Endogenous Income. The consumption-leisure model

Transcription

1 Endogenous Income The consumtion-leisure model

2 Modifing consumer s roblem For the moment, assume there is no additional eogenous income Consumer s income is the market value of her initial endowment, (, ) Given market rices and, the consumer s budget constraint is + +

3 Budget constraint + +

4 An increase in Changes in rices

5 An increase in Changes in rices

6 Changes in rices The imact of rice changes in the consumer s budget set is now more subtle: the increase of a rice ma make the consumer relativel richer if she is relativel rich in that good (i.e., if her endowment of this good is large) Notice that, indeendentl of market rices, the endowment is alwas a feasible bundle (consumer alwas can avoid trade and consume her own endowment)

7 Consumer Demand Assume derivable and the sstem u(, ) + MRS(, ) ields an interior solution to the consumer s roblem ~ (, ), ~ (, ) + That is, ~ ( ~ (,, ) ) *( *(,,,, + + ) ) where the functions on the RHS are the ordinar demands.

8 Consumer Demand: eamle ; Calculate ordinar demands: Hence, u ), ( ) (2,1 ), ( I I I I 3 ),, *( 3 2 ),, *( ), ~ ( ) 2(2 ), ~ (

9 Income and substitution effects Stud the effect of an increase in for utilit function and endowment (, ) u(, ) B A C u 2 u 1

10 Income and substitution effects Substitution effect is negative SE B - A < 0 Income effect is ositve IE C - B > 0 In this case, income effect is larger than substitution effect, so the total effect is ositive: an increase in leads the consumer to increase its consumtion of both goods TE SE + IE C - A > 0

11 Income and substitution effects Suose an increase in. The substitution effect is the same as in the case with eogenous income. The income effect of a rice increase now involves two comonents: (1) It makes the consumer oorer as It becomes more eensive buing the otimal amount of the good (ordinar income effect), and (2) It modifies the value of the consumer s endowment (endowment effect). The sign of the (total) income effect deends on whether the consumer is a net seller (negative) or a net buer (ositive) of the good whose rice has increased.

12 Income and substitution effects Total effect substitution effect + ordinar income effect + endowment effect substitution effect + (total) income effect Formall, ) ( * * * * * ~ I I I cte u cte u

13 Income and substitution effects When is the total income effect ositive (the consumer is richer)? If the good is normal and the consumer is a net seller of the good * * - ( - ) > 0 if > 0 and - < I I 0 In an case, notice that ositive income effect ( richer consumer ) does not mean more consumtion: we have to take into account the substitution effect

14 Income and substitution effects Eogenous income Endogenous income B B C C A u 1 u0 A u 2 u 1

15 The consumtion-leisure model. Labor sul Two goods: leisure (-ais) and consumtion (-ais) - Leisure, denoted b h and measured in hours. The wage er hour (or rice of leisure) is denoted b w. - Consumtion, denoted b c and measured in euros. The rice of c is therefore c 1). Initial endowment is (M,H), where: - M : initial eogenous wealth (or non-labor income). - H : number of hours available for leisure and work.

16 The consumtion-leisure model. Labor sul Budget set (recall ) c 1 c c + hw wh + M wh + M hw wh + M : eenditure on leisure : monetar value of initial endowment sloe -w M H h

17 The consumtion-leisure model. Labor sul Solve the roblem as usual, but watch out for additional constraints Ma, u( c, h) st c h c + hw wh + 0 h H c ³ 0 M

18 The consumtion-leisure model. Labor sul. Eamle Ma c, h c + 2ln h st c + wh 16 w h 16 c ³ 0 Interior solution requires And 2 MRS( h, c) w Û w Þ h( w) h h( w) h( w) 2 w 2 w ³ 0 Û " w 16 Û w > ³ w

19 The consumtion-leisure model. Labor sul. Eamle Therefore, h(w) 16 if w <1/8 2 w if w ³ 1/8 c(w) 4 if w <1/8

20 The consumtion-leisure model. Labor sul. Eamle And labor sul l( w) H - h( w) 0 if w <1/ if w ³ 1/8 w 1/8

21 The consumtion-leisure model. Labor sul. Effect of changes in wages Assume w ' < w For interior solutions, the consumer is a net sulier of leisure Total income effect: if leisure is a normal good, h / I > 0, TIE is ositive, leading the consumer to demand less leisure (or sul more labor) h - ( l - H ) > 0 I >0 <0 Substitution effect : is alwas non-ositive. Since leisure is cheaer, this effect leads the consumer to demand more leisure (or sul less labor) Total effect is ambiguous (it deends on the shae of the utilit function)

22 The consumtion-leisure model. Labor sul. Effect of changes in wages c wh + M w ' H + M A B C M H h

23 The consumtion-leisure model. Labor sul. Effect of changes in wages c wh + M w ' H + M A B M C H h

24 The consumtion-leisure model. Labor sul. Effect of changes in wages wî(0,10) For, SE dominates (leisure is more eensive and consumer offers more labor) For wî(10,20), TIE dominates (consumer is richer and does not need to work as much as before)

25 Alication: a ta on labor income Imose t Î[0,1] The new budget constraint is c + (1 -t) wh (1 -t) wh + M The ta is equivalent to a reduction of wage: its imact on leisure consumtion (or labor sul) is ambiguous Its imact on welfare is unambiguous. Of course, ta olicies have other objectives we are not considering here.

26 Alication: a ta on labor income c wh + M (1- t ) wh + M c* M h* H h

27 Alication: a ta on labor income Alternative: a non-labor income ta T. The new budget constraint is c + wh wh + ( M -T) If both goods are normal, then the introduction of T reduces their demands (increaseslabor sul, in articular)

28 Alication: a ta on labor income c wh + M wh + ( M -T ) M M -T H h

29 Alication: a ta on labor income Eercise: what if T tw(h-h*)? Hint: is (c*,h*) otimal for T? c wh + M wh + ( M -T ) (1- t ) wh + M A c* M B M -T h* h