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, CobbDouglas, 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(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.
Chapter 5
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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(.)
Chapter 5
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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(.)
Chapter 5
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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(.)
Chapter 5
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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(.)
Chapter 5
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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(.)
Chapter 5
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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(.)
Chapter 5
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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(.)
Chapter 5
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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(.)
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 AideMemoire.
• Note the important result:
• Preferences can be represented by a
utility function but this is not unique.
Exam 3 of 8 september 2008
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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
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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!!