DBJ Discussion Paper Series, No.1005

Exact microeconomic foundation for the Phillips curveunder complete markets: A Keynesian view

Masayuki Otaki(Institute of Social Science, University of Tokyo)
andYoshihiro Tamai(Faculty of Economics, Kanagawa University)

January

2011

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings, conclusions
or recommendations expressed in these papers are those of the authors and do not reflect
the views of the Institute.

Exact microeconomic foundation for the Phillips curveunder complete markets: A Keynesian view

Masayuki Otaki (University of Tokyo)
Yoshihiro Tamai (Kanagawa University)
JEL Classiﬁcation Numbers: E21; E24; E31
Keywords: Phillips Curve under Complete Markets; Intergenerational Learning by Doing; Welfare Improving Macroeconomic Policy

Abstract
Assuming that the labor productivity varies with the previous employment
level, we derive the Phillips curve based on the standard dynamic microeconomic foundation. The usage of the term standard implies that our theory
entirely excludes the assumptions unfamiliar with microeconomics such as
price or information stickiness, and money in the utility function.
We ﬁnd that when labor productivity decreases, disinﬂation advances.
This is because disinﬂation, ceteris paribus, limits the current goods supply
and increases the rate of return on money (the inverse of the inﬂation rate) inan overlapping generations (OLG) model. In addition, mass unemployment
becomes a hazard for the intergenerational skill transformation, and thus,
the higher the unemployment is, the lower the labor productivity becomesin the stationary state. Consequently, the negative correlation between inﬂation and unemployment emerges even in the dynamic general equilibriumin complete markets.
It is also noteworthy that we depend neither on linear approximations
nor on numerical methods: the method used to derive the Phillips curve ispurely analytical.

1Introduction

Every work on the Phillips curve presumes some market imperfection. For
example, Lucas (1972) assumes that the equilibrium price is a noisy signal
disturbed by monetary shocks. Calvo (1983) and Woodford (1996) stochastically conﬁne the opportunity of price realignment. Mankiw and Reis (2002)
insist that there exist some substantial diﬀusive lags that inform about thenecessity of price revision. These works imply that if there does not exist some price stickiness assumption or imperfect information (that is, themarkets are complete), the negative correlation between inﬂation and unemployment will disappear, and the vertical Phillips curve will reemerge asFriedman (1968) suggests. This also means that money is neutral in the longrun, when all adjustments are complete.
Our main concern in this article is to construct the stationary negative
correlation between inﬂation and unemployment in certainty without pricefrictions. This would not only enable us to interpret the Phillips curve as
the long-run trade oﬀ relationship like in the original work (Phillips (1958))
but also contribute toward shortening the gulf between macroeconomic andmicroeconomic theory.
The change in the labor productivity due to the learning eﬀect plays
a crucial role in this paper. Although recently, the focus with regard to
the Phillips curve has been on the responses of inﬂation and unemployment
to a monetary shock, the eﬀect of a real shock (namely, the change in thelabor productivity rate) should also be seriously considered. Hayashi and
Prescott (2002) reveal that signiﬁcant declines in the total factor productivity(TFP) and hours worked were observed in Japan during the 1990s. It is also
noteworthy that disinﬂation was prominent throughout that decade, despite
the easy monetary policy.
The decline in the labor productivity, which partially consists of the TFP,
results in disinﬂation in the deterministic overlapping generations (OLG)
model of Otaki (2007, 2009), even if we do not make any price stickiness1

assumption a priori. When the labor productivity is lowered, ceteris paribus,
the current goods supply is reduced, and this increases the rate of returnon money (the inverse of the inﬂation rate). Thus, the decline in the labor productivity is accompanied by disinﬂation. The same mechanism—
intertemporal substitution—is also adopted by Lucas (1972).
We further assume that skills nurtured through production process transmits to the child generation, and that the more employment opportunities
are oﬀered to fathers, the more productive children become. It is considered to be based on the enlargement of the educational opportunity by theincrease of income, and on, perhaps unintentional, educational eﬀect in a
family. To sum up, the current labor productivity is assumed to be theincreasing function of the previous employment level.
In stationary states, where the employment level is kept constant, thelower the employment level is, the fewer individuals are productive. Consequently, there emerges the negative correlation between the inﬂation andunemployment rate. This is our understanding on the long-run Phillips curveor the aggregate supply curve under complete markets.
The paper is organized as follows. Section 2 exhibits the basic model.
The Phillips curve is also derived. Section 3 deals with the economic welfare
implications of the expansionary ﬁscal-monetary policy. Section 4 contains
brief concluding remarks.

2

Model

2.1

Model Structure

We consider a standard two-period OLG model. There exist continuous
individuals [0, 1] × [0, 1]. They supply labor at their discretion only when
they are young. Each ﬁrm monopolistically produces the diﬀerentiated goodthat is distributed within [0, 1]. The monopoly rent is equally distributed
among the young regardless of their employment status.
2Fiat money is the only store of value. The government ﬁnances its expenditure by seigniorage. For simplicity, we assume that the goods purchasedby the government bear no additional utility to the individuals.

