Donald derives utility from only two goods, carrots (Qc) and donuts (Qd). His ut

Question

Donald derives utility from only two goods, carrots (Qc) and donuts (Qd). His utility function is as follows: U(Qc,Qd) = (Qc)(Qd) The marginal utility that Donald receives from carrots (MUc) and donuts (MUd) are given as follows:

MUc = Qd MUd = Qc Donald has an income (I) of $120 and the price of carrots (Pc) and donuts (Pd) are both $1.

a. What is Donald's budget line?

b. What is Donald's income-consumption curve?

c. What quantities of Qc and Qd will maximize Donald's utility?

d. Holding Donald's income and Pd constant at $120 and $1 respectively, what is Donald's demand curve for carrots?

e. Suppose that a tax of $1 per unit is levied on donuts. How will this alter Donald's utility maximizing market basket of goods?

f. Suppose that, instead of the per unit tax in (e), a lump sum tax of the same dollar amount is levied on Donald. What is Donald's utility maximizing market basket?

g. The taxes in (e) and (f) both collect exactly the same amount of revenue for the government, which of the two taxes would Donald prefer? Show your answer numerically and explain why Donald prefers the per unit tax over the lump sum tax, or vice versa, or why he is indifferent between the two taxes.

Explanation / Answer

Solution:

a) Budget Line is basically all those combinations of two goods, which the consumer can buy spending his given money income on the two goods at their given prices.

It can be written as

BL= Pc Qc+ Pd Qd

120= 1Qc + 1Qd

120= Qc + Qd

So, budget line will be 120.

b) As the price of both the goods i.e. carrot and donuts is $1. They are perfect substitute for each other. The IC curve will be a straight line with negative slope because the marginal utility is constant. The value of this slope is throughout minus 1, and MRScd=1

So, IC curve will satisfy this equation

MUc/MUd= Pc/Pd

Given, MUc = Qc and MUd= Qd

Qc/Qd=1/1

Qc/Qd = 1

c)  From a we can derive the budget line

120= Qc + Qd

120= Qc + Qc    ------------- From (b)

120 = 2Qc

So, Qc = 60 and Qd= Qc = 60

So 60 units of both the goods will satisfy the Donald’s utility.

d) Given the budget line

120= Pc Qc+ Pd Qd

120= Pc Qc+ 1 Qd

120= Pc Qc+ 1 Qc    (As Qd= Qc)

120= Qc(Pc+1)

So quantity demanded (Qc) = 120/ (Pc+1)

e)

After the tax the price of donut will become $2. This will change IC curve. The new IC curve will be

MUc/MUd= Pc/Pd

Qc/Qd=1/2

Qc=2Qd

Now putting it into budget line

120= Pc Qc+ Pd Qd

120= 1Qc+ 2Qd

120= 2Qd+2Qd

4Qd=120

Qd=30

So, Qc= 120-2x30 = 120- 60 = 60

f) After tax the total quantity of donuts purchased by Donald is 30. So, the lumpsum tax levied on donald will be $30. This will shift the budget line as his income will be reduced due to lump sum tax.

The new budget line will be 120-30 = 90

Now Utility market basket will be satisfied by the new budget line.

BL= Pc Qc+ Pd Qd

90= 1Qc + 1Qd

90= Qc + Qd

As, Qc=Qd

90= Qc + Qc

2Qc=90

Qc=45

Qd=Qc=45

g) When donald pay per unit tax he gets 60 units of carrot and 30 units of donuts. The total utility in this case is 60x30 = 1800

When donald pay lump sum tax he gets 45 units of carrot and 45 units of donuts. The total utility in this case is 45x45 = 2025

His utility is greater when he pays lump sum tax. So he will prefers lump sum tax over the per unit tax.

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