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
c.
Holding Donald's income and Pd constant at $120 and $1 respectively, what is Donald's
demand curve for carrots?
d. Suppose that a tax of $1 per unit is levied on donuts. How will this alter Donald's utility
maximizing market basket of goods?
e.
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?
f.
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

Answer 1

$c)$ Substituting the information into the budget line

$120= Qc + Qc$

$120 = 2Qc$

$Qc =\frac{120}{2} = 60$

$Qd = 60$

d) First is to rewrite budget line

$120= PcQc + Qd$

We substitute the information into the budget line

$120 = PcQc+Qc$

$Qc=120( Pc + 1)$

$e)$

$1 tax on the donuts tends to increase the after-tax to 2 dollars.

Income-consumption curve ends up being; $\frac{MU_c}{MU_d} = \frac{P_d}{P_c}$

When I substitute $MU_c, MU_d, P_d and P_c$

$\frac{Q_c}{Q_d} = 2$ or $Qc = 2Q_d$

Budget line; $120 = Q_c+ 2Q_d$

Eliminating Qc through Substituting income-consumption curve in budget line

$120= 2Q_d+2Q_d$

$120 = 4Q_d$

$Q_d = 30$

$Q_c = 60$

f) when Donald buys 30 donuts he will pay 30 dollars in the tax. When Donald pays 30 dollars in lump sum tax, then his income is going to be $90

Resolving Utility-maximization problem through $I = 90 , P_c = P_d = I$

Utility maximizing basket will be

$Q_d = Q_c =45$