Transcript utility

Consumer Theory: Objectives
Derive and understand:
1.How Rational People make Choices
2. How this should Guide our own Decisions
Consumer Behavior
• We are now studying the foundations of demand
theory.
- Why do demand curves slope downward?
- Why do they shift with changes in prices for
substitutes and complements?
- Why do they shift with changes in income?
- What normative significance can we give to
demand based on underlying consumer
preferences?
Preferences
• Preferences are complete if for any two
consumption points x and x', either x ≤x' (x
is at least as good as x') or x' ≥ x (x' is at
least as good as x), or both.
• Preferences are reflexive if for all x, x ≥ x (x
is at least as good as itself).
Preferences
• Preferences are transitive if x ≥ x' and x' ≥ x''
implies that x ≥ x''.
• Preferences are strongly monotonic if for any two
commodity points x = (x1, x2) and x' = (x'1, x'2) if
x1 ≤ x'1, x2 ≤ x'2, and x ≠ x', then x' is preferred to
x.
• Preferences are continuous if the set of all choices
that are at least as good as a choice x' and the set
of all choices that are no better than x' are both
closed sets.
From Preferences to Utility Function
Representation Theorem:
If a consumer has a preference relation that
is complete, reflexive, transitive, strongly
monotonic, and continuous, then these
preferences can be represented by a
continuous utility function u(x) such that
u(x) > u(x') if and only if x > x'.
• Consider the following (ordinal) utility function
for Food (F) and Clothing ( C ) for Emily:
U(F,C) = FC
• Each indifference curve gives the combinations of F and C that yield the same
level of satisfaction to Emily (e.g. 25, 50, 100).
• Think of slicing the utility function at different levels and projecting into F,C
space.
• Her marginal rate of substitution of Food for Clothing is given by the slope at
any point on an indifference curve
MRSFC = - dC/dF|U = a constant
[MRSFC is the amount of Clothing Emily is willing to give up for one addition unit
of food, holding utility constant]
• Marginal rate of substitution of F for C is equal to the marginal
utility of F divided by the marginal utility of C at a point on an
indifference curve, i.e holding utility constant (dU = 0)
dU(F,C) = UFdF + UCdC = 0
Where UF = Marginal utility of F
UC = Marginal utility of C
UF/Uc = -dC/dF|U = constant = MRSFC
• For U = FC: UF = C, UC = F -> MRSFC = C/F
• Set U = 25: C = 5, F = 5; MRSFC = 5/5 = 1
C = 2.5, F= 10; MRSFC = 2.5/10 = 0.25
Declining MRS of F for C as F increases holding U constant
Declining marginal utility of Food as F increases U constant
Budget Constraint
Budget constraints limit an individual’s ability to consume in
light of the prices they must pay for various goods and
services. This is where scarcity comes in and what helps to
make economics the dismal science
• The Budget Line
– The budget line indicates all combinations of two
commodities for which total money spent equals a given fixed
level of income (ignore borrowing and saving for now)
Consumers choose a combination of goods that
will maximize the satisfaction they can achieve,
given the limited budget available to them.
• The maximizing market basket must satisfy two
coditions:
1) It must be located on the budget line.
2) Must give the consumer the most preferred
combination of goods and services.
Max U(q1, q2, q3,…qn)
S.T. Σpiqi = I
•
•
•
•
•
Let: U = FC
I = PFF + PCC
Write C in terms of F and substitute
C = I/PC – PFF/PC
MaxF U = F(I/PC – PFF/PC) = FI/PC –
PFF^2/Pc
• dU/dF = I/Pc – 2PFF/Pc = 0
• F = I/2PF
• C = I/2PC -> MRS = C/F = PF/PC
• A Corner Solution
– If the MRS is, in fact, significantly greater than the
price ratio, then a small decrease in the price of
frozen yogurt will not alter the consumer’s market
basket.
