Please help me with it, asap, thanks!

Suppose a farmer has a risk of losing her barn to a fire. The farmer has initial wealth w and utility function u(x) = -e-*. Suppose that the probability of losing the barn is 1 -p. The money value of a barn is L, so the farmer's wealth if the barn burns is w - L. (i) Suppose that the farmer can buy insurance from an insurance company. The farmer can buy any amount of coverage C at price TC, so that 7 is the cost of coverage. Write down the optimization problem of a farmer choosing how much coverage to buy. Write down the first order conditions. (ii) Using the first order conditions you found in (i), solve for how much coverage the farmer will purchase (i.e., find the optimal value of C, as a function of the parameters of the model). Hint: to solve for the optimal level of coverage C, you need to use the following properties of exponential functions: (1) u'(x) = e-*, (2) for any x, In(e") = x, and (3) for any r and y, In(x x y) = In(x) + In(y). (iii) Use your answer to part (ii) to show that the level of coverage C that the farmer buys does not change with her wealth w