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Quick Reminder of the Theory of Consumer Choice

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Title: Quick Reminder of the Theory of Consumer Choice


1
Quick Reminder of the Theory of Consumer Choice
  • Professor Roberto Chang
  • Rutgers University
  • January 2007

2
  • Reminder of Theory of Consumer Choice, as given
    by Mankiw, Principles of Economics, chapter 21,
    and other elementary textbooks.

3
A Canonical Problem
  • Consider the problem of a consumer that may
    choose to buy apples (x) or bananas (y)
  • Suppose the price of apples is px and the price
    of bananas is py.
  • Finally, suppose that he has I dollars to spend.

4
The Budget Set
  • The budget set is the set of options (here,
    combinations of x and y) open to the consumer.
  • Given our assumptions, the total expenditure on
    apples and bananas cannot exceed income, i.e.
  • px x py y I

5
  • Rewrite
  • px x py y I
  • as
  • y I/py (px/py) x
  • This is the budget line

6
Bananas (y)
I/py
O
I/px
Apples (x)
7
Bananas (y)
I/py
O
I/px
Apples (x)
8
Bananas (y)
I/py
Budget Line px x py y I (Slope - px/py)
O
I/px
Apples (x)
9
  • If I increases, the new budget line is higher and
    parallel to the old one.

10
Bananas (y)
I/py
O
I/px
Apples (x)
11
Bananas (y)
I/py
I gt I
I/py
I/px
O
I/px
Apples (x)
12
  • If px increases, the budget line retains the same
    vertical intercept, but the horizontal intercept
    shrinks

13
Bananas (y)
I/py
O
I/px
Apples (x)
14
Bananas (y)
I/py
px gt px
O
I/px
I/ px
Apples (x)
15
Preferences
  • Now that we have identified the options open to
    the consumer, which one will he choose?
  • The choice will depend on his preferences, i.e.
    his relative taste for apples or bananas.
  • In Economics, preferences are usually assumed to
    be given by a utility function.

16
Utility Functions
  • In this case, a utility function is a function U
    U(x,y) , where U is the level of satisfaction
    derived from consumption of (x,y).
  • For example, one may assume that
  • U log x log y
  • or that
  • U xy

17
Indifference Curves
  • It is useful to identify indifference curves. An
    indifference curve is a set of pairs (x,y) that
    yield the same level of utility.
  • For example, for U xy, an indifference curve is
    given by setting U 1, i.e.
  • 1 xy
  • A different indifference curve is given by the
    pairs (x,y) such that U 2, i.e. 2 xy

18
y
Three Indifference Curves
Utility u0
x
19
y
Three Indifference Curves Here u1 gt u0
Utility u1
Utility u0
x
20
y
Three Indifference Curves Here u2 gt u1 gt u0
Utility u2
Utility u1
Utility u0
x
21
Properties of Indifference Curves
  • Higher indifference curves represent higher
    levels of utility
  • Indifference curves slope down
  • They do not cross
  • They bow inward

22
Optimal Consumption
  • In Economics we assume that the consumer will
    pick the best feasible combination of apples and
    bananas.
  • Feasible means that (x,y) must be in the
    budget set
  • Best means that (x,y) must attain the highest
    possible indifference curve

23
Bananas (y)
Consumer Optimum
I/py
O
I/px
Apples (x)
24
Bananas (y)
Consumer Optimum
I/py
C
y
x
O
I/px
Apples (x)
25
Bananas (y)
Consumer Optimum
I/py
C
y
x
O
I/px
Apples (x)
26
Key Optimality Condition
  • Note that the optimal choice has the property
    that the indifference curve must be tangent to
    the budget line.
  • In technical jargon, the slope of the
    indifference curve at the optimum must be equal
    to the slope of the budget line.

27
The Marginal Rate of Substitution
  • The slope of an indifference curve is called the
    marginal rate of substitution, and is given by
    the ratio of the marginal utilities of x and y
  • MRSxy MUx/ MUy
  • Recall that the marginal utility of x is given by
    ?U/?x

28
  • Quick derivation the set of all pairs (x,y) that
    give the same utility level z must satisfy U(x,y)
    z, or U(x,y) z 0. This equation defines y
    implicitly as a function of x (the graph of such
    implicit function is the indifference curve). The
    Implicit Function Theorem then implies the rest.

29
  • Intuition suppose that consumption of x
    increases by ?x and consumption of y falls by ?y.
    How are ?x and ?y to be related for utility to
    stay the same?
  • Increase in utility due to higher x consumption
    is approx. ?x times MUx
  • Fall in utility due to lower y consumption -?y
    times MUy
  • Utility is the same if MUx ?x - MUy ?y, i.e.
    ?y/ ?x - MUx/ MUy

30
  • For example, with U xy,
  • MUx ?U/?x y
  • MUy ?U/?y x
  • and
  • MRSxy MUx/ MUy y/x
  • Exercise Find marginal utilities and MRSxy if U
    log x log y

31
  • Back to our consumer problem, we knew that the
    slope of the budget line is equal to the ratio of
    the prices of x and y, px/py. Hence the optimal
    choice of the consumer must satisfy
  • MUx/ MUy px/py

32
Numerical Example
  • Let U xy again, and suppose px 3, py 3, and
    I 12.
  • The budget line is given by
  • 3x 3y 12
  • Optimal choice requires MRSxy px/py, that is,
  • y/x 3/3 1
  • The solution is, naturally, x y 2.

33
Changes in Income
  • Suppose that income doubles, i.e. I 24. Then
    the budget line becomes
  • 3x 3y 24
  • The MRS px/py condition is the same, so now
  • x y 4

34
y
I/py
C
O
I/px
x
35
y
I/py
An increase in income I gt I
I/py
C
O
I/px
x
I/px
36
y
I/py
An increase in income I gt I
I/py
C
C
O
I/px
x
I/px
37
  • In the precious slide, both goods are normal. But
    it is possible that one of the goods be inferior.

38
y
I/py
An increase in income, Good y inferior I gt I
I/py
C
C
O
I/px
x
I/px
39
Changes in Prices
  • In the previous example, suppose that px falls to
    1.
  • The budget line and optimality conditions change
    to
  • x 3 y 12
  • y/x 1/3
  • Solution x 6, y 2.

40
y
I/py
C
O
I/px
x
41
y
Effects of a fall in px px gt px
I/py
C
O
I/px
I/px
x
42
y
Effects of a fall in px px gt px
I/py
C
C
O
I/px
I/px
x
43
  • If x is a normal good, a fall in its price will
    result in an increase in the quantity purchased
    (this is the Law of Demand)
  • This is because the so called substitution and
    income effects reinforce each other.

44
y
I/py
C
C
O
I/px
I/px
x
45
y
Substitution vs Income Effects
I/py
C
C
C
O
I/px
I/px
x
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