where ct is consumption in real terms in period t, mt = Mt/Pt, and M is the nominal stock
of money.7
The budget constraint for the households, in nominal terms, may be written as
P,(y - txl ) = P,c, + M,,1 - M, + B^- - Bt (9)
1 + it
where y is the output, assumed constant, ,x, are the lump-sum taxes paid in ,, B, stands
for one period government debt securities, outstanding in period ,, and i is the nominal
interest rate.
The previous constraint may also be presented in real terms as
M 1P1 M B 1P1 1 B
(10)
y - txt = ct + —t+1 —--- + —t+1—t+1----t-
t tP P P PP 1 + i P
t t+1 1 t t t 1I+1 11 ɪ + zt 1 t
Defining bt = Bt/Pt and multiplying both members of the last equation by (Pt/Pt+1), the
utility optimization problem of the households is then given by
Max U(ct, mt ) = (1 - σ ) 1A1 ct 1σ + (1 - η ) 1A 2 mt 1η
Pt Pt Pt 1 Pt , (11)
s. a (y-txt) — = —ct + mt+1 - — mt +1—г bt+1 - — bt
Pt+1 Pt+1 Pt+1 1 + it Pt+1
with the following first order condition for the optimum solution, assuming that
Pe = P
1 t+1 1 t+1
P 1 + i. ,
Uc ( Ct, mt ) = -P- -r-t+1 Uc ( Ct+1, mt+1 ), (12)
Pt+1 1 + r
which is the usual Euler equation, now depicting addionally the use of money in the
households utility function.
7 The utility function used here (inspired in McCallum (1999a)) is basically a parametric version of
the general formulations used by Leeper (1991) and Woodford (1995).