178.307 Markets, Firms and
Consumers
Lecture 4- Capital and
the Firm
Overview
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Readings
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2
7: 209-211
Chapter 14
We begin with the
theory of making ‘risky
decisions’.
We conclude with
examining methods of
dealing with risk.
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Key Concepts
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Expected Utility Theory
Risk Aversion
EU Paradoxes
CAPM Model
Bank Loans and
collateral
Expected Utility Theory
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Replaced Expected
Wealth theory
St Petersberg Paradox
refuted EW theory.
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3
Based on a gamble
(tossing heads)
Fall in odds matched by
rise in payoff
Odds of Winning
St Petersberg Paradox
Wealth from Bet
4
Expected Wealth
Subjective Expected Utility

Payoffs are evaluated
in subjective terms.
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5
Choices are represented
as lotteries.
vN-M Utility function is a
cardinal, weighted sum
of utilities of lottery
payoffs.
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Axioms
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Certainty
Independence of Order
Compounding
Independence
Continuity
Montonicity
Risk Aversion
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Risk aversion is implied
Map utility against
wealth
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6
Implies that a certainequivalent gives a higher
payoff than lottery.
Arrow-Pratt Risk Aversion Coefficient
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Risk aversion can be
inferred from the slope
of u(x).
Arrow-Pratt formula
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CARA
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U(x)=a-b exp(-rX)
Two other forms are–
CRRA

 u( x)
r ( x) 
u( x)
7

–
U(x)= a-bX1-β if β >0
U(x)= ln X if β =1
Quadratic

U(x)= a+bX-cX2
Constant Absolute Risk Aversion
u ( x )  e
rx
 u( x)  re
rx
 u( x)  r e
  r 2 e rx 
r ( x) 
rx
re
r ( x)  r
8
2 rx
CARA
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Assume that x is
normally distributed
with mean μ and
standard deviation of σ.
The EU CARA function
can be derived (via a
Taylor approximation)
EU ( x) 
a  b r  2 r 
or
max(   2  )
1
9
2
1
2
2

Violations of EU Theory
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Two main paradoxes emerge
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10
Common Consequence Effect
Common Ratio Effect
Allais Paradox is earliest example of
Common Consequence effect.
Allais Paradox
a1
a3
1.00 chance of $1m
0.10 of $5m
a4
a2
0.10 of $5m
11
0.89 of $1m
0.90 of 0
0.01 of 0
0.11 of $1m
0.89 of 0
Preference Reversal
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12
Most players prefer a1 in the first game.
They prefer a3 in the second game.
Game 1 establishes that 0.11U(1m) >
0.10U(5m)
Game 2 establishes that 0.10U(5m) >
0.11U(1m).
Kahneman and Tverskey:
Common Ratio Effect
c1
c3
1.00 chance of $3000
0.25 of $3000
0.75 of 0
c2
c4
0.80 of $4000
0.20 of 0
0.20 of $4000
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0.80 of 0
Capital Asset Pricing Model
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Suppose a firm wishes
to raise capital for an
investment.
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14
The systematic risk
should be incorporated
in the cost of capital.
The idiosyncratic risk
should not be.
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The measure of
systematic risk is the
beta β.
Firm’s whose
systematic risk is
greater than market
mutual fund pay
premium.
Security Market Line
Rm
r
0
1
Beta
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Market for Loans
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The problem of
asymmetric information.
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Borrower has private
information about the risk
of the project.
Bank cannot distinguish
‘high risks’ from ‘low
risks’.
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Bank can charge
average interest rate
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Penalises low risk,
benefits high risk.
May create adverse
selection rpoblem
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Low risk firms get
alternate finance
Bank left with high risk
projects
Market may collapse
Collateral
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Suppose risks are
binomial
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Borrowers are either high
risk θH or low risk θL.
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Assume borrowers can
put down collateral C.
Cashflow equals 0 or y
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If y=0, borrower loses C
and bank gains δC.
If y>0, bank gets Rk,
borrower gets y-Rk.
Their reserve repayment is
either RH or RL
 All bargaining power
H
L
R >R
lies with the bank.
Collateral as a sorting device
R
U  (1   )( y  R ) 
RH
 C
H  L
RL
k
k
k
k
C
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Conclusions
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With asymmetric
information, all firms will
claim to be low risk.
The Bank can offer two
contracts.
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High risk firms unwilling to
bet their collateral
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No collateral but repayment
of RH
Require collateral and lower
repayment schedule.
Low risk firms prepared to
bet their collateral for lower
repayments.
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They select second contract
(weakly dominates)
Collateral is used to sort the
two types of firms
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Select the first contract.
Bank does not need to
know each firm’s type.