Intermediate Micro Homework
Q1. A survey shows that the typical Brown undergraduate has $6,000 to spend each year on books and entertainment. That typical student spends $2000 on books and the remaining $4000 on entertainment. The Brown dean finds it outrageous that the undergraduates must spend 1/3 of their disposable income on books and so the dean sets up a program to give each undergraduate a subsidy of $500. If the income elasticity of books of the typical student is 1.5 then will the subsidy program achieve its goal of lowering the percentage of income spent on books?
Q2. Frank buys hot dogs (x) and pretzels (y) at the ball park. His preferences over these two goods can be represented by the utility function U(x, y) = lnx + 3lny where x represents the number of hot dogs and y represents the number of pretzels.
a) Given his preferences find his demand functions for hot dogs (x) and pretzels (y).
b) Suppose that the price of a pretzel is $1 and that Frank has $40 to spend on hot dogs and pretzels. Write Frank's demand curve for hot dogs. Illustrate his demand curve.
c) Suppose that the price of a hot dog is $5 (the price of pretzels and income remain $1 and $40, resp.). Use your demand functions to find his best bundle. In an indifference curve diagram illustrate his best bundle at these prices.
For the remainder of the question assume that the price of a pretzel rises to $1.25 and that the price of a hot dog and his income are unchanged at Px = $5 and I = $40.
d) Use your demand functions to find his new best bundle. Illustrate the new budget line and the new best bundle that you found above in your diagram for part (c). Be sure to indicate the slopes of both budget lines.
e) What is the cross price elasticity of hot dogs in this case?
Q3. Phyllis drinks both coffee (x) and milk (y). Her preferences over these two goods can be represented by the utility function U(x,y) = x + 3y½ where x represents the number of pounds of coffee and y represents the number of quarts of milk.
a) Given her preferences find her demand functions for coffee (x) and milk(y).
b) Suppose that the price of a pound of coffee is $4 and that she has $56 to spend on coffee and milk. Write her demand curve for milk. Illustrate her demand curve.
c) Suppose that the price of a quart of milk is $1 (the price of coffee and income remain $4 and $56, resp.). Use your demand functions to find her best bundle. In an indifference curve diagram illustrate her best bundle at these prices.
For the remainder of the question assume that her income rises to $60 and that the prices of coffee and milk are unchanged at Px=$4 and Py=1.
d) Use your demand functions to find her new best bundle.
e) Illustrate the new budget line and the new best bundle that you found above in your diagram for part (c). Be sure to indicate the slopes of both budget lines.
f) What is the income elasticity of milk for these preferences (as a function of prices and income)?
Q4. Anna is indifferent between bread and rolls for her sandwiches. She believes that 2 slices of bread are always just as good as 1 roll. Let bread be the x-good measured in slices and rolls be the y-good measured in number of rolls. Suppose Anna has $1 to spend on bread and rolls.
a) Given our assumptions on preferences and the information above, illustrate indifference curves that represent Anna's preferences. What is her MRS?
b) If the price of a slice of bread is $.10 and the price of a roll is $.25 then what bundle will she choose? Show that her MRS is not equal to the price ratio at that bundle. Illustrate your answer in your diagram for part (a).
c) If the price of a slice of bread rises to $.20 then what bundle will she choose? Show that her MRS is not equal the price ratio at that bundle. Illustrate her new best bundle.
d) Briefly explain why her MRS is not equal to her best bundles in parts (b) and (c).
e) Illustrate her demand curve for bread.
Q5. Clive enjoys visiting the scenic National Park. Each month he divides his $100 income between visits to the park and picnic baskets. The cost of a picnic basket is $10 and the National Park charges an entrance fee of $20 per visit. The two goods are NOT perfect complements for Clive.
a) Illustrate Clive's budget set. What is the opportunity cost of visiting the park?
Due to the rising costs of park maintenance, the government charge a monthly membership fee to visit the parks. Specifically, to visit a park, Clive will have to buy a membership which costs $20. In addition to the membership fee, he will still have to pay $20 each time he visits the park. Suppose that Clive chooses to buy the membership and that he chooses to go to the park twice per month.
b) Illustrate Clive's new budget set in your diagram above. If Clive goes to the park twice per month then how many picnic baskets can he afford? Illustrate Clive's best bundle and an indifference curve through that bundle.
c) Illustrate the revenue measured in picnic baskets raised from the membership fee.
Suppose that the instead of charging a membership fee, the government decides to simply raise the entrance fee to the park to P > 20 (so there is no membership fee with this is price). Assume that Clive prefers paying the higher price P (and no membership fee) to paying the membership fee (and the original price).
d) In your diagram, illustrate a budget line associated with a price P such that Clive prefers paying the higher price P to paying the original price and the membership fee.
e) Briefly explain why it is that if Clive prefers the price increase to the membership fee then the revenue raised from the price increase must be less than the revenue raised from the membership fee. You may use your diagram to explain this.
Bonus: Suppose that x > 0 and y = 0 at the solution to the consumer's choice problem. Write the first order conditions that characterize this solution. Illustrate it in an indifference curve diagram. Alternatively suppose that x = 0 and y > 0 at the solution to the consumer's problem. Write the first order conditions that characterize this solution. Use your answer to write the demand functions associated with the utility function U(x,y) = x + 3y.