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MICROECONOMICS I

PROGRAM AND SET OF PROBLEMS

Universitat de ValènciaYear 2004-05

PROGRAM

MICROECONOMICS I (12146)

This course concentrates on microeconomic theory at an intermediate level. Students should have completed a course on introductory economics. The program includes topics that cover half the material normally treated in micro courses. The other half is included in Microeconomics II. Roughly, the program covers consumer theory and extensions, and an introduction to welfare and general equilibrium economics.

The main text used in the course will be:

Intermediate Microeconomics. A Modern Approach. By Hal R. Varian. Fourth Edition. Norton International Student Edition. New York, 1996. A Spanish translation exists under the title Microeconomía Intermedia (Second Edition). Editorial A. Bosch. Barcelona.

Other texts which cover the same material are:

Microeconomics and Behavior. By Robert A. Frank. Third Edition. Mc Graw-Hill. New York, 1997

Microeconomics. By Michael L. Katz and Harvey S. Rosen. Mc Graw-Hill. New York, 1998

The reference to chapters given below each lesson, corresponds to Varian’s book.

LESSION 1: BASIC IDEAS The concept of a modelOptimization and equilibriumDemand, Supply and Market EquilibriumPractical applications

Chapter: 1

LESSION 2: CONSUMER THEORY (I): BUDGET CONSTRAINTThe budget constraintChanges in the budget constraintTaxes, subsidies and rationingExamples

Chapter: 2

LESSON 3: CONSUMER THEORY (II): PREFERENCESConsumer preferencesIndifference curvesExamples of preferencesThe Marginal Rate of Substitution

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Chapter: 3

LESSON 4: CONSUMER THEORY (III): UTILITYCardinal utilityConstructing a utility functionExamples of utility functionsMarginal utility and the Marginal Rate of Substitution

Chapter: 4

LESSON 5: CONSUMER THEORY (IV): CONSUMER EQUILIBRIUMOptimal choiceThe Substitution effectThe Income effectConsumer demandPerfect Substitutes and Perfect ComplementsImplications of the Marginal Rate of Substitution conditions

Chapters: 4, 6 and 8

LESSON 6: EXTENSIONS AND APPLICATIONS OF CONSUMER THEORYThe approach of revealed preferenceBudget constraints with fixed and changing endowmentsThe application of consumer theory to labour supplyBudget constraints and preferences over timeChoice and the interest rate


LESSON 7: CONSUMER’S SURPLUSThe concept of consumer’s surplusInterpretation of changes in consumer surplusCompensating and Equivalent VariationsCalculating Gains and Losses

Chapter: 14

LESSON 8: MARKET DEMANDFrom individual to market demandThe inverse demand functionThe concept of elasticityElasticity and marginal revenueIncome elasticity

Chapter: 15

LESSON 9: MARKET EQUILIBRIUMMarket equilibriumComparative statisticsTaxes. The deadweight loss of taxation

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Pareto efficiency

Chapter: 16

LESSON 10: EXCHANGEThe Edgeworth BoxPareto efficient allocationsEquilibrium and efficiencyThe two welfare theorems

Chapter: 28

LESSON 11: WELFAREAggregation of preferencesSocial welfare functionsWelfare maximization

Chapter: 30

LESSON 12: MARKET FAILURESExternalities. The Coase theoremPublic Goods. Public and private provision of public goodsAsymmetric information


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MICROECONOMICS I

Problems

Lesson 1: Basic Ideas

1. Say whether you agree or disagree with the following statements and explain why:

a) The economic character of a problem appears only when the three following elements -diversity of objectives, transferability of resources and scarcity of resources- are present at the same time.

b) The Robbins’ definition of economics -“the study of the allocation of scarce means to satisfy competing ends”- incorporates the three elements mentioned in question a) above.

c) The model of supply and demand is a way of simplifying the world. It is a classification scheme, by means of which we decide which variables matter in demand, supply of both.

d) The “ceteris paribus” clause is just an indication of the variables that are held constant in the particular analysis of a problem.

e) In the normal terminology used by economists, a “change in the quantity demanded” means the same as a “change in the demand curve”.

2. Suppose that the market for butter and margarine can be represented by the following implicit functions

Butter Margarine

SupplyDemand

a) What other relevant variables could be considered in each of these four functions?

b) How would you represent in each of these two markets the respective conditions of equilibrium?

c) If you had to represent graphically the market for butter, what variables would you include in the “ceteris paribus” clause? Why?

d) If you had to represent graphically the market for margarine, what variables would you include in the “ceteris paribus” clause? Why?

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e) Represent graphically these two markets. (Put prices in the vertical axis and quantities in the horizontal axis).

f) Show the direction of the effect of each of the variables on the corresponding dependent variable by means of a plus (+) or minus (-) sign below the variable in question. Suppose, in carrying out this exercise, that butter and margarine are substitute goods.

g) Suppose that because of a change in technology, the production of margarine becomes significantly cheaper. What effect will this event have on the price and quantity of margarine? Will the market for butter be affected?

h) Suppose that because of the entry of a group of vegetarian tourists during the summer there is an increase in the demand for margarine. What effect will this event have on the markets of margarine and butter?

3. Suppose you are given by an econometrician the explicit parameters that define the functions considered in the previous exercise.

a) Represent graphically the two markets. (Put prices in the vertical axis and quantities in the horizontal axis).

b) Find out the equilibrium price and quantity of both butter and margarine.

c) Suppose that because of an adverse change in technology, the supply function of margarine becomes . All other functions remain the same. How will this event affect the equilibrium of both markets?

d) Return to the original specification. Suppose now that because of a change in tastes, the demand function for butter becomes . All other functions remain the same. How will this event affect the equilibrium of both markets?

