Question

Diogo has a utility function,U(q1, q2) = q1 0.8 q2 0.2,where q1 is chocolate candy and q2 is slices of pie. If the price of slices of pie, p2, is $1.00, the price of chocolate candy, p1, is $0.50, and income, Y, is $100, what is Diogo's optimal bundle?The optimal value3 of good q1 isq = units. (Enter your response rounded to two decimal places.)1 The optimal value of good q2 isq2 = units. (Enter your response rounded to two decimal places.)

Answer 1

Answer:

$(0.5 \times 8q_2)+q_2=100\\\\5q_2=100\\\\q_2=20$

since $q_2 = 20$

$q_1 = 8*20\\\\q_1=160$

Explanation:

U(q₁ q₂)

$q_1^{0.8}q_2^{0.2}\\\\P_1= \ 0.5 \ P_2=\ 1 \ Y=100$

Budget law can be given by

$P_1q_1+P_2q_2=Y\\\\0.5q_1+q_2=100$

Lagrangian function can be given by

$L=q_1^{0.8}q_2^{0.2}+ \lambda (100-0.5q_1-q_2)$

First order condition csn be given by

$\frac{dL}{dq} =0.8q_1^{-0.2}q_2^{0.2}-0.5 \lambda=0\\\\0.5 \lambda=0.8q_1^{-0.2}q_2^{0.2}---(i)$

$\frac{dL}{dq} =0.2q_1^{0.8}q_2^{-0.8}- \lambda=0\\\\ \lambda=0.2q_1^{0.8}q_2^{-0.8}---(ii)$

$\frac{dL}{d \lambda} =100-0.5q_1-q_2=0\\\\0.5q_1+q_2=100---(iii)$

From eqn (i) and eqn (ii) we have

$\frac{0.5 \lambda}{\lambda} =\frac{0.8q_1^{-0.2}q_2^{0.2}}{0.2q_1^{0.8}q_2^{-0.8}} \\\\0.5=\frac{4q_2}{q_1}\\\\q_1=8q_2}$

Putting $q_1=8q_2$ in euqtion (iii) we have

$(0.5 \times 8q_2)+q_2=100\\\\5q_2=100\\\\q_2=20$

since $q_2 = 20$

$q_1 = 8*20\\\\q_1=160$