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
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.
To find the quantities of goods X and Y that will maximize utility, we need to maximize the utility function subject to the budget constraint.
Given:
Utility function U = X^2 * Y^2
Price of X = 1 Birr
Price of Y = 4 Birr
Budget = 10 Birr
a) To maximize utility, we can use the Lagrangian method:
L = X^2 * Y^2 - λ(1X + 4Y - 10)
Taking partial derivatives and setting them equal to zero, we get:
∂L/∂X = 2XY^2 - λ = 0
∂L/∂Y = 2X^2Y - 4λ = 0
∂L/∂λ = 1X + 4Y - 10 = 0
Solving these three equations simultaneously will give us the optimal quantities of X and Y.
b) To find the total utility at the optimum point, substitute the optimal quantities of X and Y into the utility function U=X^2Y^2.
c) The marginal rate of substitution (MRS) at the optimum point is given by the ratio of the partial derivatives of the utility function with respect to X and Y (∂U/∂X)/(∂U/∂Y). Evaluate this ratio at the optimal point.
d) To show the optimum point graphically, we will plot the budget constraint and the utility function in a graph where X and Y are the axes. The optimum point will be the intersection of the utility function and the budget constraint.
Please note that the exact numerical values are required to calculate the solutions for parts a, b, and c.
To find the quantities of goods X and Y that will maximize utility, we can use the concept of consumer optimization, which involves maximizing utility subject to a budget constraint.
a) Maximizing utility subject to a budget constraint:
1. Write down the utility function: U = X^2 * Y^2
2. Write down the budget constraint: P_X * X + P_Y * Y = Income, where P_X and P_Y are the prices of goods X and Y, and Income is the consumer's budget.
In this case, P_X = 1, P_Y = 4, and Income = 10.
So, the budget constraint equation becomes: 1 * X + 4 * Y = 10
3. Substitute the budget constraint equation into the utility function: U = X^2 * Y^2
4. Rewrite the utility function in terms of a single variable: U = (10 - 4Y)^2 * Y^2
5. Maximize the rewritten utility function by taking the derivative with respect to Y and setting it to zero: dU/dY = 0
Solve for Y.
6. Once you have found the value of Y, substitute it back into the budget constraint equation to find the corresponding value of X.
b) Total utility at the optimum point:
Once you have determined the quantities of X and Y at the optimum point, substitute these values into the utility function U = X^2 * Y^2 to calculate the corresponding total utility.
c) Marginal Rate of Substitution (MRS) at the optimum point:
To find the MRS at the optimum point, 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) at the optimum point.
d) Showing the optimum point graphically:
Plot the budget constraint line on a graph with X on the horizontal axis and Y on the vertical axis. The slope of the budget constraint line will be -P_X/P_Y. Then, plot the utility function on the graph with various combinations of X and Y. The utility function should have a shape similar to a concave function. The optimum point will be at the intersection of the utility function with the budget constraint line.