Question

14. Tapha, the city finance commissioner, needs to raise additional tax revenue. The typical city resident has an income of 300 and utility function U = 10x0.6 y0.9  

Prices are Px = $4 and Py = $5.

a. What is the typical resident’s consumption and utility?

b. Tapha is considering putting a per-unit tax (t) on good X, so the city would collect a tax of t on each unit of X purchased in the market. Eventually the per-unit tax of t was imposed. What is the new price of X after tax if Tapha puts a unit tax of $2 on good X (t = 2)? What would be the typical resident’s consumption and utility? What tax revenue would be collected?

c. Instead of a unit tax on good X, Tapha now considers a lump-sum tax (θ). The lump-sum tax of θ was imposed. If Tapha put a lump-sum tax of $40 (θ = $40) instead of putting a unit tax on good X, what would be the typical resident’s consumption and utility?

d. Which tax, per-unit or lump-sum, seems to be the better option to raise tax revenue?

Answer 1

ANSWER:

Given:

Tapha the city finance commissioner needs to raise additional tax revenue.

U = 10 x0.6 0.9
; $P_{x}$ = $4 ;  $P_{y}$ = $5 ; Income = M = $ 300

(a)What is the typical resident consumption and utility

For a typical resident , we know that, his budget constraint would be:

M = Pc+Pyy

300 = 4.c + 5y..............

Now we know that for finding optimal consumption and utility , the income / budget line is tangent to an indifference curve.

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The slope of line and the slope of the indifference curve at pt. A are equal.

We know that,

Slop of in difference curve , is the MRS where

$MRS=\frac{-MU_{x}}{MU_{y}}$

U = 10 x0.6 0.9

$MU_{x}=6\times \frac{y^{0.9}}{x^{0.4}}$

$MU_{y}=9\times \frac{y^{0.6}}{x^{0.1}}$

$MRS =\frac{-6y^{0.9}}{x^{0.4}}\times \frac{1}{9}\times \frac{y^{0.1}}{x^{0.6}}$

$= \frac{-2}{3}\frac{y}{x}$

The slope of budget line = -4/5

$= \frac{-4}{5}=\frac{-2}{3}\frac{y}{x}$

$=\frac{6x}{5}=y$ ................(2)

Substituting equation 2 in 1 , we get

,

Hence the typical consumer 30 units of x and 36 unit of y and receive a total utility of approximately 1936.2.

(b) Tapha is considering putting a per - unit tax (t) on good X, so the city would collect a tax t on each unit of X.

If tapha rates a per unit tax of $2 a good X, the price of X will change from $4 to $6.

Hence Now budget constraint would be,

We consec that the in difference curve that is tangent to the budget line charges .

Now equilibrium is at pt.B

To find optimum consumption and utility

RA SA ūo=19363 Tū= 1518.3

Slope of budget line = slope of in difference curve at pt.B

$\frac{-P_{x}}{P_{y}}=\frac{-MU_{x}}{MU_{y}}$

$\frac{-6}{5}= \frac{-2}{3}\frac{y}{x}$

Using equation 3 in equation a , we get

$300=15x\Rightarrow x=20 , y=36$

$U = 10\times 20^{0.6}\times 36^{0.9}$

$1518.1\Rightarrow \bar{U_{1}}=1518.1$

Total tax = tax = 2 *20= $40 Total tax = $ 40

(c)

If Taphs puts a lumpsum tax of $\theta$ = $40, then the income of the consumer reduce to 260. Now budget constraint is

If we redo the analysis of the slopes being equal (2) pt C, we get

$\frac{-4}{5}= \frac{-2}{3}\frac{y}{x}$

$\Rightarrow 6x=5y$

$\Rightarrow y=\frac{6}{5}x............(4)$

Using equation (2) in euqation (b) we get

$\Rightarrow 10x=260$

$x=26$ ,

$U = 10\times 26^{0.6}\times (31.2)^{0.9}$

$\bar{U_{2}}=1562.2$

(d)

We can see that in both scenarios , tax collected is $40. But we can sec that

$\bar{U_{2}}> \bar{U_{1}}$.

Hence consumer receive more utility from lumpsum tax than a per unit tax.

Hence, a lumpsum tax seems to bbe a better option to raise tax revenue.