Answer:
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.