2.2

Individuals

Each individual maximizes the following lifetime utility function:2
U = u(Ct1 , Ct+1
) − δt · α,

Cji ≡

[∫ 1
0

−1

]

1
−1

cij (z)1−η dz 1−η ,(1)where cij (z) denotes the consumption of good z by the individuals in theith stage of life during period j. α is the disutility of labor. δt is a deﬁnition function that takes the values one (when employed) and zero (whenunemployed). In addition, we assume that u( · ) is a linear homogenous
function.
Then, the indirect utility function H of u is represented as
H(Pt , Pt+1 , δt Wt + Πt ) =[δt Wt + Πt ],h(Pt , Pt+1 )

Pt ≡

[∫ 1
0

] 1

pt (z)1−η dz 1−η ,(2)where Wt is the nominal wage and Πt is the aggregate nominal monopoly
rent. We must note that h is also linear homogenous.
Using (2), we can calculate the nominal reservation wage WtR as
WtR = α · h(Pt , Pt+1 ).

(3)

Since our main concern is the imperfect employment equilibrium where someindividuals are always unemployed and possess no bargaining power, WtR
becomes the equilibrium nominal wage. However, the nominal wage might
exceed the reservation wage when the individuals possess collective bargaining power. Nevertheless, as long as the nominal wage negotiation is eﬃcientin the sense of McDonald and Solow (1981), there is no need to discuss any
modiﬁcation. See Otaki (2009) for the precise proof.
It is also noteworthy that the following aggregate consumption functionof the younger generation Ct1 is obtained from the linear homogeneity of u:
Ct1 ≡ c(ρ)[

Yt
WtR Lt Πt+ ] ≡ c(ρ) ,
Pt
Pt
Pt
3

(4)

where ρ(≡ Pt+1 /Pt ) denotes the gross inﬂation rate. Lt is the aggregate
employment level.

2.3

Firms

From (1), each ﬁrm faces the following identical aggregate demand function
Dt (z):( p (z) )−η W R L

Πt
],
(5)
Pt
Pt
Pt
where Lt is the aggregate employment level. Furthermore, we assume that
Dt (z) =

t

[

t

t

+

each ﬁrm z has the following identical production function yts (z):yts (z) = γ(Lt−1 )Lt (z),γ ′ > 0.

(6)

γ is the function of the productivity of labor, which plays a key role in ourcomparative statics. (6) implies that there is a socially signiﬁcant learningeﬀect in the productivity of labor. Namely, when the more individuals who
belong to the previous generation are employed and more skills for production are socially accumulated, the easier they are transmitted to the current
generation.This is considered to be an overlapping-generations model version
of Arrow’s (1962) learning by doing.
The proﬁt maximization problem leads us to the following optimal pricing
rule:
pt (z) =WtR
α · h(Pt , Pt+1 )=.
−1γ(Lt−1 )[1 − η ]γ(Lt−1 )[1 − η −1 ](7)Aggregating (7) on z, we obtain the following important diﬀerence equation:Pt =α · h(Pt , Pt+1 ).
γ(Lt−1 )[1 − η −1 ](8)

From the linear homogeneity of h, Eq.(8) determines the equilibrium inﬂation rate ρ independent of the nominal money supply. Diﬀerentiating (8)logarithmically and using Roy’s identity, we obtaindρLγ ′ dL
dL(1 − c) =
≡ν ,
ρ
γ L
L
4

0 < c < 1,(9)

where ν is the elasticity of the labor productivity to the employment level.
The left-hand side of Eq.(9) indicates the number of future goods that canbe substituted for present goods when inﬂation occurs.
From Eq.(9), it is clear that the inﬂation rate ρ is determined so asdL
to equalize the additional present aggregate supply νto the additionalLdρdemand for present goods (1 − c) . Namely, an improvement in the laborρproductivity implies a potential increase in the current consumption level;
this advances inﬂation and reduces the rate of return on money to decreasesavings.

2.4

Government

In this model, the role of the government is very simple. It issues new money,
Mt − Mt−1 , to ﬁnance wasteful expenditure Gt . Avoiding any diversion fromthe stationary equilibrium, we assume that the expenditure is controlled soMt
as to keep the real money stock m ≡, constant over time. Hence, thePtbudget constraint of the government can be written asg≡

Gt+j
Mt+j − Mt+j−1
m
=
=m− .
Pt+j
Pt+j
ρ

2.5

Market Equilibrium

2.5.1

The Aggregate Demand Function(10)

Since our attention is conﬁned to the imperfect employment equilibrium instationary state, the labor markets are in equilibrium whenever Wt = WtR
and Lt = Lt−1 . The real aggregate demand y is deﬁned as
y = c(ρ)y + g +m= c(ρ)y + m,
ρ

y≡

Y
,
P

(11)

where the third term of Eq.(11) is the consumption demand of the old individuals. Because every equilibrium is stationary in the sense that it isindependent of time t, we abbreviate the subscripts. This is the dynamically extended multiplier process developed by Otaki (2007, 2009). Solving5

