Demand-side and supply-side equilibria, old and new

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BEBR

FACULTY WORKING
PAPER NO. 90-1658

T^i^V^

Demand-Side and Supply-Side Equilibria,

Old and New

TrJti Library of the

Unlvsrslty of Illinois
0? Urtsna-Ctimpugn

Hans Brems

College of Commerce arid Business Administration
Bureau of Economic and Business Research
University of Illinois Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 90-1658

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

June 1990

Demand-Side and Supply-Side Equilibria, Old and New

Hans Brems

Department of Economics
University of Illinois at Urbana-Champaign

Abstract

Between the last half of the seventeenth century and the mid-
eighteenth century macroeconoraic theory reversed itself from a demand-
side to a supply-side equilibrium. Policy conclusions reversed them-
selves correspondingly. For good measure the reversal repeated itself
in our own century. The purpose of the paper is to derive such a
reversal of one equilibrium into Che other as rigorously and as suc-
cinctly as possible.

INTRODUCTION

1 . Macroeconomics — The Oldest Part of Our Building

The term "macroeconomics" was coined in the early twentieth cen-
tury. But the unnamed concept had been in use since the last half of
the seventeenth century. Our founding fathers, i.e., mercantilists
like Petty (1662), Mun (1664), and Yarranton (1677), practiced macro-
economics. In the mid-eighteenth century Cantillon (1755) and Quesnay
(1759) gave us the first half of microeconomics, i.e., allocation
theory. In the last fourth of the eighteenth century Smith (1776)
gave us the second half, i.e., price theory.

-2-

Two distinct, indeed opposed, forms of macroeconomic equilibria
are found in the oldest part of our building.

2 . Early Demand-Side Equilibria

The concern of the mercantilists was unemployment. Petty [1662 •
(1899: 30)] estimated it at ten percent and analyzed it within the
framework of a demand-side equilibrium. Here physical output was seen
as bounded by demand. Supply was no problem: demand would always
create its own supply. There was always excess capacity. The rate of
interest was determined by the demand for and the supply of money
hence could be affected by the money supply. Petty thought that ample
money had reduced the rate of interest to six percent. Yarranton
[1677 (1854: 38)] believed that the use of paper money would reduce
it to four percent. Petty [1662 (1899: 29-31)] also recommended
public works "of much labour, and little art." In short: monetary
or fiscal policy could raise physical output and employment. Capital-
ism left to itself might be incapable of utilizing its own resources.
Government action was the remedy.

Within less than a century such a demand-side equilibrium was to
reverse itself.

3 . Early Supply-Side Equilibria

Hume's concern was Inflation, and he analyzed it within the frame-
work of a supply-side equilibrium. Here physical output was seen as
bounded by supply. Demand was no problem: supply would always create
its own demand. There was never excess capacity. The rate of inter-
est was determined by saving and investment. As a result, Hume [1752
(1875: 321-322)] and Turgot [1769-1770 (1922: 74-76)] agreed,
doubling the money supply would not reduce the rate of interest. Hume
realized that doubling the money supply of a not fully monetized econ-
omy could widen the scope for division of labor hence expand the goods
supply. But doubling the money supply of a fully monetized economy,
Hume [1752 (1875: 333)] insisted, would merely double prices.
Monetary stimuli would simply generate inflation and fiscal stimuli
simply crowding-out. Capitalism was entirely capable of utilizing
its own resources. Government action, however well meant, was the
problem.

A . Purpose

What interests us as historians of theory is not so much the
equilibria per se. Taking a shorter view, textbooks past and present
will identify the particular equilibrium cherished at the time.

-4-

What interests us is the reversal — all the more so since it
occurred twice in three centuries: it repeated itself in our own
century. This time the denand-side equilibria of Keynes (1936) and
Hansen (1941) reversed themselves into the supply-side equilibria of
Friedman (1968), Lucas (1972), and Sargent (1973).

Can we derive the reversal of a demand-side equilibrium, whether
Mercantilist or Keynesian, into a supply-side equilibrium, whether
vintage Hume or modern? The purpose of the present paper is to do so
as rigorously and as succinctly as possible. We shall use the follow-
ing notation.

