Right now you are in business and paying a supplier of widgets $500 per widget. You are risk neutral but your supplier is risk-averse and her utility from income I is the square root of I.. The suppliers costs vary . They are either $340 or $460 with equal probability. Thus her income is either 500- 340 or 500 - 460 with equal probability. Design a cost-plus contract that leaves her utility unaffected and lowers your expected price of a widget. That is, the contract will have the form P = C + z. where P is the price, C is the suppliers cost and z is a constant that you are going to determine to meet the conditions above. Note that under the contract the supplier always receives net income equal to $z.

The solution was to note that the suppliers certainty equivalent income is $90. The expected monetary value of his income right now is $100, so the risk premium is $10. By setting z = $90, so that the suppliers income is a certain $90, your expected payment goes from $500 to .5($430) + .5(550) =$490, so you save an amount equal to the risk premium. You are effectively providing the supplier with insurance, for which they are paying a premium of $10!

Now lets suppose that if the supplier expends effort which costs one unit of utility in every state of the world, the probability that costs are low (=$340) increases to 3/4, with the probability that they are high (=$460) decreasing to 1/4. If you could contract for this effort from the supplier, would you do so? Explain why or why not. If the effort is not contractible, design a contract that offers cost + y when costs are low and cost + x when costs are high which gives the supplier the incentive to make the effort and gives him at least as much utility as he is getting now. Would this be profitable for you? Compare the increase in your profits when effort is contractible to the increase when it is not. Why do these numbers differ?