Title: Quick Reminder of the Theory of Consumer Choice
1Quick 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.
3A 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.
4The 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
6Bananas (y)
I/py
O
I/px
Apples (x)
7Bananas (y)
I/py
O
I/px
Apples (x)
8Bananas (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.
10Bananas (y)
I/py
O
I/px
Apples (x)
11Bananas (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
13Bananas (y)
I/py
O
I/px
Apples (x)
14Bananas (y)
I/py
px gt px
O
I/px
I/ px
Apples (x)
15Preferences
- 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.
16Utility 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
17Indifference 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
18y
Three Indifference Curves
Utility u0
x
19y
Three Indifference Curves Here u1 gt u0
Utility u1
Utility u0
x
20y
Three Indifference Curves Here u2 gt u1 gt u0
Utility u2
Utility u1
Utility u0
x
21Properties of Indifference Curves
- Higher indifference curves represent higher
levels of utility - Indifference curves slope down
- They do not cross
- They bow inward
22Optimal 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
23Bananas (y)
Consumer Optimum
I/py
O
I/px
Apples (x)
24Bananas (y)
Consumer Optimum
I/py
C
y
x
O
I/px
Apples (x)
25Bananas (y)
Consumer Optimum
I/py
C
y
x
O
I/px
Apples (x)
26Key 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.
27The 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
32Numerical 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.
33Changes 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
34y
I/py
C
O
I/px
x
35y
I/py
An increase in income I gt I
I/py
C
O
I/px
x
I/px
36y
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.
38y
I/py
An increase in income, Good y inferior I gt I
I/py
C
C
O
I/px
x
I/px
39Changes 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.
40y
I/py
C
O
I/px
x
41y
Effects of a fall in px px gt px
I/py
C
O
I/px
I/px
x
42y
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.
44y
I/py
C
C
O
I/px
I/px
x
45y
Substitution vs Income Effects
I/py
C
C
C
O
I/px
I/px
x