Question

Diogo has a utility​ function, ​U(q 1​, q 2​)equalsq 1 Superscript 0.8 Baseline q 2 Superscript 0.2​, where q 1 is chocolate candy and q 2 is slices of pie. If the price of slices of​ pie, p 2​, is ​$4.00​, the price of chocolate​ candy, p 1​, is ​$8.00​, and​ income, Y, is ​$100​, what is​ Diogo's optimal​ bundle? The optimal valueLOADING... of good q 1 is

Answer 1

The value of "
`\bold{q_1=2.5}`".

Step-by-step explanation:

Given value:

`U= Max \ q_1^(0.8) \ q_2^(0.2)\\\\`

Differentiate the above equation with respect of
`q_1`, which will give
`MUq_1` as follows:

`MUq_1= q_2^(0.2)((0.2)/(q_1^(0.8)))\\\\`

`=0.2(( q_2^(0.2))/(q_1^(0.8)))`

Differentiate the equation with respect of
`q_2`, which will give
`MUq_2` as follows:

`MUq_2= q_1^(0.8)((0.8)/(q_1^(0.8)))\\\\`

`=0.8(( q_1^(0.8))/(q_2^(0.2)))}{}`

for balancing the equation

`(MUq_1)/(P_1)=(MUq_2)/(P_2)\\\\(MUq_1)/(MUq_2)=(P_1)/(P_2)\\\\`

`\frac{0.2(( q_2^(0.2))/(q_1^(0.8)))} {0.8(( q_1^(0.8))/(q_2^(0.2)))}}= (8)/(4)\\\\\frac{(( q_2^(0.2))/(q_1^(0.8)))} {4(( q_1^(0.8))/(q_2^(0.2)))}}= (2)/(1)\\\\\frac{(( q_2^(0.2))/(q_1^(0.8)))} {(( q_1^(0.8))/(q_2^(0.2)))}}= 8\\\\(q_2)/(q_1)=8\\\\q_2=8q_1\\\\`

Calculate the value of
`q_1` and
`q_2` as follows:

`100 =p_1q_1+P_2q_2\\\\100= 8q_1+4(8q_1)\\\\100=8q_1+32q_1\\\\100=40q_1\\\\q_1=(100)/(40)\\\\q_1=2.5`

`q_2=8q_1\\\\\therefore q_1=2.5\\\\q_2=8* 2.5\\\\q_2=20.0\\\\q_2=20`