A Theory of Preferences
1. True/False
1. An indifference curve is defined as a set of bundles that a consumer with a given income can afford, and among which she or he is indifferent.
2. More is preferred to less means that if the total number of goods in bundle A exceeds the total quantity in B, than A is preferred to B.
3. The assumptions of completeness, two-term consistency, transitivity and continuity are necessary for constructing a utility function over a set of preferences.
4. The utility function V(x1, x2) = 5[U(x1, x2)]/2+7 represents the same preference ordering as the utility function U(x1, x2).
5. A diminishing marginal rate of substitution implies that an individual requires increasing amounts of one good as he gives up more and more of the other good to remain at the same utility level.
6. More is preferred to less implies that two bundles with different amounts of either good 1 or good 2 and the same amount of the other good cannot be on the same indifference curve.
7. Since utility is ordinal and not cardinal, interpersonal comparisons cannot be made.
8. The MRS for indifference curve x1 + x2 =c is diminishing.
9. The indifference curve between garbage and ice cream would be positively sloped.
10. If Alfred's indifference curve between income and leisure is positively sloped and convex, then the additional income required to induce Alfred to work additional hours is constant and equal to his current wage.
Answers: 1.F 2.F 3.T 4.T 5.T
6.T 7.T 8.F 9.T 10.F
2. Short Questions.
1. Draw an indifference curve which is not convex.
2. Consider the following list of statements. Each statement in the list means the same thing as one of the other statements. Identify the pairs of statements which are equivalent:
a) Consumers always prefer to have more of a good
b) Consumers' preferences are complete
c) Consumers' preferences are transitive
d) Every market basket has an indifference curve associated with it
e) Indifference curves are convex
f) Indifference curves are downwards sloping
g) Indifference curves do not cross
h) A diminishing marginal rate of substitution is a characteristic of consumer preferencesAnswers: 2.f) and h), c) and g), b) and d), a) and e).
3. Long Questions
1. Consider the following utility function:
U(x1, x2) = 2x2 + x11/3
a) Draw the indifference curve associated with utility numbers 8, 27 and 64.
b) How does MRS changes as you move from one indifference curve to another along the vertical line x1=2? Or x1=3?2. Consider the following utility functions:
(i) U(x1, x2) = x1 x21/3
(ii) U(x1, x2) = 10x1x2
(iii) U(x1, x2) = 3x1+4x2
(iv) U(x1, x2) = 2x1+ln(x2)
(v) U(x1, x2) = x13a) Construct an indifference curve for each of these functions.
b) Calculate the MRS for each of these functions.
Answers: 1. b).MRS = (1/6)x1(-2/3) doesn't change as you move along a vertical line because does not depend on x2.
2. b). i) MRS=3x2/x1; ii) MRS=x2/x1; iii) MRS=3/4; iv) MRS=2x2 ; v) MRS=infinity
Demand Theory
1. True/False
1. Since the real price for automobiles has increased but the quantity demanded has also increased over recent years, the demand curve for automobiles must be positively sloped.
2. There is no money illusion for the demand curve x1=5M(p2/p1)1/2.
3. All inferior goods are Giffen goods.
4. All Giffen goods are inferior goods.
5. If the income elasticity of a good is negative, the demand curve of that curve must be negatively sloped.
6. If, at the utility maximizing bundle of good 2 and good 1, the MRS of good 2 for good 1 is greater than p1/p2, then good 1 is an inessential good.
7. Suppose that an individual consumes only two goods, good 1 and good 2. If the price of good 1 rises, with all else constant, and the price elasticity of demand for good 1 is -0.7, then the quantity of good 2 will increase.
8. If an individual's income is the same before and after an excise tax is imposed on good 1 and the demand for good 1 is elastic, the excise tax will lead to an increase in spending on other consumer goods.
9. Since a decrease in the price of one unit along a linear demand curve will result in an increase in the quantity demanded by a constant amount, the price elasticity is constant for all quantities along a linear demand curve.
10. Let good 1 be on the horizontal axis, and let the quantity of the composite commodity be on the vertical axis. Then, if the (absolute value of the) price elasticity of demand for good 1 is less than 1, the price consumption line is negatively sloped.
11. Along a consumer's budget line, money income is constant.
12. A commodity bundle lying below a consumer's budget line must be inferior to all bundles lying on the budget line.
13. Assume that a certain individual consumes only goods 1 and 2. If the prices of good 1 and good 2 double and his income doubles, then the quantity demanded for goods 1 and 2 will not change.
14. If a decline in the prices of agricultural products results in a reduction in the consumption of these goods by farmers, then these products must be inferior goods.
Answers: 1.F 2.F 3.F 4.T 5.F 6.F 7.F
8.T 9.F 10.F 11.T 12.F 13.T 14.F
2. Short Questions.
1. Sally's demand for apples is given by x1=M/p1. Graph her demand curve, and calculate the price elasticity of demand for apples.
2. Suppose that a consumer's preferences are given by U = ln x1 + x2, and his budget constraint is given by p1x1 + p2x2 = m where x1 and x2 are goods, p1 is the price of x1, and m is the consumer's income. Find the demand functions for x1 and x2. Is x1 an ordinary good? Why or why not?
3. Cari Obeck's demand function for tuna is given by: q=2m/5p where q is the number of cans of tuna demanded, p is the price of tuna, and m is her income. Suppose Cari's income is $100. If the price of tuna falls from $5 to $4, what happens to the demand for tuna?
