The St' Petersburg Paradox

1
The St. Petersburg Paradox

• The St. Petersburg Paradox is a gamble of tossing
  a fair coin, where the payoff doubles for every
  consecutive head that appears. The expected
  monetary value of this gamble is 2(.5)
  4(.25) 8(.125) 16(.0625) ... 1
  1 1 ... ?.
• But no one would be willing to wager all he or
  she owns to get into this bet. It must be that
  people make decisions by criteria other than
  maximizing expected monetary payoff.

2
Expected Utility Analysisto Compare Risks

• Utility is satisfaction
• Each payoff has a utility
• As payoffs rise, utility rises

• Risk Neutral -- if indifferent between risk a
  fair bet

.5U(10) .5U(20) is a fair bet for 15
U
U(15)
10 15 20
3
Risk Averse Risk Seeking

• Prefer a certain amount to a fair bet

• Prefer a fair bet to a certain amount

U U
certain
risky
risky
certain
10 15 20
10 15 20
4
Expected Utility an example

• Suppose we are given a quadratic utility
  function
• U .09 X - .00002 X2
• Gamble 30 probability of getting 100 30 of
  getting 200 and a 40 probability of getting
  400.
• Versus a certain 150?
• U(150) 13.05 (plug X150 into utility function)
• Find Expected Utility of the gamble
• EU ? pi U(Xi)
• EU .30(8.8) .30(17.2) .40( 32.8) 20.92

5
Risk Adjusted Discount Rates

• Riskier projects should be discounted at higher
  discount rates
• PV ????t / ( 1 k) t where k varies with risk
  and ??t are cash flows.
• kA gt kB as in diagram since A is riskier

B
A
6
Sources of Risk Adjusted Discount Rates

• Market-based rates
• Look at equivalent risky projects, use that rate
• Is it like a Bond, Stock, Venture Capital?
• Capital Asset Pricing Model (CAPM)
• Projects beta and the market return

7
z-Values

• z is the number of standard deviations away from
  the mean
• z (r - r )/??
• 68 of the time within 1 standard deviation
• 95 of the time within 2 standard deviations
• 99 of the time within 3 standard deviations
• Problem income has a mean of 1,000 and a
  standard deviation of 500.
• Whats the chance of losing money?

8
Diversification

• The expected return on a portfolio is the
  weighted average of expected returns in the
  portfolio.
• Portfolio risk depends on the weights, standard
  deviations of the securities in the portfolio,
  and on the correlation coefficients between
  securities.
• The risk of a two-security portfolio is
• ?p ?(WA2?A2 WB2?B2 2WAWB?AB?A?B
  )
• If the correlation coefficient, ?AB, equals one,
  no risk reduction is achieved.
• When ?AB lt 1, then ?p lt wA?A wB?B. Hence,
  portfolio risk is less than the weighted average
  of the standard deviations in the portfolio.

9
DEMAND ANALYSIS

• Demand Relationships
• Demand Elasticities
• Income Elasticities
• Cross Elasticities of Demand
• Indifference Curves

10
Health Care Cigarettes

• Raising cigarette taxes reduces smoking
• In Canada, 4 for a pack of cigarettes reduced
  smoking 38 in a decade
• But cigarette taxes also helps fund health care
  initiatives
• The issue then, should we find a tax rate that
  maximizes tax revenues?
• Or a tax rate that reduces smoking?

11
Demand Analysis

• An important contributor to firm risk arises from
  sudden shifts in demand for the product or
  service.
• Demand analysis serves two managerial objectives
  
• (1) it provides the insights necessary for
  effective management of demand, and
• (2) it aids in forecasting sales and revenues.

12
Demand Curves

• Individual Demand Curve the greatest quantity of
  a good demanded at each price the consumers are
  Willing to Buy, ceteris paribus.

/Q
Unwilling to Buy
Willing to Buy
Q/time unit
13
Sam Diane Market

• The Market Demand Curve is the horizontal sum of
  the individual demand curves.
• The Demand Function includes all variables that
  influence the quantity demanded

4 3
7
Q f( P, Ps, Pc, I, W, E)
- ? ?
14
Supply Curves

• Firm Supply Curve - the greatest quantity of a
  good supplied at each price the firm is
  profitably able to supply, ceteris paribus.

Able to Produce
/Q
Unable to Produce
Q/time unit
15
Acme Universal Market

• The Market Supply Curve is the horizontal sum of
  the firm supply curves.
• The Supply Function includes all variables that
  influence the quantity supplied

4 3
7
Q g( P, W, R, TC)
- -
16
Equilibrium No Tendency to Change
S
P

• Superimpose demand and supply
• If No Excess Demand
• and No Excess Supply
• No tendency to change

willing able
Pe
D
Q
17
Downward Slope

• Reasons that price and quantity are negatively
  related include
• income effect--as the price of a good declines,
  the consumer can purchase more of all goods since
  his or her real income increased.
• substitution effect--as the price declines, the
  good becomes relatively cheaper. A rational
  consumer maximizes satisfaction by reorganizing
  consumption until the marginal utility in each
  good per dollar is equal
• Optimality Condition is MUA/PA MUB/PB MUC/PC
  ...
• If MU per dollar in A and B differ, the consumer
  can improve utility by purchasing more of the one
  with higher MU per dollar.

18
Comparative Statics and the Supply-Demand Model
P

• Suppose a shift in Income, and the good is a
  normal good
• Does Demand or Supply Shift?
• Suppose wages rose, what then?

