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ADVANCED MICROECONOMICS
Preferences and Utility
Axioms of the rational choice:
Completeness
The preferences of the costumers are assumed to be complete : the consumer is always
able to say if he prefers A, B or he’s indifferent between A and B.
Transitivity
The preferences of the costumers are assumed to be transitive : if the consumer prefers A
to B and B to C, he will prefer A to C too.
Continuity
The preferences of the costumers are assumed to be continuous : if the consumer prefers
A to B, then solutions similar to A will be preferred to B as well
Assuming completeness, transitivity, and continuity, people are able to rank all possible situations
from the least desirable to the most: this ranking is called utility.
What matters in utility rankings is the order, not the difference between the utility of commodity
bundles.
It is impossible to compare utilities between people.
Individuals’ preferences are represented by a utility function of the form
$
&
(
where x
1
, x
2
, ..., x
n
are the quantities of n goods that might be consumed in a period.
This function is unique and up to an order-preserving transformation.
Utility is affected by
o Consumption of physical commodities
o Psychological attitudes
o Peer group pressures
o Personal experiences
o General cultural environment
Ceteris paribus assumption: keeping constant all the other variables that affects behavior,
considering only the choices among quantifiable options.
Utility from consumption of goods
For two goods, x and y:
Utility = U(x,y)
Arguments of utility functions
- U(W) = utility from real wealth (W);
- U(c,h) = utility from consumption (c) and leisure (h);
- U(c
1
,c
2
) = utility from consumption in two different periods.
The more of any specific good x i
,
the better.
Trades and Substitution
The indifference curve shows a set of consumption bundles about which the individual is
indifferent, i.e. all the bundles provide the same level of utility.
The bundles are preferred to be “well-balanced”, not heavily weighted toward one good.
The marginal rate of substitution ( MRS ) is the negative of the slope of an indifference curve (U
at some point which reflects the individual’s willingness to trade y for x.
The MRS changes as x and y change.
232 4
There exist many indifference curves, whose level of utility increases moving northeast.
Indifference curves are convex (diminishing MRS).
Indifference curves cannot intersect (transitivity axiom).
CES Utility (Constant elasticity of substitution)
𝑈(𝑥, 𝑦) = H𝑥
I
$
where d £ 1, d ¹ 0
For d = 1 U(x,y) = perfect substitutes case.
For d → 0 U(x,y) ∼ Cobb-Douglas.
For d → - ¥ U(x,y) ∼ perfect complements case.
Monotonic transformation
∗
= [𝑈
]
$
The elasticity of substitution (𝛿)
CES utility Þ 𝛿 =
$
$?>
Perfect substitutes Þ 𝛿 = ¥
Perfect complements Þ 𝛿 = 0
The utility function is homothetic if the MRS depends only on the ratio of the amounts of the two
goods not on the total quantities of the good.
MRS is the same at every point
a
b
a
b
a
b
- General Cobb-Douglas function
MRS depends only on the ratio y/x
5
Y
7 ?$
8
7
8 ?$
Some utility functions are not homothetic.
= 𝑥 + ln
MRS is independent from the quantity of x consumed.
5
Y
Several Good Case
The utility function
$
&
(
defines an indifference surface in n dimensions with the same level of utility (convex surface).
&
$
2
( 5 4
, 5 \
,…, 5 ]
) 3 ^
5
4
$
&
(
5
\
$
&
(
Christopher Snyder 12
Utility Maximization and Choice
The economic approach is criticized because:
It pretends people had made calculations to maximize their utility before choosing.
It treats people as selfish, since they only look at their interests.
Utility maximization
Given a fixed amount of income to spend, maximizing utility means that the individual has to
choose the quantities of goods to buy for which the MRS is equal to the rate at which x can be
traded one for y in the marketplace, or say the ratio of the price of x to the price of y (p
x
/p
y
The Two-Good Case
Income: 𝐼
Price of x: 𝑝
5
Price of y: 𝑝
Y
Budget constraint: 𝑝
5
Y
Slope = −
a
b
a
c
First-order conditions for a maximum: tangency between budget constraint and indifference curve.
Slope of the budget constraint = −
a
b
a
c
Slope of indifference curve =
dY
d
e
23 fg(hij(i
a
b
a
c
dY
d
e
23 fg(hij(i
If
tℒ
t 5
4
t
t 5
u
r
< 0 then x i
= 0 and 𝑝
r
t 2
b u
tv
Any good whose price exceeds its marginal value to the consumer will not be purchased.
Indirect Utility Function
It is often possible to manipulate first-order conditions to solve for optimal values of x 1,
x 2
,…, x n
The optimal values will be:
$
$
$
&
(
&
&
$
&
(
(
(
$
&
(
We can use the optimal values of the x’s to find the indirect utility function, that will indirectly
depend on prices and income.
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 = 𝑈[𝑥
$
$
&
(
&
$
&
(
(
$
&
(
, 𝐼)]
$
&
(
Expenditure Minimization
How to allocate the income to reach a fixed level of utility (Ū) with the minimal expenditure.
