To solve this problem, we need to find the utility-maximizing quantities of X and Y for the given budget and prices. We can do this by applying the Lagrange multiplier method.
a) Set up the following Lagrangian function:
L = X^2Y^2 - λ(PxX + PyY - M)
Where:
L = Lagrangian function
X = quantity of good X
Y = quantity of good Y
λ = Lagrange multiplier
Px = price of good X (1 Birr)
Py = price of good Y (4 Birr)
M = budget (10 Birr)
Now, take the partial derivatives of L with respect to X, Y, and λ and set them equal to zero:
∂L/∂X = 2XY^2 - λPx = 0
∂L/∂Y = 2X^2Y - λPy = 0
∂L/∂λ = PxX + PyY - M = 0
Solving these equations simultaneously, we get:
2XY^2 = λPx -- (1)
2X^2Y = λPy -- (2)
PxX + PyY = M -- (3)
Now, divide equation (1) by equation (2) to get:
Y/X = Px/Py
Y = (Px/Py)X -- (4)
Substitute equation (4) into equation (3):
PxX + Py((Px/Py)X) = M
PxX + PxX = M
2PxX = M
X = M / (2Px)
Substitute the value of X into equation (4):
Y = (Px/Py)(M / (2Px))
Y = M / (2Py)
Substitute the given values:
Px = 1 Birr
Py = 4 Birr
M = 10 Birr
X = 10 / (2 * 1) = 5
Y = 10 / (2 * 4) = 1.25
So, the quantities of goods X and Y that will maximize utility are X = 5 and Y = 1.25.
b) To find the total utility at the optimum point, substitute the values of X and Y into the utility function:
U = X^2Y^2
U = 5^2 * 1.25^2
U = 31.25
Therefore, the total utility at the optimum point is 31.25.
c) To find the marginal rate of substitution (MRS) at the optimum point, take the partial derivatives of the utility function with respect to X and Y:
∂U/∂X = 2XY^2 = 2 * 5 * 1.25^2 = 15.625
∂U/∂Y = 2X^2Y = 2 * 5^2 * 1.25 = 62.5
MRS = ∂U/∂X / ∂U/∂Y
MRS = 15.625 / 62.5
MRS = 0.25
Therefore, the MRS at the optimum point is 0.25.
d) To show the optimum point graphically, we can plot the budget constraint and the utility function on a graph with X on the x-axis and Y on the y-axis.
First, plot the budget constraint:
PxX + PyY = M
1X + 4Y = 10
Y = (10 - X) / 4
Next, plot the utility function:
U = X^2Y^2
Graphically, the optimum point is where the budget constraint is tangent to the utility function. The coordinates of this point are X = 5 and Y = 1.25.
By plotting the budget constraint and the utility function on the same graph, we can visually show the optimum point.
A consumer consuming two commodities X and Y has the following utility function U=X^2Y^2.If the price of the two commodities are Birr 1 and 4 respectively and his/her budget is birr 10.
a)Find the quantities of good X and Y which will maximize utility.
b)Total utility at optimum point
c)Find the MRS x,y at optimum point
d)Show optimum point graphicall
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