This paper presents a simple general equilibrium model in which the only non-Walrasian feature is imperfect competition in the goods market. The model is shown to exhibit various Keynesian characteristics. In particular, as competition in the goods market becomes less perfect, the fiscal policy multipliers approach the values implied by the textbook Keynesian cross.

N. Gregory Mankiw

English

Research Papers in Economics

NBER WORKING PAPER SERIES
IMPERFECT COMPETITION
AND THE KEYNESIAN CROSS
N. Gregory Mankiw
Working Paper No. 2386
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
September 1987
I am grateful to David Romer and the participants in the NBER Summer Institute
on Industrial Organization and Macroeconomics for helpful comments, and to the
National Science Foundation and the Olin Foundation for financial support.
The research reported here is part of the NBER's research program in Economic
Fluctuations. Any opinions expressed are those of the author and not those of
the National Bureau of Economic Research.NBER Working Paper #2386
September 1987
Imperfect Competition and the Keynesian Cross
ABSTRACT
This paper presents a simple generalequilibrium model in which the only
non-Wairasian feature is imperfect competition in thegoods market. The model
is shown to exhibit various Keynesian characteristics.In particular, as
competition in the goods market becomes less perfect, thefiscal policy
multipliers approach the values implied by the textbookKeynesian cross.
N. Gregory Mankiw
NBE R
1050 Massachusetts Avenue
Cambridge, MA 02138
(617)868-3900I. INTRODUCTION
Fiscal policy multipliers are central to Keynesian macroeconomics.
In this paper I explore a possible microeconomic foundation for one
fundamental theory of income determination, the "Keynesian cross." My
model deviates from a Wairasian equilibrium model only by the assumption of
imperfect competition in the goods market.I show that textbook fiscal
policy multipliers arise as a limiting case.1
Under imperfect competition, firms are always eager to sell an
additional unit of output, since price exceeds marginal cost. This profit
margin creates the potential for the multiplier. An expansionary change in
fiscal policy increases aggregate expenditure, which increases profits,
which in turn increases expenditure, and so on.
The theme that imperfect competition may be crucial to macroeconomic
issues is increasingly prevalent. See, for example, the work of Weitzman
[1982], Hart [1982], Solow [1984], Blanchard and Kiyotaki [1985], and Startz
[1986]. The purpose of the model presented here is partly pedagogical. I
therefore do not hesitate making strong (yet not implausible) assumptions
about the economic structure: Cobb-Douglas utility, constant marginal cost,
and constant mark-up pricing. There is no reason tosuppose, however, that
the sorts of effects highlighted here are specific to these assumptions.
While the model is in some ways surprisingly similar to the
standard Keynesian model, in other ways it differs greatly. In
particular, it incorporates both an equilibrium labor market and a static
environment. These features are chosen for simplicity rather than realism.
The goal is not to provide a complete reformulation of Keynesian economics,—2--
but only to illustrate what sort of Keynesian results one can obtain with a
small movement away from Walrasian equilibrium in the direction of imperfect
competition.
II. THE ECONOMY
This section describes the economy. The following section discusses the
economy's response to changes in fiscal policy.
People
All people are the same. The representative person maximizes a
Cobb-Douglas utility function over consumption of the single produced good
(C) and leisure (L):
U =alog C +(1—a)log L. (1)
Leisure is the nunieraire.If w is the endowment of time, then w-L is labor
income. Total after—tax income is(o—L) +fl—T,where 11 is profits and T
is the lump-sum tax levied by the government. The individual's budget
constraint is therefore
PC =(w-L)+fl-T (2)
PC +L=w+11-T,
where P is the price of the consumption good.
The Cobb—Douglas utility function implies a constant share a of "full
income" is devoted to consumption. That is,
PC =a(w+fl-1). (3)
Equation (3) is the consumption function, and a is the marginal propensity to
consume.—3-
Government
The revenue raised by the government is used for twopurposes. An
amount G is used to purchase the produced good, and Wgovernment workers are
hired. The government budget constraint requires thatgovernment spending
equals revenue. That is,
T =G+W.
(4)
Total expenditure on the produced good is
Y =PC+G.
(5)
Using equation (3) to substitute into equation (5), we find
V =a(w+II-1)+6.
(6)
Expenditure therefore depends positively on profits andgovernment purchases
and negatively on taxes.
Firms
There are N firms producing the single good. Theindustry takes
expenditure in the economy as given.2 That is, theindustry demand function
is unit elastic:
Q =YIP
(7)
where Q is total output.
