the model and describe the theoretical role of a theory of price dynamics. In section 4, I ..... in u reduces q by the same amount on the left and right hand sides ...
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working paper department of economics
UNEMPLOYMENT AND PRICE DYNAMICS IN A MONETARY-FISCAL POLICY MODEL by
Duncan K. Foley
Number 80 - OLpLuuW- r 1971
massachusetts institute of
technology 50 memorial drive Cambridge, mass. 02139
imii^m I
f^ASS. ir43T. TtCH.
NOV 2
1S7i
DEWEY 'LIBRARY
UNEMPLOYMENT AND PRICE DYNAMICS IN A MONETARY-FISCAL POLICY MODEL
by Duncan K. Foley
Number 80 - OLpluiiW- r 1971
My work on this paper was supported by the National Science Foundation (grant number GS 2966) While I have had fruitful discussions of these ideas with Franco Modigliani, Jeremy Siegel, Robert So low and several M.I.T. graduate students, responsibility for errors lies with me. .
UNEMPLOYMENT AND PRICE DYNAMICS IN A MONETARY-FISCAL POLICY MODEL
In a recent paper [1] and book [2] Miguel Sidrauski and
I
developed a
full employment continuous- time model of an economy controlled indirectly by
monetary and fiscal policy. Theory of Keynes
[4]
The close relation of this model to the General
and to some traditional versions of macroeconomic theory
was somewhat obscured by the fact that we analyzed only a full employment
version of the model.
In this paper it is my purpose to extend that model
to include unemployment.
The strict treatment of stocks and flows in con-
tinuous time characteristic of the original monetary-fiscal policy model has
consequences for unemployment as well; it unravels some logical confusions about the role of price and wage flexibility in unemployment models and illuminates
the theoretical function of a theory of price dynamics. In section 1,1 outline the basic monetary-fiscal policy model; this review
assumes some familiarity with Part
I
of our earlier book.
In section 2,
I
dis-
cuss the problem of static or instantaneous unemployment equilibrium, the dis-
tinction between disequilibrivira of relative prices and disequilibrium of money prices, the consequences of unemployment in assets markets and what might be
called the static accelerator.
In section 3, I discuss the dynamic version of
the model and describe the theoretical role of a theory of price dynamics.
section
4,
I
In
begin to discuss the formation of a theory of price dynamics based
-2-
on the notions of rational expectations and stochastic process theory.
Section
1:
The Basic Model
The monetary-fiscal policy model follows a central macroeconomic
tradition (most familiar in the ideas of the IS-LM analysis [3]) in analyzing
macroeconomic equilibrium as the interaction of two kinds of markets: assets
markets for existing stocks of money, bonds, and capital; and a consumption market for the flow of output from the consumption sector. The production model is the standard two-factor two-sector production model, with constant-returns to scale production functions in investment and consumption sectors.
The first degree homogeneity permits me to write
all market-clearing conditions in terms of intensive per capita quantities.
At each instant, rentals to capital and wages must be the same in each sector given the relative price,
Throughout
p,
,
of investment goods in consumption goods units.
use consumption goods as numeraire.
I
Just as
of investment goods in terms of consumption goods so p
p,
is the price
is the price of money
in terms of consianption goods, the inverse of the price level.
The wage w
and rental r are both measured in consumption good units, so that a pure number,
the own rate of return to capital.
(1.1)
r - f '(k^) . p^ f;(k^)
(1.2)
w = f^(k^)
- k^ f^(k^)
=
V^if^(\)
-
4
f;(kj))
r/p,
is
-3-
where
f
,
are the intensive production functions in the consumption
f
and investment sectors respectively, and k
and k^ are the capital in-
These equations suffice to determine k
tensities in each sector.
and k^,
but the actual outputs depend on the distribution. of labor between the two sectors, which is determined by market clearing in the labor and capital
markets
+ 1^ kj = k
1^ k^
(1.3)
1
(1.4)
c
we always assume
where
1
,
+ 1^ = I
k
1,
> k^ c
I
are the proportions of the labor force employed in each sector,
1
and k is the total capital stock divided by the total labor force.
Since
these equations imply full employment they will have to be revised when
I
come to introduce unemployment.
The solution of these equations can be summed up as supply functions q
(k,
k and
Pi^)
p,
,
for given
>
q-r(k,
p,
showing the rate of output in each sector for given
)
and a function p,
showing the own rate of return to capital
r(p, )/p,
.
In the assets market the real supplies of the three assets are fixed
at any instant, and the demands depend on desired portfolio balance among them, which in turn involve non-hiiman total net worth,
a, = p
g +
p.
k,
where g is the nominal value of the government debt, output q(k,
p, )
= q
(k,
p
)
+
p
q^(k,
p,
)
which represents the transactions demand
for money, and expected real rates of return to the assets.
