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1. Helen is a small business owner. She has one employee, Gene, whom she pays $1000. Gene’s salary is the only cost of running her business (so profits = revenues – Gene’s salary). Her revenues vary with market conditions. If demand is HIGH then her revenues are RH = 4600. If demand is LOW then her revenues fall to RL = 1900. Each state (high or low demand) is equally likely. Helen is a risk averse individual and her utility function can be written as u(π) = ln(π) where π is her profits from the business.
a) What is her profit if demand is high? If demand is low? What is her expected profit?
b) What is the certainty equivalent? What is the risk premium associated?
c) Illustrate her expected utility, expected profit, certainty equivalent, and the risk premium.
Suppose that Helen offers Gene a profit sharing contract. Hence, instead of paying him a flat wage of 1000, she offers to pay him an amount that varies with the profitability of the company. Specifically, she will pay him WH when revenues are high and WL when revenues are low.
d) If WH > 1000 > WL and 4600 – WH > 1900 – WL then in what way is Helen’s profit sharing offer like an insurance contract (and who is getting the insurance)? Briefly explain.
For parts (e) through (g) below, assume that the contract that Helen offers Gene is of the form WJ = 350 + 0.2RJ where J is either H or L.
e) What is Gene’s expected wage from this contract? If Gene accepts this contract, then what is Helen’s expected profit?
f) Assuming that Gene accepts the contract then illustrate Helen’s expected utility in your diagram above. Will Helen be better or worse off if Gene accepts this contract?
g) If Gene is risk averse then will he accept this contract (ie does he prefer this contract to his flat wage of 1000)? Briefly explain.
Suppose that Gene rejects Helen’s contract offer. Gene counteroffers a contract where she pays him WH and WL such that Helen is fully insured. In other words, Gene offers a pair of wages such that
h) What would Helen’s expected profit be if she accepted this proposal (the answer is a function of the wages)?
i) What is the highest wage demands (WH and WL) that Gene can make that Helen will accept? Remember that the wages must satisfy the equation 4600 – WH = 1900 – WL.
j) What is Gene’s expected wage at the wages that you determined in part (j)? You can write his expected wages as equal to the sum of his original flat wage and a value, x. In words what is x?
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