Introductory Microeconomics
(ES10001)
Topic 3: Risk and Uncertainty
1
1. Introduction

We have so far assumed that the world is certain

This is a a (very) strong assumtion

This world is inherently uncertain

The same people who insure their cars and houses
also but lottery tickets and play bingo! Why?
2
2. Uncertainty

Assume that there are two states of the world
State 1: Wealth = w1
State 2: Wealth = w2 = w1 - L

where L > 0 occurs with probability p > 0

Expected wealth:
(
)
w = 1- p w1 + pw2
3
2.

Risk and Uncertainty
Expected wealth:
w = (1 - p)w1 + pw2
Þ
w = (1 - p)w1 + p (w1 - L )
Þ
w = w1 - pw1 + pw1 - pL
Þ
w = w1 - pL
4
2. Risk and 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 individual’s utility function is u = u(w)

Individual’s objective is to maximise expected
utility, not expected wealth!
5
2. Risk and Uncertainty

Utility function:
( )
u=u w

We assume that total utility increases with wealth
such that marginal utility is positive:
( )
du
= u¢ w > 0
dw
6
2. Risk and Uncertainty

Expected Utility:
u = (1 - p)u (w1 ) + pu (w2 )

Add and subtract u(w2)
u = u (w2 ) + (1 - p)u (w1 ) + pu (w2 ) - u (w2 )
Þ
u = u (w2 ) + (1 - p)Þ
Þu (w1 ) - u (w2 )Þ
Þ
7
2. Risk and Uncertainty

Multiply and divide second term by (w1 – w2)
u = u (w2 ) + (1 - p)Þ
Þu (w1 ) - u (w2 )Þ
Þ
Þ
Þu (w1 ) - u (w2 )Þ
u = u (w2 ) + Þ
Þ(1 - p)(w1 - w2 )
Þ (w1 - w2 ) Þ
8
2. Risk and Uncertainty

Consider final term (1- p)(w1 – w2)
(1 - p)(w1 - w2 )
Þ
pw2 + (1 - p)w1 - w2
Þ
Þ
Þpw2 + (1 - p)w1 Þ
Þ- w2
Þ
w - w2
9
2. Risk and Uncertainty

Thus:
Þu (w1 ) - u (w2 )Þ
u = u (w2 ) + Þ
Þ(1 - p)(w1 - w2 )
Þ (w1 - w2 ) Þ
Þ
Þu (w1 ) - u (w2 )Þ
u = u (w2 ) + Þ
Þ(w - w2 )
Þ (w1 - w2 ) Þ
10
2. Risk and Uncertainty

This is the equation of a straight line!
( ) ( ) éé( w- w )
(
) é
éu w1 - u w2
u = u w2 + é
w1 - w2
é
a
( )
2
x
b

Consider the following:
11
u(w)
Figure 1: Risk Averseness
u(w)
0
w
12
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u (w2 )
0
u(w)
A
w2
w1
w
13
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u (w2 )
0
u(w)
A
w2
w
w1
w
14
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u(w)
u (w1 ) - u (w2 )
u (w2 )
A
(w1 - w2 )
0
w2
w1
w
15
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u(w)
u (w1 ) - u (w2 )
u (w2 )
A
(w1 - w2 )
0
w2
w1
w
16
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u(w)
u (w1 ) - u (w2 )
u (w2 )
A
b
(w1 - w2 )
0
w2
w1
w
17
( ) ( ) éé
(
) é
u(w)
éu w1 - u w2
é
é w1 - w2
b
u (w1 )
u (w2 )
0
Figure 1: Risk Averseness
A
w2
D
u(w)
b
w1
w
18
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u (w2 )
0
A
w2
u(w)
b
w1
w
19
u(w)
Figure 1: Risk Averseness
( ) ( ) éé( w- w )
(
) é
éu w1 - u w2
u = u w2 + é
é w1 - w2
a
( )
b
u (w1 )
2
x
D
u(w)
E
u (w2 )
A
b
(w - w2 )
0
w2
w
w1
w
20
u(w)
Figure 1: Risk Averseness
( ) ( ) éé( w- w )
(
) é
éu w1 - u w2
u = u w2 + é
é w1 - w2
a
( )
b
u (w1 )
u
u (w2 )
2
x
D
u(w)
E
A
b
(w - w2 )
0
w2
w
w1
w
21
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u
u (w2 )
0
u(w)
E
A
w2
w
w1
w
22
2. Uncertainty