– At point B, the MRS of ice cream for frozen yogurt
is greater than the slope of the budget line.
– This suggests that if the consumer could give up
more frozen yogurt for ice cream he would do so.
– However, there is no more frozen yogurt to give up
since all of her income is going to ice cream
How Rational Consumers Choose: A Graphical
Approach
2 ingredients:
Good 2
Budget Line: p1x1 + p2x2 = I
Slope: Ratio of Prices (p1/p2)
Direction of increasing Utility
Optimal
bundle
-Preferences: for each
consumer, tells us tastes of
consumers
-Budget constraints: tells us
prices of goods and a
consumer’s income
Slope: Marginal Rate of Substitution
of Good 1 for Good 2 (M.R.S.)
MRS =
p1/p2
Indifference Curve: tells us between
which bundles the consumer is indifferent
0
Good 1
Budget Set: tells us what the consumer can afford,
given his income I and prices p1, p2
Graphical Example
Price of Beer: pB = 6, Price of Tacos: pT = 1, Income: I = 18
Tacos
18
If you spend $1 more on beer
rather than on tacos:
a/b = M.R.S.
-you lose -1/pT units of Tacos
12
-you gain 1/pB units of beer
pB/pT = price ratio
-1/pT
1/pB
-a
b
1
3
Indifference curve corresponding to
a utility of 5 (say)
Beer
Numerical Example
x = Units of Tacos, y = Units of Beer
M.R.S.(x,y) = .5x/y : MRS of Beer for Tacos
Given formula, say
Price of Beer: 6, Price of Tacos: 1, Income: 18
Need to Solve: .5x/y = 6 and x + 6y = 18
Which gives: x =12, y =1
Applications
1. You have just bought a mansion in Middletown.
Suddenly, after payment, the price of all houses in
your new neighborhood increase. Are you better off
after the change? Would you be better off if the price
had decreased instead (ignore transaction costs,
fees…)?
2. The government decided that potato consumption
should be subsidized: for each pound bought, you get
$1 from the government. The government then
decides, to balance its budget, to levy the lowest tax per capita- that achieves this goal. Are you overall
better off – or worse off?
y= All others goods ($)
B
I
I’
: All indifference
curves to follow are for
sake of illustration!
X’
(they are not derived)
X
B’
x= Housing (sq.)
(Housing) price increase
X = (x,y)-bundle chosen before price increase
B = initial budget line; I = initial indifference curve
X = (x,y)-bundle chosen after price increase
B’= final budget line; I = final indifference curve
y= All others goods ($)
I
I’
B
B’
X
X’
x= Housing (sq.)
(Housing) price decrease
X= (x,y)-bundle chosen before price decrease
B= initial budget line; I= initial indifference curve
X= (x,y)-bundle chosen after price decrease
B’= final budget line; I= final indifference curve
B : Income: I = $150; (potato) price: $3
X : chosen bundle (initial situation)
B’ : Income: I = $150; price: $2 = 3 - 1($1 subsidy)
X’ : chosen bundle (when subsidy is applied, but not the tax)
B’’: Income: I = 150 - Tax; price: $2
(Total Subsidy = Tax)
X’’: chosen bundle (eventual situation)
y= all other goods ($)
150
B’
B
120
B’’
90
I
X
I’
X’
70
Initial budget line
X’’
60
Final budget line
I’’
20
30
40
50
75
x= potatoes (pds)
Income and Substitution Effects
• Substitution Effect
• When the price of a good decreases relative to others
goods, holding purchasing power constant, consumers
substitute toward the good that is relatively cheaper.
• Note that relative prices change, so intuitively, the
slope of the budget line will change.
• Note that purchasing power is held constant.
• Thus, we hold the consumer to the same level of utility
(i.e., consumer can only purchase the same level of
utility as before).
Suppose the price of CD’s
falls, rotating the budget
line out.
Rotating the Budget line
Movie
• Remember the slope of this
budget line is -PC/PM.
