Doubt : Work-Leisure trade off

[neha:)]

Please help with this question!
A consumer has a total of H hours per day. He either enjoys leisure for L hours or does work (H-L hours) and earns wage rate W per hour.  He earns a real dividend income P , and pays lump-sum tax T. He spends his entire income on Good X, such that when he earns W, it gives him W units of good X. His preferences are given by U(C,L)= Min [ C, 2L].

a. What's his budget constraint?
b. Find Optimal Consumption bundle ( C,L )
c. If Taxes T fall, compute the change in optimal bundle (c',L'). From you answer, determine whether the good consumed by him is a normal good or not?
d. If Wage W rises, compute change in optimal bundle (C", L"). What will be the substitution effect and income effect in this case?

[Amit Goyal]

Budget constraint is C <= w(H-L) + P - T
Given that utility function is min{C, 2L},
Optimal bundle is
(C, L) = (P - T, H) if P - T >= 2H
(C, L) = (2(wH+P-T)/(w+2), (wH+P-T)/(w+2)) if P - T < 2H
Do the comparative statics yourself now.

[neha:)]

Thank you sir.
 One query: why are we comparing P-T with 2H?

[Amit Goyal]

Because we need consider the case when his non-wage income is sufficient to take care of his consumption needs even if he doesn't work at all. Technical way of saying this is P - T > 2H.

[Nupur]

Neha I'm actually getting very confused with the "real" dividend income thing! Can you check and tell me if the answers of the first two parts are correct, because only then its reasonable to proceed to other parts!!- Here are my answers:
Budget Line: C+WL=(P-T)+WH
And the optimum bundle is coming as: L*={(P-T)+WH}/(2+W)
and C*={2/(2+W)}*{(P+T)+WH}

[neha:)]

Got it! Thank you sir :)

[neha:)]

Yes Nupur, that's correct!

[Nupur]

Yeah, just a small typo I made in writing C* that sign should be negative in (P-T) which is wrote as positive!!

[neha:)]

Oh I overlooked that ... Yes it should be P-T.

[Shefali]

NUPUR PLS TELL HOW YOU FOUND OPTIMAL BUNDLE

[Nupur]

See Budget line is: C+WL=(P-T)+WH--(i)
Now optimality condition whew utility fn is of type min{C,2L}
Is C=2L---(ii)
Substitute (ii) in budget line (i) and you get the answers as above!! You can also see the solution below but my writing might not be so clear:

[neha:)]

Any idea how to go about the subsequent parts?

[Dreyfus]

c. dC/dT=-2/(2+w) so change in consumption of good due to change in tax is dC=(-2/(2+w))*dT
But taxes are falling here so dT<0, therefore dC>0
Decreasing taxes is equivalent of increasing income, and change in consumption is positive therefore good is normal

[Shefali]

this quest was asked in snu xam ryt,if i'm not mistaken?

[neha:)]

Thanks Vaibhav!
@ Shefali : Yes, it was asked in snu.

[Shefali]

How are you prepaing for interview of it?
N hw was ur xam?i attempted 4 1/2 of section 2..some quest were tricky!!

[SoniaKapoor]

nupur....a very stupid quest
how did you get 1st line of budget eqn ie pC+wL=p(P-T)+wH

[Nupur]

Sonia, if you see in Varian, after all the derivation under the topic of labor supply in chapter 9, the final form of budget line is given as: p.C+w.R=p.C bar +w.Rbar (RHS is the value of the endowments: non labor money income measure din terms of what it can buy {=p.C bar} is one endowment and the other endowment is the time, opportunity cost of which is the wage rate {=w.R bar}). But in this question the notations are given different from Varian. See the notations on the top of my working in that snapshot! Now its just a matter of plugging the notations that are given in the question in the above formula! Just be careful that what is R (leisure)in Budget Line in bold above is given as L in the question; non labor income {p.Cbar} is given as P-T in the question and total time which is Rbar above is given as H in the question! So its just a matter of putting the notations used in the question correctly in the formula of Budget Line. You need not even learn the Varian formula, its very intuitive; afterall its just saying the value of total consumption of a good and leisure should equal the value of the total endowments of time and non labor income.

[neha:)]

I found the eco part to be quite tough... For the interview I'm revising those concepts which were tested in the exam.