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ECON3107 Lecture 8v-4 - Uncertainty An Example ECON...

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ECON 3107/5106 Economics of Finance Lectures 8-9: Pricing Basic Arrow Securities Valentyn Panchenko Uncertainty: An Example Two periods: today (time 0) and future (time 1) Today’s state of nature s 0 is known. Alternatively, the probability that s 0 occurs π ( s 0 ) = 1; Set of possible future events - good weather (GW) and bad weather (BW): S = { GW , BW } . GW occurs with probability π ( s 1 = GW ) ; BW with probability π ( s 1 = BW ) . Notation (state-contingent consumption plans): c ( s 0 ) - consumption at time 0. c ( s 1 = GW ) consumption at time 1 if next period’s state is GW; c ( s 1 = BW ) consumption at time 1 if next period’s state is BW; Expected Utility: an introduction The representative consumer has a time and state separable utility function over consumption c ( s 0 ) and c ( s 1 ) . Consumers discount future expected utility with time discount factor β ( 0, 1 ) The lower the β , the more impatient are the consumers The period utility function u ( c ) is assumed to be strictly increasing and concave; The consumer maximizes expected utility, U , given by U = u ( c ( s 0 )) + β [ π ( GW ) · u ( c ( GW )) + π ( BW ) · u ( c ( BW ))] expected discounted future utility More on the instantaneous utility: u ( c ) is assumed to be strictly increasing and concave: u ( c ) > 0, u ( c ) 0 An example often used in Asset Pricing literature: CRRA utility function has marginal utility of the form: u ( c ) = c γ , where γ > 0 is the coe cient of relative risk aversion. The functional forms for CRRA utility function u ( c ) = c 1 γ 1 γ , for γ > 0, and γ = 1 u ( c ) = log c , for γ = 1
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Endowments: There is no production; There is an exogenously given supply of a non-storable consumption good at each time and state; As of date t = 0, the second period endowment is random. The consumer does not know which state will realize. Notation (Endowments): e ( s 0 ) - the initial endowment of consumption good; e ( s 1 = GW ) - the quantity of the consumption good consumer receives (say apples from a tree) at time 1 if the realized state is Good Weather; e ( s 1 = GW ) - the endowment available at time 1 in the Bad Weather state; Market structure: The consumer can freely borrow or lend in a complete set of basic Arrow securities. We assume the existence of two securities: Bad Weather security and Good Weather security. One unit of ’GW security’ sells at time 0 at a price q ( s 0 , s 1 = GW ) and pays one unit of consumption at time 1 if state ’GW’ occurs and nothing otherwise. One unit of ’BW security’ sells at time 0 at a price q ( s 0 , s 1 = BW ) and pays one unit of consumption in state ’BW’ only.
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