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Lesson 2: Budget constraint


a) The budget constraint permits to identify those bundles of goods which are affordable by a given consumer from those bundles which are not affordable.

b) The budget line is the set of bundles goods which are affordable by the consumer.

c) If the prices of goods increase by the same proportion as income, the budget constraint remains unchanged.

d) If, other things equal, the prices of goods double, the slope of the budget line will also double.

e) If the price of one good doubles and income also doubles, the slope of the budget line will remain the same.

f) If the consumer’s income increases with no change in relative prices, the budget line will move parallel to itself.

g) If the consumer’s income increases by 10% and the prices of goods x and y both increase by 10%, the consumer will buy 10% more of each of the two goods. (Exam, July 04)

2. You have an income of € 40 to spend on two commodities. Commodity x costs € 10 per unit and commodity y costs € 5 per unit.

a) Write down your budget equation and represent it graphically. (Put commodity x on the horizontal axis and commodity y on the vertical axis). Call this budget A.

b) Suppose that the price of commodity x falls to € 5 while everything else stays the same. Write down and represent graphically your new budget equations. Call this budget B.

c) Suppose the amount you are allowed to spend falls to € 30, while the price of both commodities remain at € 5. Write down your budget equation and represent it graphically. Call this budget C.

d) Which of the three budgets gives the consumer the largest set of consumption opportunities?

e) Compare budgets A and C. At which bundle of commodities do they cross? Shade the area representing commodity bundles affordable with budget C but

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not affordable with budget A. Shade the area of commodity bundles affordable with budget A but not affordable with budget C.

3. Your budget is such that if you spent your entire income, you could afford either 4 units of x and 6 units of y or 12 units of x and 2 units of y.

a) Draw this budget line.

b) What is the ratio of the price of x to the price of y?

c) If you spent all your income on x, how many units of x could you buy?

d) If you spent all your income on y, how many units of y could you buy?

e) You are told that the price of x is € 1. Write down the budget equation that gives you the budget line of this problem.

f) You are told that the price of x is € 3. Write down the budget equation that gives you the same budget line.

4. Rosa was consuming 100 units of x and 50 units of y. The price of x rose from € 2 to € 3. How much would Rosa’s income have to rise so that she can still exactly afford 100 units of x and 50 units of y?

5. If Pepe spent his entire allowance, he could afford 8 units of x and 8 units of y a week. He could also afford 10 units of x and 4 units of y a week. The price of x is € 0.50. What is the price of y and Pepe’s weekly allowance?

6. Marta is preparing for the only two exams left to finish her degree in economics: Microeconomics and Econometrics. She has time to read 40 pages of Microeconomics and 30 pages of Econometrics. In the same amount of time she could also read 30 pages of Microeconomics and 60 pages of Econometrics.

a) Assuming that the number of pages per hour that she can read of either subject does not depend on how she allocates her time, how many pages of Econometrics could she read if she decided to spend all her time on this subject and none on Microeconomics?

b) How many pages of Microeconomics could she read if she decided to spend all her time reading Microeconomics?

7. A consumer’s weekly income is € 200, and the prices of commodities x1 and x2 are respectively p1 = € 4 and p2 = € 2. Additionally, a law prohibits people to buy more than 30 units of x1 per week. What is the budget set of this consumer? What is the equation of his budget line?

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8. The government wants to promote the consumption of x and penalize the consumption of y. To this end, it passes a law according to which consumers, to obtain goods, in addition to the payment of the corresponding nominal prices of each good in money ( ), will have to pay a coupon price for

each good ( ). A consumer has a nominal weekly income of € 200 ( ) and the government gives him 200 non transferable coupons every week.

a) Draw the budget constraint of this consumer and derive the equation of the budget line.

b) Suppose that at the equilibrium position, the consumer buys 40 units of x and 20 units of y. What will the new equilibrium be if the number of weekly coupons that the government gives him rises to 400? (Suppose the weekly income of the consumer remains the same at ).

c) Return to the original situation, with and 200 coupons, and suppose that the equilibrium position is units of x and units of y (

= 33.333...). Suppose now that the nominal weakly income of the consumer rises to , while the government maintains unchanged at 200 the number of weekly coupons. As a result of this increase in income, the consumption of x goes up to and the consumption of y goes down to . This shows that y is an inferior good for this consumer. Do you agree? Explain.

(Exam, January 03)

9. A consumer spends all his weekly income of € 200 (m = 200) in two goods x and y. While y can be bought at a constant price, x is sold under a discount system if the purchase is high. Specifically, the price of y is € 2 (py = 2); and the price of x is 4 (px

= 4) for the first 10 units of x and 2 (px = 2) for units in excess of 10. Draw the budget set of this consumer and derive the budget equation.

10. A consumer spends all his weekly income of € 200 (m = 200) in two goods x and y, the prices of which are respectively € 2 and € 4 (px = 4 and py = 2).

a) Draw the budget set and derive the budget equation.

b) How is the budget set modified if the consumer receives a weekly subsidy of € 50 which is non-transferable and can only be spent in good x.

c) Return to the initial budget. How is it modified if the government imposes a tax of € 1 per unit of commodity y bought?

