$6.$ A farmer can make a profit of $1000$ if there is no damage to his crop. But if there is damage then his profit is only $300.$ The probability of the crop getting damaged is $0\mathrm{.}25.$ The farmer$$s utility function is U $=$ ln Y $,$ where Y is profit. The farmer can purchase insurance against loss q at premium rate $\backslash$pi $.[$Premium rate $\backslash$pi means: to insure $1$ worth of income the individual pays a premium of $\backslash$pi $.]$

$($a$)$ Find the highest premium rate $\backslash$pi $$ at which the farmer would buy full insurance $($against the entire loss$).$

$($b$)$ Suppose there is only one insurance firm. Find the premium rate $\backslash$pi that maximizes the firm$$s expected profit.

$($c$)$ Suppose that a new insurance firm enters the market. Let $\backslash$pi $1$ denote the premium rate offered by the incumbent firm and $\backslash$pi $2$ by the new firm. The farmer chooses the firm that offers the lowest premium rate. Find the equilibrium premium rates.