lecture14 - Economics 202A Lecture Outline, October...

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Economics 202A Lecture Outline, October 30-November 1, 2007 (version 2.0) Maurice Obstfeld Optimal Consumption in a Frictionless World: Complete Markets To understand consumption under uncertainty, we start with the benchmark case of complete markets. Complete asset markets e/ectively allow consumers to buy insurance against any contingency (or to sell insurance). This is possi- ble because there exist assets with returns di/erentiated across every state of nature, and, subject to an overall budget constraint, individuals can purchase any (positive or negative) amount of such assets. derivative products in recent years is as an evolution of real-world markets toward the ideal of completeness. Why, then, consider this case? Because the availability of this benchmark handle on more complex problems. For example, Newton±s law F = ma is counterintuitive until one learns to abstract from the force exerted by friction. Assumptions. Let±s start with a pure endowment model (no investment or production). There are two periods. On date 1, individual i ±s endowment is y i . From the perspective of date 1, however, the date 2 endowment is a random variable. There are also only two possible states of nature on date 2. In state 1 the endowment is y i (1) , in state 2 it is y 1 (2) : Let c i denote the individual±s date 1 consumption, c i (1) and c i (2) the indi- vidual±s contingency plans for consumption on date 2. The plans are contingent on the state that actually occurs on date 2. The probability that state s occurs is ( s ) , where, summing over all states s , s ( s ) = 1 : A key hypothesis is that the individual chooses the consumption plan that maximizes average lifetime utility, U i = (1) u ( c i ) + ±u ± c i (1) ²³ + (2) u ( c i ) + ±u ± c i (2) ²³ = u ( c i ) + ± (1) u ± c i (1) ² + (2) u ± c i (2) ²³ = u ( c i ) + ± E u ± c i ( s ) ² ; where c ( s ) denotes consumption in state s . This is the von Neumann-Morgenstern expected utility criterion and, being linear in probabilities, it is somewhat spe- cial. One of its consequences (as we shall see) is that it forces the intertemporal substitution elasticity to equal the (inverse) coe¢ cient of absolute risk aversion for isoelastic utility. We shall de²ne the risk aversion coe¢ cient later. A basic Arrow-Debreu security for state s pays its owner 1 unit of output on date 2 if state s occurs and nothing otherwise. (In contrast, a riskless bond pays its owner the same amount of output in every state.) Let r be the rate of interest on a bond. We de²ne r by the de²nition that 1 = (1 + r ) is the price (all prices are in terms of date 1 consumption) of a bond 1
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paying its owner 1 unit of output on date 2 regardless of the state of nature.
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This note was uploaded on 08/01/2008 for the course ECON 202A taught by Professor Akerlof during the Fall '07 term at University of California, Berkeley.

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lecture14 - Economics 202A Lecture Outline, October...

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