p(t) = 4-[m*(t-i + l) +
m*(t-i + 2) + . . .m*(t)] (6)
Here the first term represents prices set in the
oldest поп-expired contract. Thus, in contrast
to the earlier models, the contract model al-
lows for a lag between prices and permanent
money—as we will see below, a lag between
prices and actual money will be observed even
if this is not the case. This offers a potential
explanation of the Keran-Zeldes finding of an
apparently shorter lag between money and ex-
change rates than between money and prices.
Exchange rates are apt to respond immediately
to permanent money—i.e., transient money
changes will tend to be speculated out—while
the price response can be delayed because of
contracts.
Finally, the existence of different contract
lengths does not greatly alter conclusions
based upon (6). To each contract length there
corresponds a particular horizon for the cal-
culation of permanent money. When there are
different contract lengths, the current aggre-
gate price level will be a weighted average of
current and past values of these alternative
permanent-money aggregates.
Implications of Models
When prices depend upon forecasts of future
money, as in the above, the usual forecasting
relation between p(t) and current and past m()
depends upon how individuals use current in-
formation to predict money. To see this, sup-
pose that the price-permanent money relation
is as shown in (6). Assume first that money
follows a (invertable) stationary process—
known to individuals and used by them to fore-
cast—described by,
A(L)m(t) = u(t) , A(L) ≡
1 + a,L + a2L2 + . . . (7)
where u(t) is a white-noise disturbance and
A(L) is a polynomial in the lag operator L. To
simplify matters, suppose that the horizon over
which permanent money is forecast is two pe-
riods (i = 2). If individuals use only the past
history of money in forecasting its future—that
is, if they employ (7)—permanent money can
be written as,
m*(t) = ½[(1 - a1)m(t) - a2m(t~l)
~ a3m(t-2) + . . .] (8)
since m(t + l) = u(t + l) - alm(t) - a2m(t-
1). . . , so tmc(t + l) = -a1m(t) -
a2m(t-l). . . . Then substituting in (6)
p(t) = ½[m*(t) + m*(t-l)] =
l∕4[(l-a,)m(t) + (l-a1-a2)m(t-l)
- (a2 + a3)m(t-2) + . . .] (9)
Relation (9) is the standard relation of prices
45
and current and past money given in the text.
Notice that the long-run impact of money on
prices measured from this—which is
½(l-a,-a2 . . .)—depends upon the coeffi-
cients of the process (7) generating money, and
generally will not be equal to unity.
To see this more specifically, suppose first
that money is known and expected to follow
a given path with random but temporary de-
viations:
m(t) = m + u(t) (10-a)
where u(t) is a white noise. In effect the au-
thorities are expected to correct any “base-
drift” in the next period, since tme(t + 1) = rh.
Then the price-money relation is,
p(t) = m + ¼[(m(t) — fh) +
(m(t-l) ........... m)] (ll-a)
so that the long-run impact of money upon
prices is ½; this will be smaller of course for
longer forecasting horizons. In contrast, sup-
pose that money changes are purely random,
that is “base-drift” is not corrected,
m(t) = m(t- 1) + u (t) (10-b)
Then at any time the forecast of future money
is simply today’s observed level: thus m*(t)
= m(t) and,
p(t) = ½(m(t) + m(t-l)). (ll-b)
The long-run effect here is unity. It is easy to
show that when current money changes are