Question

3. Donald derives utility from only two goods, carrots (Qc) and donuts (Qd). (Qc and Qd are the consumption quantity of carrots and donuts). His utility function is as follows: U(Qc,Qd) = (2Qc)(Qd). 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 quantities of Qc and Qd will maximize Donald’s utility?

Answer 1

a) Budget line

Budget line shows the combination of two products , which consumer can consume depending on his given income.

Budget line equation for Donald:

I= (Price of Carrots*Quantity of Carrots) + ((Price of Donuts*Quantity of Donuts)

$I=\lparen{P_c*Q_c}\rparen+\lparen{P_d*Q_d}\rparen$

as $P_c=P_d=1$ and Income =120,

$120=\lparen{1*Q_c}\rparen+\lparen{1*Q_d}\rparen$

$\boxed{120=Qc+Qd}$

b) Utility Maximisation

Utility function of Donald: $U\lparen Q_c, Q_d \rparen = \lparen 2Q_c \rparen \lparen Q_d \rparen$

Consumer maximisation condition: $MRS= \cfrac {P_c} {P_d}$

Marginal Rate of Satisfaction (MRS) is the rate, the consumer willing to substitute one good for another. $\cfrac {P_c} {P_d}$ is the price ratio of carrots and donuts.

Setting MRS equal to the price ratio:

$MRS= \frac{Marginal utility of carrots} {Marginal utility of donuts}$

$MRS= \frac{MU_c} {MU_d}$ ($MU_c$ and $MU_d$ is remaining constant)

Then, $MU_c= 2Q_d$ and $MU_d= 2Q_c$

$MRS= \frac {\not2Q_d} {\not2Q_c}$

So,

$MRS= \frac{Q_d} {Q_c}$

When setting MRS equal to price ratio: $MRS= \frac{Q_d} {Q_c} = \frac{P_d} {P_c}$

$\therefore MRS = \frac{Q_d} {Q_c} =\frac{1} {1}$

$Q_d = Q_c$

Substituting $Q_d = Q_c$ in to the Budget line: ${120=Qc+Qd}$

since $Q_d = Q_c$ , substitute $Q_c for Q_d$

$120=Qc+Qc$

$120=2Qc$

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

$\frac{{\not120}} {\not2}= \frac {\not2Qc} {\not2}$

$\boxed{60= Q_c}$

since $Q_d = Q_c$ ,

$\boxed{Q_d = 60}$

Quantities which will maximise the Donald's utility is $Q_c = 60$ and $Q_d = 60$ .