The utility that a consumer obtains by consuming goods A and B is given by U(Qa,Qb) = Qb^0.1Qa^0.1.

Question

A. With the assumption that , Pa = 10,Pb = 6 and Y = 100, calculate the optimal Qa and Qb .
B. Determine the MRSba at the optimum level

Answer 1

A. To find the optimal Qa and Qb, we need to maximize the utility function subject to the budget constraint.

The budget constraint is given by: Pa * Qa + Pb * Qb = Y, where Pa is the price of good A, Pb is the price of good B, and Y is the consumer's income.

In this case, Pa = 10, Pb = 6, and Y = 100.

The utility function is U(Qa, Qb) = Qb^0.1 * Qa^0.1.

To solve this problem, we can use the method of Lagrange multipliers.

First, set up the Lagrangian function:
L(Qa, Qb, λ) = Qb^0.1 * Qa^0.1 + λ * (Pa * Qa + Pb * Qb - Y)

Then, take the partial derivatives with respect to Qa, Qb, and λ, and set them equal to zero:

∂L/∂Qa = 0.1 * Qb^0.1 * Qa^(-0.9) + λ * Pa = 0
∂L/∂Qb = 0.1 * Qb^(-0.9) * Qa^0.1 + λ * Pb = 0
∂L/∂λ = Pa * Qa + Pb * Qb - Y = 0

Simplifying these equations, we get:

0.1 * Qb^0.1 * Qa^(-0.9) + λ * Pa = 0
0.1 * Qb^(-0.9) * Qa^0.1 + λ * Pb = 0
Pa * Qa + Pb * Qb = Y

Substituting the given values, we have:

0.1 * Qb^0.1 * Qa^(-0.9) + 10 * λ = 0 ...(1)
0.1 * Qb^(-0.9) * Qa^0.1 + 6 * λ = 0 ...(2)
10 * Qa + 6 * Qb = 100 ...(3)

To solve these equations, we can solve equations (1) and (2) simultaneously to find the values of Qa, Qb, and λ.

From equations (1) and (2), we can eliminate λ:

0.1 * Qb^0.1 * Qa^(-0.9) + 10 * λ = 0
0.1 * Qb^(-0.9) * Qa^0.1 + 6 * λ = 0

Multiplying the first equation by 6 and the second equation by 10, we get:

0.6 * Qb^0.1 * Qa^(-0.9) + 60 * λ = 0 ...(4)
1 * Qb^(-0.9) * Qa^0.1 + 60 * λ = 0 ...(5)

Now, subtract equation (5) from equation (4):

0.6 * Qb^0.1 * Qa^(-0.9) - 1 * Qb^(-0.9) * Qa^0.1 = 0

Rearranging the terms, we get:

0.6 * Qb^0.1 / Qb^(-0.9) = Qa^0.1 / Qa^(-0.9)

Simplifying further, we have:

0.6 * Qb^(0.1 + 0.9) = Qa^(0.1 - 0.9)
0.6 * Qb = Qa^(-0.8)
Qa = (0.6/Qb)^(1/0.8)

Substituting this value of Qa in equation (3), we can solve for Qb:

10 * ((0.6/Qb)^(1/0.8)) + 6 * Qb = 100

To solve this equation, we can use numerical methods such as trial and error or Newton's method to find the value of Qb.

Once we have found Qb, we can substitute it back into the equation Qa = (0.6/Qb)^(1/0.8) to find the value of Qa.

B. The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade one good for another while keeping utility constant.

MRSba = ∂U/∂Qb / ∂U/∂Qa

Differentiating the utility function with respect to Qb, we have:

∂U/∂Qb = 0.1 * Qb^(-0.9) * Qa^0.1

Differentiating the utility function with respect to Qa, we have:

∂U/∂Qa = 0.1 * Qb^0.1 * Qa^(-0.9)

Now, we can calculate MRSba at the optimal level by substituting the values of Qa and Qb obtained in part A into these derivatives and calculating the ratio.