The individual’s problem is to choose x 1,
x 2
,…, x n
to minimize
$
$
&
&
(
(
subject to the utility constraint
𝑈𝑡𝑖𝑙𝑖𝑡𝑦 = Ū = U(x
$
, x
&
, … , x
Expenditure function
The expenditure function and the indirect utility function are inversely related; both depend on
market prices and involves constraint (utility and income respectively).
5
Y
Properties:
o Homogeneous of degree 1: doubling all prices will double the value of required
expenditures.
o Nondecreasing in prices:
t
ta
u
≥ 0 for every good i.
o Concave in prices: functions that always lie below tangents to them.
Income and Substitution Effects
Demand Functions
The demand functions for only two goods can be expressed as:
∗
5
Y
∗
5
Y
With prices and income exogenous.
The demand function is homogeneous of degree 0, because doubling all prices and income, since
the budget constraint doesn’t change, the optimal quantities demanded remain the same.
Changes in income
The budget constraint moves parallelly and the slope (p x
/ p y
) doesn’t change. The MRS stays
constant as the individual moves to higher satisfaction.
If the income increases, the consumption increases as well.
If the income increases, the consumption decreases.
Changes in a Good’s Price
The slope of the budget constraint and the MRS changes. There are two effects:
The consumer will buy more of the good which is now cheaper.
We move along the indifference curve reaching the new intersection with the budget constraint.
The consumer’s real income changes and so he will move to a new level of utility.
When p ¯: Substitution effect Þ demand (more purchasing power)
Income effect Þ demand (higher utility level)
When p : Substitution effect Þ ¯ demand (less purchasing power)
Income effect Þ ¯ demand (lower utility level)
When p ¯: Substitution effect Þ demand
Income effect Þ ¯ demand
When p : Substitution effect Þ ¯ demand
Income effect Þ demand
Giffen’s paradox
For inferior goods, we cannot predict which effect will be predominant.
If the income effect is stronger than the substitution one, it will be the Giffen’s paradox case.
An increase in price leads to a drop in real income and, since the good is inferior, this causes a
rise in the quantity demanded.
Compensated (HICKSIAN) Demand Curves and Functions
The Compensated ( Hicksian ) demand curve shows the relationship between the price of a good
and the quantity purchased, keeping other prices and utility constant.
It keeps real income (or utility) constant while examining reactions to changes in p x
. The effects of
the price change are “compensated” so as to force the individual to remain on the same
indifference curve. Reactions to price changes include only substitution effects.
A 2-D representation of the compensated demand function is:
∗
f
5
Y
Shephard’s lemma
Compensated demand function for a good can always be found from the expenditure function by
differentiation with respect to the good’s price.
5
Y
𝑦 + 𝜆[𝑈
− Ū]
It produces the expenditure function: 𝐸(𝑝
5
Y
5
Y
5
5
f
5
Y
For normal goods, the compensated demand curve is less responsive to price changes than the
uncompensated one because:
- Uncompensated demand curve reflects both income and substitution effects.
- Compensated demand curve reflects only substitution effects.
A Mathematical Development of Response to Price Changes
The utility-maximization model explain how the demand for good x changes when p
x
changes,
calculating
t
ta
b
Comparative static methods by differentiating the three first order conditions for a maximum with
respect to p x
. It provides little economic insight.
It assumes there are only two goods, x and y, and focuses on the compensated demand function
f
5
Y
, 𝐼) and its relationship to the ordinary demand 𝑥(𝑝
5
Y
f
5
Y
, 𝐼 = 𝑥[𝑝
5
Y
5
Y
, 𝑈]
f
5
5
5
5
f
5
5
Substitution effect: the slope of the compensated demand curve:
f
5
Income effect: the way in which changes in p
x
affect the demand for x through changes in
purchasing power.
5
Income elasticity of demand, e x,I
measures the sensibility of demand function to a percentage
change in income.
5 ,a
b
5
Y
Cross-price elasticity of demand e x,py
measures the sensibility of the demand function of x to a
percentage change in the price of y.
5 ,a
b
Y
Y
Y
Y
5
Y
Y
Y
Consumer Surplus
It measures the change in welfare an individual’s experiences if the price of good x increases from
p
0
x
to p
1
x
to reach U 0
0
x
5
Y
1
x
5
$
Y
The compensating variation (CV) is the change of the expenditure to maintain the same level of
utility after a price change:
5
$
Y
5
Y
The compensated demand function is the derivative of the expenditure function with respect to p x
f
5
Y
5
Y
5
Then, the amount of CV required will be the area below the compensated demand function
between px
0
and px
1
5
Y
5
5
f
5
Y
5
a b
4
a
b
a b
4
a
b
The consumer surplus is the area below the compensated demand curve and above the market
price and it represents the extra benefit the person receives by being able to make market
transactions at the prevailing market price.
Also, the consumer surplus is the area below the Marshallian demand curve and above price and it
shows what an individual would pay for the right to make voluntary transactions at this price.
Changes in consumer surplus measure the welfare effects of price changes.
