2. Self study note
• This note is prepared to solve some of your puzzles, if any, in the topic
of uncertainty.
• Basically, I hope it will help you understand more about the diagrams in
this topic.
• What is involved is, however, not about economics but some simple
math and how diagrams express the math relations involved.
• Often, neither textbooks nor teachers will explain these (perhaps)
because they think this is not about economics but just some math and
diagram (though students may still be confused without some
explanations).
3. Why the straight line joining two points on the utility curve can
be used to identify expected utility?
4. Answer
• Any points on a straight line can be represented
by a linear equation U = α+βW.
• At point A, U(a) = α+βa.
• At point B, U(b) = α+βb.
• If two outcomes, a and b may be realized at a
respective probability of p and 1-p, EV = pa+(1-
p)b.
• By definition, EU = pU(a)+(1-p)U(b).
• By substitution, EU = p(α+βa)+(1-p)(α+βb) =
α+β[pa+(1-p)b] = α+β(EV).
• Thus, we can find out the EU by tracking along
the line joining A and B. The value of EU is
identified by the point on the straight line that
corresponds to EV.
5. How can the probability associated with one possible wealth outcome be
reflected in the length of the straight line on the utility curve?
• Two possible wealth levels: v and v’.
• v is closer to H than v’.
• Hence, the probability of H is higher for the case
of v than v’.
• Point B corresponds to an expected value v = pH
+ (1-p)K, where p is probability for H to be
realized.
• Rearranging, we have p = (K-v)/(K-H).
• Draw the right triangle at point f and point B
respectively.
• cos(angle f) = cos(angle B).
• Thus, ef/(K-H) = Be/(K-v).
• Rearranging, (K-v)/(K-H) = Be/ef.
• Therefore, p = Be/ef.
• Similarly, 1-p = (v-H)/(K-H) = Bf/ef.
• So, we have p/(1-p) = Be/Bf.