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Lesson 3: Preferences


a) The set of bundles weakly preferred is convex if a line segment connecting any two points in that set, lies entirely in the set.

b) Convex preferences is just one way of saying that people prefer a bundle that mixes commodities to a bundle that concentrates on one commodity.

c) The assumption of convexity is reasonable because people, when consuming, do not specialize in only one good.

d) A computer “hacker” is offered a choice between 10 floppy disks and 5 software manuals, or 9 floppy disks and 20 software manuals. If this individual’s preferences satisfy the usual assumptions, we can safety predict that he will choose the second option.

e) Perfect complements are goods that have to be consumed in fixed proportions.

f) The downward slope of indifference curves is a consequence of the diminishing marginal rate of substitution.

2. Carlos likes both apples xa and bananas xb. In fact, he consumes nothing else. We know he is indifferent between the bundle formed by 20 apples and 5 bananas (20, 5) and any other bundle such that xaxb = 100. We also know that if we place him in the bundle (10, 15), he will show that he is indifferent between this bundle and any other such that xaxb = 150.

a) Graph the indifference curve that passes through point (20, 5), and the indifference curve that passes through point (10, 15).

b) Say whether the following four statements are correct or incorrect:

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c) Is the set of bundles that Carlos weakly prefers to (20, 5) a convex set?

d) Is the set of bundles that Carlos considers inferior to (20, 5) a convex set?

e) Find Carlos’ marginal rate of substitution around the bundle (10, 10).

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f) Find Carlos’ marginal rate of substitution around the bundle (5, 20) and around the bundle (20, 5). Do Carlos’s preferences exhibit diminishing marginal rate of substitution? What is the meaning of this result?

3. For Alex, coffee and tea are substitutes, but not perfect substitutes. Likewise, he regards butter and toast as complements, but not perfect complements.

a) Draw Alex’s indifference curves between coffee and tea.

b) Draw Alex’s indifference curves between butter and toast.

c) With the information given, is it possible to identify any significant difference between the two set of curves in a) and b)?

4. Eva likes apples but hates pears. If apples and pears are the only two goods available, draw her indifference curves.

5. Juan likes food but dislikes cigarette smoke. The more food he has, the more food he would be willing to give up to achieve a given reduction in cigarette smoke. If food and cigarette smoke are the only two goods, draw Juan’s indifference curves.

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Lesson 4: Utility


a) A utility function that represents a person’s preferences is a function that assigns a utility number to each commodity bundle.

b) Once preferences are represented by utility functions, it is very easy to find the marginal rate of substitution because the slope of an indifference curve is minus the ratio of the marginal utility of one good to the marginal utility of the other.

c) If two consumer’s utility functions are monotonic increasing transformations of each other, then these consumers must have the same map of indifference curves.

2. Complete the following table:

U = ( x 1, x2) MU1 ( x 1, x 2) MU2 ( x 1, x 2) MRS ( x 1, x 2)

2x1 + 3x2 - - -

ax1 + bx2 - - -

lnx1 + x2 - - -

x1x2 - - -

- - -

3. Carlos’ utility function is U = xaxb, where xa are the number of apples and xb the number of bananas.

a) Carlos has 40 apples and 5 bananas. Find the equation of the indifference curve that passes through the bundle (40, 5).

b) Berta offers to give Carlos 15 bananas if he will give her 25 apples. Would Carlos have a bundle that he likes better than (40, 5) if he makes this trade? What is the largest number of apples that Berta could demand from Carlos in return for 15 bananas if she expects him to be willing to trade or at least to be indifferent about trading?

4. Out of the whole map of preferences of a consumer, we know the formulas of two of them: x2 = 40 – 4 , and x2 = 24 – 4 .

a) Find out the utility function of this consumer. Is this a quasilinear utility function?

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b) The consumer’s initial consumption is 9 units of x1, and 10 units of x2. If his consumption of x1 is reduced to 4 units, how many units of x2 will he have to consume to be as well-off as before.

c) The consumer is indifferent between the bundle (9, 10) and the bundle (25, 2). If you double the amount of each in each bundle, you would have bundle (18, 20) and bundle (50, 4). Are those two bundles on the same indifference curve?

d) What is the marginal rate of substitution (MRS) when he is consuming the bundle (9, 10)? And when he is consuming the bundle (9, 20)?

e) Can you write a general expression for the consumer’s MRS? Does it depend on the variables x1 and x2? What significance has this fact?

5. Lucia’s utility function is U = max{x, 2y}.

a) Graph the function x = 10 and also the function 2y = 10.

b) Find the value of U if x = 10 and 2y<10. Find the value of U if x>10 and 2y = 10.

c) Now draw the indifference curve along which U = 10.

d) Does Lucia have convex preferences?

6. Antonio’s utility functions is , where x1 is the number of apples and x2 the number of bananas consumed.

a) Fund out the slope of Antonio’s indifference curve through the bundle (4, 6).

b) Three bundles that belong to this indifference curve are: (-, 0); (7, -); and (2, -). Complete the blanks.

c) Find out the equation for the indifference curve through (4, 6).

d) Juana offers Antonio 9 bananas in exchange for 3 apples. Will Antonio accept the offer?

e) Juana says to Antonio “Antonio, I do not understand you. Your MRS is –2. That means that an extra apple is worth only twice as much to you as an extra banana. I offered you 3 bananas for every apple you gave me. If I offer to give you more than your MRS you should want to trade with me”. Is Juana right?

