# Econ 100A – Problem Set #1

University of California at Irvine

Econ 100A – Problem Set #1 – Amjad Toukan – Summer I 2021

Due on Friday July 2^{nd}, 2021

1. Draw indifference curves that represent the following individuals’ preferences for hamburgers and soft drinks. Indicate the direction in which the individuals’ satisfaction (or utility) is increasing.

a. Joe has convex preferences and dislikes both hamburgers and soft drinks.

b. Jane loves hamburgers and dislikes soft drinks. If she is served a soft drink, she will pour it down the drain rather than drink it.

c. Bob loves hamburgers and dislikes soft drinks. If he is served a soft drink, he will drink it to be polite.

d. Molly loves hamburgers and soft drinks, but insists on consuming exactly one soft drink for every two hamburgers that she eats.

e. Bill likes hamburgers, but neither likes nor dislikes soft drinks.

f. Mary always gets twice as much satisfaction from an extra hamburger as she does from an extra soft drink.

2. If Jane is currently willing to trade 4 movie tickets for 1 basketball ticket, then she must like basketball better than movies. True or false? Explain.

3. Janelle and Brian each plan to spend $20,000 on the styling and gas mileage features of a new car. They can each choose all styling, all gas mileage, or some combination of the two. Janelle does not care at all about styling and wants the best gas mileage possible. Brian likes both equally and wants to spend an equal amount on each. Using indifference curves and budget lines, illustrate the choice that each person will make.

4. Suppose that Bridget and Erin spend their incomes on two goods, food (*F*) and clothing (*C*). Bridget’s preferences are represented by the utility function , while Erin’s preferences are represented by the utility function .

a. With food on the horizontal axis and clothing on the vertical axis, identify on a graph the set of points that give Bridget the same level of utility as the bundle (10,5). Do the same for Erin on a separate graph.

b. On the same two graphs, identify the set of bundles that give Bridget and Erin the same level of utility as the bundle (15,8).

c. Do you think Bridget and Erin have the same preferences or different preferences? Explain.

5. The price of DVDs (*D*) is $20 and the price of CDs (*C*) is $10. Philip has a budget of $100 to spend on the two goods. Suppose that he has already bought one DVD and one CD. In addition there are 3 more DVDs and 5 more CDs that he would really like to buy.

a. Given the above prices and income, draw his budget line on a graph with CDs on the horizontal axis.

b. Considering what he has already purchased and what he still wants to purchase, identify the three different bundles of CDs and DVDs that he could choose. For this part of the question, assume that he cannot purchase fractional units.

6. Debra usually buys a soft drink when she goes to a movie theater, where she has a choice of three sizes: the 8-ounce drink costs $1.50, the 12-ounce drink $2.00, and the 16-ounce drink $2.25. Describe the budget constraint that Debra faces when deciding how many ounces of the drink to purchase. (Assume that Debra can costlessly dispose of any of the soft drink that she does not want.)

7. Consumers in Georgia pay twice as much for avocados as they do for peaches. However, avocados and peaches are the same price in California. If consumers in both states maximize utility, will the marginal rate of substitution of peaches for avocados be the same for consumers in both states? If not, which will be higher?

8. Brenda wants to buy a new car and has a budget of $25,000. She has just found a magazine that assigns each car an index for styling and an index for gas mileage. Each index runs from 1 to 10, with 10 representing either the most styling or the best gas mileage. While looking at the list of cars, Brenda observes that on average, as the style index increases by one unit, the price of the car increases by $5000. She also observes that as the gas-mileage index rises by one unit, the price of the car increases by $2500.

a. Illustrate the various combinations of style (*S*) and gas mileage (*G*) that Brenda could select with her $25,000 budget. Place gas mileage on the horizontal axis.

b. Suppose Brenda’s preferences are such that she always receives three times as much satisfaction from an extra unit of styling as she does from gas mileage. What type of car will Brenda choose?

c. Suppose that Brenda’s marginal rate of substitution (of gas mileage for styling) is equal to *S*/(4*G*). What value of each index would she like to have in her car?

d. Suppose that Brenda’s marginal rate of substitution (of gas mileage for styling) is equal to (3*S*)/*G*. What value of each index would she like to have in her car?

9. Connie has a monthly income of $200 that she allocates among two goods: meat and potatoes.

a. Suppose meat costs $4 per pound and potatoes $2 per pound. Draw her budget constraint.

b. Suppose also that her utility function is given by the equation *U*(*M*, *P*) = 2*M* + *P*. What combination of meat and potatoes should she buy to maximize her utility? (*Hint*: Meat

and potatoes are perfect substitutes.)

c. Connie’s supermarket has a special promotion. If she buys 20 pounds of potatoes (at $2 per pound), she gets the next 10 pounds for free. This offer applies only to the first 20 pounds she buys. All potatoes in excess of the first 20 pounds (excluding bonus potatoes) are still

$2 per pound. Draw her budget constraint.

d. An outbreak of potato rot raises the price of potatoes to $4 per pound. The supermarket ends its promotion. What does her budget constraint look like now? What combination of meat and potatoes maximizes her utility?

