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A caslkcredit economy. Consider a variation of the basic CIA model we have studied. Suppose that there are two "types" of consumption goods, "cash...

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I don't know how to start part b. I think I've got part a right, but I can't combine 3 FOCs to a single condition.

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4. A caslkcredit economy. Consider a variation of the basic CIA model we have studied.
Suppose that there are two “types” of consumption goods, “cash goods” and “credit
goods.” Cash goods, denoted by Cu, are goods whose purchase requires money, while credit goods, denoted by C2,, are goods whose purchase does not require money (i.e.,
they can be bought “on credit”). Optimal Monetary Policy 281 The representative consumer wants to consume both cash and credit goods: his instan-
taneous utility function is u(c,,, Ca)- Note that there is no leisure in this utility flinc-
tion: the economy is an endowment economy, in which the real endowment y, in each
period t is outside the control of anyone in the economy. The resource constraint of the
economy is C“ +c2. = y.. Furthermore the nominal price of each type of good is the
same, P,. The only asset that consumers have to accumulate is money (no bonds, no stock), and
let MH be the nominal amount of money that consumers begin period I with. Hence
the period-t budget constraint of the consumer is Bel, + Bag, + M, = By, + M,., + t},
where, as always, 1', is a iump-sum transfer from (or to) the central bank. The cash-in-
advance constraint in this economy applies only to ch: in period t, the cash-in—advance
constraint is Rcl, = M, (note the subscript!). Finally, as usual, the transfer depends on
some growth rate of money, r, = g,M,_1. In period t the consumer chooses C1” C2,, and M,. a. Setting up an appropriate Lagrangian, obtain the consumer’s first-order condi-
tions with respect to c“, c2" and M,. b. Combine the three FOCs you derived in part a in a single condition that involves
only marginal utilities and inflation, and then do the following: impose steady
state, and express the final steady-state condition in terms of "1(ED E2.) =
“2(as 52) where the (. . .) is for you to figure out. (Note the order in which you should do things
here.) c. For this cash—credit model, solve for the optimal g, showing the important steps
in your work.

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