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Lesson 5: Consumer equilibrium


a) The model of consumer behaviour postulated by economists is very simple. It just says that consumers choose the most preferred bundle from their budget sets.

b) Whenever the MRS is different from the price ratio, the consumer cannot be at his or her optimal choice.

c) If a consumer has a utility function , the fraction of her income that she will spend on good 2 is ¼.

d) If two goods are perfect substitutes, the demand for good 2 will be zero whenever .

e) In perfectly competitive markets for goods 1 and 2, all consumers of these two goods will have the same marginal rate of substitution between them.

f) At the point of equilibrium, the increment in utility obtained by the last euro spent on a good, must be the same for all goods.

g) If goods x and y are perfect substitutes, the demand for good x will be zero whenever the price of x is greater than the price of y. (Exam, January 03)

h) Ana is a non typical consumer because, for her, coffee and tea are complements: when the price of coffee goes up, her demand for tea goes down. This must necessarily mean that, for her, tea is a normal good. (Exam, January 03)

i) A man initially spends half his income on “food” and the other half on “non-food”. The price of “food” rises by 10% and income by 5%. As a result of these changes, this man, whose indifference curves are well behaved, will be neither better nor worse off. (Exam, January 03)

2. CDs cost € 10 each and tapes € 2 each. In equilibrium, Victoria consumes both CD’s and tapes; Alberto consumes only tapes. What can you infer about Victoria’s marginal rate of substitution of CD’s for tapes. What about Alberto’s? Draw two diagrams depicting the situation of each consumer.

3. A student spends 8 hours per day listening to music. M hours are devoted to Mozart and B hours to Beethoven. The student’s utility function is , where U measures utility.

a) Find the equation of the indifference curves corresponding to utility levels of 4 and 5.

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b) Write an equation for the student’s budget constraint.

c) Draw on a diagram your answers to a) and b).

d) On the basis of your diagram, what are the approximate utility-maximizing quantities of M and B?

e) Using calculus, find the exact equilibrium quantities of M and B.

4. A consumer’s utility function is . The consumer’s income is € 40 per week and the prices of the two goods are p1 = 1 and p2 = 2. Find the optimal weekly consumption of this consumer.

5. Consider again the consumer of question 4. His utility functions is . Now we want to find out general expressions for his demand function for good 1,

, and his demand function for good 2, , where p1 and p2

are the prices of the two goods and m is the consumer’s income.

a) Derive the equation that equates the slope of the budget line to the marginal rate of substitution.

b) The equation derived in a) plus the budget line gives you a system of two equations with two unknowns, x1 and x2. Solve this system and find the consumer’s demand for good 1, , and the consumer’s demand for

good 2, .

c) In general, the demand for both commodities will depend on the price of both commodities and on income. But for this utility function, the demand for x1

depends only on income and the price of x1, and the demand for x2 depends only on income and the price of x2. This means that the consumer always spends the same fraction of his incomes on x1. What is this fraction?

d) Repeat a), b) and c) for the utility functions , , and .

6. Consider yet again the same consumer as in the previous to questions. His utility function is . Suppose now that it has an income of € 24 (that is, m = 24). Initially p1 = 3 and p2 = 1.

a) Using the demand functions derived in problem 5, find out what are the consumption of the two goods which maximize the consumer’s utility.[Notation: following the notation used in the theory class, we will denote the optimal consumption of, say, x1 for prices p1 and p2, and income m as

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. Therefore, what this question asks is to find out the values of

and ].

b) Now suppose that the price of good 1 falls to € 2 and that the price of good 2 remains at € 1. Find out the new levels of consumption. That is, find out

and .

c) After the fall in the price of good 1, what level of income would the consumer need in order to be able to buy the old bundle of goods? Call this new level of income m’.

d) Find out the bundle of goods that the consumer would choose at the new set of prices (p1 = 2 and p2 = 1) but with income m’. That is, find our and

.

e) Following the notation used in the theory class the total effect of this price change for the two goods is

Total effect for x1 = -

Total effect for x2 = -

Define, using the same type of notation, the substitution effect and the income effect for the two goods.

f) Find out quantitatively the total, substitution and income effects of this price change for both goods, and check that the Slutsky equation holds for both of them.

7. Julia is trying to decide how to allocate her time in studying for her economics course. She has to take two tests: Test 1 and Test 2. Her overall score for the course will be the minimum of her scores on the two examinations. She has decided to devote a total of 1,200 minutes to studying for these two exams, and she wants to get as high an overall score as possible. For every 10 minutes that she spends studying for Test 1, she will increase her score by 1 point (suppose that if she does not study at all the score is zero and that scores are in principle ilimited; that is, the minimum is 0 and the maximum can go above 100). For every 20 minutes she spends studying for Test 2, she will increase her score by one point.

a) Draw the “budget line” of this problem and derive its equation. (Hint: the two goods are the score of Test 1, S1, and the score of Test 2, S2; “income” is the amount of minutes Julia has, and “prices” are the number of minutes needed in each test to rise the score by one point).

b) Write the “utility function” of this problem and draw some “indifference curves” (Hint: the “utility function” in this case is the rule whereby the scores of the two tests determine the overall score).

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c) Draw a straight line that goes through the kinks of Julia’s indifference curves. Write the equation of this line.

d) Find the S1 and S2 that will maximize the overall score and state how will she distribute her 1,200 minutes of study between the two tests.

e) How would your answers to this problem change if the overall score was determined not as the minimum of the two scores, but as the maximum of the two scores?

