Diogo has a utility function, U(q1, q2) = (q1^0.2)(q2^0.8) Where q1 is chocolate candy and q2 is slices of pie. If the price of slices of pie, p2 is $4.00, the price of chocolate candy, p1 is $ 2.00, and income, Y is $100, what is Diogo's optimal bundle? That is the optimal value of good q1= units

Explanation:

We have:

U (q1, q2) = q1^(0.2) q2^(0.8)

Given in the question above, we have:

p1 = $2

p2 = $4

Y = $ 100

Diogo's budget constraint is:

p1q1 + p2q2 = Y

2q1 + 4q2 = 100

Dividing through the equation by 2, we have:

q1 + 2q2 = 50

Now, we will use Langrange's theorem to calculate the optimal bundle of Diogo:

L = U (q1,q2) − λ (q1 + 2q2 − 50)

L = q1^(0.2) q2^(0.8) − λ (q1 + 2q2 − 50)

We will now calculate the first order derivatives and equate them to 0:

∂L/∂q1 =0.2q1^(−0.8) q2^(0.8) − λ = 0

Therefore:

λ = 0.2q1^(−0.8) q2^(0.8)

∂L/∂q2=0.8q1^(0.2) q2^(−0.2) − λ = 0

Therefore:

λ = 0.8q1^(0.2) q2^(−0.2)

∂L/∂λ = q1 + 2q2 − 50 = 0

This gives us:

q1 + 2q2 = 50

Now, we will equate the values of lambda (λ):

0.2q1^(−0.8) q2^(0.8) = 0.8q1^(0.2) q2^(−0.2)

q2/q1 = 4

q2 = 4q1

We now have the value of q2 to be 4q1 and we will put this value of q2 in the budget constraint to calculate the optimal bundle:

q1 + 2q2 − 50 = 0

q1 + 2(4q1) = 50

q1 + 8q1 = 50

9q1 = 50

q1 = 50/9

q1 = 5.55

Now that we have q1, we can put the value of q1 into the budget constraint equation to calculate for q2, we have:

q1 + 2q2 − 50 = 0

5.55 + 2q2 = 50

2q2 = 50 - 5.55

2q2 = 44.45

q2 = 44.45/2

q2 = 22.22

Therefore:

q1 = 5.55 units

q2 = 22.22 units

From the calculations above, we can see that the optimal value of good q1 is 5.55 units.

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