Managerial economics guide 6

Risk aversion Risk neutral Risk lover

If an individual is presented with a lottery and a chance to get the mean of the lottery for sure

̅ ̅ ̃) (̅) (̅) (̃) (̅) (̅) (̃)

)

( = ( ) > ( = = = <

If he prefers to get the mean of the lottery If he’s indifferent between getting the mean If he prefers the lottery itself to getting the

for sure then he’s said to be risk averse of the lottery for sure and facing the lottery mean for sure then he is said to be a risk

itself then he is said to be risk neutral. lover.

We restrict ourselves to two distinct state of nature (two markets) X1 and X2. X1 represents a simple random variable.

On the fraction we can track u(X1) and u(X2) by reading along the curve.

The mean is a point on the horizontal axis : (̃) = +

1 1 2 2

If p1=p2=1/2 then the mean will be the midpoint of the AB segment,

(̅) (̅)

=

The utility of the mean (or the instantaneous utility of the mean) can be found on the function, it is also the expected utility of the symbol

lottery payoffs.

The expected utility of the lottery will be somewhere on the segment of the arc, in between ux1 and ux2.

Proof: (̃) ) )

= ( + (

1 1 2 2

The convex combination of the two points are points in the segment joints these two points.

→ ∈ [0; 1]

st nd

Suppose we call the 1 point A and the 2 point B, the convex combination of A and B is:

(1

+ − )

The convex combination will lie along the arc connecting A & B. Their coordinates can be expressed as:

))

= ( , (

1 1 (̃)

))

= ( , ( =

2 2

p represents coordinates between 0 and 1, these coordinates can also be interpreted as probability. Probabilities are normally range from 0

to 1 and sum up to 1, which is exactly what you need for making convex combination: coordinates which are positive whose sum equal to 1.

Therefore, we can write the coordinates as follows: )) ))

= ( , ( + (1 − )( , (

1 1 2 2

(1 ] [( ) (1 )]

= {[ + − ) ; + − )( }

⏟ ⏟

1 2 1 2

1 2

The expected utility will lie on the arc that’s joining the two points

The corresponding point on the function is the utility of the mean.

Another important concept is certainty equivalent of the lottery , in a word, it’s the utility of something that you get for sure