Post on 19-Apr-2022
1
ENDOWMENTS OF GOODS
[See Lecture Notes]
Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.
2
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.
3
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)
4
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.
5
Budget Constraint• Agent endowed with lots of good 1.
• Buys good 2 and sells good 1.
6
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.
7
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.
8
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]
10
Labor Supply
• Utility: u(x1,x2) = 2x11/2 + 2x2
1/2
– Good 1 = leisure
– Good 2 = general consumption good.
• Endowments
– Income m
– T hours for work/leisure.
• Let p1=w (wage) and p2=1 (normalisation).
11
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
12
Solving the Problem
• Lagrangian
L = 2x11/2 + 2x2
1/2 + [wT+ m - wx1 - x2]
• FOCs
x1-1/2 = w and x2
-1/2 =
• Rearranging, x1w2 = x2.
• Using budget constraint:
)()1(
and )()1(
121 mwT
w
wxmwT
wwx
13
APPLICATION: INTERTEMPORAL
OPTIMIZATION
[See Notes or p. 595–600 in book]
14
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
15
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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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
17
Solving the Problem
18
Solving the Problem
• Lagrangian
L = ln(x1)+(1+β)-1ln(x2) + [(m1-x1)+ (1+r)-1(m2-x2)]
• FOCs
x1-1 = and (1+β)-1x2
-1 = (1+r)- 1
• Rearranging, (1+r)x1 = (1+β)x2.
• Using budget constraint:
])1([2
1
1
1 and ])1([
2
12
1
122
1
11 mrmr
xmrmx
19
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.
20
OWN PRICE EFFECTS
21
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.
22
Change in Prices• Agent initially rich in good 1.
• Suppose p2 rises (or p1 falls).
Budget line pivots
around endowment
Fall in p1 makes agent poorer.
Moves from A to B, on lower IC
Also substitutes towards good 1
23
Slutsky Equation• Suppose p1 increases by ∆p1.
1. Substitution Effect.
– Holding utility constant, relative prices change.
– Increases demand for x1 by
2. Income Effect
– Agent’s income rises by (ω1-x*1)×∆p1.
– Increases demand by
1
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24
Slutsky Equation
• Fix prices (p1,p2).
• Let m= p1ω1 + 2p2ω2 and u = v(p1,p2,m).
• Then
• 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
),,()),,((),,(),,,( 21
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25
Example: u(x1,x2)=x1x2• From UMP
• Given endowments, demand is
• From EMP
• LHS of Slutsky:
21
2
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14
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26
Example: u(x1,x2)=x1x2
• RHS of Slutsky:
• Summing, this yields –p2ω2/2p12, as on the LHS
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27
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.