Question

Given the following marginal utility schedule for good X and good Y for an individual A, given that the price of X and the price of Y are both $10, and that the individual spends all his income of $70 on X and Y,

Q x          1         2       3      4      5      6        7
MUX      15       11       9      6      4      3        1
Q y           6        5       4      3      2      1        0
MUY       12        9       8      6      5      2        1

1. Provide the slope of the budget line
2. Estimate the MRS at the optimum

3. Indicate how much of X and Y the individual should purchase to maximize utility.
Select one:
a. 1. Provide the slope of the budget line: -10
2. Estimate the MRS at the optimum: -10

3. Indicate how much of X and Y the individual should purchase to maximize utility. 4X and 3Y
b. 1. Provide the slope of the budget line: -1
2. Estimate the MRS at the optimum:-1

3. Indicate how much of X and Y the individual should purchase to maximize utility: 6X and 6Y
c. 1. Provide the slope of the budget line: -10/70
2. Estimate the MRS at the optimum: -10/70

3. Indicate how much of X and Y the individual should purchase to maximize utility. 1X and 6Y
d. 1. Provide the slope of the budget line: -1
2. Estimate the MRS at the optimum: -1

3. Indicate how much of X and Y the individual should purchase to maximize utility. 7X and 0Y
e. 1. Provide the slope of the budget line: -1
2. Estimate the MRS at the optimum: -1

3. Indicate how much of X and Y the individual should purchase to maximize utility: 4X and 3Y

Answer 1

It is given that the price of X and the price of Y are both $10, and that the individual spends all his income of $70 on X and Y.

Hence, Px = $10 and Py = $10 and Income is

M = $70

Hence, the equation of the budget line is,

Px.X + Py.Y = M

or, 10.X + 10.Y = 70.........(1)

This is the budget line. If we plot X on the horizontal axis and plot Y on thr vertical axis, then the slope of the budget line in equation (1) is

Slope of the Budget Line = -10/10 = (-1)

1. Provide the slope of the budget line: -1.

Hence, we eliminate options (a) and (d) as the slopes are not correct in these two options.

Now, the following table shows the marginal utility schedule for good X and good Y for an individual A.

We also add a column for Marginal Rate of Substitution between X and Y i.e. MRSxy.

MRSxy is defined as

MRSxy = -(MUX/MUY) for each combination X and Y.

Qx	MUX	Qy	MUy	MRSxy=-(MUX/MUY)
1	15	6	12	-1.25
2	11	5	9	-1.22
3	9	4	8	-1.125
4	6	3	6	-1
5	4	2	5	-0.8
6	3	1	2	-0.67
7	1	0	1	-1

From the above table, we can see that MRS is decreasing for increasing values of Qx. Hence, MRS is diminishing. So we can say, the Utility Function is Convex. And, there is no corner solution. There is an interior solution.

According to the utility maximization theory, at the optimum level of consumption, the MRS must be equal to the slope of the budget line.

Hence, at optimum,

MRS = Slope of the Budget Line

Now, using the value from part (1) we can write,

MRS = -1

2. Estimate the MRS at the optimum: -1.

Now, from the table, we will look for the combination of (X, Y) for which MRS = -1.

Now, we find two combinations of (X, Y) for which MRS = -1.

We can see from the table

✓ When the individual buys 4X and 3Y, MRS = -1.

And, also

✓ When the individual buys 7X and 0Y, MRS = -1.

Hence, we eliminate option (b), as the MRS is not -1 for the given combination of X and Y i.e. (6X, 6Y).

Now, we found that, the utility function is Convex and there is no corner solution.

Hence, we can see that (7X, 0Y) is a corner solution. Hence, we eliminate option (d) as it gives a corner solution.

Finally, we choose the combination (4X, 3Y) for which the MRS =-1 and it is an interior solution.

3. Indicate how much of X and Y the individual should purchase to maximize utility: 4X and 3Y.

Hence, the answer is Option (e).

Hope the solution is clear to you my friend.