Demand Relationships among Goods
The Two-Good Case
How the quantity of x consumed is affected by a change in the price of y.
If p y
decreases, the budget constraint moves outward, the quantity of good y increases and
- For gross complements goods: small substitution effect. X and p y
moves in opposite
directions.
- For gross substitute goods: large substitution effect. X and p y
moves in the same direction.
Slutsky-type equation
5
Y
Y
Y
23 fg(hij(i
Substitution Income
effect (+) effect (-)
Generalize the Slutsky equation for any two goods x i
, x j
r
$
s
r
s
23 fg(hij(i
s
r
Substitutes goods: one good may, as conditions change, replace the other.
Complements goods: goods that ‘‘go together’’.
Gross substitutes: an increase in the price of one good causes more of the other good to be
bought:
r
s
Gross complements: an increase in the price of one good causes less of the other good to be
bought:
r
s
Net (Hicksian) substitutes:
r
s
23 fg(hij(i
Net (Hicksian) complements:
r
s
23 fg(hij(i
Gross substitutes and gross complements are asymmetric: it can be that, at the same time, x 1
is a
substitute of x 2
and x 2
is a complement of x 1
Net substitutes and net complements are symmetric:
t 5 u
ta
23 fg(hij(i
t 5
ta u
e
23 fg(hij(i
Fair Gambles and The Expected Utility Hypothesis
A ‘‘fair’’ gamble is a specified set of prizes and associated probabilities whose expected value is
zero.
It is observed that people would prefer not to play fair games
St. Petersburg Paradox
A coin is flipped until a head appears.
If a head appears on the n
th
flip, the player is paid $
n
$
&
(
(
The probability of getting of getting a head on the i
th
trial is
$
&
r
$
&
(
(
The expected value of the St. Petersburg paradox game is infinite
r
r
(
r 3 $
r
r
(
r 3 $
Because no player would pay a lot to play this game, it is not worth its infinite expected value
Individuals do not care directly about the dollar values of the prizes but about the utility that the
dollars provide.
If we assume diminishing marginal utility of wealth, the game may converge to a finite expected
utility value; this would measure how much the game is worth to the individual.
The expected utility can be calculated in the same manner as the expected value:
r
r
(
r 3 $
Utility may rise less rapidly than the dollar value of the prizes: expected utility can be less than the
monetary expected value.
Risk Aversion
The risk is the variability of the outcomes of some uncertain activity.
Two lotteries may have the same expected value but differ in their riskiness
With the same expected value, individuals will usually choose the one with lower risk.
Assuming decreasing marginal utility of wealth.
W
0
: individual’s current wealth.
U(W): utility function of wealth.
In the gamble A, there is a 50% chance of winning/losing $ h. The expected utility will be:
[𝑈
] =
Due to uncertainty, the expected value of accepting the gamble A is less than the utility of keeping
the current wealth (not taking part to the gamble). So the person will refuse to bet. The sum he
would accept to risk in the gamble would be only CEA (< W 0 ):
If individuals exhibit a diminishing marginal utility of wealth, they will be risk averse, i.e. that always
refuses fair bets. Then, they will be willing to pay something to avoid taking fair bets.
Game Theory
A person may not always have an obvious choice of what is best but it may depend on the actions
of another person.
Using game theory to analyse an economic situation solves the problem in an easier way.
A game is an abstract model of a strategic situation, formed by 3 elements:
- Players: decision makers.
- Strategies: courses s i
of action open to a player.
r
$
&
(
- Payoffs: levels of utility obtained by the players.
r
$
&
(
Best Response:
r
r
?r
r
£
r
?r
r
£
r
Given s i
the best response of the player i to rivals’ strategies s
- i
, denoted BR i
(s
- i
Nash Equilibrium: strategy profile
$
∗
&
∗
(
∗
Such that every s
i
is the best response to the others’ strategies.
Prisoners’ Dilemma
Two suspects are arrested for a crime. The police want to extract a confession so he offers each a
deal:
- Only one fink on the other: the first 1 - year sentence and the other a 4-year sentence;
- Both fink on each other: 3-year sentence.
- Nobody finks: 2-year sentence.
Nash Equilibrium:
If player 2 finks, the BR of the player 1 will be finking as well. The same holds for player 2: the two
responses are symmetric.
Dominant strategy : a strategy that is a best response to any strategy the other players might
choose (e.g. finking is a dominant strategy for both players).
r
∗
r
?r
?r
When a dominant strategy exists, it is the unique Nash equilibrium
Example: Battle of the Sexes
A wife and husband may either go to the ballet or to a boxing match. Both prefer spending time
together than alone; the wife prefers ballet and the husband boxing.
There are two Nash equilibria:
- Both going to the ballet
- Both going to boxing
There is no dominant strategy.
Example: Rock, Paper, Scissors
Two players simultaneously display one of three hand signals
- Rock breaks scissors
- Scissors cut paper
- Paper covers rock
None of the nine boxes represents a Nash equilibrium because any strategy pair is unstable
because it makes at least one of the players want to deviate.
There is no Nash Equilibrium.