4. Pat Impecunious spends all her money on two goods: bologna and white bread. Can both of them be inferior? Explain.
5. Susan Ng's preferences over bowls of french fries (x1) and pints of ice cream (x2) are given by: U=x12+x22
The price of french fries is $2 a bowl while the price of ice cream is $3 a pint. Her income is $8. At her optimal choice, how many bowls of fries will she buy? How many pints of ice cream?6. Explain what is meant by the terms demand function and Engel curve and for each of the following state whether the function is a demand function, an Engel curve, both or neither.
x1 = m - p1/p2
m = p1x1
x2 = m2 - x1/p1p27. Sally consumes two goods, X and Y. Her utility function is given by the expression U(X,Y) = X0.4Y0.6
The current market price of X is $20 per unit, while the market price of Y is $100 per unit. Sally's current income is $10,000.
a). Calculate Sally's demand for X as a function of prices and incomes. Is good X normal? Please show all work.
b). How much X and Y will Sally demand at current market prices? What will Sally's utility be with that consumption?
c). Find the income elasticity and the price elasticity of the Marshallian demand for X.8. If the compensated demand curve for peanuts is steeper than the ordinary demand curve, then
i). peanuts are a normal good
ii). peanuts are an inferior good
iii). we do not have enough information to decide whether peanuts are normal or inferior.
Select one of (i), (ii) or (iii) and support your answer with diagrams and an explanation.Answers: 1. E11=-1
2. x1*=p2/p1; x2*=(m-p2)/p2; Yes because the demand is downward sloping.
3. The demand for tuna will increase from 8 to 10 cans.
4. No. If Pat's income increases she cannot decrease the consumption of both goods. The Budget costraint will not be satisfied. At least one of the goods must be normal.
5. x1*=4 and x2*=0. Note that the indifference curves are not convex, therefore the tangency is not the optimal solution. In this case we have a corner solution.
6. demand curve, Engel curve, neither (because of the x1 in the formula)
7. a) x=0.4M/px; y=0.6M/py; Yes. b) x=200, y=60. c) E11=-1, E1M=1.
8. Normal good.If the compensated demand curve for peanuts is steeper than the ordinary demand curve, this means that the TE is larger than the SE., and both are negative. Therefore IE is also negative.
3. Long Questions.
1. Mary has an income of $10 per week, which she spends on Marmite which costs $4 a jar, and bread, which costs $2 per loaf.
a) Draw Mary's budget line? What is the slope of her budget line?
b) Show the effect on the budget line of halving the price of Marmite to $2 per jar.
c) Show that a 50 per cent reduction in the price of both Marmite and bread (to $2 a jar and $1 per loaf) has the same effect on Mary's budget line as a doubling of Mary's income to $20 per week.
d) Suppose the government, due to a national shortage of brewers yeast (the main ingredient in Marmite) decides to ration Marmite consumption to one jar per week. Show the effect of the rationing on the original (part a) budget constraint.2. Geoff Howard's well-being depends upon his consumption of milk (in pints, x1) and video games (x2). His utility function is given by: U = 3lnx1+x2. Calculate his demand for milk, his demand for video games, and his optimal choice when p1=$3/pint, p2=$2/game and m=$14.
Now suppose the government subsidized Geoff's milk by $2 per pint, so the price of milk fell from $3 to $1.
a) Calculate
(i) how much milk Geoff would demand at the subsidized price
(ii) How many video games he would demand and
(iii) His level of utility
(iv) The cost of the subsidy to the government.
b) If the government gave Geoff the amount calculated in part (a - iv) in cash (leaving the price of milk at its original level of $3) how much milk would he demand, how many video games, and what would his utility be?
c) Draw a diagram showing the budget constraint with the subsidy, the budget constraint with the cash transfer, Geoff's optimal choice under the two schemes, and the cost of each program..Answers:
1. a) slope is -2 (with Marmite on the horizontal axis); b) rotates outward; d) the budget line will consist of two parts: for any quantity of Marmite lass than 1 will be the same as before rationing and for then at 1 will be a vertical line. Basically any quantity more than 1 is not available anymore.
2. x1=3p2/p1 = 2; x2=(M-3p2)/p2 = 4. a) i)6, ii)4, iii)3ln6+4, iv)$12. b) milk=2, video=10, U=3ln6+10.More Demand Theory
1. True/ False
1. An exact measure of the individual's willingness to pay for the opportunity to purchase an automobile at some price is the consumer surplus.
2. If the substitution effect is zero, then an individual will be indifferent between a subsidy on each unit consumed of some good and a lump-sum cash subsidy that costs the government an equal amount of money.
3. A two-part tariff on some good that laves a consumer at the same level of utility as received under a simple per-unit price is always more profitable for the firm selling the product.
4. If the government can subsidize a low-income family with a subsidy on clothing or with a lump-sum cash transfer, and both programs are equally costly to the government, then the family will be better off with the subsidy on clothing.
5. If the government wants to increase its revenue through a tax on some good, then it can collect more revenue through a lump sum tax than through a per-unit tax that would leave the individual at the same utility level.
6. The higher the price of a good, the larger the Consumer Surplus associated with that good will be.
Answers: 1.F 2.T 3.T 4.F 5.T 6.F
2. Short Questions
1. Explain using a diagram how to calculate the equivalent variation associated with introducing a new good.
3. Long Questions
1. Mohammed spends all of his income on two goods, left shoes and right shoes.
His utility function is given by:
U(x1,x2)=min(x1,x2).
The price of right shoes is $10 and the price of left shoes is $20. His income is $60.
Suppose the price of left shoes falls to $10. Measure the change in Mohammed's well-being(i) Using compensating variation
(ii) Using equivalent variationAnswers: 1. Show it in the diagram. In this case there is no substitution effect. CV=20, EV=30. Be careful when you calculate CV and EV. It is not the vertical distance between two budget lines because the price of right shoes is not 1 but 10.