ENDOWMENTS OF GOODS
[See Lecture Notes]
Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.
1
Endowments as Income
• So far assume agent endowed with income m.
• Where does income come from?
– Supplying labor.
– Selling other goods and services.
• Endowment model
– Agent endowed with goods.
– Sells some goods to buy others.
• Aim: We wish to make model self-contained.
So we can look at the entire economy.
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Model
•
•
•
•
•
There are N goods.
Consumer has endowments {ω1,…,ωN}.
Consumer faces linear prices {p1,p2,…,pN}.
Preferences obey usual axioms.
Consumer’s problem: Choose {x1,x2,…,xN} to
maximize utility u(x1,x2,…,xN) subject to budget
constraint and xi ≥ 0.
• Marshallian demand denoted by
x*i(p1,p2,…,pN; ω1,…,ωN)
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1
Budget Constraint
• We can suppose the agent makes
choices in two steps:
1. Sells all her endowment. This generates
income
m = p1ω1 + p2ω2 + … + pNωN
2. Given income, she chooses {x1,x2,…,xN}
to solve the UMP, as before.
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Budget Constraint
• Agent endowed with lots of good 1.
• Buys good 2 and sells good 1.
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What’s the Big Deal?
• Calculating optimal consumption is
same as before.
• Changes in prices now affect value of
endowments, and thus income of agent.
– This is important: income taxes affect
household wealth.
• Agents may also face kinks in budget at
endowment.
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2
Example: u(x1,x2)=x1x2
• UMP implies demand is
x*1(p1,p2,m) = m/2p1
• Endowment is (ω1, ω2), yielding income
m = p1ω1 + p2ω2
• Demand is x*1(p1,p2,ω1,ω1) = (p1ω1+ p2ω2)/2p1
• Hence an increase in p2 increases the demand
for x1.
• In comparison, when income is exogenous,
x*1(p1,p2,m) is independent of p2.
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Two Applications
1. Labor supply
–
–
–
–
Workers trade off work and consumption
What does the labor supply function look like?
What is the effect of income tax?
Model used in labor economics and public finance
2. Intertemporal Optimisation
–
–
–
–
Agents trade off consumption today and tomorrow
How does this depend on when receive income?
What is the effect of a change in interest rates?
Model used in public finance, macro, finance.
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APPLICATION: LABOR SUPPLY
[See Notes or p. 573-580 in book]
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3
Labor Supply
• Utility: u(x1,x2) = 2x11/2 + 2x21/2
– Good 1 = leisure
– Good 2 = general consumption good.
• Endowments
– Income m
– T hours for work/leisure.
• Let p1=w (wage) and p2=1 (normalisation).
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Budget Constraint
• The agent’s spending must be less that her
income
wx1 + x2 ≤ wT + m
• Equivalently, her spending on good x2 must
be less than her labor income
x2 ≤ w(T-x1) + m
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Solving the Problem
• Lagrangian
L = 2x11/2 + 2x21/2 + λ[wT+ m - wx1 - x2]
• FOCs
x1-1/2 = λw and x2-1/2 = λ
• Rearranging, x1w2 = x2.
• Using budget constraint:
x1 =
1
w
( wT + m) and x2 =
( wT + m)
w(1 + w)
(1 + w)
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4
APPLICATION: INTERTEMPORAL
OPTIMIZATION
[See Notes or p. 595–600 in book]
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Intertemporal Consumption
• An agent chooses to allocate consumption
across two periods.
• Example: College Student
–
–
–
–
Low income when at college.
High income when graduate.
How much debt should you accumulate?
How does this depend on the interest rate?
• Model crucial to understand savings decisions.
• Treat two periods exactly like two goods
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Preferences
• Two periods: t = 1,2.
• Consumption is (x1,x2).
• Utility
