There are two goods 1 and 2. Denote by (x1; x2) the consumption bundle consisting of x1
units of good 1 and x2 units of good 2. A consumer has preferences that described by the
linear utility function:
u(x1; x2) = x1 + 2x2:
1. Write down the equation for the indi�erence curve that passes through the consumption
bundle (2; 1). Draw that indi�erence curve in a diagram with x1 on the horizontal axis
and x2 on the vertical axis. In the same diagram, draw the indi�erence curve along
which the consumer's utility is 2. Determine the marginal rate of substitution (MRS).
To write down the equation for the indifference curve passing through the consumption bundle (2, 1), we need to set the utility function equal to a constant value, which represents the level of satisfaction or utility for the consumer.
Given that the utility function is u(x1, x2) = x1 + 2x2, we can set it equal to a constant, say k. So, the equation becomes:
x1 + 2x2 = k
Now, let's find the value of k using the consumption bundle (2, 1):
2 + 2(1) = k
2 + 2 = k
k = 4
Therefore, the equation for the indifference curve passing through the consumption bundle (2, 1) is x1 + 2x2 = 4.
To draw the indifference curve, we can use the equation x1 + 2x2 = 4 and plot a range of values for x1 and x2. Assuming the ranges for x1 and x2 are from 0 to 4, we can choose different values within that range and plot the corresponding points on a diagram.
Now, let's draw the indifference curve where the utility is equal to 2. To find its equation, we set the utility function equal to 2:
x1 + 2x2 = 2
Using the same range of values for x1 and x2, we can plot the corresponding points on the diagram.
The marginal rate of substitution (MRS) is the rate at which a consumer is willing to exchange one good for another while keeping the same level of satisfaction. In the case of a linear utility function, the MRS is constant along the indifference curve.
To calculate the MRS, we can use the following formula:
MRS = - (MUx1 / MUx2)
Where MUx1 is the marginal utility of good 1 and MUx2 is the marginal utility of good 2.
In this case, the marginal utility of good 1 (MUx1) is always 1, and the marginal utility of good 2 (MUx2) is always 2.
Therefore, the MRS is calculated as:
MRS = - (1/2) = -0.5
This means that the consumer is willing to give up 0.5 units of good 2 for each additional unit of good 1, while keeping the same level of satisfaction.