Introductory Microeconomics (ES10001)

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Introductory Microeconomics (ES10001)

• Topic 7 Risk

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I. Introduction

• We have so far assumed that the world is certain
• This is a bit silly?
• This world is inherently uncertain
• The same people who insure their cars and houses
  also but lottery tickets and play bingo! Why?

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II. Uncertainty

• Assume that there are two states of the world
• State 1 Wealth w1
• State 2 Wealth w2 w1 - L
• where L gt 0 occurs with probability p gt 0
• Expected wealth

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II. Uncertainty

• Expected wealth

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II. Uncertainty

• Individuals are not interested in wealth per se,
  but in the utility of wealth
• This is an important distinction an increase in
  wealth of 100 is unlikely to change the utility
  of a prince (David Beckham?) and a pauper (me!)
  by the same amount
• Assume individuals utility function is u u(w)
• Individuals objective is to maximise expected
  utility, not expected wealth!

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II. Uncertainty

• Utility function
• We assume that total utility increases with
  wealth such that marginal utility is positive

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II. Uncertainty

• Expected Utility
• Add and subtract u(w2)

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II. Uncertainty

• Multiply and divide second term by (w1 w2)

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II. Uncertainty

• Consider final term (1- p)(w1 w2)

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II. Uncertainty

• Thus

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II. Uncertainty

• This is the equation of a straight line!
• Consider the following

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Figure 1 Risk Averseness
u(w)
u(w)
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
C
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
E
A
w
0

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II. Uncertainty

• Note that expected utility, , is equal to the
  utility of wealth with certainty i.e.,
• We define as individual's certainty
  equivalent level of wealth vis. level of wealth
  that allows him the same utility as he could
  expect if he faces a (1 - p) chance of w1 and a p
  chance of w2
• This, is the maximum
  premium the individual would be prepared to pay
  for insurance

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II. Uncertainty

• Under an insurance contract, ? (r, L), the
  individual pays a premium, r (in both states of
  the world) and in return the insurance company
  contracts to reimburse the individual should he
  suffer the state 2 loss, L.
• Thus, individual's state contingent wealth under
  an insurance contract is
• State 1
• State 2

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Figure 1 Risk Averseness
u(w)
D
u(w)
B
E
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
C
B
E
A
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
C
B
E
A
rmax
w
0

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II. Uncertainty

• If insurance company agrees to compensate
  individual, then it can expect to face costs of
• Thus, rmin , the minimum premium the insurance
  company would be prepared to accept, is equal to

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II. Uncertainty
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Figure 1 Risk Averseness
u(w)
D
u(w)
C
B
E
A
rmin
rmax
w
0

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Figure 1 Risk Averseness
u(w)
D
u(w)
C
B
E
A
The Market for Insurance
w
0

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II. Uncertainty

• Note that
• Since , there is a Pareto-improving market
  for insurance i.e. because the individuals
  utility function is concave, he is willing to pay
  to insure against risk.
• Such an individual is said to be risk averse

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II. Uncertainty

• N.B. Change in marginal utility with respect to
  wealth
• (1) Risk Averse u?(w) lt 0 ? rmax gt rmin
• (2) Risk Neutral u?(w) 0 ? rmax rmin
• (3) Risk Loving u?(w) gt 0 ? rmax lt rmin

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II. Uncertainty

• In words
• Risk averse individuals are prepared to pay a
  premium to avoid risk
• Risk neutral individuals are indifferent to
  paying a premium and not paying a premium to
  avoid risk.
• Risk loving individuals are prepared to pay a
  premium to take risk

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Figure 2 Risk Loving
u(w)
u(w)
rmax

rmin
w
0
w1
w2

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Figure 3 Risk Neutral
u(w)
u(w)
rmin rmax

w
0
w1
w2

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Figure 1 Risk Averseness
u(w)
D
u(w)
C
B
E
A
rmin
rmax
w
0

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II. Uncertainty

• If insurance company agrees to compensate
  individual, then it can expect to face costs of
• Thus

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II. Uncertainty

• Where rmin is the minimum premium the insurance
  company would be prepared to accept