8. Suppose that Telefónica allows consumers to choose between to different pricing plans. Plan A: for a fixed fee of € 12 per month you can make as many local phone calls as you want, at no additional charge per call. Plan B: you can pay a fixed fee of € 8 per month and be charged € 0.05 for each local phone call that you make. Suppose that you have € 20 per month to spend.

a) Draw the budget lines of Plan A and Plan B in a graph in which the horizontal axis measures the number of local phone calls and the vertical axis money (or, what is the same, consumption of other goods).

b) Where do the two budget lines cross?

c) If your preferences are well behaved (smooth, convex, downward sloping indifference curves) what Plan would you tend to choose?

d) What sort of preferences would make you prefer Plan B?

9. Consider this text of Alice in Wonderland by Lewis Carroll (1832-1898):

“I should like to buy an egg, please” she said timidly. “How do you sell them?”“Five pence farthing for one – two pence for two,” the Sheep replied.“Then two are cheaper than one?” Alice said, taking out her purse.“Only you must eat them both if you buy two,” said the Sheep.“Then I’ll have one please”, said Alice (...)

a) Draw Alice’s budget set, under the assumption that she has a total of 8 pence to spend. Notice that a farthing is of a penny. Consider that she can buy either 0, 1 or 2 eggs, but not fractional eggs. Then her budget set consists of only three points. Put “eggs” on the horizontal axis and “money” on the vertical axis.

b) Identify the point chosen by Alice.

c) Draw indifference curves for Alice that are consistent with this behaviour and explain the meaning of her utility function.

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Lesson 6: Extensions and applications of consumer theory


a) If the bundle of goods A is chosen when the bundle B was available, B will only be chosen when A is not available.

b) The prices are , and the consumer is currently consuming

. Now prices change to . Since he cannot afford his old bundle at the new prices, the consumer is clearly worse off.

c) An increase in the wage rate is a bad thing for consumers because when the wage increases leisure becomes more expensive.

d) The US currently imports about half of the petroleum it uses. The rest of its needs are met by domestic production. The price of oil could rise so much that the US could be made better off.

e) Anita’s consumption of two goods happens to be equal to her endowment. Her preferences fulfill the usual assumptions and she is endowed with positive amounts of both goods. Any change in the relative price of the two goods will make Anita better off.

f) If a person is a lender and the interest rate rises, he or she will remain a lender.

g) If a person is a borrower and the interest rate rises, he or she will remain a borrower.

h) If after a price change the Laspeyre’s quantity index of a given consumer is less than 1, this consumer must be better off as a result of this price change. (Exam, January 03)

2. When prices are a consumer demands , and when

prices are this consumer demands .

a) Draw a graphic depicting the two budget constraints and the two choices.

b) Is this behaviour consistent with the model of maximizing behaviour?

3. Suppose that between 1960 and 1985, the price of all goods exactly doubled while every consumer’s income tripled. Would the Laspeyres price index for 1985, with base year 1960 be less than 2, grater than 2, or exactly equal to 2. What about the Paasche index? Can we say anything about how the welfare of consumers evolved between 1960 and 1985?

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4. Luis consumes only loafs of bred and bottles of wine. Last week the prices of bread and wine were € 1 each, and Luis consumed 15 units of each.

a) What was Luis’ income last week?

b) Draw last week Luis’ budget line and label his consumption bundle with the letter A.

c) This week the price of bread increases to € 2 per loaf, and the price of wine remains at € 1 per bottle. By chance, Luis’ income changes so that he can just afford his old consumption bundle (15, 15) at the new prices. Draw the new budget line. Does it go through point A? What is the slope of this line?

d) Can Luis, after the price change, afford to consume more bread than last week? Will he consume more bread?

5. Mario produces lettuces and tomatoes. He is both a consumer and a seller of the production obtained. The two vegetables are perfect complements for him and he consumes them always at a 1:1 ratio (expressed in kilograms).

a) In the past week production was 30 kgs. of lettuce and 10 kgs. of tomatoes, and the prices of each vegetable was € 5 per kg., what was the monetary value of Mario’s endowment during that week?

b) Draw Mario’s budget line and calculate its consumption. Label this bundle A. How much lettuce and tomato was Mario selling/buying in the market?

c) This week the price of tomatoes rises to € 15 per kg., while the price of lettuce stays at € 5 per kg. What will be his new consumption bundle and how much produce will he sell to the market?

d) Is there any set of prices that would make Mario a net buyer of lettuce? Why or why not?

6. Julio has an income of € 2,000 a year, and expects an income of € 1,100 next year. He can borrow and lend money at an interest rate of 10%. Consumption goods cost € 1 per unit this year and there is no inflation.

a) What is the present value of Julio’s endowment? What is the future value of Julio’s endowment? Draw his budget line and write an expression for the slope of this line.

b) Suppose that Julio has the utility function . Write an expression for Julio’s marginal rate of substitution between consumption this year ( ) and consumption next year ( ).

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c) Find out the equilibrium position, and state whether Julio will be a borrower or a lender; and by how much.

d) How would your answers to a), b) and c) change if the interest rate rose to 20%?

7. A consumer has the utility function . The consumer has income € 100 in period 1 and € 121 in period 2.

a) Find out the optimal consumption of this consumer for the two periods if the interest rate is 10%. Will he be a borrower or a lender?

b) Suppose now that there is an inflation rate of 6% and that the price of first period goods is € 1. What is now the budget line? How will the existence of inflation affect the equilibrium of the consumer?

8. When the price of x is 1 and the price of y is 1, the consumer buys 6 units of x and 4 units of y.

a) Draw his budget constraint and find out the equation for the budget line.

b) The price of x rises to 2 and the price of y falls to 0.5, and at the same time the consumer income falls so that he is now buying 3 units of x and 5 units of y. Draw the new budget constraint and find the equation for the new budget line.

c) Is the behaviour of this consumer consistent with the Weak Axiom of Revealed Preference?

