Each week, Bill, Mary and Jane select the quantity of two goods, x1 and x2, that they will consume in order to maximize their respective utilities. They each spent their entire weekly income on these two goods.
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 Bills utility increase or decrease between week one and week 2? Between week 1 and week 3? Explain using a graph to support your answer.
Without a doubt, between week-1 and week-2, Bill's utility went down. He consumes less of everything (both x1 and x2 go down).
Without a doubt, between week-2 and week-3, Bill's utility went up. He consumes more of everything.
To determine whether Bill's utility increased or decreased between week one and week two, and between week one and week three, we need to analyze his choices based on his preferences and budget constraints.
First, let's understand the given information:
- x1 and x2 represent the quantities of two goods that Bill consumes.
- P1 and P2 represent the prices of goods x1 and x2, respectively.
- I represents Bill's income, which he spends entirely on these two goods.
To analyze these choices, we need to understand Bill's preferences, which are not explicitly given in the information. Utility is a subjective measure of satisfaction, and without the knowledge of Bill's specific utility function, we cannot directly determine whether his utility increased or decreased.
However, we can make some observations by comparing the quantities of goods consumed and the budget constraints. Let's create a graph to understand these choices visually:
1. Graph construction:
On the x-axis, plot the quantity of good x1 consumed by Bill.
On the y-axis, plot the quantity of good x2 consumed by Bill.
Plot the given choices for each week on the graph using different symbols or colors. Connect the dots representing each week's choices to visualize the path traced by Bill's consumption bundle over time.
2. Interpretation:
By analyzing the graph, we can observe the following:
a. Week 1 to Week 2:
If Bill's utility increased between week one and week two, we would expect his consumption bundle to move towards a higher level of satisfaction or a higher indifference curve. This would mean that he would consume more of both goods x1 and x2.
If the graph shows that Bill's consumption bundle in week two is above or on a higher indifference curve compared to week one, then his utility has increased. If the consumption bundle is on the same indifference curve or below the one from week one, then his utility has either remained the same or decreased.
b. Week 1 to Week 3:
Using a similar analysis, we can compare the consumption bundle between week one and week three. If Bill's utility increased, we would expect his consumption bundle to move towards an even higher level of satisfaction or a higher indifference curve. Again, if the consumption bundle is on a higher indifference curve, his utility increased; otherwise, it remained the same or decreased.
However, keep in mind that without precise details about Bill's preferences and utility function, determining the exact change in utility is not possible. We can only make assumptions based on observed changes in consumption bundles.
In conclusion, to determine whether Bill's utility increased or decreased between week one and week two, and between week one and week three, we need to analyze his choices using a graph and compare the movement of his consumption bundles between different weeks.