lecture20 - Economics 202A Lecture Outline, December 6,...

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Economics 202A Lecture Outline, December 6, 2007 (Version 1.0) Maurice Obstfeld Financial Instability In this lecture I describe the Diamond-Dybvig model of bank runs, from Journal of Political Economy intermediaries promote risk sharing among individuals, but they are subject to arbitrary panics. The model There are three periods, T = 0 ; 1 ; 2 : There are two possible technologies on date 0, short and long. Investment of 1 unit of output in the short technology at T = 0 yields 1 unit of output in period 1 and 0 in period 2. Investment of 1 unit of output in the long technology at T = 0 yields 0 units of output in period 1 and R > 1 units in period 2. Individuals need not specify the technology they are choosing ex ante . They opt for the short or long technology simply by \harvesting" the yield either on date 1 or 2, respectively. The idea is that more roundabout technologies are more productive. At time 0, a depositor does not know his/her \type," patient or impatient. Depositors are indexed by the unit interval, [0 ; 1]. at the start of period 1, a fraction p is revealed to be of type 1, or impatient. The rest (of measure 1 p ) are of type 2, patient. An agent has an endowment 1 in period 0 and consumes in period 1 and/or 2. The utility functions of types 1 and 2 are U ( c 1 ; c 2 ; 1) = u ( c 1 ) ; U ( c 1 ; c 2 ; 2) = u ( c 1 + c 2 ) ; where lim c ! 0 u 0 ( c ) = 1 , lim c !1 u 0 ( c ) = 0, and cu 00 ( c ) =u 0 ( c ) > 1. 1
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First consider an autarkic individual. That person will pick c 1 = 1 if he/she turns out to be impatient, c 2 = R if patient. That person’s ex ante expected utility is an average over the utilities of the two types: E U = pu (1) + (1 p ) u ( R ) : People can do better than this, however, if there are &nancial intermedi- aries. Social optimum A benevolent and omnipotent planner would withdraw an amount 1 x from investment on T = 1 so as to maximize the expected utility of a representative individual pu c 1 1 ± + (1 p ) u ( c 2 1 + c 2 2 ) subject to the aggregate resource constraints pc 1 1 + (1 p ) c 2 1 = 1 x; (1 p ) c 2 2 = Rx: Here, c i j is the amount type i consumes in period
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lecture20 - Economics 202A Lecture Outline, December 6,...

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