Answer to Question #242416 in Microeconomics for Mendie

Question

if u=q1 q2 and p1 = 2 Naira per kg, p2 = 5 naira per kg and y = p1 q1 + p2 q2.
calculate the quantities of q1 and q2 purchased by the consumer assuming that he was able to maximise his satisfaction if y=200 naira

Answer 1

given Information

$U = q_1q_2$

P1 = Price of good 1 = 200

P2 = Price of good 2 = 5

 

Budget Constraint is given as 

$p_1 q_1 + p_2 q_2 = 200$

Maximum Utility is achieved where the slope of IC = Sloe if budget constraint.

i.e. $MRS = \frac{P_1}{p_2}$

 

Slope of IC curve

$MRS =\frac{ MU_{q_1}}{MU_{q_2}} \\MU_{q_1} = \frac{∂u}{∂q_1} = q_2\\ MU_{q_2} = \frac{∂u}{∂q_2} = q_1\\MRS = \frac{q_2}{q_1}$

Now Put $MRS = \frac{p_1}{p_2}$ , we

$\frac{q_2}{q_1} =\frac{200}{5}\\\frac{ q_2}{q_1} =40\\q_2 = 40q_1\\Now\space Put \space q_2 = 40q_1 \space in\space Budget\space Costraint,\\ we\space get\\p_1 q_1 + p_2 q_2 = 200\\200\times q_1 + 5\times (40q_1) = 200\\200q_1 + 200q_1 = 200\\400q_1 = 200\\q_1 = \frac{1}{2}\\$

Now Put $q_1 = \frac{1}{2}$ in budget constraint to calculate the quantity of good 2

$p_1 q_1 + p_2 q_2 = 200\\200\times \frac{1}{2} + 5\times q_2 = 200\\100 + 5q_2 = 200\\5q_2 = 100\\q_2 = 20$

Therefore utility is Maximize when a half unit of good 1 is purchased and 20 units of good 2.