5. Variables

C = physical consumption

D = desired holding of money

E = excess demand in goods market

I = physical investment

L = labor employed

R = tax revenue

r = rate of interest

w = money wage rate

-5-

X = physical output

Y = money national income

y = money disposable income

6. Parameters

A = autonomous consumption

a = joint factor productivity

a = elasticity of output with respect to labor

B = autonomous investment

b H inducement to invest

S = elasticity of output with respect to capital

c ^ marginal propensity to consume

F ^ available labor force

f i inducement to hold speculative money

G H physical government purchase of goods

J = autonomous demand for money

j = propensity to hold transaction money

A = "natural" fraction of labor force employed

M = supply of money

S = physical capital stock

T = tax rate

-6-

The price P of goods will be a parameter in a demand-side equi-
librium but a variable in a supply-side equilibrium.

II. DEMAND-SIDE EQUILIBRIUM

1 . Demand-Side Equilibrium: Solution

A Keynes-Hansen demand-side equilibrium encompassed two markets,
a goods market and a money market, and two equilibrating variables,
physical output and the rate of interest. It was a partial equi-
librium having neither enough markets nor enough equilibrating var-
iables. We write it as follows.

Let us ignore capital consumption allowances. We nay then define
national income as the market value of physical output:

Y 5 PX (1)

Let all such national income be distributed to persons and be
taxed once and at the rate T. Then tax revenue is

-7-

R = TY

(2)

where 0 < T < 1.

Define disposable income as national income minus tax revenue:

y = Y - R

(3)

Let consumption be a function of disposable real income:

C = A + cy/'P

(4)

where A > 0 and 0 < c < 1.

Let investment be a function of the rate of interest:

I = B - br

(5)

where B > 0 and b > 0.

Let desired real money holdings be a function of the rate of
interest as well as physical output:

D/P = J - fr + jX

(6)

where J > 0, f > 0, and j > 0.

-8-

Equllibriutn in the goods and money markets requires supply to
equal demand:

X = C + I + G (7)

M = D (8)

Equations (1) through (8) is a linear system of eight equations
in the eight variables C, D, 1, R, r, X, Y, and y. Basic algebra
easily delivers solutions for physical output and the rate of inter-
est :

(A + B + G)f + b(M/P - J)

X = (9)

bj + [1 - c(l - T)]f

(A + B + G)j - [1 - c(l - T)](M/P - J)

r= (10)

bj + [1 - c(l - T)]f

2 . Demand-Side Equilibrium: Policy Conclusions

How sensitive are our demand-side equilibria (9) and (10) to
fiscal and monetary policy?

-9-

Fiscal-policy instruments are government purchases G and the tax
rate T. So take the partial derivatives of (9) and (10) with respect
to G:

3X

8G bj + [1 - c(l - T)]f

> 0

(11)

3r

> 0

30 bj + [1 - c(l - T)]f

(12)

So if physical governnent purchase G is up, so is physical output
(9) and the rate of interest (10). The higher rate of interest will
discourage investment. Consequently there is some crowding-out. Next
take the partial derivatives of (9) and (10) with respect to T. On
the latter use (6) with (8) inserted:

3X cfX

3T bj + [1 - c(l - T)]f

< 0

(13)

3r cjX

3T bj + [1 - c(l - T)]f

< 0

(14)

-10-

So if the tax rate T is down, both physical output (9) and the
rate of interest (10) are up. Again there is some crowding-out.

The monetary-policy instrument is the money supply M. So take the
partial derivatives of (9) and (10) with respect to M:

9X

b/P

> 0

(15)

3M bj + [1 - c(l - T)]f

3r [1 - c(l - T)]/P
3M bj + [1 - c(l - T)]f

< 0

(16)

So if the money supply is up, physical output (9) is up but the
rate of interest (10) is down. We turn to supply-side equilibrium.

III. SUPPLY-SIDE EQUILIBRIUM

1 . The Natural Supply of Goods

Modern supply-side equilibria added the missing market and the
missing equilibrating variable. The missing market was the labor

-11-

market. Here entrepreneurs are demanding labor and are facing
diminishing returns to it. Let their production function be of Cobb-
Douglas form:

X = aL^S^

(17)

where 0<a<l;0<B<l;a+6=l; and a > 0.

At a frozen capital stock S let purely competitive entrepreneurs
maximize their profits with respect to labor employed. Then the real

wage rate equals the physical marginal productivity of labor:

— = — = aaL S
P 3L

(18)

Raise both sides to the power -1/6 and find demand for labor

-1/S
w

L = (aa)^/^(-) S

(19)

Facing such a demand function, how does labor respond? Friedman's
answer (1968) was his "natural" rate of unemployment to which current

-12-

labor-market literature adds institutional color: Lindbeck and Snower
(1986) and Blanchard and Summers (1988) distinguish between "insiders,"
who are employed hence decision-making, and "outsiders," who are unem-
ployed hence disenfranchised. Let insiders accept the "natural"
employment rate X where 0 < X <_ 1 . In other words, if L > XF insiders
will insist on a higher real wage rate. If

L = XF

(20)

they will be happy with the existing one. If L < XF they will settle
for a lower one.

