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.
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]
To find the optimal quantities of goods A and B, we need to maximize the utility function subject to the budget constraint.
Given:
Price of good A (Pa) = 10
Price of good B (Pb) = 6
Income (Y) = 100
The budget constraint can be expressed as:
Pa * Qa + Pb * Qb = Y
Substituting the given values:
10 * Qa + 6 * Qb = 100
To solve for Qa and Qb, we need to use the Lagrange multiplier method. The Lagrangian function is:
L(Qa, Qb, λ) = U(Qa, Qb) - λ(Pa * Qa + Pb * Qb - Y)
Taking the partial derivatives with respect to Qa, Qb, and λ, and setting them equal to 0, we have:
dL/dQa = 0.1 * Qb^0.1 * Qa^-0.9 - λ * Pa = 0
dL/dQb = 0.1 * Qa^0.1 * Qb^-0.9 - λ * Pb = 0
dL/dλ = Pa * Qa + Pb * Qb - Y = 0
Simplifying these equations, we have:
0.1 * Qb^0.1 * Qa^-0.9 - λ * 10 = 0 -> (1)
0.1 * Qa^0.1 * Qb^-0.9 - λ * 6 = 0 -> (2)
10 * Qa + 6 * Qb - 100 = 0 -> (3)
From equation (3), we can solve for Qa in terms of Qb:
Qa = (100 - 6 * Qb) / 10 -> (4)
Substituting this value of Qa into equation (1), we have:
0.1 * Qb^0.1 * (100 - 6 * Qb)^-0.9 - λ * 10 = 0
Rearranging and simplifying, we get:
Qb^0.1 * (100 - 6 * Qb)^-0.9 = 10 * λ
Similarly, substituting the value of Qa from equation (4) into equation (2), we have:
0.1 * (100 - 6 * Qb)^0.1 * Qb^-0.9 - λ * 6 = 0
Rearranging and simplifying, we get:
(100 - 6 * Qb)^0.1 * Qb^-0.9 = 6 * λ/10
Simplifying further, we have:
(100 - 6 * Qb)^0.1 / Qb = 6 * λ/10
Squaring both sides, we get:
(100 - 6 * Qb)^0.2 / Qb^2 = (6 * λ/10)^2
(100 - 6 * Qb)^0.2 = (6 * λ/10)^2 * Qb^2
Taking the 5th root of both sides, we have:
(100 - 6 * Qb)^(0.2/5) = (6 * λ/10)^2 * Qb^(2/5)
Simplifying further:
(100 - 6 * Qb)^(0.04) = (36 * λ^2/100) * Qb^(2/5)
Now, we have a relationship between Qb and λ. We can solve this equation numerically using a calculator or software to find the optimal Qb. Once we have Qb, we can substitute it back into equation (4) to find the optimal Qa.
To calculate MRSba (Marginal Rate of Substitution of B for A) at the optimum, we can differentiate the utility function with respect to Qa and Qb:
∂U/∂Qa = 0.1 * Qb^0.1 * Qa^-0.9
∂U/∂Qb = 0.1 * Qa^0.1 * Qb^-0.9
The MRSba is the ratio of these two derivatives:
MRSba = (∂U/∂Qa) / (∂U/∂Qb)
Substituting the optimal Qa and Qb values, we can calculate MRSba.
To find the optimal quantities of goods A and B and the Marginal Rate of Substitution (MRS) at the optimum level, we need to maximize the utility function U(Qa,Qb) while satisfying the given conditions.
Step 1: Write down the budget constraint
The budget constraint represents the total amount of money (Y) available to spend on goods A and B. In this case, Y = 100.
Step 2: Set up the Lagrangian function
The Lagrangian function incorporates both the utility function and the budget constraint. It is defined as L(Qa, Qb, λ) = U(Qa,Qb) - λ(Y - PaQa - PbQb), where λ is the Lagrange multiplier.
In this case, the Lagrangian function is:
L(Qa, Qb, λ) = Qb^0.1Qa^0.1 - λ(100 - 10Qa - 6Qb)
Step 3: Find the first-order conditions
To find the optimal quantities, we need to take partial derivatives of the Lagrangian with respect to Qa, Qb, and λ and set them equal to zero:
∂L/∂Qa = 0: 0.1Qb^0.1Qa^(-0.9) - λ(-10) = 0 (equation 1)
∂L/∂Qb = 0: 0.1Qa^0.1Qb^(-0.9) - λ(-6) = 0 (equation 2)
∂L/∂λ = 0: 100 - PaQa - PbQb = 0 (equation 3)
Step 4: Solve the equations simultaneously
Solve the three equations (equations 1, 2, and 3) simultaneously to find the optimal values of Qa, Qb, and λ.
a) With the assumption that Pa = 10, Pb = 6, and Y = 100, we can substitute these values into the equations and solve for Qa, Qb, and λ:
From equation 1:
0.1Qb^0.1Qa^(-0.9) + 10λ = 0
From equation 2:
0.1Qa^0.1Qb^(-0.9) + 6λ = 0
From equation 3:
100 - 10Qa - 6Qb = 0
Solve these equations simultaneously to find the values of Qa, Qb, and λ.
Step 5: Calculate the Marginal Rate of Substitution (MRS)
The MRS represents the rate at which a consumer is willing to trade one good for another while maintaining the same level of satisfaction/utility.
The MRS is given by the ratio of the marginal utilities of goods A and B: MRSba = (∂U/∂Qa) / (∂U/∂Qb).
Differentiate the utility function U(Qa, Qb) with respect to Qa and Qb separately to find their partial derivatives.
MRSba = (0.1Qb^0.1Qa^(-0.9)) / (0.1Qa^0.1Qb^(-0.9))
Finally, evaluate the MRSba at the optimal level of Qa and Qb to get the answer.