• When PC falls, the slope becomes
less negative.
• First rotate budget line out,
(i.e., from white line to blue
line).
• Then parallel shift it back to the
green line.
CD
Substitution Effect
• Next draw in your original
indifference curve.
• The original point is C1.
• When the price of CD’s falls,
the substitution effect causes
consumption to increase to C2.
• The substitution effect looks
the same for all three cases.
Movie
C1
C2
CD
Income Effect
• The direction of the income effect depends on
whether the good is a normal good (i.e., increased
purchasing power increases consumption) or an
inferior good (i.e., increased purchasing power
decreases consumption).
• The income effects help to distinguish our three
cases.
Case I: Income Effect:
Normal Good
Movie
• Increased purchasing power
increases consumption from C2
to C3.
U2
U1
C2 C3
CD
Income Effect:
• Increased purchasing power
increases consumption from C2
to C3.
• Put C3 to the right of C2.
Movie
U2
U1
C1
C2 C3
CD
Total Movement is C1 to C3
Movie
Note: Utility
rises as price falls
PCD
P1
u2
P2
u1
C1 C2 C3
D
CD
CD
C
C
1
3
Case II: Inferior Good.
Substitution Effect is Stronger
• Derive the Substitution Effect
just as before:
•i.e., from C1 to C2
Movie
U1
C1
C2
CD
Income Effect:
Movie
Be careful that
your indiff.
curves don’t
cross!
U2
• Increased
purchasing power
decreases
consumption from
C2 to C3.
• Put C3 to the left
of C2 but to the
right of C1.
U1
C1 C3 C2
CD
Derivation of Demand:
Total effect is C1 to C3
Movie
Note: Again,
utility rises as
price falls
PCD
Demand still
downward
sloping.
P1
u2
P2
u1
C1 C3 C 2
CD
D
C C
1
3
CD
Case III: Super-Inferior (Giffen) Good
Income Effect is Stronger
• Derive the Substitution Effect
just as before:
•i.e., from C1 to C2
Movie
U1
C1
C2
CD
Income Effect:
Movie
Be careful that
your indiff.
curves don’t
cross!
U2
• Increased
purchasing power
decreases
consumption from
C2 to C3.
•Put C3 to the left of
C2 and to the left of
C1.
U1
C3 C1
C2
CD
Derivation of Demand:
Total effect is C1 to C3
Movie
Note: Again,
utility rises as
price falls.
PCD
u2
Here, we get
an upward
sloping
Demand
P1
P2
u1
C3 C 1 C 2
CD
D
CD
C C
3
1
• Many public programs to help the poor
“target” subsidies at particular goods and
services (food stamps, housing subsidies)
• These programs increase the individual’s
real income, but the money must be spent
on the targeted goods and services
• If the gov’t transferred an equivalent
amount of income without restrictions, the
individuals would be better off
• However, these programs are paternalistic
and reflect concerns that the funds will be
wasted on “beer”
Points to remember:
• In the economic model of the utility-maximizing consumer, the
consumer’s utility function associates a numerical value to each
conceivable choice. Given prices and monetary resources, the
consumer chooses the best (utility-maximizing) bundle from among
all those she can afford.
– This optimal choice occurs when, given his choice, what an additional unit of a
commodity is worth to him equals the price of the commodity
• The model of the utility-maximizing consumer is rationalized by
economists as an as if model. No one believes that consumers actually
maximize a utility function. But if the consumer’s choice behavior
conforms to relatively simple rules, the consumer acts as if she
maximizes utility. And important consequences follow.
– Unhappily, systematic violations of these simple rules can be observed in real life.
Consumer marketers and advertising executives are well compensated for their skills in
manipulating how consumers frame their choice
– Economists continue to use the model of the utility-maximizing consumer, in the belief
that the violations are usually insignificant or in the hope that the conclusions drawn from
models so constructed are not grossly affected by violations