(Exam, January 03)

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Lesson 7: Consumer Surplus


a) The area between the demand curve of a consumer and the price line will measure the true gain in welfare of this consumer as a result of being able to consume the corresponding good, only if the income effect is nil. However, if the income effect is relatively small, this area will be a good approximation of this gain in welfare.

b) The inverse demand curve of a consumer has a height of 100 when q = 0 (q is the number of units bought by the consumer) and a height of 50 when q = 5. The consumer would need a compensation of at least € 125 to induce him not to purchase the 5 units that he is currently consuming.

c) Suppose that two goods are perfect complements for a consumer. Then for this consumer the compensating variation of a price change will be the same whether we measure it “a la Hicks” or “a la Slutsky”.

d) If the percentage rise in income between two years is the same as the percentage rise in the Laspeyres’ price index, the consumer will be worse off.

2. The following table shows the reservation price for apartments of eight persons named A to H.

Person: A B C D E F G HReservation price: 40 25 30 35 10 18 15 5

a) If the equilibrium rent for an apartment turns out to be € 20, which consumers will get apartments?

b) If the equilibrium rent is € 20, what is the consumer’s surplus generated in this market for person A? And for person D? And for the whole market?

c) If the rent declines to € 19, by how much does the total consumer’s surplus increase? What about if the rent declines to € 10?

3. Jordi’s inverse demand curve is .

a) Suppose Jordi currently has 5 units of the good. How much money would you have to pay him to compensate him for surrendering the 5 units of x that he has?

b) Suppose now that Jordi has to pay a price of € 50 per unit to purchase the 5 units of x. If you require him to reduce his purchases to zero, how much money would be necessary to compensate him?

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4. Berta’s utility function is . Initially she faces prices and has an income of € 10.

a) What is the bundle of goods that Berta consumes at her equilibrium point?

b) What is the utility that she obtains at this equilibrium point?

c) Suppose that prices change to . Calculate Berta’s new equilibrium point and utility level.

d) Calculate the compensating and equivalent variations corresponding to the above price change.

5. When the price of gasoline is € 1 per litre, you consume 1,000 litres per year. Then two things happen: a) the price of gasoline rises to € 2 per litre; and b) you inherit € 1,000 a year.

a) Are you better off than before?

b) You are asked to calculate the compensating variation (a la Slutsky) of this price change; do you need any additional information? If yes, say what; if not, find the value of the compensating variation.

c) You are asked to calculate the compensating variation (a la Hicks) of this price change; do you need any additional information? If yes, say what; if not, find out the value of the compensating variation.

6. Suppose that a consumer has a utility function . He originally faces prices (1,1) and has income 100. Then the price of good 1 increases to 2 while the price of good 2 remains constant at 1. What are the compensating and equivalent variations of this price change?

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Lesson 8: Market demand


a) In a two good model, if one good is an inferior good the other good must be a luxury good.

b) If a demand curve is inelastic, revenue will increase when price goes down.

c) As a result of a 5% increase in the price of a good, market revenue goes down by 3%. Therefore the demand curve for this good is inelastic.

d) If the market demand curve is , revenue will be maximized when the price equals 3.

2. Find an expression for the price elasticity of the following demand functions. In all cases where the elasticity is not constant, express this elasticity as a function of the price.

3. For each of the following demand curves, compute the inverse demand curve.

4. In a city there are two kinds of gasoline consumers: Clio owners and Mondeo owners. Every Clio owner has a demand function for gasoline . Every Mondeo owner has a demand function for gasoline . Quantities are litres per week and prices are € per litre. Suppose that in the city there are 100 Clio owners and 50 Mondeo owners.

a) If the price is € 3, what is the gasoline demand by an individual Clio owner? And by an individual Mondeo owner?

b) What is the total amount demanded by all Clio owners in the city? And by all Mondeo owners?

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c) What is the total amount of gasoline demanded in the city if ?

d) Represent graphically the total demand by all Clio owners, the total demand by all Mondeo owners, and the total demand for gasoline in the city.

e) At what price has the demand curve have kinks.

f) When the price is € 1 per litre, how much does weekly demand fall when price rises by 10 cents?

g) Repeat your answer for question f) when the price is € 4.50 per litre and when the price is € 10 per litre. (Remember that the individual curves are defined only for positive values of q).

5. The demand for bottles of wine is .

a) At what price will total revenue realized from the sale of wine be at its maximum?

b) How many bottles of wine will be sold at that price?

6. The demand function for football tickets for a Spanish first division club is . The financial director of the club sets the price of tickets so

as to maximize revenue. The football stadium holds 100,000 spectators.

a) Write down the inverse demand function.

b) Write expressions for total revenue and marginal revenue as a function of the number of tickets sold.

c) Draw in a graph the inverse demand function, the marginal revenue function and a vertical line representing the capacity of the stadium.

d) What price will generate maximum revenue? What quantity will be sold at that price?

e) At this quantity, what is marginal revenue? At this quantity, what is the price elasticity of demand? Will the stadium be full?

f) Repeat your answers to a), b), c), d) and e) above if the demand function is .

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Lesson 9: Market equilibrium


a) The equilibrium of a competitive market is efficient in the sense that it maximises the sum of consumers’ and producers’ surplus.

b) The equilibrium of a competitive market is efficient in the sense that it maximises the difference between the value to buyers and the cost to sellers of the quantity produced and sold.

c) A situation is Pareto efficient if there is no way to make some group of people better off without making some other group worse off.

d) When a good is taxed, there will always be two prices: the price paid by the demanders and the price received by the suppliers. The difference between the two represents the amount of tax.

e) If the demand curve is vertical while the supply curve slopes upward, a tax imposed in this market will end up being paid totally by producers.

f) If the supply curve is vertical, the deadweight loss of a tax is nil.

