ECONOMIC RESEARCH CENTERDISCUSSION PAPERE-Series

No.E05-7

Coordination Failure under Perfect Competition-A Micro Foundation of Keynes-type Consumption Function-byShin Kawai

July 2005

ECONOMIC RESEARCH CENTERSCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Coordination Failure under Perfect Competition– A Micro Foundation of Keynes-type Consumption Function –Shin Kawai∗
July 4, 2005

Abstract
This paper considers the coordination failure under perfect competition in an overlapping generations model. If the marginal propensity
to consume decreases, then the outputs decrease. There is a continuumof Pareto-ranked equilibria with respect to the marginal propensity toconsume.

JEL classification: E12, E21, D41, D50,
Keywords: Coordination Failure, Eﬀective Demand, Overlapping Generations, Perfect Competition∗Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya,464-8601, Japan. Tel. & Fax.: +81-52-789-5456. E-mail: skawai@soec.nagoya-u.ac.jp1

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Introduction

This paper considers the coordination failure under perfect competition inan overlapping generations model. The coordination failure, commonly, hasbeen studied in the model of imperfect competition (e.g., Hart 1982, Heller
1987, Blanchard and Kiyotaki 1987, Cooper and John 1988)1 . The imperfectcompetition is regarded as a necessary factor to generate the Pareto rankablemultiple equilibria since firms’ strategic interactions might be important.
On the other hand, the coordination failure stems from consumer side
has not been well studied theoretically so far. This paper focuses on thispoint. That is, if the marginal propensity to consume decreases, then theoutputs decrease. It is shown that these equilibria can be Pareto ranked.
In section 2, we present the model. In section 3, the general equilibriumis derived. The welfare analysis is presented in section 4. We conclude thepaper in section 5.

2The Model

2.1

Consumer’s problem

Consider an overlapping-generations economy. There are a identical consumers in each generation. They live only two periods. Their preferences
are defined by(c1 )γ (c2 )1−γ − kl1e ,0 < γ < 1, k > 0, e ≥ 1,(1)where c1 and c2 is the consumption when they are young and old respectively,
l1 is the labor supply when they are young. This type of utility function
is commonly used in macro models with imperfect competition (see Dixonand Rankin 1994; Cooper 1999). For simplicity, suppose that e = 12 .
12See Dixon and Rankin (1994) and Cooper (1999) for comprehensive discussion.
The main property does not change even if e > 1.

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Then, the representative consumer’s problem is(c1 )γ (c2 )1−γ − kl1max

c1 ,c2 ,l1

s.t. p1 c1 + m1 = w1 l1 + r1 ,pe2 c2 = m1 ,(2)nwhere r1 ≡1Xπi .
ai=1From (2), the individual demand function in which the income is regarded
as given can be derived toγI1,p1(1 − γ)I1c2 (pe2 , I1 ) =,pe2
c1 (p1 , I1 ) =where I1 ≡ w1 l1 + r1 .

Substituting the demand function into the utility function (1), we obtainthe following form:v(l1 ) = Γ

µp1
pe2

¶1−γ(w1 l1 + r1 )− kl1 ,
p1(3)where Γ ≡ γ γ (1 − γ)1−γ . Hence, the optimal condition for labor supply iskw1=.
p1Γ (p1 /pe2 )1−γ2.2(4)Eﬀective demand

Since there are a consumers in each generation, the consumption demandsof generation t are the following:
¶γI1,
p1µ
¶
(1 − γ)I1t
C2 (p2 , I1 ) = a.
pe2

C1t (p1 , I1 ) = aµThus, the market demand function of period t becomes
Ct = C1t (pt , It ) + C2t−1 (pt , It−1 )a= [γIt + (1 − γ)It−1 ]pt3(5)

Imposing the equilibrium condition of labor market, i.e., alt = nLt , into
It , and from the definition of rt , the following Income-Expenditure identity
is held in equilibrium:aIt∗ = npt qt∗ .

(6)

Substituting (6) into (5), we obtain the eﬀective demand, or Keynes-type
consomption function:
Ct∗ = n∙pt−1 ∗
γqt∗ + (1 − γ)
· qt−1pt¸(7)From (7), it is found that γ represents the marginal propensity to consume,
since the real income at period t is nqt∗ from (6). The second term can be
regarded as the basic consumption at period t.

2.3

Firm’s problem

There are n identical firms. The production function in period t is given by
qt = αLβt ,α > 0, 0 < β < 1.

(8)

From the price-taking assumption and no capital, the firm’s problem is reduced to the each period’s profit maximization. Thus, the optimal labordemand at period t can be derived toLdt =3µwt
αβpt¶− 11−β.

