Microeconomics: Problem set 1 (Due on April 27 period 4 in the TA recitation)
Instructor: Toshihiro Ichida
Page 1 of 7
Chapter 1
1.4
Suppose the supply curve for wool is given by Qs = P,whereQs is the quantityoffered for sale when the price isP. The demand for wool is given by Qd =10 – P +I, whereQd is the quantity of wool demanded when the price is P and the level of income is I. Assume I is an exogenous variable.
a)Suppose the level of income is I = 20. Graph the supply and demand relationships, and indicate the equilibrium levels of price and quantity on your graph.
b)Explain why the market for wool would not be in equilibrium if the price of wool were 18.
c)Explain why the market for wool would not be in equilibrium if the price of wool were 14.
1.5
Consider the market for wool described by the supply and demand equation in Problem 1.4.
Suppose income rise from.
a)Using comparative statics analysis, find the impact of the change in income on the equilibrium price of wool.
b)Using comparative statics analysis, find the impact of the change in income on the equilibrium quantity of wool.
1.6
You are the Video Acquisitions Officer for your dormitory. The other officers of your dorm will tell you how many videos they would like to rent during the year. Your job is to find the least expensive way of renting the required number of videos. After researching the options, you have found that there are three rental plans from which you can choose.
Plan A: Pay $3 per video, with no additional fees.
Plan B: Join the Frequent Viewer Club. Here you pay a yearly membership fee of $50, with an additional charge of $2 for each video rented.
Plan C: Join the Very Frequent Viewer Club. In this club you pay a yearly membership fee of $150, with an additional charge of $1 for each video rented.
a)Which plan would you select if your instructions are to rent 75 movies a year at the lowest possible cost?
b)Which plan would you select if your instructions are to rent 125 movies a year at the lowest possible cost?
c)In this exercise, is the number of videos rented endogenous or exogenous? Explain.
d)Is the choice of plan (A, B, or C) endogenous or exogenous? Explain.
e)Are total expenditures on videos endogenous or exogenous? Explain.
1.7
Reconsider the problem of the Video Acquisitions Officer in Problem 1.6. Suppose the officers of your dorm give you a specified amount of money to spend, and want you to maximize the number of videos you can rent with that budget. You can choose from the same three plans (A, B, and C) available in Problem 1.6.
a)Which plan would you select if your instructions are to rent the most movies possible while spending $125 per year?
b)Which plan would you select if your instructions are to rent the most movies possible while spending $300 per year?
c)In this exercise, is the number of videos rented endogenous or exogenous? Explain.
d)Is the choice of plan (A, B, and C) endogenous or exogenous? Explain.
e)Are total expenditures on videos endogenous or exogenous? Explain.
Chapter 2
2.1
The demand for beer in Japan is given by the following equation:
Qd = 700 – 2P – PN + 0.1·I
where P is the price of beer, PN is the price of nuts, and I is average consumer income.
a) What happens to the demand for beer when the price of nuts goes up? Are beer and nuts demand substitutes or demand complements?
b) What happens to the demand for beer when average consumer income rises?
c) Graph the demand curve for beer when =100 and.
2.2.
Suppose the supply of steel is given by the following equation:
where P is the price of a steel, andw is the wage rate of steel workers.
a)What happens to the supply of steel when the wage rate of steel workers goes up?
b)Graph the supply curve for steel when w=16. (Remember, )
2.3
The demand and supply curves for coffee are given by
a)Plot the supply and demand curves on a graph and show the equilibrium occurs.
b)Using algebra, determine the market equilibrium price and quantity of coffee.
2.5.
You have decided to study the market for fresh-picked cherries. You learn that over the last 10 years, cherry prices have risen, while the quantity of cherries purchased has also risen. This seems puzzling because you learned in microeconomics that an increase in price usually decreases the quantity demanded. What might explain this seemingly strange pattern of prices and consumption levels?
2.8.
Consider a liner demand curve, .
a)Derive the inverse demand curve corresponding to this demand curve.
b)What is the choke price?
c)What is the price elasticity of demand at?
2.9
Suppose that the market for air travel between Chicago and Dallas is served by just two airlines, United and American. An economist has studied this market and has estimated that the demand curves for round-trip tickets for each airline are as follows:
= 10000―100＋99 (United’s demand)
= 10000―100 + 99 (American’s demand)
Where is the price changed by United, and is the price charged by American.
a)Suppose that both American and United charge a price of $300each for a round-trip ticket between Chicago and Dallas. What is the price elasticity of demand for United flights between Chicago and Dallas?
b)What is the market-level price elasticity of demand for air travel between Chicago and Dallas when both airlines charge a price of $300? (Hint; Because United and American are the only two airlines serving the Chicago-Dallas market, what is the equation for the total demand for air travel between Chicago and Dallas, assuming that the airlines charge the same price?)
2.10. You are given the following information:
Price elasticity of demand for cigarettes at current prices is －0.5.
Current price of cigarettes is $0.05 per cigarette.
Cigarettes are being purchased at a rate of 10 million per year.
a) Find a linear demand that fits this information, and graph that demand curve.
b) Find a constant elasticity demand curve that fits this information, and graph that demand curve.
2.11
Consider the following sequence of events in the U.S. market for strawberries during the years 1998-2000
1998: Uneventful. The market price was $5.00 per bushel, and 4 million bushels were sold.
