Question

3. Consider a person who chooses an amount of consumption c and non-working or leisure time R to maximize the utility function U(R,c) = 100R – R2 + c subject to the constraint c+wR=wL+M, where L is the maximum amount of time available (i.e., the maximum amount of leisure and labor supply possible) and M is the initial income that the consumer receives whether he works or not. Assume I = 24 and M = 42. 1 You can assume that m > P2. 1 3.1 [15 points) Solve this person's utility-maximization problem to find the demand functions for consumption and leisure as a function of w. 3.2 (7.5 points) Now suppose the an income tax is imposed on this person. Hence, the take-home wage rate is reduced to (1 – t)w, where t is the tax rate. The amount of tax collected is given by tw(L-R), since this is the tax rate times the amount of money the person earns. Formulate the new budget constraint for the utility maximization problem and find the new demand functions for consumption and leisure. 3.3 [5 points) Assuming w = 3/2, what is the amount of tax collected if the person faces an income tax rate t = 1/3? 3.4 (7.5 points] Suppose alternatively that a lump-sum tax of size T < M is imposed. Under this tax, the person in question must pay 7 in taxes, regardless of how much they earn or what they consume. Write the new budget constraint and find the new demand functions for consumption and leisure. 3.5 (10 points) Putting leisure on the horizontal axis and consumption on the vertical axis, plot (a) the budget constraint when the person faces a wage rate w = 3/2 and an income tax rate t = 1/3; (b) the budget constraint when he faces a wage rate w = 3/2 and a lump-sum tax that raises the same amount as the tax in 3.3 (that is, a lump-sum tax equal to the the amount of tax collected in 3.3) (c) On the diagram, identify the two post-tax consumption bundles. Hint: Check if the consumption bundle in (a) lies on the budget constraint in (b). 3.6 [5 points) Which tax makes the person better off? Why?

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Answer 1

Solution:-

Given that

U(R,C) = 100R – R? +

c = consuption

R = leisure

Budget constraint :  $C+wR=w\bar{L}+M$ ........(1)

$\bar{L}$ = Max time available = 24

M = inital Income = 42

(3.1)

From (1),

$C=w\bar{L}+M-wR$

Put value of c in utility function and differentiate with respect to R

to get optimal value of R

U(R) = 100R – R2 + wL+M - WR

dU(R) dR 100 – 2R – W w

$\frac{dU(R)}{dR}=0$

$100-2R-w=0$

$2R=100-w$

$R^*=\frac{100-w}{2}=50-\frac{w}{2}$ .......(2)

and  

$C=w\bar{L}+M-wR$

C = wL+M - w(50 – 2

$C=w\bar{L}+M-50w+\frac{w^2}{2}$

$C^*=(\bar{L}-50)w+\frac{w^2}{2}+M$ ......(3)

Put $\bar{L}$ = 24 & M = 42 in eqn (3)

$C^*=(24-50)w+\frac{w^2}{2}+42$

$C^*=42-26w+\frac{w^2}{2}$

(3.2)

After tax,

Wage rate w' = (1 - t)w

Amount of tax collected = $tw(\bar{L}-R)$

New Budget constraint

C+ w'R= w'L+M

$C+(1-t)wR=(1-t)w\bar{L}+M$

$C=(1-t)w\bar{L}+M-(1-t)wR$

U(R,C) = 100R – R? +

$U(R)=100R-R^2+(1-t)w\bar{L}+M-(1-t)wR$

$\frac{dU(R)}{dR}=100-2R-(1-t)w$

$\frac{dU(R)}{dR}=0$

$100-2R-(1-t)w=0$

$2R=100-(1-t)w$

$R_1=\frac{100-(1-t)w}{2}$ New R

&  $C=(1-t)w\bar{L}+M-(1-t)wR$

$C=(1-t)w\bar{L}+M-(1-t)w(\frac{100-(1-t)w}{2})$

$C=(1-t)w\bar{L}+M-50(1-t)w+\frac{(1-t)^2w^2}{2}$

Put $\bar{L}$ = 24, M = 42

$C=24(1-t)w+42-50(1-t)w+\frac{(1-t)^2w^2}{2}$

$C_1=42+(24-50)(1-t)w+\frac{(1-t)^2w^2}{2}$

$C_1=42-26(1-t)w+\frac{(1-t)^2w^2}{2}$

(3.3)

Amount of tax collected is given by T

$T=tw(\bar{L}-R)$

when  $t=\frac{1}{3},\ w=\frac{3}{2},\ \bar{L}=24,\ R=50-\frac{w}{2}$

$T=\frac{1}{3}\times \frac{3}{2}[24-(50-\frac{3}{2}\times \frac{1}{2})]$

$T= \frac{1}{2}[24-(\frac{200-3}{4})]$

$T= \frac{1}{2}[24-\frac{197}{4}]$

$T= \frac{1}{2}[\frac{96-197}{4}]$

$T=\frac{-101}{8}$

$T=-12.625$

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