Question

Each week, Bill, Mary, and Jane select the quantity of two goods,x1 andx2, that they will consume in order to maximize their respective utilities. They each spend their entire weekly income on these two goods.

a. Suppose you are given the following information about the choices that Bill makes over a three-week period:

	X1	X2	P1	P2	I
Week 1	10	20	2	1	40
Week 2	7	19	3	1	40
Week 3	8	31	3	1	55

Did Bill’s utility increase or decrease between week1 and week 2? Between week 1 and week 3? Explain using a graph to support your answer.

b. Now consider the following information about the choices that Mary makes:

	X1	X2	P1	P2	I
Week 1	10	20	2	1	40
Week 2	6	14	3	2	40
Week 3	20	10	3	2	60

Did Mary’s utility increase or decrease between week 1 and week 3? Does Mary consider both goods to be normal goods? Explain.

c. Finally, examine the following information about Janie's choices:

	X1	X2	P1	P2	I
Week 1	12	24	2	1	48
Week 2	16	32	1	1	48
Week 3	12	24	1	1	36

Draw a budget line-indifference curve graph that illustrates Jane’s three chosen bundles. What can you say about Jane’s preferences in this case? Identify the income and substitution effects that result from a change in the price of good x1.

Short answer

a. Bill’s utility will fall between week 1 and 2 and rise between week 1 and week 3.

The following is the graph showing the changes in consumption over the weeks:

b. Mary’s utility went up between week 1 and week 3 since she consumed more of all goods in week 3. X1 is a normal good for her since she increased the consumption of this good as her income increased, and X2 is inferior good for her since its consumption decreased as her income increased.

c. Jane’s indifference curve and the budget line is given below:

The indifference curves are L-shaped, saying that Jane prefers to consume two goods in a fixed ratio.

The substitution effect is zero since the ratio between goods consumed is not changed.

The income effect can be traced from the table, as the income fell to 36 when Jane consumed lesser units of X1 and X2. This is purely an income effect since prices are the same.

Step-by-step solution

Explanation of part (a)

In the graph below, good 1, i.e., X₁ is taken on the x-axis, and good 2, i.e., X₂ on the y-axis:

The utility derived from the consumption of the commodities is given by U₁, U₂, and U₃for each week. You can see that U₂ is less than U₁. This is due to the increase in the price of good 1 from $2 to $3, which resulted in the inward movement of the budget line on the x-axis. The new optimum consumption is achieved when the U₂curve intersects this new budget line.Therefore, you can conclude that the utility fell between week 1 and week 2.

You can see that the income increases from week 1 ($40) to week 3 ($55). Thus, B3 represents the new budget line (parallel shift in budget line) from B₂. The new optimum consumption is achieved when the U₃ curve intersects this new budget line, and since U₃ is more than U₁, the utility has increased between week 1 and week 3.

Explanation of part (b)

Mary’s income increased from $40 to $60 from week 1 to week 3. This means her utility level has increased (parallel forward shift in budget line).

At the same time, her consumption of good 1 or X₁ increased from 10 units to 20 units, while her consumption of good 2 or X₂ decreased from 20 units to 10 units. This means she considers X₁ as a normal good since, in the case of such goods, the demand increases with an increase in income. She considers X₂ as an inferior good since, in the case of such goods, the demand falls with an increase in income.

Explanation of part (c)

In the graph below, X1 is taken along the x-axis and SX2 along the y-axis.

The pink lines on the graph show the budget lines, and the blue lines show the indifference curves, which are L-shaped. The table shows that the ratio between X1 and X2 is 2:1 for all the weeks. This signifies that they are complementary goods, and jane prefers to consume them this way.

Since it is established that the goods are complementary, one can conclude that the substitution effect is absent. The income effect can be traced down from the table where, with the reduction in income in week 3 from 46-38, the consumption of both the goods reduced from 16-12 and 32-24, even though prices are constant. Depicting that demand decreased due to a decrease in income.