1999: There was a scare over the possibility of contaminated strawberries from Michigan. The market price was $4.50 per bushel, and 2.5 million bushels were sold.
2000: By the beginning of the year, the scare over contaminated strawberries ended when the media reported that the initial reports about the contamination were a hoax. A series of floods in the Midwest, however, destroyedsignificant portions of the strawberry fields in Iowa, Illinois, and Missouri. The market price was $8.00 per bushel, and 3.5 million bushels ware sold.
Find linear demand and supply curves that are consistent with this information.
Chapter 3
3.1
Consider the utility function with the marginal utilities and .
a)Does the consumer believe that more is better for each good?
b)Do the consumer’s preferences exhibit a diminishing marginal utility of ? Is the marginal utility of diminishing?
3.2
For the following sets of goods draw two indifference curves,U1 and U2,with U2 ＞U1 .
Draw each graph placing the amount of the first good on the horizontal axis.
a)Hot dogs and chili (the consumer likes both and has a diminishing marginal rate of
substitution of hot dogs for chili)
b)Sugar and Sweet’N Low (the consumer likes both and will accept an ounce of Sweet’N Low
or an ounce of sugar with equal satisfaction)
c).Peanut butter and jelly (the consumer likes exactly 2 ounces of peanut butter for
every ounce of jelly)
d) Nuts (which consumer neither likes nor dislikes) and ice cream (which the consumer likes)
e) Apples (which the consumer likes) and liver (which the consumer dislikes)
3.3
The utility that Julie receives by consuming food F and clothingC is given by U (F, C) =FC.
For this utility function, the marginal utilities are MUF=CandMUC=F.
a)On a graph with F on the horizontal axis and C on the vertical axis, draw indifference curves for U=12, U=18,U=24.
b)Do the shapes of these indifference curves suggest that Julie has a diminishing marginal rate of substitution of food for clothing? Explain.
c)Using the marginal utilities, show that the MRSF, C=C/F. What is the slope of the indifference curve U=12, at the basket with 2 units of food and 6 units of clothing? What is the slope at the basket with 4 units of food and 3 units of clothing? Do the slopes of the indifference curves indicate that Julie has a diminishing marginal rate of substitution of food for clothing? (Make sure your answers to parts (b) and (c) are consistent!)
3.4.
Sandy consumes only hamburgers (H) and milkshakes (M). At basket A, containing 2 hamburgers and 10 milkshakes, his MRSH,M is 8. At basket B, containing 6 hamburgers and 4 milkshakes, his MRSH,M is 1/2. Both basket A and B are on the same indifference curve. Draw the indifference curve, using information about the MRSH,Mto make sure that the curvature of the indifference curve is accurately depicted.
3.5
James Bond likes his vodka martinis to contain exactly 10 parts of vodka for every 1part of vermouth. On a graph with the amount of vodka on one axis and the amount of vermouth on the other axis, draw two of his indifference curves, U₁and U₂, with U₂U₁
3.6
Draw indifference curves to represent the following types of consumer preferences.
a)I like both peanut butter and jelly, and always get the same additional satisfaction from an ounce of peanut butter as I do from 2 ounces of jelly.
b)I like peanut butter, but neither like nor dislike jelly.
c)I like peanut butter, but dislike jelly.
d)I like peanut butter and jelly, but I only want 2 ounces of peanut butter for every ounce of jelly.
3.7
Dr. Strangetaste buys only food (F) and clothing (C) out of his income. He has positive marginal utilities for both goods, and his MRSF,C isincreasing. Draw two of Dr. Strangetaste’s indifference curves, U₁and U₂, with U₂U₁.
The following exercises will give you practice in working with a variety of utility functions and marginal utilities, and to help you understand how to graph indifference curves.
3.8
Consider the utility function U(x,y) = 3x + y, with MUx = 3 and MUy = 1.
a)Is the assumption that more is better satisfied for both goods?
b)Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain.
c)What is the MRSx,y?
d)Is the MRSx,y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve?
e)On a graph with x on the horizontal axis and y on the vertical axis, draw a typical indifference curve (it need not be exactly to scale, but it needs to reflect accurately whether there is a diminishing MRSx,y). Also indicate on your graph whether the indifference curve will intersect either or both axes. Label the curve U1.
f)On the same graph draw a second indifference curve U2, with U2 > U1.
3.9
3.10
3.11
3.12
3.13
3.14
Suppose a consumer’s preferences for two goods can be represented by the Cobb-Douglas utility
function U = Axαyβ, where A, α, and β are positive constants. The marginal utilities are
MUx = αAxα-1yβ and MUy =βAxαyβ-1. Answer all parts of Problem 3.8 for this utility function.
3.15.
Suppose a consumer has preferences over two goods that can be represented by the quasi-linear utility function The marginal utilities are
d)Is the assumption that more is better satisfied for both goods?
e)Does the marginal utility ofdiminish, remain constant, or increase as the consumer buys more? Explain.
f)What is the expression for?
g)Is thediminishing, constant, or increasing as the consumer substitutes more for along an indifference curve?
h)On a graph withon the horizontal axis andon the vertical axis, draw a typical indifference curve (it need not be exactly to scale, but it should accurately reflect whether there is a diminishing). Indicate on your graph whether the indifference curve will intersect either or both axes.
i)Show that the slope of every indifference curve will be the same when What is the value of that slope?