Microeconomics Corso E John Hey

Chapter 5 We know that the indifference curves of an individual are given by the preferences of that individual. We know that the demand and supply curves depend upon the preferences. Up till now we have assumed a particular kind of preferences – quasi-linear – where the indifference curves are vertically parallel.

Chapter 5 Today we study other types of preference. Economists have made a catalogue of the types that we observe in reality. We cannot study all these types. We make an important selection: Perfect Substitutes, Perfect Complements, Cobb- Douglas, Stone-Geary. The important thing: demand and supply depend on the preferences.

Chapter 5 We first make a small generalisation: we work with two goods (instead of one good and money): the quantity of good 1 on the horizontal axis and the quantity of good 2 on the vertical axis. Of course, a special case is when good 2 is money (and then its price is 1).

Chapter 5 Representation of preferences with utility functions. Suppose indifference curves are given by g(q 1,q 2 ) = constant where the higher the constant the happier the individual. Then we can represent these preferences by the utility function: U(q 1,q 2 ) = g(q 1,q 2 ) or by: U(q 1,q 2 ) = f[g(q 1,q 2 )] for any increasing function f[.]. Note that this utility function is not unique.

Chapter 5 Perfect substitutes 1:1 An indifference curve is given by: q 1 + q 2 = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = q 1 + q 2 Or U(q 1, q 2 ) = f(q 1 + q 2 ) for any f(.)

Chapter 5 Perfect substitutes 1:2 An indifference curve is given by: q 1 + q 2 /2 = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = q 1 + q 2 /2 Or U(q 1, q 2 ) = f(q 1 + q 2 /2) for any f(.)

Chapter 5 Perfect substitutes 1:a An indifference curve is given by: q 1 + q 2 /a = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = q 1 + q 2 /a Or U(q 1, q 2 ) = f(q 1 + q 2 /a) for any f(.)

Chapter 5 Perfect complements 1 with 1 An indifference curve is given by: min(q 1, q 2 ) = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = min(q 1, q 2 ) Or U(q 1, q 2 ) = f[min(q 1, q 2 )] for any f(.)

Chapter 5 Perfect complements 1 with 2 An indifference curve is given by: min(q 1, q 2 /2) = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = min(q 1, q 2 /2) Or U(q 1, q 2 ) = f[min(q 1, q 2 /2)] for any f(.)

Chapter 5 Perfect complements1 with a An indifference curve is given by: min(q 1, q 2 /a) = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = min(q 1, q 2 /a) Or U(q 1, q 2 ) = f[min(q 1, q 2 /a)] for any f(.)

Chapter 5 Cobb-Douglas with parameter a An indifference curve is given by: q 1 a q 2 (1-a) = constant Or by: a ln(q 1 )+ (1-a) ln(q 2 ) = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = q 1 a q 2 (1-a) or U(q 1, q 2 ) = a ln(q 1 )+ (1-a) ln(q 2 ) or U(q 1, q 2 ) = f(q 1 a q 2 (1-a) ) for any f(.)

Chapter 5 Stone-Geary with parameters a, s 1 and s 2 An indifference curve is given by: (q 1 -s 1 ) a (q 2 –s 2 ) (1-a) = constant Or by: a ln(q 1 –s 1 )+ (1-a) ln(q 2 –s 2 ) = constant Hence a utility function which represents these preferences is U(q 1, q 2 ) = (q 1 –s 1 ) a (q 2 –s 2 ) (1-a) or U(q 1, q 2 ) = a ln(q 1 –s 1 )+ (1-a) ln(q 2 ) or U(q 1, q 2 ) = f[(q 1 –s 1 ) a (q 2 –s 2 ) (1-a) ] for any f(.)

Chapter 5 In the book you can find all the formula. It is not necessary to remember the formulas… … in the exams there will be an Aide- Memoire. Note the important result: Preferences can be represented by a utility function but this is not unique.

Exam 3 of 8 september 2008 Consider a market for a hypothetical good in which there are a number of buyers and sellers, each of which wants to buy or sell one unit of the good. Assume that a buyer who is indifferent about buying always buys and a seller who is indifferent about selling always sells. The reservation prices are given below, first for the buyers and then for the sellers. Buyers: 10, 10, 8, 5, 4. Sellers: 4, 5, 5, 7, 2, 4. Question 1: What is the competitive equilibrium price (specify a range if more than one equilibrium price)? Question 2: What is the quantity exchanged in the competitive equilibrium? Question 3: What is the maximum total surplus generated in the market? Question 4: What is the maximum number of trades (not necessarily with the same price)?

Exam 3 of 8 september 2008 Consider a market for a hypothetical good in which there are a number of buyers and sellers, each of which wants to buy or sell one unit of the good. Assume that a buyer who is indifferent about buying always buys and a seller who is indifferent about selling always sells. The reservation prices are given below, first for the buyers and then for the sellers. Buyers: 10, 10, 8, 5, 4. Sellers: 4, 5, 5, 7, 2, 4. Question 1: What is the competitive equilibrium price (specify a range if more than one equilibrium price)? Answer 5. Question 2: What is the quantity exchanged in the competitive equilibrium? Answer 4. Question 3: What is the maximum total surplus generated in the market? Answer 18. Question 4: What is the maximum number of trades (not necessarily with the same price)? Answer 5.

Chapter 5 Goodbye!!