A firm produces umbrellas at a cost of C(q) = 0.5q² + 5q + 100. It chooses its output quantity at the beginning of the week. If it turns out to be a rainy week, umbrellas sell at a price of $30. If it's a sunny week, they sell for $20. The probability of it being a rainy week is 0.5 and the probability of a sunny week is 0.5.

If profit_(rain)(q) is the firm's profit on a rainy week, and profit_sunny(q) be the firm's profit on a sunny week. The firm's expected profit is Eprofit(q) = 0.5(profit_rain(q)) + 0.5(profit_sunny(q)) Write out the expression for Eprofit(q) and simplify.

What quantity q maximizes expected profit?

Suppose the owner has a utility for money given by u(x) = x^0.5. Thus, her utility on a rainy week is u(profit_rainy(q)) = (profit_rain(q))^0.5. Her expected utility is defined by Eu(q) = 0.5(profit_rain(q))₊ 0.5(profitsunny(q)) If she produces the expected-profit-maximizing quantity q from the last problem, what will be her expected utility.

Give the first order condition for maximizing the owner's expected utility.

If the firm knew the weather in the coming week with certainty, what q would maximize the owner's utility when it's rainy? What q would maximize her utility when it is sunny?