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

`(0.5 * 8q_2)+q_2=100\\\\5q_2=100\\\\q_2=20`

since
`q_2 = 20`

`q_1 = 8*20\\\\q_1=160`

Step-by-step 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

`(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)`

`(dL)/(dq) =0.2q_1^(0.8)q_2^(-0.8)- \lambda=0\\\\ \lambda=0.2q_1^(0.8)q_2^(-0.8)---(ii)`

`(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

`(0.5 \lambda)/(\lambda) =(0.8q_1^(-0.2)q_2^(0.2))/(0.2q_1^(0.8)q_2^(-0.8)) \\\\0.5=(4q_2)/(q_1)\\\\q_1=8q_2}`

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

`(0.5 * 8q_2)+q_2=100\\\\5q_2=100\\\\q_2=20`

since
`q_2 = 20`

`q_1 = 8*20\\\\q_1=160`