For money the
-4-
real rate is
tt
,
the expected rate of deflation, for bonds
i
+
it
,
where i is the interest rate, since we assume bonds to be instantly redeemable at a fixed money price, like a savings account, and for capital
+
r(p, )/p,
TT
,
where
it
is the escpected rate of change in p
.
We write the asset market clearing in three equations: (1.5)
Pjn
(1.6)
Pjn
(1.7)
Pj,
" L(a,
TT^,
i
+
tt^,
r(pj^)/pj^
+
tt^^)
,
TT^,
i
+
tt^,
r(pj^)/pj^
+
tt^^)
k = J(a, q(k, pj^),
TT^,
i
+
7T^,
r(pj^)/pj^
+
tt^^)
°i
q(k, pj^),
h = H(a, q(k,
p^^)
where h is the nominal supply of government bonds and H the net demand of the private sector to hold bonds.
These demands must satisfy a wealth
constraint: L + H + J = a,
(1.8)
since at any instant in revising portfolios people can only buy one asset by selling another. For a given p
clearing
p.
and i.
,
equations (1.5) - (1.7) can be solved to find market
The relation between p
and
p,
that clears the assets
markets we call the aa schedule: under mild assumptions it is upward sloping (see [2] Ch. 3).
The aa schedule is a more general analogue of the LM
schedule.
The instantaneous market clearing model is closed by the addition of a
consumption function relating, disposable income, wealth, and government expenditures to total demand for the flow of consumption goods. clearing condition In this flow market is:
The market
.
.
,
-5-
q^(k,
(1.9)
pj^)
= c (a, q(k, P^) + P^
'^
" ^
"*
^r^j
P^ g +
TTj^
Pj^
k)
+
e.
The reader can quickly verify that if e is government expenditures (assumed to be all consumption goods) and d is the nominal deficit then p
d - e
TT,
p,
is net real transfers less taxes.
The terms
'^m
k
m p*^m "g
it
and
represent anticipated capital gains, which we include in disposable
income Again, for given p
,
(1.9) can be solved for the
that just induces
p,
a flow supply of consumption goods equal to the flow demand.
between p
and p
This relation
we call the cc schedule: it is frequently downward- s lop ing
The analogy is to the IS curve.
The total instantaneous equilibrium of the economy can be pictured as the Intersection of the aa and cc curves (see Figure 1
m
)
m
:
-6-
The situation pictured in Figure
can be described in words as
1
follows: at time t, given that the government is running a deficit d with
expenditures
e,
that the monetary authorities have divided the government
debt in a supply m of high-powered money and b of bonds, that the accumulated capital stock is k and that expectations of changes in the relative
prices of money and capital are p*
m
and the price of capital
and
it
p*, k
it,
,
if the price of money were
asset holders would be content with
their portfolios and the supply and demand for the flow of consumption goods
would be equal.
Government policy, by changing d and
e,
can shift the cc, or by open
market operations can alter m and b and shift the aa.
Over time, investment
will alter k, the deficit will change the sum (m + b) and experience will modify ^
IT
m
and
ir,
k
;
as a result both cc and aa will move.
It is not
difficult to set up a system of differential equations to represent the dynamics of this system.
For convenience
I
will summarize them (where n
is the labor force growth rate)
(1.10)
m - L(a, q(k, p^) p„ m
(1.11)
Pj^
(1.12)
q^(k,
(1.13)
k = q^Ck,
(I.IA)
g = d - n
k = J(a, q(k,
pj^)
= c'^(a,
p, )
tt^,
i
+
ir^,
r(pj^)(pj^
+
tt^^)
pj^),
TT^,
i
+
TT^,
r(pj^)(pj^
+
-n^)
y'^)
+ e
,
-|
Instantaneous Equilibrium Conditions
- n k
Laws of Accumulation
To these five must be added two equations determining the formation of expectations, that is, of
it
m
and
tt,
K
,
and three representing either paths
-7-
(since there are three government tools,
or goals for government policies
the level of expenditures, the size of the deficit, and the composition
For a detailed analysis of several
of the debt between money and bonds.)
such systems, see [2].
Section
Unemployment
2:
Let me call the value of
employment price of money
.
p
m
that satisfies (1.10) - (1.12) the full-
By assuming that the actual price of money,
is at every instant equal to the full-employment price of money,
p
,
we rule
p*,
out any difficulties of maintaining full employment and any discrepancy be-
tween full-employment plans for savings and investment.
The actual price of
money will at every instant reconcile these plans.
Suppose now that for some reason Whether we should expect it to or not
We have two possibilities, a *^
p
'^m
p*
m
as in Figure 2-2.
does not instantaneously equal p*.
p I
below
will discuss in succeeding sections. p*,
m
as in Figure 2-1 or a
p
m
above
-8-
Given: m, b, k, d, e,
tt
m
,
it,
k
y -^^^
Pko
--
^^-^
^^
>/*-aa
N.
-(-cc
-•'
....,1.,
i/y
'Vi'^feiiT
'S/*^
•.'(
1
Date Due
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