class4_uncertainty%2Bge_

class4_uncertainty%2Bge_ - Econ 51, Winter 2011 Class #4...

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1 Econ 51, Winter 2011 Class #4 • Sections on Friday: Sign up on Coursework, if questions, ask Head TA, Anqi Li. • PS #1 posted today: Due Friday 21st in Section. – It’s neither easy nor short … so don’t wait until the last minute to get started. – Don’t spend your life trying to solve it. It’s only a problem set after all. If after a reasonable amount of effort, you feel stuck, wait for the answer key (will be provided). Today: Decision under uncertainty and start general equilibrium. Thursday, January 13, 2011
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2 Time and Uncertainty • We learned concepts related to decision under uncertainty, finishing with risk premium. We finish this part of the course by studying applications. Thursday, January 13, 2011
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3 A Very Short Summary of the Last Class Uncertainty is represented by states of the world s=1,2,. ..,S . Individuals have utility u(x 1 ,x 2 ,...,x S ). In most cases, an individual has a von Neumann-Morgenstern (expected) utility representation: u(x 1 ,x 2 ,...,x S ) = π 1 v(x 1 )+ π 2 v(x 2 )+. ..+ π S v(x S ) That is, the individual takes an expectation of utilities . The individual’s decision under uncertainty can be derived just as in the case for apples and bananas (utility maximization problem, using the expected utility). Thursday, January 13, 2011
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4 Applications Thursday, January 13, 2011
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5 Application - portfolio allocation An individual’s vNM utility function is v(x)=ln(x). The individual has A dollars to invest over the next year. In the end of that year she retires and spends the rest of her life in Hawaii. The A dollars can be allocated to either a savings account, which pays 10% per year, or towards purchasing California bonds, which expire in a year. If things go well (with probability 0.9), a California bond pays $1.20 in the end of the year. If things do not go well, California defaults on its bonds (i.e. produces $0). How many bonds (denoted q ) does the individual demand as a function of the current bond price p ? Thursday, January 13, 2011
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6 Portfolio choice (cont.) The individual solves: max q (9/10)ln[1.2q+1.1(A-pq)]+(1/10)ln[0+1.1(A-pq)], The F.O.C. is (9/10) × (1.2-1.1p)/ (1.2q+1.1(A-pq))+(1/10) × (-1.1p)/(1.1 (A-pq)) =0 Solving for q to obtain the optimum q* , q*=(A/p) × (10.8-11p)/(12-11p) Thursday, January 13, 2011
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7 Portfolio choice (cont.) Looking at the optimum q* from the last slide, q*=(A/p) × (10.8-11p)/(12-11p), We can also express it as the fraction of wealth invested in the risky asset: q*p/A=(10.8-11p)/(12-11p) Thursday, January 13, 2011
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8 Portfolio choice (cont.) A related question: What would a risk neutral individual do? A risk-neutral individual only cares about the expected payment of
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class4_uncertainty%2Bge_ - Econ 51, Winter 2011 Class #4...

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