2. The maximum price consumers of beer are willing to pay is 5 € per pack (that is, at 5 € or above the demand for beer is zero). The minimum price at which producers will supply beer is 0.50 € per pack (that is, at 0.50 € or below the supply of beer is zero). The market equilibrium is achieved at a quantity of 1,000 packs per day and a price of 2.5 € per pack (suppose that both demand and supply are linear functions).

i) A tax of 0.50 € per pack is imposed on consumers and, as a result, the price consumers end up paying is 2.78 € per pack of beer. Find out the price producers get in the new equilibrium and explain your answer. Would this answer change if the tax, instead of being imposed on consumers, had been imposed on producers?

ii) As a result of the tax, the new quantity sold is 889 packs per day. Find out the change in consumers’ and producers’ surplus, and the tax revenue collected by the government. Has the tax lowered the efficiency of this market? What is the efficiency cost of this tax?

iii) How would your answers to i) and ii) above change if in the situation previous to the tax the supply curve was completely elastic at 2.5 € per pack and the equilibrium quantity after the imposition of the tax was 800 packs per day?

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3. The demand equation for ski lessons is given by and the supply curve is given by .

a) What is the equilibrium price? What is the equilibrium quantity?

b) If a tax is introduced, the price paid by demanders, , will not coincide with the price received by suppliers, . Therefore, in this situation, the demand equation is , and the supply equation . Suppose that indeed a tax of € 10 per ski lesson is imposed on consumers. Write an equation that relates the price paid by demanders to the price received by suppliers.

c) Write a system of four equations formed by: i) the demand equation after the tax is imposed; ii) the supply equation after the tax is imposed; iii) the equation that relates and derived in b) above; and iv) the condition of market equilibrium.

d) Solve this system for the four unknowns: , , and .

e) What is the deadweight loss of this tax?

4. A politician suggests that although ski lesson consumers are rich and deserve to be taxed, ski instructors are poor and deserve a subsidy. He proposes a subsidy of € 6 per lesson to be given to suppliers, while at the same time maintaining the € 10 tax per lesson on consumers.

a) How will the subsidy move the supply curve?

b) What will be the number of lessons, the price paid by consumers and the price received by suppliers?

c) Would this policy have any different effects for suppliers or for demanders than a tax of € 4 per lesson imposed on demanders?

5. The demand equation for salted codfish is and the supply equation .

a) Draw the demand and supply curve.

b) A quantity tax of € 2 per unit sold is placed on suppliers. Draw the new supply curve.

c) Find out the new equilibrium price paid by demanders, the new price received by suppliers, and the equilibrium quantity.

d) Calculate the deadweight loss due to this tax.

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6. The price elasticity of demand for books is constant and equal to –1. When the price is € 10 per unit, the total amount demanded is 6,000 units.

a) Write an equation for the demand function.

b) Graph this demand function for prices between € 20 and € 5.

c) If the supply is perfectly inelastic at 5,000 units, what is the equilibrium price? Show the supply curve in the graph.

d) Suppose that the demand curve shifts outward by 10%. Write down the new demand function. Suppose that the supply function remains vertical but shifts to the right by 5%. Solve for the new equilibrium price and quantity. By what percentage approximately did the equilibrium price rise?

e) Suppose that in this problem the demand curve shifts outward by and the supply curve shifts by . By approximately what percentage will the equilibrium price rise?

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Lesson 10: Exchange


a) A Pareto efficient allocation is one in which there is no feasible reallocation of the goods that would make all consumers at least as well-off and at least one consumer strictly better off.

b) A given allocation can be Pareto efficient even if in that allocation everyone is worse off than at a non Pareto efficient allocation.

c) If we know the contract curve, then we know the outcome of any trading.

d) The fact that the value of excess demand in 8 out of 10 markets is equal to zero, does not say anything about what is the value of excess demand in the remaining two markets.

e) A competitive equilibrium is Pareto efficient.

2. Suppose there a two agents in an economy named A and B, and two goods named x and y. The endowment of A is and that of B .

a) Represent the corresponding Edgeworth box of this economy (put good x on the horizontal axis and good y on the vertical axis) and identify the endowment point.

b) You are told that after trading, the respective position of the two agents is and . Describe who sells what and who buys

what.

c) Suppose that the trading above has been done in the context of a market which generates prices for the two goods, and that you are told that the price of good x is 1. What is the price of y?

3. Antonio (A) and Belen (B) are the only two agents in an economy. There are only two goods: x and y. Antonio has an initial endowment of 60 units of x and 10 units of y; . Belen’s initial endowment is . For Antonio, the two goods are perfect substitutes, so his utility function is

. Belen, on the other hand, has a Cobb-Douglas utility function .

a) Draw the Edgeworth box of this economy and identify the initial endowment point.

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b) Find out Antonio’s MRS, and Belen’s MRS when she consumes the bundle . What is the equation of the locus of Pareto optimal allocations

(“contract curve”)?

c) Given the form of Antonio’s utility function, can you deduce what will be the equilibrium relative price of this economy? (Assume that both agents and up consuming some of x and some of y).

d) If we make good x the numeraire, what is the equilibrium price of good y?

e) At the equilibrium prices just found, what is the value of Belen’s initial endowment? At these prices, what will be the bundle Belen will choose to consume? If Antonio consumes all x and all y not consumed by Belen, what will be Antonio’s equilibrium consumption bundle?

f) Calculate Antonio’s income at the equilibrium prices. Can Antonio at these prices afford a bundle that he likes better that his equilibrium consumption bundle?

