Microeconomics Corso E

1. MicroeconomicsCorso E John Hey

2. 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.

3. 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.

4. 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).

5. Chapter 5 • Representation of preferences with utility functions. • Suppose indifference curves are given by g(q1,q2) = constant • where the higher the constant the happier the individual. Then we can represent these preferences by the utility function: • U(q1,q2) = g(q1,q2) or by: • U(q1,q2) = f[g(q1,q2)] for any increasing function f[.]. • Note that this utility function is not unique.

6. Chapter 5 • Perfect substitutes 1:1 • An indifference curve is given by: • q1 + q2 = constant • Hence a utility function which represents these preferences is • U(q1 , q2) = q1 + q2 • Or • U(q1 , q2) = f(q1 + q2) for any f(.)

7. Chapter 5 • Perfect substitutes 1:2 • An indifference curve is given by: • q1 + q2/2 = constant • Hence a utility function which represents these preferences is • U(q1 , q2) = q1 + q2/2 • Or • U(q1 , q2) = f(q1 + q2/2) for any f(.)

8. Chapter 5 • Perfect substitutes 1:a • An indifference curve is given by: • q1 + q2/a = constant • Hence a utility function which represents these preferences is • U(q1 , q2) = q1 + q2/a • Or • U(q1 , q2) = f(q1 + q2/a) for any f(.)

9. Chapter 5 • Perfect complements 1 with 1 • An indifference curve is given by: • min(q1, q2) = constant • Hence a utility function which represents these preferences is • U(q1 , q2) = min(q1, q2) • Or • U(q1 , q2) = f[min(q1, q2)] for any f(.)

10. Chapter 5 • Perfect complements 1 with 2 • An indifference curve is given by: • min(q1, q2/2) = constant • Hence a utility function which represents these preferences is • U(q1, q2) = min(q1, q2/2) • Or • U(q1, q2) = f[min(q1, q2/2)] for any f(.)

11. Chapter 5 • Perfect complements1 with a • An indifference curve is given by: • min(q1, q2/a) = constant • Hence a utility function which represents these preferences is • U(q1, q2) = min(q1, q2/a) • Or • U(q1, q2) = f[min(q1, q2/a)] for any f(.)

12. Chapter 5 • Cobb-Douglas with parameter a • An indifference curve is given by: • q1a q2(1-a) = constant • Or by: • a ln(q1 )+ (1-a) ln(q2 ) = constant • Hence a utility function which represents these preferences is • U(q1 , q2) = q1a q2(1-a) • or • U(q1 , q2) = a ln(q1 )+ (1-a) ln(q2 ) • or • U(q1 , q2) = f(q1a q2(1-a)) for any f(.)

13. Chapter 5 • Stone-Geary with parameters a, s1 and s2 • An indifference curve is given by: • (q1-s1)a(q2 –s2)(1-a) = constant • Or by: • a ln(q1–s1)+ (1-a) ln(q2 –s2) = constant • Hence a utility function which represents these preferences is • U(q1 , q2) = (q1–s1)a (q2 –s2 )(1-a) • or • U(q1 , q2) = a ln(q1–s1)+ (1-a) ln(q2 ) • or • U(q1 , q2) = f[(q1–s1)a (q2–s2)(1-a)] for any f(.)

14. 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.

15. 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)?

17. 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.

18. Chapter 5 • Goodbye!!