S
e1
D
Q
19
Elasticity as Sensitivity

• Elasticity is measure of responsiveness or
  sensitivity
• Beware of using Slopes

Slopes change with a change in units of
measure
price price per per bu.
bu
bushels hundred tons
20
Price Elasticity

• E P change in Q / change in P
• Shortcut notation E P ?Q / ?P
• A percentage change from 100 to 150
• A percentage change from 150 to 100
• Arc Price Elasticity -- averages over the two
  points

arc price elasticity
D
21
Arc Price Elasticity Example

• Q 1000 at a price of 10
• Then Q 1200 when the price was cut to 6
• Find the price elasticity
• Solution E P ?Q/ ?P 200/1100
  - 4 / 8
• or -.3636. The answer is a number. A 1
  increase in price reduces quantity by .36 percent.

22
Point Price Elasticity Example

• Need a demand curve or demand function to find
  the price elasticity at a point.
• E P ?Q/ ?P (??Q/?P)(P/Q)
• If Q 500 - 5P, find the point
  price elasticity at P 30 P 50 and P 80
• E QP (??Q/?P)(P/Q) - 5(30/350) - .43
• E QP (??Q/?P)(P/Q) - 5(50/250) - 1.0
• E QP (??Q/?P)(P/Q) - 5(80/100) - 4.0

23
Price Elasticity (both point price and arc
elasticity )

• If E P -1, unit elastic
• If E P gt -1, inelastic, e.g., - 0.43
• If E P lt -1, elastic, e.g., -4.0

Straight line demand curve
price
elastic region unit elastic
inelastic region
24
TR and Price Elasticities

• If you raise price, does TR rise?
• Suppose demand is elastic, and raise price. TR
  PQ, so, ?TR ?P ?Q
• If elastic, P , but Q a lot
• Hence TR FALLS !!!
• Suppose demand is inelastic, and we decide to
  raise price. What happens to TR and TC and
  profit?

25
Another Way to Remember
Elastic Unit Elastic
Inelastic

• Linear demand curve
• TR on other curve
• Look at arrows to see movement in TR

Q Q
TR
26
1979 Deregulation of Airfares

• Prices declined
• Passengers increased
• Total Revenue Increased
• What does this imply about the price elasticity
  of air travel ?

27
Determinants of the Price Elasticity

• The number of close substitutes
• more substitutes, more elastic
• The proportion of the budget
• larger proportion, more elastic
• The longer the time period permitted
• more time, generally, more elastic
• consider examples of business travel versus
  vacation travel for all three above.

28
Income Elasticity

• E I ?Q/ ?I (??Q/??I)( I / Q)
• arc income elasticity
• suppose dollar quantity of food expenditures of
  families of 20,000 is 5,200 and food
  expenditures rises to 6,760 for families earning
  30,000.
• Find the income elasticity of food
• ?Q/ ?I (1560/5980)(10,000/25,000) .652

29
Definitions

• If E I is positive, then it is a normal or
  income superior good
• some goods are Luxuries E I gt 1
• some goods are Necessities E I lt 1
• If E QI is negative, then its an inferior good
• consider
• Expenditures on automobiles
• Expenditures on Chevrolets
• Expenditures on 1993 Chevy Cavalier

30
Point Income Elasticity Problem

• Suppose the demand function is
• Q 10 - 2P 3I
• find the income and price elasticities at a price
  of P 2, and income I 10
• So Q 10 -2(2) 3(10) 36
• E I (??Q/??I)( I/Q) 3( 10/ 36) .833
• E P (??Q/??P)(P/Q) -2(2/ 36) -.111
• Characterize this demand curve !

31
Cross Price Elasticities

• E X ?Qx / ?Py (??Qx/??Py)(Py / Qx)
• Substitutes have positive cross price
  elasticities Butter Margarine
• Complements have negative cross price
  elasticities VCR machines and the rental price
  of tapes
• When the cross price elasticity is zero or
  insignificant, the products are not related

32

• HOMEWORK PROBLEM
• Find the point price elasticity, the point
  income elasticity, and the point cross-price
  elasticity at P10, I20, and Ps9, if the demand
  function were estimated to be Qd 90 - 8P
  2I 2Ps.
• Is the demand for this product elastic or
  inelastic? Is it a luxury or a necessity? Does
  this product have a close substitute or
  complement? Find the point elasticities of
  demand.

33
Indifference Curve AnalysisAppendix 3A
U

• Consumers attempt to max happiness, or utility
  U(X, Y)
• Subject to an income constraint
• I PxX PyY
• Graph in 3-dimensions

Uo
Y
Uo
X
34

• Consumer Choice - assume consumers can rank
  preferences, that more is better than less
  (nonsatiation), that preferences are transitive,
  and that individuals have diminishing marginal
  rates of substitution.
• Then indifference curves slope down, never
  intersect, and are convex to the origin.

U2
U1
X
Uo
9 7 6
convex
5 6 7
Y
give up 2X for a Y
35
Uo U1
X
Indifference Curves
We can "derive" a demand curve graphically
from maximization of utility subject to a budget
constraint. As price falls, we tend to buy more
due to (i) the Income Effect and (ii) the
Substitution Effect.
c
a
b
Y
Py
demand
Y
36
When p1 falls the consumer buys more x1 as it's
relatively cheaper (the substitution effect, S,
from 0 to 1) and more of both goods (if they're
normal) as her real income has risen (the income
effect, I, from 1 to 2).
x2
I/p2
p11 lt p10
.0
.2
.1
I/p10
I/p11
x1
S
I
37
I72, p21, p101, p114. Initial
consumption bundle A. Final consumption bundle
C. Substitution effect represented by
hypothetical movement from A to B. (Shift from B
to C is the income effect.) Note the COMPENSATED
BUDGET LINE
x2
.B
72
.A
.C
36
x1
36
18
9
72
S
I
38
Income and substitution effects for inferior
good work in different directions
x2
I/p2
. 2
p11 lt p10
.0
. 1
x1
S
I