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.

2 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) 3 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.

4 Budget Constraint • Agent endowed with lots of good 1. • Buys good 2 and sells good 1.

5 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.

6 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.

7 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. 8 APPLICATION: LABOR SUPPLY

[See Notes or p. 573-580 in book] 9 Labor Supply

1/2 1/2 • Utility: u(x1,x2) = 2x1 + 2x2 – Good 1 = leisure – Good 2 = general consumption good. • Endowments – Income m – T hours for work/leisure.

• Let p1=w (wage) and p2=1 (normalisation).

10 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

11 Solving the Problem • Lagrangian 1/2 1/2 L = 2x1 + 2x2 + [wT+ m - wx1 - x2] • FOCs -1/2 -1/2 x1 = w and x2 =  2 • Rearranging, x1w = x2. • Using budget constraint: 1 w x  (wT  m) and x  (wT  m) 1 w(1 w) 2 (1 w)

12 APPLICATION: INTERTEMPORAL OPTIMIZATION

[See Notes or p. 595–600 in book] 13 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

14 Preferences • Two periods: t = 1,2.

• Consumption is (x1,x2). • Utility -1 u(x1,x2) = ln(x1) + (1+β) ln(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.

15 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) -1 -1 m1 + (1+r) m2 = x1 + (1+r) x2 LHS = lifetime income RHS = lifetime consumption

16 Solving the Problem

17 Solving the Problem • Lagrangian -1 -1 L = ln(x1)+(1+β) ln(x2) + [(m1-x1)+ (1+r) (m2-x2)] • FOCs -1 -1 -1 - 1 x1 =  and (1+β) x2 = (1+r)

• Rearranging, (1+r)x1 = (1+β)x2. • Using budget constraint: 1 1  1 x  [m  (1 r)1m ] and x  [m  (1 r)1m ] 1 2   1 2 2 1 r 2   1 2

18 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.

19 OWN PRICE EFFECTS

20 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.

21 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

22 Slutsky Equation

• Suppose p1 increases by ∆p1. 1. Substitution Effect. – Holding utility constant, relative prices change.

h1 – Increases demand for x1 by p1 p1 2. Income Effect

– Agent’s income rises by (ω1-x*1)×∆p1. x* ( p , p ,m) – Increases demand by (  x* ) 1 1 2 p 1 1 m 1 23 Slutsky Equation

• Fix prices (p1,p2).

• Let m= p1ω1 + 2p2ω2 and u = v(p1,p2,m). • Then

 *  *  * x1 ( p1, p2 ,1, 2 )  h1( p1, p2 ,u)  (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

24 Example: u(x1,x2)=x1x2 • From UMP 2 * m m x1 ( p1, p2 ,m)  and v( p1, p2 ,m)  2p1 4p1 p2 • Given endowments, demand is

* p11  p2 2 x1 ( p1, p2 ,1, 2 )  2p1

• From EMP 1/2  p   2  1/ 2 h1( p1, p2 ,u)   u and e( p1, p2 ,u)  2(up1 p2 )  p1 

• LHS of Slutsky:  * p2 2 x1 ( p1, p2 ,1, 2 )   2 p1 2 p1

25 Example: u(x1,x2)=x1x2

• RHS of Slutsky:

 1 1/ 2 3/ 2 1/ 2 1 1 h1( p1, p2 ,u)   u p1 p2   2 m  2 [ p11  p2 2 ] p1 2 4p1 4p1

  m  1 * *   1  x1 ( p1, p2 ,m) x1 ( p1, p2 ,m)  1   m  2 p1  2 p1  p  p   1  1 1 2 2   1    2 p1  2 p1 1  2 [ p11  p2 2 ] 4 p1

2 • Summing, this yields –p2ω2/2p1 , as on the LHS 26 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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