Example: A consumer consuming two commodities X and Y has the utility function U (X ,Y ) = XY + 2X . The prices of the two commodities are 4 birr and 2 birr respectively. The consumer has a total income of 60 birr to be spent on the two goods.
Find the utility maximizing quantities of good X and Y.
Find marginal rate of substitution x to y.
To find the utility-maximizing quantities of goods X and Y, we can use the concept of marginal utility and the budget constraint.
First, let's find the marginal utility of X and Y:
The marginal utility of X (MUx) is the partial derivative of the utility function with respect to X:
MUx = ∂U/∂X = Y + 2
The marginal utility of Y (MUy) is the partial derivative of the utility function with respect to Y:
MUy = ∂U/∂Y = X
To maximize utility, the consumer should allocate their budget in such a way that the marginal utility per birr spent on each good is equal.
Next, let's find the consumer's budget constraint:
The consumer's income (I) is 60 birr, and the prices of X (Px) and Y (Py) are 4 birr and 2 birr respectively.
The budget constraint can be expressed as:
I = Px * X + Py * Y
60 = 4X + 2Y
Now, we can find the utility-maximizing quantities of X and Y by setting the marginal utility per birr spent on X equal to the marginal utility per birr spent on Y:
MUx / Px = MUy / Py
(Y + 2) / 4 = X / 2
Simplifying the equation, we have:
(Y + 2) / 4 = X / 2
2(Y + 2) = 4X
Y + 2 = 2X
Since we have two unknowns (X and Y), we need another equation to solve for their values. We can use the budget constraint equation:
60 = 4X + 2Y
Substitute Y + 2 = 2X into the budget constraint equation:
60 = 4X + 2(2X - 2)
60 = 4X + 4X - 4
60 = 8X - 4
64 = 8X
X = 8 birr
Substitute the value of X back into Y + 2 = 2X:
Y + 2 = 2(8)
Y + 2 = 16
Y = 14 birr
Therefore, the utility-maximizing quantities of goods X and Y are X = 8 birr and Y = 14 birr.
To find the marginal rate of substitution (MRS) of X to Y, we can derive the utility function with respect to X and Y:
MRS = ∂U/∂X / ∂U/∂Y
MRS = (Y + 2) / X
MRS = (14 + 2) / 8
MRS = 1.75
So, the marginal rate of substitution of X to Y is 1.75.
To find the utility maximizing quantities of goods X and Y, we'll use the concept of marginal utility and the budget constraint.
Step 1: Calculate the marginal utilities of X and Y.
The marginal utility of X is calculated by taking the derivative of the utility function with respect to X: MUx = dU/dX = Y + 2.
The marginal utility of Y is calculated by taking the derivative of the utility function with respect to Y: MUy = dU/dY = X.
Step 2: Set up the budget constraint.
The budget constraint is given by the total income of 60 birr, which is spent on the two goods. The prices of commodity X and Y are 4 birr and 2 birr respectively. Therefore, the budget constraint can be expressed as: 4X + 2Y = 60.
Step 3: Set up the condition for utility maximization.
To maximize utility, the consumer allocates their budget in such a way that the marginal utility per birr spent on both goods is equal. Mathematically, this can be expressed as: MUx / Px = MUy / Py, where Px and Py are the prices of X and Y respectively.
Step 4: Find the utility maximizing quantities.
Substituting the marginal utilities from step 1 and the prices into the condition for utility maximization, we get: (Y + 2) / 4 = X / 2.
Simplifying the equation, we have: Y + 2 = 2X.
From the budget constraint in step 2, we can express Y in terms of X: Y = (60 - 4X) / 2.
Substituting this into the equation from the condition of utility maximization, we have: (60 - 4X) / 2 + 2 = 2X.
Simplifying the equation, we get: 60 - 4X + 4 = 4X.
Continuing to simplify, we have: 8X = 56.
Solving for X, we get: X = 7.
Substituting this into the equation for Y, we have: Y = (60 - 4(7)) / 2 = 17.
Therefore, the utility maximizing quantities of X and Y are X = 7 and Y = 17.
Step 5: Find the marginal rate of substitution (MRS) of X to Y.
The marginal rate of substitution is the ratio of the marginal utilities of X and Y. From step 1, we found that MUx = Y + 2 and MUy = X.
Therefore, MRSx,y = MUx / MUy = (Y + 2) / X.
Substituting the utility maximizing quantities from step 4, we get: MRSx,y = (17 + 2) / 7 = 19/7.
The utility maximizing quantities of goods X and Y are X = 7 and Y = 17, and the marginal rate of substitution (MRS) of X to Y is 19/7.