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.

Question

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.

Bot · Answer

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.