To find the optimal quantity of goods A and B, we need to maximize the utility function U(Qa, Qb) subject to the budget constraint.
The budget constraint is given by: PbQb + PaQa = Y
Substituting the given values: 6Qb + 10Qa = 100
To solve for the optimal quantities, we can use the Lagrange Multiplier method. We set up the following equation:
L(Qa, Qb, λ) = U(Qa, Qb) - λ(PbQb + PaQa - Y)
Taking the partial derivatives of L with respect to Qa, Qb, and λ:
∂L/∂Qa = 0.1Qb^0.1 * 0.1Qa^-0.9 - λPa = 0 (1)
∂L/∂Qb = 0.1Qa^0.1 * 0.1Qb^-0.9 - λPb = 0 (2)
∂L/∂λ = -(PbQb + PaQa - Y) = 0 (3)
Solving equations (1) and (2) simultaneously, we can find the values of Qa and Qb:
0.1Qb^0.1 * 0.1Qa^-0.9 - λPa = 0
0.1Qa^0.1 * 0.1Qb^-0.9 - λPb = 0
0.01Qb^0.1 * Qa^-0.9 = λPa
0.01Qa^0.1 * Qb^-0.9 = λPb
Dividing the two equations:
(Qb/Qa)^0.1 = (Pa/Pb)
Taking both sides to the power of 10:
Qb/Qa = (Pa/Pb)^10
Now we substitute the given values: Pa = 10 and Pb = 6
Qb/Qa = (10/6)^10
Qb/Qa = (5/3)^10
Taking the 10th root of both sides:
Qb/Qa = (5/3)^1
Now, we solve for Qb/Qa:
Qb/Qa = 5/3
We can rewrite the budget constraint as:
6Qb + 10Qa = 100
Qb = (100 - 10Qa)/6
Substituting Qb/Qa with 5/3:
(100 - 10Qa)/6 = (5/3)Qa
Multiplying both sides by 6 to remove the denominator:
100 - 10Qa = 10Qa
Simplifying:
20Qa = 100
Dividing by 20:
Qa = 5
Plugging Qa = 5 into the budget constraint:
Qb = (100 - 10(5))/6
Qb = 5/3
So, the optimal quantities are Qa = 5 and Qb = 5/3.
To find the MRSba at the optimum level, we need to find the marginal utility of A (MUa) and the marginal utility of B (MUb) and take the ratio:
MUa = ∂U/∂Qa = 0.1Qb^0.1 * 0.1Qa^-0.9
MUa = 0.1(5/3)^0.1 * 0.1(5)^-0.9
MUb = ∂U/∂Qb = 0.1Qa^0.1 * 0.1Qb^-0.9
MUb = 0.1(5)^0.1 * 0.1(5/3)^-0.9
MRSba = MUa / MUb
MRSba = [0.1(5/3)^0.1 * 0.1(5)^-0.9] / [0.1(5)^0.1 * 0.1(5/3)^-0.9]
1)The utility that a consumer obtains by consuming goods A and B is given by U(Qa,Qb)= Qb^0.1Qa^0.1.
a)With the assumption that Pb=6 and Pa=10, Y = 100, calculate the optimal Qa and Qb .
Determine the MRSba at the optimum level.
1 answer