cia - Cash-In-Advance Randall Wright 1 Basic Assumptions We...

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Cash-In-Advance Randall Wright 1 Basic Assumptions We begin with a very simple model: an endowment economy with homoge- neous agents. The representative agent chooses a sequence for consumption c t to solve max X t =0 β t u ( c t ) , subject to recursive budget and CIA constraints p t c t = p t e t + m t + T t m t +1 p t c t m t + T t , where p t is the nominal price level, e t is the endowment, m t is money hodlings at the begining of period t ,and T t is a transfer of money from the government, that could be negative and that the agent regards as lump sum. The CIA constraint requires that consumption be f nancedoutofcashonhandatthe start of the period, including the transfer. 1 In particular, one cannot use the p t e t dollars one receives from the sale of one’s endowment at t to f nance c t ; it must be carried forward and used for c t +1 . There are di f erent interpretations of this assumption. One that is similar to a story Lucas proposed is as follows. First, each household consists of a pair, say a “worker” and a “shopper.” Second, there are several types of 1 Alternatively, one can assume T t is not available that period, so that the CIA con- straint would be p t c t m t ;seebe low
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households, and each type lives in a distinct physical location, say an island. To motivate gains from trade, assume that the consumption good comes in K varieties, say di f erent colors, and that each type produces good k but wants to consume good k +1(mod K ) . Soeachper iod“shoppers”o feach type k simultaneously take cash to the next island in order to purchase color k +1 consumption goods, while their “worker” partners — perhaps better called “vendors,” really — stay home waiting to sell their endowment of color k goods for money. Goods cannot be bartered directly if K> 2 . 2 Clearly, cash aquired from today’s sales cannot be used until next period, since the shopper leaves before the money rolls in. This motivates the CIA constraint. What is relevant is the timing, not that households come in pairs; e.g., the “vendor” agent can be replaced by “vending machine” that collects cash while an individual is out shopping. We can rewrite the CIA contraint using the budget constraint as m t +1 p t e t . Hence, the assumption can be reinterpretted as saying that agents are forced to hold at least the nominal value of their endowment as money from each period t into t +1 . Although not usually stated this way, this makes it prettyobviouswhattheCIAconstraintisdoing—simplyimposingademand for money. The supply of money at t is denoted M t ,wherethein it ia lstock M 0 > 0 is given exogenously. The governm en tbudg e tcon s t ra in th e r ei s T = M 0 M . Formulating the problem in dynamic programing terms, after eliminating c t , Bellman’s equation can be written V ( m )= max m 0 s.t. m 0 pe ½ u μ e +
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cia - Cash-In-Advance Randall Wright 1 Basic Assumptions We...

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