Economics 300: Intermediate Microeconomic Theory
University of Delaware – Spring 2023
Section 010
Solutions to Problem Set 3
Due Monday, March 13 by 11:59 pm
Total Points = 90 points
Give clear, well-written answers to the questions below. Please make your graphs legible and label all parts of the
graph (e.g., axes, lines, etc.). Assume that goods may be consumed in partial quantities (e.g., 3.54 espressos). Please
round your answers to two decimal places.
1. (30 points) Siblings Ritvik and Roshan each have $50 to spend on Xbox games (X) and all other goods (Y).
Games cost $25 each, and all other goods (AOG) have a composite price of $1 each. A friend offers them a $25
Xbox gift card or $25 cash.
a. (15 pts) Ritvik loves to play Xbox and finds AOG to be useless. Using an indifference curve and budget line
analysis, show Ritvik’s optimal choice for each option (Xbox gift card and cash gift) as well as the initial
choice. Which gift would Ritvik prefer? Does your graph match your intuition? Explain.
Step 1: Graph the budget lines for the initial choice and for the two gifts.
Initial Budget: 25X + Y = 50
$25 Cash Gift Budget: 25X + Y = 75
Gift of 1 Xbox game ($25 gift card) Budget: If he spends all his income ($50) on AOG and gets the gift, he will
have 1 game and 50 units of AOG. If he spends all his income on games and gets the gift, he will have 3
games. His budget line connects these two extreme points. Note that the slope of this budget line has not
changed because the prices of X and Y have not changed. Between these two points, this budget line is the
same as the budget line for the cash gift.
Step 2: Indifference Curves
Ritvik finds AOG to be useless. Games are driving his utility. His indifference curves are vertical.
Step 3: Identify Optimal Choice for each scenario.
Initial Optimal Choice: X* = 2 and Y* = 0. Utility denoted U1.
$30 Cash Gift Optimal Choice: Xcash = 3 and Ycash = 0. Utility denoted Ucash.
Gift of 1 game Optimal Choice: Xgame = 3 and Ygame = 0. Utility denoted Ugame.
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As the graph indicates, Ritvik receives the same utility from the gift card as the cash. He would be happy with
either gift. Ucash = Ugame
The graph matches intuition. Since Ritvik finds AOG useless, the cash will not increase his utility. Games
drive his utility. He can spend the cash on another game or receive the gift card. Either gift will give him the
same increase in utility (getting one more game).
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b. (15 pts) Roshan likes both gaming and AOG and will choose to consume positive amounts of both goods. His
preferences are convex, and he prefers AOG to gaming. Using a separate graph from part (a), show Roshan’s
optimal choice for each option (Xbox gift card and cash gift) as well as the initial choice. Which gift would
Roshan prefer? Does your graph match your intuition? Explain.
Note that the budget lines for Roshan are identical to those for Ritvik. See part (a) for budget line
explanations.
Ritvik’s indifference curves are convex, and he likes AOG. His indifference curves are relatively flat,
indicating that he places a relatively higher value on AOG.
Optimal choice – see graph.
As the graph indicates, Roshan will prefer the cash since he can then reach a higher indifference curve with
the cash than the gift card. (Ucash > Ugame).
The graph matched intuition because the cash gift allows Roshan to get more AOG, which is his relatively
stronger preference. He cannot get more AOG with the gift card. His utility will be higher under the cash gift.
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2. (10 points) Suppose Joy buys food (X) and clothing (Y) with their income of $500 per month. The prices that Joy
faces in month one are $2 per unit of food and $4 per unit of clothing. Their utility-maximizing choice is 100
food units and 75 clothing units. Their preferences are convex.
a. (4 pts) Use an indifference curve-budget line analysis to identify their choices in month one. Put food on the
horizontal axis and clothing on the vertical axis.
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b. (4 pts) Now assume that in month two, Joy's income increases to $545, the price of food increases to $2.30,
and the price of clothing increases to $4.20. Their preferences (family of indifference curves) have not
changed. Note that Joy can afford their month one choice in month two. However, is the month one choice
(X* = 100 and Y* = 75) the optimal choice in month two? Using your graph from part (a), clearly identify
Joy’s optimal choice in month two as (X**, Y**). (Note that you do not have enough information to give a
numerical answer for Joy’s month two choice.)
Joy’s month one choice is not the optimal choice in month two.
In month two, the MRS of X for Y at the month one choice (X*, Y*) is the absolute value of the slope of
U1(red). The relative price of X in terms of Y is the absolute value of the slope of I2 (blue). The MRS of X for
Y is smaller (flatter) than Px/Py (steeper). Therefore, Joy can be better off if they change their choice to more
Y, less X.
c. (2 pts) Have the income and price changes made Joy better off or worse off in month two compared to month
one? Explain.
