20d_CAPM_Quadratic

20d_CAPM_Quadratic - Econ 251a John Geanakoplos CAPM with...

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Econ 251a John Geanakoplos CAPM with Quadratic Utilities 1 Arrow-Debreu Equilibrium Given uncertainty about the future, Ken Arrow and Gerard Debreu (both members of the Cowles Foundation, both future Nobel Prize winners, though Debreu did not get promoted to tenure at Yale) suggested that the economy could be understood just like any trading economy, by interpreting the same commodity in two di f erent states of nature as two di f erent commodities. Instead of trading apples for oranges, one could trade apples today for the Arrow security promising apples in state of nature s tomorrow. The utility W h ( x 0 ,x 1 2 3 ) to agent h of the consumption bundle ( x 0 1 2 3 ) can be taken to be the expected utility of consumption u h 0 ( x 0 )+ P S s =1 u h ( x s ) γ h s , where γ h s is the probability h assigns to state s, and u h ( x ) is his von Neumann Morgenstern utility of consumption. The price of an Arrow security s is determined just as the price of any commodity in an exchange economy, by its marginal utility of consumption. The marginal utility of W h is the probability γ s multiplied by the vNM marginal utility of consumption in state s. The latter rises as consumption declines. Thus we get the conclusion that Arrow prices are determined by probablities, but states where consumption is small get weighted by more than their probability, while states where consumption is relatively large get weighted by less than their probabilities. By no arbitrage, the price of an asset is just its payo f s times the corresponding Arrowprices.Itfollowsfromwhatwejustsaidthatassetswhichpayo f when people are poor will be worth more than assets that have the same expected payo f , but tend to pay in states when people are rich. 2 Quadratic Utility and CAPM Tjalling Koopmans, the director of the Cowles Foundation, suggested that his PhD student Harold Markowitz work on the problem of f nance by assuming that all the vNM utilities are taken to be quadratic and that everyone agree on the probablities. 3 Example with no aggregate risk Consider a world with 3 states of nature next period, three f rms and objective prob- abilities of each state. Two consumers, A and B have endowments in period zero of e α 0 = 133 . 5 e β 0 =6 6 . 5 1
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Individual α owns f rm A and β owns f rms B and C . Period 0 Period 1 ABC Totals . 25 50 150 300 500 % −→ & . 25 100 180 220 500 . 5 75 365 60 500 Firm A has a payo f of 50 if state 1 is realized, 100 if state 2 occurs and 75 if state 3 is realized. The probabilities of the three states are .25,.25,.5, respectively. We wish to answer the question: How much is each f rm worth? For example, what should each f rm’s stock market valuation be?
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20d_CAPM_Quadratic - Econ 251a John Geanakoplos CAPM with...

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