Question

Consider an individual making choices over two goods, x and y with prices px = 3 and py = 4, and who has the income I = 120 and her preferences can be represented by the utility function U(x,y) = (x^2)(y^2). Suppose the government imposes a sales tax of $1 per unit on good x:

(a) Calculate the substitution effect and Income effect (on good x) after the price change. Also Illustrate on a graph.

(b) Find the government tax revenues (T), the equivalent lump sum tax (L), and the dead weight loss (DWL). (Hint: Expenditure minimization at the final level of utility.)

(c) Is good x a normal good? Does law of demand hold for good x? Are good x and good y substitutes or complements?

Answer 1

px= 3 py=4 income,I= 120 U(x,y) = (x^2)(y^2)

BUDGET LINE : px*x + py*y = I

3x+4y=120

At optimal consumption we have, MRS=px/py

MRS = mux/muy = 2x(y^2) / 2y(x^2) = y/x

px/py= 3/4

y/x=3/4 => y=3x/4

putting value of y=3x/4 in budget line,we get

3x + 4(3x/4) =120 => 3x+3x=120 =>6x=120 => x=20

y=3x/4=60/4 = 15

Now the government imposes a sales tax of $1 per unit on good x

price of good x will change to px+$1 = 3+1=4

let the new price of good x be px' = 4

NEW BUDGET LINE : px'*x+py*y = 120

4x+4y=120

At optimal consumption , y/x = 4/4 => y=x

putting this y=x in new budget line we get , x=y= 15

Therefore change in quatity of good x due to imposition of sales tax is 5 units.

Now we have to find out how much of this change is due to substitution effect and how much due to income effect.

we will break the price movement into two steps: first let the relative price change and adjust money income so as to hold purchasing power constant, then we will let purchasing power adjust while holding the relative prices constant.

we can break this movement of the budget line up into two steps: first pivot the budget line around the original demanded bundle and then shift the pivoted line out to the new demanded bundle.

The purchasing power of the consumer has remained constant in the sense that the original bundle of goods is just affordable at the new pivoted line.

Change in money income necessary to make the old bundle affordable at the new prices is just the original amount of consumption of good times the change in prices.

I'-I = x (px'-px) =15*1 =15

Income level necessary to keep purchasing power constant is I' = 120+15 = 135

According to Slutsky , substitution effect = x(px',I') - x(px,I)

demand for good x at price level px' and income level I' .

px' x + py y =I'

4x+4y=135 => x=y=16.9

therefore, substitution effect of good x = 16.9 - 20= -3.1

Income effect is the change in the demand for good when we change income from I' to I ,holding the price of good fixed at px' .

Income effect= x(px',I) - x(px',I')

=15 - 16.9 = -1.9

(b) Government tax revenue = t * q = $1 * 15= $15

Deadweight loss(DWL) = (1/2)* t*(20-15) = 0.5*1*5 =2.5

(c) Good x is a normal good because increase in price of good x due to tax decrease the quantity demanded of good x.

Law of demand states that If the demand for a good increases when income increases , then the demand for that good must decrease when its price increases. Law of demand hold for good x as increase in price decrease quantity demanded for good x and inrease in income increase quantity demanded.

good x and good y are neither substitute nor complement .