The N firms have the same increasing returns to scaletechnology. The
technology requires F units of overhead labor. After the plant is setup,
one unit of output requires c Units of labor. The cost functionof each firm
is therefore
TC(q) =F+cq, (8)
where costs are measured in terms of the numeraire, leisure.
The N firms play some oligopoly game, the details of whichI do not need
to specify. This game determines the profitmargin—4-
=(P-c)/P. (9)
As an example, if the firms act as Cournot oligopolists, then ji= 1/N.More
generally, a conjectural variation equilibrium allows all possibilites
between Bertrand competition (ji= 0)and perfect collusion (gi- 1);in each
case, jidependsonly on N and the conjectural variation.I therefore take
the profit margin iasgiven for any fixed number of firms N.3
Note the relation between output and expenditure:
Q =E(1-s)/c]V. (10)
For given values of the profit margin iandmarginal cost c, expenditure on
the produced good and output are proportional. Government workers W are not
included in expenditure V or output Q; hence, these measures are analogous to
industrial production rather than GNP.
Total profits are revenue less costs:
fl =PQ—NF—cQ. (11)
Using equations (7) and (9), aggregate profits can be expressed in terms in
terms of expenditure V and the profit margin JA:
11=.tV-NF. (12)
Hence, higher aggregate expenditure implies higher aggregate profits.
The Labor Market
The above discussion centers on the goods market. Walras's Law ensures
that the labor market clears if the above relations are satisfied. To see
that this is true, note that labor supply is the time endowment less the
demand for leisure:
Labor Supply =w-(1
-a)(w+fl—T) (13)
= — (1-a)(1I—T).—5-
Labor demand •is the sum of firms' demand, NF +cQ,and government demand, W.
Thus,
Labor Demand =(NF+cQ)+W, (14)
=(V-H)+(T-
= (a(w+11-T)+G-ii)+(1-
= aw—(1
-a)(fl—1).
Hence, goods market equilibrium (including the governmentbudget constraint)
implies that supply equals demand in the labor market as well.
Summary
The two key equations are (6) and (12):
Y =a(w+IT-T)+0, (6)
fl=iY-NF.
(12)
Expenditure depends on profits and the fiscal policy variables, whileprofits
depend on expenditure.
III. FISCAL POLICY
This section addresses the impact of fiscal policy. Theanalysis is
short—run in that the number of firms N and thus theprofit margin giareheld
fixed.
The Balanced Budget Multiplier
Consider first an equal increase in government purchases 0 andtaxes T.
Equations (6) and (12) imply that
-1-a
(15) dotdT=dG
-
1-
Themultiplier thus depends on both the marginal propensity to consume a and
the profit margin t.Underperfect competition (ji= 0),the balanced budget-6-
multiplier is 1-a. In the limiting case in which the revenue from the
marginal unit goes entirely to profit (i= 1),the balanced budget multiplier
is unity.
The story that accompanies this multiplier is in many ways standard.
Initially, the increase in government purchases of tG raises expenditure by
tG, while the equal tax increase lowers private expenditure by aG. The net
increase in expenditure is thus (1-a)tG, which raises profits by ji(1-a)tG.
The increase in profits in turn raises expenditure by a(1-a)AG, which again
raises profits, and so on. This multiplier process yields the infinite
series,
(1-a) +ai(1-a)+a2112(1_a)+a3j13(1—a)+
whichequals the balanced budget multiplier in equation (15). Imperfect
competition plays a key role here, for if the profit margin were zero, the
process would end after the initial increase in expenditure.
The Tax Multiplier
Consider now an increase in taxes T, holding constant the level of
government purchases G. The government budget constraint (4) implies that
the amount of labor purchased by the government W must increase by T. The
extra labor income received by government employees exactly equals the extra
taxes paid; on net, individuals give up their time but receive no additional
income. This policy intervention is thus equivalent to a reduction in the
endowment w of T.
In standard analysis, tax increases are coupled with reductions -in
government debt. Government debt serves the function of transferring—7-
resources from future generations to the current generation.Hence, a tax
increase is an endowment reduction to the currentgeneration. In this
sense, a tax increase in the static model of this paper is analogous toa
debt-financed tax increase in intertemporal (finite horizon) models.
Equations (6) and (12) imply that the tax multiplier is
dv--a
(16) dl -
1-
Underperfect competition (ii= 0),the tax multiplier is —a. As competition
becomes less perfect (,i-+ 1),the tax multiplier approaches —a/(1-a). Again,
the multiplier process works through profits. The tax increaselowers
expenditure, which lowers profits, which lowers expenditure, andso on.
The Government Purchases Multiplier
Consider now an increase in government purchases G,holding constant the
level of taxes 1.In standard analysis, future generationspay for a
debt-financed increase in purchases. Here, the increase inpurchases is
