An investor will invest in two. The payoff is as follow: When the market is bull, probability $0\mathrm{.}35,$ investment $1$ will have a payoff of $60000,$ investment $2$ will have a payoff of $100000$or$.$ When the market is bear, probability $0\mathrm{.}65,$ investment $1$ will have a payoff of $30000$ and investment $2$ will have a payoff of $15000.$

The investor has the option of doing a market research to indicate whether the market conditions will be bull or bear. The market research will cost him an additional $5000.$

a$)$ What should be the gain in the expected value of information if the market research could provide perfect information?

In reality, the market research the investor conducts will not provide perfect information. The table below summarizes the probability that the market research returns a positive$/$negative result given the true market conditions. These probabilities were estimated from past marker research. Note that a positive result from the market research indicates belief that the market conditions will be high.

True market conditions Negative Positive

Low $0\mathrm{.}70\mathrm{.}3$

High $0\mathrm{.}250\mathrm{.}75$

b$)$ what would be the gain in information given the above track record of the market research?

c$)$ What is the efficiency of information