Answer to Question #153120 in Microeconomics for Asmaa

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?

Answer 1

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

$BL= P_c \times Q_c+ P_d \times Q_d \\ 120= 1 \times Q_c + 1 \times Q_d \\ 120= Q_c + Q_d$

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

$\frac{MU_c}{MU_d}= \frac{P_c}{P_d} \\ MU_c = Q_c \\ MU_d= Q_d \\ \frac{Q_c}{Q_d}= \frac{1}{1} \\ \frac{Q_c}{Q_d} = 1$

c. From a we can derive the budget line

$120= Q_c + Q_d \\ 120= Q_c + Q_c \\ 120 = 2Q_c \\ Q_c = 60 \\ Q_d= Q_c = 60$

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

d. Given the budget line

$120= P_c \times Q_c+ P_d \times Q_d \\ 120= P_c \times Q_c+ 1 \times Q_d As Q_d= Q_c 120= P_c \times Q_c+ 1 \times Q_c \\ 120= Q_c(P_c+1)$

So quantity demanded

$Q_c = \frac{120}{P_c+1}$

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

$\frac{MU_c}{MU_d}= \frac{P_c}{P_d} \\ \frac{Q_c}{Q_d}= \frac{1}{2} \\ Q_c=2Q_d$

Now putting it into budget line

$120= P_c \times Q_c+ P_d \times Q_d \\ 120= 1 \times Q_c+ 2 \times Q_d \\ 120= 2 \times Q_d+2 \times Q_d \\ 4Q_d=120 \\ Q_d=30 \\ Q_c= 120-2 \times 30 = 120- 60 = 60$