financed by a reduction in W. In both cases, there isno immediate impact on
the current individuals' budget constraint (2).
Equations (6) and (12) imply that the government purchases multiplier is
dY- 1
(17) dG -
Underperfect competition, dY/do is unity. As the profit margin approaches
one, dY/dC approaches the standard Keynesian value of 1/(1-a).
Figure 1 shows how to demonstrate the multiplier graphically.
Expenditure V is a linear function of profits II, with a slope of themarginal
propensity to consume a. Profits are also a linear function of expenditure;-8-
the slope of this line is In the limiting case in which gi =1,this
locus becomes the 450 line of the Keynesian cross. An increase in government
purchases shifts the expenditure function upward by tG, which causes a
multiplied increase in total expenditure.
Welfare Analysis
Here I examine the effect of fiscal policy on the welfare of the
representative person, as judged by his utility function (1). Government
purchases are assumed not to affect utility directly. A complete evaluation
of fiscal policy would also take account of the benefit received from public
expenditure. The analysis here is thus limited in scope.
An individual's utility increases only if his budget set, as defined by
equation (2), is expanded. Since relative prices are constant, profits less
taxes, 11 —1,are sufficient for utility. The impacts of the fiscal policy
changes on II -Tare
PkflL)J --(1-i.) 18)
dG !dT=dG —lagi
d(fl-T) — —1
dT
—1-au
d(fl—T) = (20)
dG
A balanced budget fiscal stimulus in general reduces welfare. In the
limiting case in which gi= 1,however, a balanced budget increase has no
impact on welfare. As the textbook Keynesian cross suggests, the increase in
government purchases has no social cost. The increase in income (here,
profits) exactly offsets the higher tax bill.
Both increases in government purchases and reductions in taxes increase
welfare. In standard analysis, increases in G or reductions in T are-9-
financed by future generations. Here, these changes are financed by
reductions in government workers W.In neither case is it surprising that
the welfare of current individuals increases.
IV. CONCLUSION
The model examined here is surprisingly similar to both Walrasian
models of general equilibrium and Keynesian models of income determination.
It deviates from a standard general equilibrium model only by the assumption
of imperfect competition in the goods market. As competition in the goods
market becomes less perfect, the fiscal policy multipliers approach the
values implied by the Keynesian cross.
The model could be usefully extended in several directions. First, the
labor market might be made less classical. One could posit imperfect
competition among workers, for example. Some of the rents generated by
expansionary fiscal policy would therefore accrue as labor income. The
multiplier would thus work through both labor income and firm profits.5
Second, the model could be made intertemporal. The impact of
debt-financed fiscal policy obviously cannot be studied in a static model.
That saving and inC'estment play an important role in standard Keynesian
analysis also suggests extending this model to a dynamic setting.-10-
NOTES
1.For an exposition of the Keynes-ian cross, see Samuelson [1948) or almost
any introductory text.
2.One might object that this assumption is not reasonable because
expenditure depends on industry profits. The model could easily be amended,
however, to include a continuum of industries; the demand curve of each
industry would depend on aggregate profits.
3.One could also imagine that each firm produces a differentiated product.
In this case, the profit margin ti depends on each firm's elasticity of
demand, which could plausibly be assumed constant.
4.Note that the second line is always steeper than the first,
since a <1<1/p.
5.Alternatively, the labor market could be characterized by efficiency
wages.—11—
REFERENCES
Blanchard, Olivier J., and Nobuhiro K-iyotaki, 1985, "MonopolisticCompetition,
Aggregate Demand Externalities, and the Real Effects of NominalMoney,"
NBER Working Paper No. 1770.
Hart, Oliver, 1982, "A Model of Imperfect Competition withKeynesian
Features," Quarterly Journal of Economics 87, 109-38.
Samuelson, Paul A., 1948, "The Simple Mathematics of IncomeDetermination,"
in Income, Employment and Policy, New York: W.W.Norton, reprinted in
The Collected Scientific Papers of Paul A. Samuelson.
Solow, Robert M., 1984, "Monopolistic Competition and theMultiplier," M.I.T.
Startz, Richard, 1986, "Monopolistic Competition as a Foundation forKeynesian
Macroeconomic Models," University of Washington.
Weitzman, Martin L., 1982, "Increasing Returns and the Foundations of
Unemployment Theory," Economic Journal 92, 787-804.l-a
V
I
Figure 1
A New Keynesian Cross
ri
(fl +NE)
Yct(wfl-T)+ G

A Model of Imperfect Competition withKeynesian Features,&quot;

Imperfect Competition and the Keynesian Cross

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