Question

Let’s assume that a group of risk-averse individuals are debating whether to play a risky gamble. For each individual, he/she is endowed with wealth level W0. If one chooses to invest in the gamble, he/she invest her entire wealth and receive either Ws if the individual wins or Wf if the individual loses. The outcome is either win or lose and the probability associated with winning is given as 1−p. Assume that Wf < W0 < Ws and p·Wf +(1−p)·Ws < W0 holds true. What would be the proportion of people investing in the gamble?

NOTE: In order to show this, use properties of risk-averse utility functions. Do not use just sentences to prove your result, but prove that your result is true by using utilites and expected values and inequality signs.

Answer 1

Given agents in the economy are risk averse i.e U(wealth)=w^0.5(Example)

Each agent is endowed with W0 and if the agent chooses to invest in a lottery then her wealth is Ws if she wins with probability 1-p else Wf.with probability p.

Wf<W0<Ws.........(Given)

pWf+(1-p)Ws<W0......(Given)

The expected return from a lottery is pWf+(1-p)Ws

Expected Utility of an agent i if he chooses to gamble = [ pWf+(1-p)Ws]^0.5

The expected return from not gambling is Wo

Expected Utility of an agent i if he chooses not to gamble=[Wo]^0.5

Clearly,Since

pWf+(1-p)Ws<W0......(Given)

U( ) = W*0.5 is a monotonically increasing function

Hence,Expected Utility From Gambling for an agent <Expected Utility from not gambling

Therefore,proportion of people investing in a lottery would be zero.