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Introductory Microeconomics (ES10001)
Topic 3: Risk and Uncertainty
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. 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:
2. Risk and Uncertainty
Expected wealth:
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!
2. Risk and Uncertainty
Utility function:
We assume that total utility increases with wealth such that marginal utility is positive:
2. Risk and Uncertainty
Expected Utility:
Add and subtract u(w2)
2. Risk and Uncertainty
Multiply and divide second term by (w1 – w2)
2. Risk and Uncertainty
Consider final term (1- p)(w1 – w2)
2. Risk and Uncertainty
Thus:
2. Risk and Uncertainty
This is the equation of a straight line!
Consider the following:
w
u(w)
0
u(w)
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
C
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
E
w
u(w)
0
u(w)
A
D
E
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
E
Figure 1: Risk Averseness
2. Uncertainty
Note that expected utility, , is equal to the utility of wealth with certainty
1.e.
w
u(w)
0
u(w)
A
D
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
E
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
D
E
Figure 1: Risk Averseness
B
2. Risk and Uncertainty
We define 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
Thus, is the maximum premium the individual would be prepared to pay for insurance
w
u(w)
0
u(w)
A
E
D
B
C
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
E
D
B
C
rmax
Figure 1: Risk Averseness
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:
State 2:
2. Risk and 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 given by:
2. Risk and Uncertainty
w
u(w)
0
u(w)
A
E
D
B
C
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
E
D
B
C
Figure 1: Risk Averseness
rmin
w
u(w)
0
u(w)
rmin
rmax
A
E
D
B
C
Figure 1: Risk Averseness
w
u(w)
0
u(w)
A
E
D
B
C
The Market for Insurance
Figure 1: Risk Averseness
rmax
rmin
w
u(w)
0
u(w)
A
E
D
B
C
The Market for Insurance
Figure 1: Risk Averseness
2. Risk and Uncertainty
Note that:
Since , 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
2. Risk and Uncertainty
N.B. Change in marginal utility with respect to wealth (second derivative):
(1) Risk Averse:
(2) Risk Neutral:
(3) Risk Loving:
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
w
u(w)
0 w2 w1
u(w)
Figure 2: Risk Neutral
rmin = rmax
w
u(w)
0 w2 w1
u(w)
Figure 3: Risk Loving
rmin
rmax