Eq.(11), we obtainy=m.
1 − c(ρ(L))(12)(12) is the aggregate demand function of our model. It can be easily seenthat an expansionary ﬁscal-monetary policy increases the real GDP as longas the inﬂation rate is kept constant.
2.5.2

The Aggregate Supply Function: The Phillips Curve

We have already established the negative correlation between inﬂation andunemployment in 2.3. Phillips curve is implicitly deﬁned by Eq. (9). Consequently, the Phillips curve is obtained as illustrated by Curve AS in Figure
1.
2.5.3

Market Equilibrium

The good market equilibrium is expressed by the solution of (8) and (12).
It is illustrated by Figure 2. Although the slope of the aggregate demand
function AD in (L, ρ) plain is ambiguous, a suﬃcient stability condition of
the stationary state E is that Curve AD cuts AS from the downward. It
implies that the intertemporal substitution rate on consumption is positive
but not too high. From empirical analyses, (For example, see Hansen andSingleton (1983) and Campbell and Mankiw (1989)) it does not seem sorestrictive assumption, and hence, we assume that the stationary state isstable.
The ﬁscal-monetary policy can only control the location of the aggregate demand curve (12). Namely, ﬁscal-monetary expansion results in arightward parallel shift in the aggregate demand curve. This implies that a
discretionary expansion in the ﬁscal-monetary policy can increase the real
GDP y together with accelerating inﬂation.

6

3

The Welfare Implication of Fiscal-Monetary
PolicySince the indirect lifetime utility is represented by Eq.(2), considering thatthe labor supply is never in surplus, the social welfare SW is deﬁned asΠtαη −1 h( · )αη −1
SW (L) ≡=L=
L.
h(Pt , Pt+1 )[1 − η −1 ]h( · )1 − η −1

(13)

Let us analyze the welfare implication of our macroeconomic policy in theimperfect employment equilibrium. The result can be easily obtained from
Figure 2 and Eq.(13). From Figure 2, it is apparent that the real money
supply m increases the real GDP y and employment level L. Accordingly,
Eq.(13) reveals that the social welfare is always enhanced by an expansionaryﬁscal-monetary policy.

4

Concluding Remarks

We have developed the microeconomic foundation for the Phillips curve undercomplete markets. The obtained results are as follows.
First, the Phillips curve can be derived not from a monetary shock but
from an endogenous structure of the economy per se. Whenever the laborproductivity increases, inﬂation rises. This is because the current consumption needs to be stimulated in accordance with the potential increase in the
production capacity. Hence, the rate of return on money decreases, andthe inﬂation rate increases. Thus, a positive correlation between the laborproductivity and inﬂation emerges.
On the other hand, the lower labor productivity is reproduced by itself.
Namely, the mass unemployment of the previous generation deprives theformal and informal educational opportunity from the current generation.
Thereby, the lower employment level causes the lower labor productivity.
Thus, the stationary negative correlation between the inﬂation and unemployment emerges even under complete markets.
7Second, as long as the stationary state is stable, an expansionary policy
can raise the real GDP and improve the social welfare. This is because theexpansionary policy increases the real GDP by the multiplier process andimproves the labor productivity through the reduction of the unemployment
level.

8

ρ
AS

O

L

Figure 1: The Phillips Curve

9

ρ
AS

AD

E

O

L

Figure 2: Market Equilibrium

10References[1] Arrow, K. J. (1962). The economic implications of learning by doing.Review of Economic Studies, 155-173.
[2] Calvo, G. A. (1983). Staggered prices in a utility-maximizing framework.
Journal of Monetary Economics, 383-398.
[3] Campbell, J. Y. and Mankiw, N. G. (1989). Consumption, income, andinterest rates: reinterpreting the time series evidence. NBER Macroeconomics Annual 4, 185-216.
[4] Friedman, M. (1968). The role of monetary policy. American Economic
Review, 1-17.
[5] Hansen, L. P. and Singleton, K. J. (1983). Stochastic consumption, eisk
aversion, and the temporal behavior of asset returns. Journal of Political
Economy, 249-265.
[6] Hayashi, F. and Prescott, E. C. (2002). The 1990s in Japan: a lost
decade. Review of Economic Dynamics, 206-235.
[7] Lucas, R.E. Jr. (1972). Expectations and the neutrality of money. Journal of Economic Theory, 103-124.
[8] Mankiw, N. G. and Reis, R. (2002). Sticky information versus sticky
prices: a proposal to replace the new Keynesian Phillips curve. Quarterly
Journal of Economics, 1295-1328.
[9] McDonald, I. M. and Solow, R. M. (1981). Wage bargaining and employment. American Economic Review, 896-908.
[10] Otaki, M. (2007). The dynamically extended Keynesian cross and thewelfare-improving ﬁscal policy. Economics Letters, 23-29.
[11] Otaki, M. (2009). A welfare economics foundation for the fullemployment policy. Economics Letters, 1-3.
11

[12] Phillips, A. W. (1958). The relation between unemployment and therate of change of money wage rates in the United Kingdom, 1861-1957.
Economica, New Series, 283-299.
[13] Woodford, M. (1996). Control of the public debt: A requirement for
price stability? NBER Working Papers: 5684.

12