Economic and Game Theory
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Consider an example of a ski resort owner investing in both snowmaking equipment and in expanding the number of ski lifts at the resort. If there is sufficient natural snow, then the investment in snowmaking equipment adds nothing to the resort’s profits, because this equipment is not used. Similarly, if there is not enough natural snow, then the investment in additional ski lifts is worthless because there is not enough snow to ski on the new runs. (Assume the resort owner cannot afford to purchase enough snowmaking equipment to cover the new runs).
Suppose that each dollar invested in snow-making equipment generates $100 in revenue if there is not enough natural snow; the more snow the resort can make, the more people will come to ski. If there is enough natural snow; the more snow the resort can make, the more people will come to ski. If there is enough natural snow, on the other hand, then the investment in snowmaking equipment is worthless, but each dollar invested in expanded lifts generates $100 in revenue. The owner knows that with probability 0.3 there will not be enough natural snow, while the probability .7 there is enough natural snow. Use the notation xs to denote the amount of money invested in snowmaking equipment and xL to denote the amount invested in new ski lifts.
b. Use lottery notation to characterize the uncertain payoffs the resort owner faces is she invests xS dollars in snow making equipment and xL dollars in new chair lifts.
c.Suppose that the ski resort owner’s utility function is u(w) = w, so her marginal utility is u’(w) = ˝ w)), the price of her snow making equipment is pS = 10 per unit, the price of expanded ski-lift capacity is PL = 30 per unit. Given her budget constraints of $210, the owner decides to invest xS = 9 and xL = 4. Is this optimal? If so, explain why. If not, then explain how the owner can increase her expected utility.
d.Explain why the (risk-averse) ski resort owner should always diversify her investment, i.e., invest in both snow making equipment and expanded lift capacity. Provide an economic explanation in terms of this specify ski resort example. Be as precise as possible in your answer. A diagram may be useful