Consider a risk-neutral farmer who can grow either maize or hybrid cotton on his plot. In the case of maize, the farmer will earn $400 if the rains are good and $0 if they are bad. If he grows cotton, the farmer will earn $300 if the rains are good and $300 if they are bad. This year, there is a 50% chance of good rains and 50% chance of bad rains. To grow either crop, the farmer must borrow $100 at an interest rate i from a lender to cover costs. If the farmer is able, he will repay the loan with interest; otherwise he will not. The farmers profit (which he seeks to maximize) is the expected value of his revenue minus loan repayments. The lender's profit is the expected value of repayments minus the initial loan. The farmer can earn $ 50 by growing vegetables that do not need loan funding. He will not borrow to plant a crop if his expected payoff is less than this amount. (i) Find, in terms of i, A. The farmer's expected net profit y(c, i) if he grows cotton B. The farmer's expected net profit y(m, i) if he grows maize. (ii) What is the highest interest rate i* for which the farmer will grow cotton? (iii) What is the highest interest rate imax at which the farmer will grow maize (remember the farmer can grow vegetables)? (iv)Assume the farmer grows cotton if the interest rate is i* or less, and switches to maize when the interest rate is higher than i*. Find the moneylender's profit (expected repayment minus the $100 loan) if he charges i*, and if he charges imax. Remember the choices the farmer will make at these interest rates. Which rate will he choose, and what crop will the farmer grow? (v) Now suppose the government digs an irrigation canal to this village. The maize crop no longer depends on the rains, and gives a revenue of $ 400 whether it rains or not. Explain how this will affect the cropping pattern and interest rates. Will the farmers benefit? Will the moneylender benefit? (You don't need to do any calculations for this part, but you may if you wish).