A consumer consuming two commodities X and Y has the following utility function U=XY+2X. If the price of the two commodities are 4 and 2 respectively and his/her budget is birr 60.
a) Find the quantities of good X and Y which will maximize utility.
b) Find the MRSxy at optimum.
a) The consumer's budget constraint can be written as PₓX + PᵧY = M, where Pₓ = 4, Pᵧ = 2, and M = 60. Solving for Y, we get:
Y = (M - PₓX)/Pᵧ = (60 - 4X)/2 = 30 - 2X
Substituting this into the utility function and differentiating with respect to X, we get:
MUₓ = Y + 2 = 32 - 2X
Setting MUₓ equal to the price of X (i.e. 4), we get:
32 - 2X = 4
2X = 28
X = 14
Substituting X = 14 into the budget constraint, we get:
PᵧY = M - PₓX
2Y = 60 - 4(14)
2Y = 12
Y = 6
Therefore, the quantities of X and Y that will maximize utility are 14 and 6, respectively.
b) The marginal rate of substitution (MRS) between X and Y is the absolute value of the slope of the indifference curve, which can be found by taking the partial derivative of U with respect to X and dividing it by the partial derivative of U with respect to Y:
MRSₓᵧ = |MUₓ/MUᵧ| = |(32 - 2X)/X|
At the optimum, X = 14, so the MRSₓᵧ is:
MRSₓᵧ = |(32 - 2(14))/14| = 1.14
Therefore, the MRSxy at optimum is 1.14.
To find the quantities of goods X and Y that will maximize utility, we first need to set up the consumer's optimization problem.
Step 1: Set up the budget constraint
The consumer's budget constraint can be expressed as:
PₓX + PᵧY ≤ M
where Pₓ and Pᵧ are the prices of goods X and Y respectively, M is the budget, and X and Y are the quantities of goods X and Y.
In this case, the prices are given as Pₓ = 4 and Pᵧ = 2, and the budget is M = 60. Thus, the budget constraint is:
4X + 2Y ≤ 60
Step 2: Set up the utility function
The utility function is given as:
U = XY + 2X
Step 3: Formulate the optimization problem
The consumer's optimization problem can be stated as:
Maximize U = XY + 2X
subject to the budget constraint: 4X + 2Y ≤ 60
Step 4: Solve the optimization problem
To solve this optimization problem, we can use the method of Lagrange multipliers.
Let L(X, Y, λ) be the Lagrangian function:
L(X, Y, λ) = XY + 2X - λ(4X + 2Y - 60)
Taking the partial derivatives of L with respect to X, Y, and λ, and set them equal to zero, we get the following equations:
∂L/∂X = Y + 2 - 4λ = 0 --(1)
∂L/∂Y = X - 2λ = 0 --(2)
∂L/∂λ = 4X + 2Y - 60 = 0 --(3)
From equation (1), we can solve for Y:
Y = 4λ - 2 --(4)
Substituting equation (4) into equation (3), we have:
4X + 2(4λ - 2) - 60 = 0
4X + 8λ - 4 - 60 = 0
4X + 8λ = 64
X + 2λ = 16 --(5)
Substituting equation (2) into equation (5), we have:
X + 2(2λ) = 16
X + 4λ = 16
X = 16 - 4λ --(6)
Now, we can substitute equation (6) into equation (4) to solve for Y:
Y = 4λ - 2
Y = 4(2λ) - 2
Y = 8λ - 2 --(7)
Substituting equations (6) and (7) into the budget constraint (4X + 2Y ≤ 60), we can solve for λ:
4(16 - 4λ) + 2(8λ - 2) ≤ 60
64 - 16λ + 16λ - 4 ≤ 60
60 - 4 ≤ 60
56 ≤ 60
Since 56 ≤ 60 is true for all values of λ, there are no restrictions on λ.
Therefore, the optimal quantities of goods X and Y that maximize utility can be obtained by substituting the values of λ into equations (6) and (7):
X = 16 - 4λ
Y = 8λ - 2
Now, we can proceed to find the MRSxy at the optimum.
To find the MRSxy (Marginal Rate of Substitution), we need to take the derivative of the utility function with respect to X (dU/dX) divided by the derivative of the utility function with respect to Y (dU/dY):
MRSᵪᵧ = (dU/dX) / (dU/dY)
Taking the derivatives of the utility function U = XY + 2X, we get:
dU/dX = Y + 2
dU/dY = X
Substituting the optimal values of X and Y that we obtained earlier:
dU/dX = (8λ - 2) + 2 = 8λ
dU/dY = 16 - 4λ
MRSxy = (dU/dX) / (dU/dY) = (8λ) / (16 - 4λ)
Thus, the MRSxy at the optimum is (8λ) / (16 - 4λ).
To find the quantities of goods X and Y that will maximize utility, we need to solve the consumer's optimization problem. In this case, the consumer wants to maximize the utility function U = XY + 2X, subject to a budget constraint.
Let's start by understanding the budget constraint. The consumer's budget is birr 60, and the prices of goods X and Y are 4 and 2 respectively. This means that the consumer's total expenditure on goods X and Y cannot exceed birr 60. Mathematically, we can express this as follows:
4X + 2Y ≤ 60
To solve this optimization problem, we use a technique called Lagrange optimization. We introduce a Lagrange multiplier (λ) to incorporate the budget constraint into the utility maximization problem. The Lagrangian function is defined as:
L(X, Y, λ) = XY + 2X + λ(60 - 4X - 2Y)
To find the quantities of goods X and Y that maximize utility, we need to solve the following system of equations:
∂L/∂X = 0 (1)
∂L/∂Y = 0 (2)
4X + 2Y = 60 (3)
Taking the partial derivative of the Lagrangian function with respect to X and setting it equal to zero, we get:
∂L/∂X = Y + 2 - 4λ = 0 (1)
Taking the partial derivative of the Lagrangian function with respect to Y and setting it equal to zero, we get:
∂L/∂Y = X - 2λ = 0 (2)
Now, we can solve equations (1) and (2) simultaneously to find the values of X, Y, and λ.
From equation (2), we have:
X = 2λ (4)
Substituting equation (4) into equation (1), we get:
Y + 2 - 4λ = 0
Y = 4λ - 2 (5)
Substituting equations (4) and (5) into equation (3) (the budget constraint), we get:
4(2λ) + 2(4λ - 2) = 60
8λ + 8λ - 4 = 60
16λ = 64
λ = 4
Now, we can find the values of X and Y using equations (4) and (5):
X = 2λ = 2(4) = 8
Y = 4λ - 2 = 4(4) - 2 = 14
Therefore, the quantities of goods X and Y that will maximize utility are X = 8 and Y = 14.
To find the marginal rate of substitution (MRS) of X for Y at the optimum, we need to calculate the partial derivatives of the utility function with respect to X and Y and divide them. The MRS is the negative ratio of the marginal utilities:
MRSxy = - (∂U/∂X) / (∂U/∂Y)
Taking the partial derivatives of the utility function, we have:
∂U/∂X = Y + 2
∂U/∂Y = X
Plugging in the values of X and Y at the optimum, we get:
∂U/∂X = 14 + 2 = 16
∂U/∂Y = 8
Now, we can calculate the MRS:
MRSxy = - (∂U/∂X) / (∂U/∂Y) = -16/8 = -2
Therefore, the MRSxy at the optimum is -2.