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.