Joy is better off in month 2 because they can reach a higher indifference curve. U2 > U1
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3. (30 points) Xavier considers lemonade (X), iced tea (Y) to be normal goods, and has convex preferences.
a. (10 pts) Design an indifference curve-budget line diagram showing the substitution and income effects created
when the price of lemonade increases. In your diagram, place lemonade on the horizontal axis and iced tea on
the vertical axis. Identify the initial optimal choice as (X*, Y*) and the optimal choice after the price increase
as (X**, Y**).
Initially, the optimal choice is represented by (X*, Y*), I1, and U1. Due to the increase in the price of
lemonade, the budget line rotates in (I2). The new optimal choice is (X**,Y**) – tangency between I2 and U2.
The SE effect is the movement along U1, from A to B. Utility is held constant. Lemonade is now more
expensive, so this person substitutes away from the relatively more expensive good (XB < X*). The MRS of X
in terms of Y at point B is the same as the MRS of X in terms of Y at point C. MRS at B = MRS at C = new
relative price of X in terms of Y (Px2/Py1).
The IE effect is the movement to a new indifference curve, from B to C. The relative price of X in terms of Y is
held constant (at the new Px2/Py1). The MRS of X in terms of Y at point C is equal to the new relative price of
X in terms of Y. Real income has fallen due the increase in price of lemonade. Since lemonade is a normal
good, the amount of lemonade will fall from XB to X**. The choice will be on a lower indifference curve.
Overall, the optimal amount of lemonade has fallen due to the increase in its price (X** < X*).
b. (4 pts) How can you tell from your diagram that lemonade and iced tea are normal goods? Explain.
Real income or purchasing power has decreased since the price of lemonade went up. This income effect is
shown in the graph as a movement from point B to point C. This person buys less lemonade and iced tea when
his income falls (X** < XB and Y**<YB). Thus, lemonade and iced tea are normal goods.
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c. (6 pts) In a new graph, draw Xavier’s demand curve for lemonade. Include both the initial choice and choice
after the price increase.
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d. (5 pts) Using your diagram in part (a), does Xavier view these two goods as substitutes or complements?
Explain. (Note: answers may vary.)
In the above graph, the quantity of iced tea increased (Y** > Y*) as the price of lemonade increased.
Therefore, this person views these two goods as substitutes. As lemonade became more expensive, the person
switched to iced tea as they considered them to be substitutes (not consumed together, consume one or the
other).
An alternative answer is shown in the graph below. Since the quantity of iced tea decreased (Y* < Y**) as the
price of lemonade increased, this person views these two goods as complements. This person likes to consume
iced tea and lemonade together. If the price of lemonade goes up, this person will consume less lemonade and
consequently less iced tea as well.
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e. (5 pts) In a new graph, draw Xavier’s initial demand curve for iced tea, including the initial choice. Assume
that it is downward sloping. Illustrate the effect of an increase in the price of lemonade on the demand curve
for iced tea in this graph.
If lemonade and iced tea are substitutes, then the demand curve for iced tea will increase.
If lemonade and iced tea are complements, then the demand curve for iced tea will decrease.
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4. (20 points) Suppose a person spends her income on a gym membership and streaming their favorite shows. Both
activities are normal, and their preferences are convex. The price of an hour at the gym is Px, the price per
streaming shows is Py, and income is I.
a. (5 pts) Use an indifference curve-budget line diagram to show their utility maximizing choice. In your
diagram, place gym hours (X) on the horizontal axis and streaming shows (Y) on the vertical axis. Denote the
initial choice as (X*, Y*).
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b. (5 pts) Now assume that the person’s preferences change. Their preferences are still convex, but they now
have a stronger preference for going to the gym than they did in part (a). Using your graph is part (a), show
their new utility-maximizing choice (X**, Y**).
The convex indifference curves are steeper since they have a stronger preference for good X.
c. (5 pts) Compare the MRS of X for Y at (X*, Y*) to the MRS of X for Y at (X**, Y**). Is the MRS from part
(a) greater than, less than, or equal to the MRS from part (b)? Briefly explain.
The MRS of X for Y is the same in both parts (a) and (b).
At the optimal choice, MRS of X for Y = Px/Py. The relative price of X in terms of Y has not changed.
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d. (5 pts) Using the information from parts (a) and (b), illustrate what happened to this person’s demand curve
for going to the gym. Clearly identify the relevant points.
Since preferences have changed, the quantity of X increases (X** > X*). Prices have not changed. When
preferences change, the demand curve for gym hours will increase (shift to the right).
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