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.

9

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.

16

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

1

1 pp

h

121

*

1*

11

),,()( p

m

mppxx

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

*

121

*

11211

1

2121

*

1

1

mppxm

mppxupphp

ppxp

25

Example: u(x1,x2)=x1x2• From UMP

• Given endowments, demand is

• From EMP

• LHS of Slutsky:

21

2

21

1

21

*

14

),,( vand 2

),,(pp

mmpp

p

mmppx

2/1

2121

1/2

1

2211 )(2),,(e and ),,( ppuuppu

p

pupph

2

1

222121

*

1

1 2),,,(

p

pppx

p

1

22112121

*

12

),,,(p

ppppx

26

Example: u(x1,x2)=x1x2

• RHS of Slutsky:

• Summing, this yields –p2ω2/2p12, as on the LHS

][4

1

4

1

2

1),,( 22112

1

2

1

2/1

2

2/3

1

2/1

211

1

ppp

mp

ppuupphp

][4

1

2

1

2

2

1

2),,(),,(

22112

1

11

22111

11

121

*

121

*

11

ppp

pp

pp

pp

mmppx

mmppx

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.