10. Jane receives utility from days spent traveling on vacation domestically (*D*) and days spent traveling on vacation in a foreign country (*F*), as given by the utility function *U*(*D*,*F*) = 10*DF*. In addition, the price of a day spent traveling domestically is $100, the price of a day spent traveling in a foreign country is $400, and Jane’s annual travel budget is $4000.

a. Illustrate the indifference curve associated with a utility of 800 and the indifference curve associated with a utility of 1200.

b. Graph Jane’s budget line on the same graph.

c. Can Jane afford any of the bundles that give her a utility of 800? What about a utility

of 1200?

d. Find Jane’s utility-maximizing choice of days spent traveling domestically and days spent in a foreign country.

Chapter 4 Problems:

- Jack, our representative consumer, consumes varying amounts of beef and rice. Assume that B = quantity of beef consumed, and that R = quantity of rice consumed. Jack’s utility function is given as:
- a. Assume further that the price of beef is $4, the price of rice is $2, and that Jack’s income is $200. How much of each product should he purchase?
- b. How are the quantities calculated in (a) above affected when the price of rice increases from $2 to $4? Calculate the substitution and income effects for rice?
- c. Is rice a normal good or an inferior good? Explain.

2. An individual consumes two goods, clothing and food. Given the information below, illustrate both the income-consumption curve and the Engel curve for clothing and food.

Price Clothing | Price Food | Quantity Clothing | Quantity Food | Income |

$10 | $2 | 6 | 20 | $100 |

$10 | $2 | 8 | 35 | $150 |

$10 | $2 | 11 | 45 | $200 |

$10 | $2 | 15 | 50 | $250 |

3. Jane always gets twice as much utility from an extra ballet ticket as she does from an extra basketball ticket, regardless of how many tickets of either type she has. Draw Jane’s income-consumption curve and her Engel curve for ballet tickets.

4. a. Orange juice and apple juice are known to be perfect substitutes. Draw the appropriate

price-consumption curve (for a variable price of orange juice) and income-consumption curve.

b. Left shoes and right shoes are perfect complements. Draw the appropriate price-consumption and income-consumption curves.

5. Each week, Bill, Mary, and Jane select the quantity of two goods, *x*_{1} and *x*_{2}, 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:

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 week 1 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:

I |
|||||

Week 1 | 10 | 20 | 2 | 1 | 40 |

Week 2 | 6 | 14 | 2 | 2 | 40 |

Week 3 | 20 | 10 | 2 | 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 Jane’s choices:

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 *x*_{1}.

6. Two individuals, Sam and Barb, derive utility from the hours of leisure (*L*) they consume and from the amount of goods (*G*) they consume. In order to maximize utility, they need to allocate the 24 hours in the day between leisure hours and work hours. Assume that all hours not spent working are leisure hours. The price of a good is equal to $1 and the price of leisure is equal to the hourly wage. We observe the following information about the choices that the two individuals make:

Sam | Barb | Sam | Barb | ||

Price of G |
Price of L |
L (hours) |
L (hours) |
G ($) |
G ($) |

1 | 8 | 16 | 14 | 64 | 80 |

1 | 9 | 15 | 14 | 81 | 90 |

1 | 10 | 14 | 15 | 100 | 90 |

1 | 11 | 14 | 16 | 110 | 88 |

Graphically illustrate Sam’s leisure demand curve and Barb’s leisure demand curve. Place price on the vertical axis and leisure on the horizontal axis. Given that they both maximize utility, how can you explain the difference in their leisure demand curves?

7. Suppose the income elasticity of demand for food is 0.5 and the price elasticity of demand is -1.0. Suppose also that Felicia spends $10,000 a year on food, the price of food is $2, and that her income is $25,000.

a. If a sales tax on food caused the price of food to increase to $2.50, what would happen to her consumption of food? (*Hint*: Because a large price change is involved, you should assume that the price elasticity measures an arc elasticity, rather than a point elasticity.)

b. Suppose that Felicia gets a tax rebate of $2500 to ease the effect of the sales tax. What would her consumption of food be now?

c. Is she better or worse off when given a rebate equal to the sales tax payments? Draw a graph and explain.

8. Suppose you are in charge of a toll bridge that costs essentially nothing to operate. The demand for bridge crossings *Q* is given by

a. Draw the demand curve for bridge crossings.

b. How many people would cross the bridge if there were no toll?

c. What is the loss of consumer surplus associated with a bridge toll of $5?

d. The toll-bridge operator is considering an increase in the toll to $7. At this higher price, how many people would cross the bridge? Would the toll-bridge revenue increase or decrease? What does your answer tell you about the elasticity of demand?

e. Find the lost consumer surplus associated with the increase in the price of the toll from

$5 to $7.

Appendix to Chapter 4 Problems:

1. Sharon has the following utility function:

where *X* is her consumption of candy bars, with price *P _{X}* = $1, and

*Y*is her consumption of espressos, with

*P*= $3.

_{Y}a. Derive Sharon’s demand for candy bars and espressos.

b. Assume that her income *I* = $100. How many candy bars and how many espressos will Sharon consume?

c. What is the marginal utility of income?

2. Maurice has the following utility function: where *X* is his consumption of CDs, with a price of $1, and *Y* is his consumption of movie videos, with a rental price of $2. He plans to spend $41 on both forms of entertainment. Determine the number of CDs and video rentals that will maximize Maurice’s utility.