(9)Market Equilibrium

Assume that consumers have perfect foresight; i.e., pet+1 = pt+1 . Substituting (4) into (9), and from (8), we obtain the equilibrium supply of representative firm with labor market clearing:qt∗ = αµαβΓθt 1−γk4β¶ 1−β,(10)where θt = pt /pt+1 .
Imposing the equilibrium condition of commodity market, i.e., Ct∗ = nqt∗ ,into (7) yields,∗qt∗ = θt−1 qt−1.

(11)

Substituting (10) into (11), we obtain,1−βγ

θt = θt−1 β(1−γ) .

(12)Under the perfect foresight assumption, the unique stationary equilibrium with money3 is θ ∗ = 1. From (10) and (12), it is found that theequilibrium relative price does not change while equilibrium outputs changewith respect to γ. Therefore, the following proposition is established.
PROPOSITION 1. There is a continuum of stationary state with respectto γ.
∗q (Γ) = αµαβΓk¶ β1−β.

(13)and if γ ∈ (0.5, 1),dq∗ (Γ)
> 0.
dγ

(14)

An example of Proposition 1 appears in Figure 1. That is the case whenthe marginal propensity to consume decreases. Notice that the consumptionof old generation plays the role of basic consumption as in Keynes-type
consumption function (7), that is, the intercept of vertical axis. Hence,the decrease of the marginal propensity to consume implies not only thedecrease of consumption of young, but also the increase of consumption ofold generation4 . Proposition 1 shows that the eﬀects of former dominatesthat of latter if γ ∈ (0.5, 1).
3

As in the standard OLG model, outside money is endowed to the generation zero. We

have omitted the description of it for the sake of brevity.
4If we focus on the period that γ is just changed, the intercept does not change at that
period because it is historically determined.

5Ct∗
nEquilibrium Condition Curveγ∗(1 − γ)qt−1

Eﬀective Demand Curve

γ45 °Oqt∗

qt

Figure 1: If the marginal propensity to consume decreases, then the equilibrium outputs decrease.

4Welfare analysis

From (13), (6), and (3), the welfare function with respect to Γ can be derivedto³ β´ 1
β11
W (Γ) = n β 1−β − β 1−β α 1−β k− 1−β Γ 1−β(15)PROPOSITION 2. From assumptions α > 0 and 0 < β < 1, if γ ∈(0.5, 1),dW (Γ)> 0.
dγ(16)

From the above discussions, we can establish the following proposition.
PROPOSITION 3. From Proposition 1 and Proposition 2, there is a continuum of Pareto-ranked equilibria with respect to the marginal propensity
to consume.

5Conclusion

We have presented a micro foundation of Keynesian features, especially theKeynes-type consumption function that is regarded as there is no micro6

foundation so far, in a simple macro economic model. It is found that acoordination failure can emerge even under perfect competition. In otherwords, imperfect competition does not necessary for it.
Next, the consumption function (7) is similar to that of Friedman’s Permanent Income Hypothesis and Modigliani’s Life Cycle Hypothesis. However, the implication is much diﬀerent. That is, the above two hypotheses
insist that consumers tend to equalize their each period’s consumption. To
the contrary, this model predicts that the large consumption at young isbetter for welfare. This is the so-called “paradox of thrift”.
Finally, notice that the equilibrium is uniquely determined if γ is fixed.
Changes in γ, however, represents the changes in mind or market psychology. It is natural to consider it as a driving force of business cycles. Noticealso that although the labor market is underemployment, it might be more
important for Keynesian features that these equilibria are Pareto-ranked.
Furthermore, the involuntary unemployment could be investigated by introducing a rationing rule into the labor market. In addition to that, introducing government or capital is a natural extension. These are for future
research.

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References
Blanchard, O., Kiyotaki, N., 1987. Monopolistic Competition and the Effects of Aggregate Demand. American Economic Review, 77, 647-666.
Cooper, R., John A., 1988. Coordinating Coordination Failures in Keynesian Models. Quarterly Journal of Economics, 103, 441-463.
Cooper, R., 1999. Coordination Games -Complementarities and Macroeconomics. Cambridge University Press.
Dixon, H., Rankin, N., 1994. Imperfect Competition and Macroeconomics:A Survey. Oxford Economic Papers, 46, 171-199.
Hart, O., 1982. A Model of Imperfect Competition with Keynesian Features. Quarterly Journal of Economics, 97, 109-138.
Heller, W., 1986. Coordination Failure Under Complete Markets with Applications to Eﬀective Demand. in Walter Heller, Ross Starr and DavidStarrett(Eds.), Equilibrium Analysis, Essays in Honor of Kenneth J,Arrow, Vol. II, Cambridge: Cambridge University Press, 1986, 155175.

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