆pi ∆pi ∆m +
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Slutsky’s Equation Revisited
x1
2
1
x2 Initial prices are (p1, p2)
How is the change in demandfrom (x1, x2) to (x1', x2') explained?
x1
x2
x2'
x1'
Final prices are (p1', p2')
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Slutsky’s Equation Revisited
x1
2
1
x2 Pure substitution effect
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Slutsky’s Equation Revisited
x1
2
1
x2 Pure substitution effectOrdinary income effect
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Slutsky’s Equation Revisited
x1
2
1
x2 Pure substitution effectOrdinary income effectEndowment income effect
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Slutsky’s Equation RevisitedOverall change in demand caused by a
change in price is the sum of:
(i) A pure substitution effectChange in demand at constant real income
(ii) An ordinary income effectChange in demand holding money income fixed
(iii) An endowment income effectChange in demand due to a change in endowment value
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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.
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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
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Labor Supply
p C w R R mc ( )
rearranges to
Cwp
Rm wR
pc c
.
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Labor SupplyC
RR
endowment
m wRpc
Cwp
Rm wR
pc c
m
slope = , the ‘real wage rate’ wpc
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Labor SupplyC
RR
endowment
Cwp
Rm wR
pc c
m
C*
R*
leisuredemanded
laborsupplied