Note that expected utility, u , is equal to the utility of
wealth w with certainty
1.e. u = u (w)
23
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u (w2 )
0
u(w)
A
w2
w1
w
24
Figure 1: Risk Averseness
u(w)
D
u (w1 )
u
u (w2 )
0
u(w)
E
A
w2
w
w1
w
25
Figure 1: Risk Averseness
u(w)
D
u (w1 )
B
u
u (w2 )
0
u(w)
E
A
w2
w
w
w1
w
26
2. Risk and Uncertainty

We define w as individual’s certainty equivalent level
of wealth

That is, the level of wealth that allows individual the
same utility as he could expect if he faces a (1 - p)
chance of w1 and a p chance of w2

max
= w1 - w is the maximum premium the
Thus, r
individual would be prepared to pay for insurance
27
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u = u (w)
u (w2 )
0
u(w)
B
E
A
w2
~
w
w
w1
w
28
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u = u (w)
u (w2 )
u(w)
B
E
A
rmax
0
w2
~
w
w
w1
w
29
2. Risk and Uncertainty

Under an insurance contract, θ = (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:
w1q = w1 - r
State 2:
w2q = w2 - r + L = w1 - L - r + L = w1 - r
30
2. Risk and Uncertainty

If insurance company agrees to compensate
individual, then it can expect to face costs of:
(
)
c = pL + 1- p 0

Thus, rmin , the minimum premium the insurance
company would be prepared to accept, is given by:
31
2. Risk and Uncertainty
(
)
(
c = pL + 1- p 0 = p w1 - w2
Þ
(
)
)
c = w1 + p w1 - w2 - w1
Þ
(
)
Þ
c = w1 - Þ
pw
+
1p
w
2
1
Þ
Þ
Þ
c = w1 - w = r min
32
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u = u (w)
u (w2 )
0
u(w)
B
E
A
w2
~
w
w
w1
w
33
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u = u (w)
u (w2 )
u(w)
B
E
A
rmin
0
w2
~
w
w
w1
w
34
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u = u (w)
u (w2 )
u(w)
B
E
A
rmin
rmax
0
w2
~
w
w
w1
w
35
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u = u (w)
u (w2 )
u(w)
B
E
A
b
The Market for Insurance
rmin
rmax
0
w2
~
w
w
w1
w
36
Figure 1: Risk Averseness
u(w)
D
u (w1 )
C
u (w)
u = u (w)
u (w2 )
u(w)
B
E
A
b
The Market for Insurance
0
w2
~
w
w
w1
w
37
2. Risk and Uncertainty

Note that:
r
max
= w1 - w > w1 - w = r
min

Since w > w, there is a Pareto-improving market for
insurance; i.e. because the individual’s utility
function is concave, he is willing to pay to insure
against risk.

Such an individual is said to be risk averse
38
2. Risk and Uncertainty

N.B. Change in marginal utility with respect to
wealth (second derivative):
( ) = u¢¢ ( w)
d 2u w
dx2
( )
(1) Risk Averse: u¢¢ w < 0 Þ w > wÞ r max > r min
(2) Risk Neutral:
( )
u¢¢ w = 0 Þ w = wÞ r max = r min
( )
(3) Risk Loving: u¢¢ w > 0 Þ w < wÞ r max < r min
39
2. Risk and 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
40
Figure 2: Risk Neutral
u(w)
u(w)
u( w1 )
( ) ( )
u=u w =u w
rmin = rmax
u( w2 )
0
w2
w= w
w
w1
41
Figure 3: Risk Loving
u(w)
u(w)
u( w1 )
( )
u( w)
u=u w
rmax
rmin
u( w2 )
0
w2
w
w
w w1
42