u(x1,x2) = ln(x1) + (1+β)-1ln(x2)
where β≥0 is agent’s discount rate.
• If β=0 then weigh consumption same in both
periods.
• If β>0 then weigh current consumption higher.
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5
Budget Constraint
• Agent has income (m1,m2) in two periods.
• Interest rate r≥0.
– $1 today is worth $(1+r) tomorrow.
– Inverting, $1 tomorrow is worth $(1+r)-1 today.
• Budget constraint (in period 1 dollars)
m1 + (1+r)-1m2 = x1 + (1+r)-1x2
LHS = lifetime income
RHS = lifetime consumption
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Solving the Problem
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Solving the Problem
• Lagrangian
L = ln(x1)+(1+β)-1ln(x2) + λ[(m1-x1)+ (1+r)-1(m2-x2)]
• FOCs
x1-1 = λ and (1+β)-1x2-1 = λ(1+r)- 1
• Rearranging, (1+r)x1 = (1+β)x2.
• Using budget constraint:
x1 =
1
1+ β 1
[m1 + (1 + r ) −1 m2 ] and x2 =
[m1 + (1 + r ) −1 m2 ]
2+ β
1+ r 2 + β
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6
Lessons
• If r=β, then x1* = x2*.
– If agent is as patient as market, then smooth
consumption over time.
• If r>β, then x1* > x2*.
– If agent more patient than market, then save and
consume more tomorrow.
• Consumption independent of how income
distributed over time, if net present value the
same.
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OWN PRICE EFFECTS
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Budget Constraint
• The agent’s budget constraint is
p1x1 + p2x2 ≤ p1ω1 + p2ω2
• If p2 doubles constraint is
p1x1 + 2p2x2 ≤ p1ω1 + 2p2ω2
• If p1 halves constraint is
½p1x1 + p2x2 ≤ ½p1ω1 + p2ω2
• These are same! Only relative prices matter.
• We can normalise p2=1 without loss.
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7
Change in Prices
• Agent initially rich in good 1.
• Suppose p2 rises (or p1 falls).
Fall in p1 makes agent poorer.
Moves from A to B, on lower IC
Also substitutes towards good 1
Budget line pivots
around endowment
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Slutsky Equation
• Suppose p1 increases by ∆p1.
1. Substitution Effect.
– Holding utility constant, relative prices change.
– Increases demand for x1 by
∂h1
∆p1
∂p1
2. Income Effect
– Agent’s income rises by (ω1-x*1)×∆p1.
∂x * ( p , p , m)
– Increases demand by (ϖ 1 − x1* ) 1 1 2
∆p1
∂m
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Slutsky Equation
• Fix prices (p1,p2).
• Let m= p1ω1 + 2p2ω2 and u = v(p1,p2,m).
• Then
∂ *
∂
∂ *
x1 ( p1 , p2 ,ϖ 1 ,ϖ 2 ) =
h1 ( p1 , p2 , u ) − (ϖ 1 − x1* ( p1 , p2 , m)) ⋅
x1 ( p1 , p2 , m)
∂p1
∂p1
∂m
• SE always negative since h1 decreasing in p1.
• IE depends on (a) whether x*1 normal/inferior,
and (b) whether ω1 is greater/less than x*1
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8
Example: u(x1,x2)=x1x2
• From UMP
x1* ( p1 , p2 , m) =
m
and
2 p1
v( p1 , p2 , m) =
m2
4 p1 p2
• Given endowments, demand is
x1* ( p1 , p2 ,ϖ 1 ,ϖ 2 ) =
• From EMP
p1ϖ 1 + p2ϖ 2
2 p1
1/2
p 
h1 ( p1 , p2 , u ) =  2 u 
 p1 
• LHS of Slutsky:
and e( p1 , p2 , u ) = 2(u p1 p2 )1/ 2
∂ *
pϖ
x1 ( p1 , p2 ,ϖ 1 ,ϖ 2 ) = − 2 22
∂p1
2 p1
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Example: u(x1,x2)=x1x2
• RHS of Slutsky:
∂
1 1/ 2
1
1
h1 ( p1 , p2 , u ) = − u p1−3 / 2 p21/ 2 = − 2 m =
[ p1ϖ 1 + p2ϖ 2 ]
∂p1
2
4 p1
4 p12
(ϖ
1
) ∂∂m x ( p , p , m) = ϖ
− x1* ( p1 , p2 , m) ⋅
*
1
1
2
m  1

1−
2 p1  2 p1


p ϖ + p2ϖ 2  1

= ϖ 1 − 1 1
2 p1

 2 p1
1
=
[
p
ϖ
−
p
ϖ
]
1
1
2
2
4 p12
• Summing, this yields –p2ω2/2p12, as on the LHS
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Endowments: Summary
• Income often comes via endowments.
• Calculating demand same as before:
– First, agent sells endowments at mkt price. This
determines income.
– Second, agent chooses consumption, as before.
• Price effect now different:
– Change in price affects value of endowment.
– This alters income effect.
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