ENDOWMENTS OF GOODS. [See Lecture Notes] Copyright 2005 by South-Western, a division of Thomson Learning. All rights reserved.

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1 ENDOWMENTS OF GOODS [See Lecture Notes] Coyright 005 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? Sulying 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 {ω ω N }. Consumer faces linear rices { N }. Preferences obey usual axioms. Consumer s roblem: Choose {x x x N } to maximize utility u(x x x N ) subject to budget constraint and x i 0. Marshallian demand denoted by x* i ( N ; ω ω N ) 3

4 Budget Constraint We can suose the agent makes choices in two stes:. Sells all her endowment. This generates income m = ω + ω + + N ω N. Given income she chooses {x x x N } to solve the UMP as before. 4

5 Budget Constraint Agent endowed with lots of good. Buys good and sells good. 5

6 What s the Big Deal? Calculating otimal consumtion is same as before. Changes in rices now affect value of endowments and thus income of agent. This is imortant: income taxes affect household wealth. Agents may also face kinks in budget at endowment. 6

7 Examle: u(x x )=x x UMP imlies demand is x* ( m) = m/ Endowment is (ω ω ) yielding income m = ω + ω Demand is x* ( ω ω ) = ( ω + ω )/ Hence an increase in increases the demand for x. In comarison when income is exogenous x* ( m) is indeendent of. 7

8 . Labor suly Two Alications Workers trade off work and consumtion What does the labor suly function look like? What is the effect of income tax? Model used in labor economics and ublic finance. Intertemoral Otimisation Agents trade off consumtion today and tomorrow How does this deend on when receive income? What is the effect of a change in interest rates? Model used in ublic finance macro finance. 8

9 APPLICATION: LABOR SUPPLY [See Notes or in book] 9

10 Labor Suly Utility: u(x x ) = x / + x / Good = leisure Good = general consumtion good. Endowments Income m T hours for work/leisure. Let =w (wage) and = (normalisation). 0

11 Budget Constraint The agent s sending must be less that her income wx + x wt + m Equivalently her sending on good x must be less than her labor income x w(t-x ) + m

12 Solving the Problem Lagrangian L = x / + x / + [wt+ m - wx - x ] FOCs x -/ = w and x -/ = Rearranging x w = x. Using budget constraint: w x ( wt m) and x ( wt m) w( w) ( w)

13 APPLICATION: INTERTEMPORAL OPTIMIZATION [See Notes or in book] 3

14 Intertemoral Consumtion An agent chooses to allocate consumtion across two eriods. Examle: College Student Low income when at college. High income when graduate. How much debt should you accumulate? How does this deend on the interest rate? Model crucial to understand savings decisions. Treat two eriods exactly like two goods 4

15 Two eriods: t =. Preferences Consumtion is (x x ). Utility u(x x ) = ln(x ) + (+β) - ln(x ) where β 0 is agent s discount rate. If β=0 then weigh consumtion same in both eriods. If β>0 then weigh current consumtion higher. 5

16 Budget Constraint Agent has income (m m ) in two eriods. Interest rate r 0. $ today is worth $(+r) tomorrow. Inverting $ tomorrow is worth $(+r) - today. Budget constraint (in eriod dollars) m + (+r) - m = x + (+r) - x LHS = lifetime income RHS = lifetime consumtion 6

17 Solving the Problem 7

18 Solving the Problem Lagrangian L = ln(x )+(+β) - ln(x ) + [(m -x )+ (+r) - (m -x )] FOCs x - = and (+β) - x - = (+r) - Rearranging (+r)x = (+β)x. Using budget constraint: x [ m ( r) m ] and x [ m ( r) m r ] 8

19 If r=β then x * = x *. Lessons If agent is as atient as market then smooth consumtion over time. If r>β then x * > x *. If agent more atient than market then save and consume more tomorrow. Consumtion indeendent of how income distributed over time if net resent value the same. 9

20 OWN PRICE EFFECTS 0

21 Budget Constraint The agent s budget constraint is x + x ω + ω If doubles constraint is x + x ω + ω If halves constraint is ½ x + x ½ ω + ω These are same! Only relative rices matter. We can normalise = without loss.

22 Change in Prices Agent initially rich in good. Suose rises (or falls). Fall in makes agent oorer. Moves from A to B on lower IC Also substitutes towards good Budget line ivots around endowment

23 Slutsky Equation Suose increases by.. Substitution Effect. Holding utility constant relative rices change. h Increases demand for x by. Income Effect Agent s income rises by (ω -x* ). * * x ( m) Increases demand by ( x ) m 3

24 4 Slutsky Equation Fix rices ( ). Let m= ω + ω and u = v( m). Then SE always negative since h decreasing in. IE deends on (a) whether x* normal/inferior and (b) whether ω is greater/less than x* ) ( )) ( ( ) ( ) ( * * * m x m m x u h x

25 5 Examle: u(x x )=x x From UMP Given endowments demand is From EMP LHS of Slutsky: * 4 ) v( and ) ( m m m m x / / ) ( ) e( and ) ( u u u u h * ) ( x * ) ( x

26 6 Examle: u(x x )=x x RHS of Slutsky: Summing this yields ω / as on the LHS ] [ 4 4 ) ( / 3/ / m u u h ] [ 4 ) ( ) ( * * m m x m m x

27 Endowments: Summary Income often comes via endowments. Calculating demand same as before: First agent sells endowments at mkt rice. This determines income. Second agent chooses consumtion as before. Price effect now different: Change in rice affects value of endowment. This alters income effect. 7