4. Consider an economy with two agents, A and B, and two goods, x and y. Initial endowments are and . A’s utility function is

. B’s utility function is .

a) Draw the Edgeworth box, identify the initial endowment and sketch a few indifference curves of A and B.

b) From your answer to question a), can you identify the locus of Pareto optimal allocations (the “contract curve”)?

c) Measure the level of utility of A and B at the initial endowment.

d) Would a trade such that the final consumption was and

be in principle acceptable to agent B?

e) Give three bundles, all of which could in principle be equilibrium consumption bundles.

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Lesson 11: Welfare


a) A welfare function is an instrument to choose among the many Pareto efficient allocations that may exist.

a) Welfare functions are simply a way to represent distributional judgements.

a) As long as the welfare function is increasing in each individual’s utility, an allocation in which welfare is maximum is also a Pareto efficient allocation.

2. A Rawlsian welfare function counts only the welfare of the worst off agent. The opposite of such function might be called a “Nietzschean” welfare function –a welfare function that says the value of an allocation depends only on the best off agent.

a) What mathematical form would a “Nietzschean” welfare function have?

b) Draw in a graph a set of social indifference curves for this “Nietzschean” welfare function.

c) Suppose that the utility possibilities set is a convex set. What kind of allocations represent welfare maxima of the “Nietzschean” welfare function?

3. A social planner has decided that she wants to allocate income between two people so as to maximize the welfare function where Yi is the amount of income that person i gets. Suppose the planner has a fixed amount of money to allocate, 100, so that the income possibility frontier has the formula .

a) Draw the income possibility frontier of this economy.

b) Find out the distribution of income that maximizes welfare.

c) Suppose that the planner likes person 2 more than person 1 and that the welfare function is now . What is the new optimal distribution of income?

d) Suppose now that person 1 is very forgetful, and every time you give him € 1, he looses € 0.50. Define the new income possibility frontier, in terms of income really held by each person. What is now the optimal distribution?

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4. Suppose that the utility possibility frontier for two individuals is given by .

a) Draw the utility frontier.

b) If the social welfare function is of the Nietzschean type, , what is the optimal distribution of utilities?

c) What is the optimal distribution if instead the welfare function is of the Rawlsian type, ?

d) Find out now the optimal distribution for the Cobb-Douglas welfare function, .

e) Draw the three results above.

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Lesson 12: Market failures


a) An explicit allocation of property rights usually eliminates the problem of externalities.

b) If property rights are clearly assigned, the equilibrium obtained will be Pareto efficient, irrespective of whether these rights are assigned to one agent or another.

c) If a public good can be provided in a variable amount, then the necessary condition for a given amount to be Pareto efficient is that the sum over all consumers of the marginal rate of substitution between the private and public good should equal the marginal cost.

d) Imperfect and asymmetric information can lead to drastic differences in the nature of market equilibrium.

e) When adverse selection or moral hazard are present some agents will want to invest in signals that will differentiate them from other agents.

2. A steel mill pumps its waste in a nearby lake. The lake is also used by a fishery. Let X be the amount of waste that the steel mill pumps every year into the lake. If the steel mill pumps X units of waste into the lake, its profits, B, will be:

. More waste, unfortunately, means less fish. The total cost, C, that the steel mill’s waste has on the fishery (that is, the amount by which the fishery’s benefits will decrease) can be measured by the following function:

.

a) Suppose that the steel mill has the legal right to pump as much waste as it wants. How many units of waste will it pump into the lake?

b) Considering the existence of the fishery, is this an efficient outcome? Why or why not?

c) By how much is the benefit of the fishery reduced by the behaviour of the steel mill?

d) Suppose now that the steel mill and the fishery can negotiate. What is the likely outcome as far as the number of units of waste that the factory will pump? Is this an efficient outcome?

e) Would your answer to d) change if the fishery had the legal right to a non polluted water in the lake? Explain.

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f) Suppose that steel mill and the fishery merge into a single firm. What will then be the optimal amount of waste pumped by the new firm? Compare this answer to that obtained in d).

3. Every March 19th three Valencian friends (Vicente, Amparo and José) hold a fireworks display for them and their families. The marginal cost of each rocket is constant at € 100. Currently, the display consists of 75 rockets. The marginal value of the 75th rocket is € 5 for Vicente, € 50 for Amparo and € 10 for José. From an efficiency point of view, is the fireworks display too big, too small or the correct size? Explain.

4. The public good Z can be provided at a constant marginal cost of € 12. Laila’s demand for Z is , and Clara’s demand is , where Z is the quantity of the public good demanded and the price per unit. What is the Pareto efficient level of Z?

5. Consider a market with 100 people who want to sell their used cars and 100 people who want to buy a used car. Everyone knows that 50% of the cars are good cars (“plums”) and 50% are bad cars (“lemons”). The current owner of each car knows its quality, but the prospective purchasers don’t know the quality of the cars.

a) If the owner of a lemon is willing to sell it for € 1,000 and the owner of a plum for € 2,000, and if the buyers are willing to pay € 1,200 for a lemon and € 2,400 for a plum, what will the likely outcome of this market be?

b) Can you suggest a mechanism whereby this outcome could be improved?

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