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 , Pa = 10,Pb = 6 and Y = 100, calculate the optimal Qa and Qb .
B. Determine the MRSba at the optimum level
To find the optimal quantities of goods A and B, we need to maximize the utility function U(Qa, Qb) subject to the budget constraint.
A.
1. The budget constraint is Px * Qx + Py * Qy = Y, where Px and Py are the prices of goods X and Y, Qx and Qy are the quantities of goods X and Y, and Y is the consumer's income.
In this case, Pa = 10, Pb = 6, and Y = 100. So, the budget constraint becomes 10 * Qa + 6 * Qb = 100.
2. To find the optimal quantities, we need to differentiate the utility function with respect to each good and set the derivatives equal to each other, i.e., ∂U/∂Qa = ∂U/∂Qb.
∂U/∂Qa = 0.1 * Qb^0.1 * 0.1 * Qa^(-0.9) = 0.01 * Qb^0.1 * Qa^(-0.9)
∂U/∂Qb = 0.1 * Qb^(-0.9) * 0.1 * Qa^0.1 = 0.01 * Qb^(-0.9) * Qa^0.1
Setting these derivatives equal to each other and solving for Qa:
0.01 * Qb^0.1 * Qa^(-0.9) = 0.01 * Qb^(-0.9) * Qa^0.1
Qb^0.1 * Qa^(-0.9) = Qb^(-0.9) * Qa^0.1
Qb^0.1 / Qb^(-0.9) = Qa^0.1 / Qa^(-0.9)
Qb^1 = Qa^1
Qb = Qa
3. Substituting Qb = Qa into the budget constraint:
10 * Qa + 6 * Qa = 100
16 * Qa = 100
Qa = 100 / 16
Qa = 6.25
4. Since Qb = Qa, Qb also equals 6.25.
Therefore, the optimal quantities of goods A and B are Qa = 6.25 and Qb = 6.25.
B.
To find the marginal rate of substitution (MRSba) at the optimum level, we need to calculate the partial derivatives of the utility function:
∂U/∂Qb = 0.1 * Qb^(-0.9) * 0.1 * Qa^0.1 = 0.01 * Qb^(-0.9) * Qa^0.1
∂U/∂Qa = 0.1 * Qb^0.1 * 0.1 * Qa^(-0.9) = 0.01 * Qb^0.1 * Qa^(-0.9)
MRSba = ∂U/∂Qb / ∂U/∂Qa = (0.01 * Qb^(-0.9) * Qa^0.1) / (0.01 * Qb^0.1 * Qa^(-0.9))
= Qb^(-0.9) * Qa^0.1 / Qb^0.1 * Qa^(-0.9)
= (Qb^(-0.9) / Qb^0.1) * (Qa^0.1 / Qa^(-0.9))
= Qb^(-1) * Qa / Qb * Qa^(-1)
= Qa / Qb
At the optimum level, Qa = 6.25 and Qb = 6.25, so
MRSba = 6.25 / 6.25
MRSba = 1
Therefore, the MRSba at the optimum level is 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.
A. To find the optimal quantities of goods A and B, we need to maximize the utility function U(Qa, Qb) subject to the budget constraint.
Given:
Pa = 10 (Price of good A)
Pb = 6 (Price of good B)
Y = 100 (Income)
To solve for the optimal quantities, we can use the Lagrange multiplier method.
Step 1: Set up the Lagrangian function.
L(Qa, Qb, λ) = Qb^0.1 * Qa^0.1 + λ(Y - Pa * Qa - Pb * Qb)
Step 2: Take the partial derivatives.
∂L/∂Qa = 0.1 * Qb^0.1 * Qa^(-0.9) - λ * Pa
∂L/∂Qb = 0.1 * Qb^(-0.9) * Qa^0.1 - λ * Pb
∂L/∂λ = Y - Pa * Qa - Pb * Qb
Step 3: Set the partial derivatives equal to zero and solve the equations simultaneously.
0.1 * Qb^0.1 * Qa^(-0.9) - λ * Pa = 0
0.1 * Qb^(-0.9) * Qa^0.1 - λ * Pb = 0
Y - Pa * Qa - Pb * Qb = 0
Step 4: Solve the equations to find the optimal quantities.
From the first equation:
0.1 * Qb^0.1 * Qa^(-0.9) = λ * Pa
Qb^0.1 * Qa^(-0.9) = 10 * λ
From the second equation:
0.1 * Qb^(-0.9) * Qa^0.1 = λ * Pb
Qb^(-0.9) * Qa^0.1 = 6 * λ
Divide the two equations to eliminate the λ:
(Qb^0.1 * Qa^(-0.9)) / (Qb^(-0.9) * Qa^0.1) = (10 * λ) / (6 * λ)
Qb^1 * Qa^(-1) = 5/3
Then simplify:
Qb / Qa = 5/3
Qb = (5/3) * Qa
Substitute this into the budget constraint equation:
Y - Pa * Qa - Pb * Qb = 0
100 - 10 * Qa - 6 * ((5/3) * Qa) = 0
100 - 10Qa - 10Qa = 0
100 - 20Qa = 0
20Qa = 100
Qa = 5
Substitute Qa back into Qb = (5/3) * Qa:
Qb = (5/3) * 5
Qb = 8.33
Therefore, the optimal quantities are Qa = 5 and Qb = 8.33.
B. The marginal rate of substitution (MRS) measures the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. It is given by the ratio of the partial derivatives of the utility function with respect to the quantities of each good.
MRSba = ∂U/∂Qb / ∂U/∂Qa
Taking partial derivatives of the utility function:
∂U/∂Qb = 0.1 * Qb^(-0.9) * Qa^0.1
∂U/∂Qa = 0.1 * Qb^0.1 * Qa^(-0.9)
Plugging in the optimal quantities:
∂U/∂Qb = 0.1 * (8.33)^(-0.9) * (5)^0.1
∂U/∂Qa = 0.1 * (8.33)^0.1 * (5)^(-0.9)
Calculating the values:
∂U/∂Qb ≈ 0.0604
∂U/∂Qa ≈ 0.0302
MRSba = (0.0604) / (0.0302) ≈ 2
Therefore, the marginal rate of substitution at the optimum level is approximately 2.