The real wage rate insiders will be happy with, given their
natural rate X of employment, might be called the "natural" one. Find
it by inserting (20) into (19) and rearranging:

w

- = aci(XF) ^S

(21)

At the frozen capital stock S, then, labor can have a 6 percent
higher natural real wage rate by accepting a one percent lower natural
rate X of employment.

-13-

May the actual real wage rate differ from the natural one (21)?
It may. For example let a random hence unanticipated expansion of the
money supply encourage demand. Let goods prices respond more readily
than does the money wage rate and let employers perceive the response
sooner than does labor. At first, then, a real wage rate lower than
(21) will be perceived by employers but not yet by labor. As a result,
actual employment will exceed the natural one (20). Vice versa, let
a random hence unanticipated contraction of the money supply discour-
age demand. At first, then, a real wage rate higher than (21) will be
perceived by employers but not yet by labor. As a result, actual
employment will fall short of the natural one (20). But, as Friedman
(1968) insisted, eventually labor will perceive and respond: new
rounds of collective bargaining will restore the equality between the
actual and the natural real wage rate, hence the equality between the
actual and the natural employment.

According to New Classicals like Lucas (1972), Sargent (1973),
and Sargent-Wallace (1975) only such random hence unanticipated var-
iations of the money supply can generate deviations of actual from
natural. Systematic variations would be anticipated by better-
informed agents acting as if they knew the structure of the model as
well as any systematic public policy applied to it. Ill-informed
agents would not survive.

-14-

At the frozen capital stock S the supply of goods correspondinc

to the natural rate X of employment might be called the "natural"

3
one. Find it by inserting (20) into (17):

X = a(XF)''s^ (22)

2 . Supply-Side Equilibrium: Solution

At a frozen price P there is no reason why such a natural supply
(22) should coincide with our demand-side equilibrium (9). There nay
well be positive or negative excess demand defined as the difference
between th'e right-hand sides of (22) and (9):

(A + B + G)f + b(M/P - J) -

EE a(\F)'^S^ (23)

bj + [1 - c(l - T)]f

But modern supply-side equilibria also added the missing equili-
brating variable enabling (22) to coincide with (9): unfreeze price
P, then positive excess demand will raise price and keep raising it
as long as E > 0, and negative excess demand will lower price and keep

-15-

lowering it as long as E < 0. Now excess demand (23) is itself a
function of price:

3E b M

— = J (24)

3P bj + [1 - c(l - T)]f P

Since (24) is negative, a higher price will reduce excess demand
and a lower price raise it. Find the price at which excess demand
will vanish by setting (23) equal to zero and solve for P:

P = bM/H, where (25)

H = a(XF)'^S^{bj + [1 - c(l - T)]f} - (A + B + G)f + bJ

The new equilibrating variable P also occurred in our demand-side
equilibrium solution (10) for the rate of interest. To find the
supply-side equilibrium solution for the rate of interest insert (25)
into (10):

A + B + G - a(XF)'^S^[l - c(l - T)]

(26)

-16-

Policy conclusions drawn from such supply-side equilibria will
reverse the policy conclusions drawn from our demand-side equilibria
(9) and (10). Let us draw them.

3. Supply-Side Equilibrium: Policy Conclusions

How sensitive are the new supply-side equilibria (22), (25), and
(26) to fiscal and monetary policy?

Fiscal-policy instruments are government purchases G and the tax
rate T. So take the partial derivatives of (22), (25), and (26) with
respect to G:

ax

— =0 (27)

3G

3P P

— = f - > 0 (28)

3G H

3r 1

— = - > 0 (29)

30 b

-17-

So if physical government purchase G is up, so is price (25) and
the rate of interest (26), but physical output (22) is unaffected.
The higher rate of interest will discourage investment — but more than
it did in the demand-side equilibrium: since physical output (22) is
unaffected in the supply-side equilibrium, investment must be down by
as much as government purchase is up. The crowding-out is complete.
Next take the partial derivatives of (22), (25), and (26) with respect
to T:

ax

— = 0 (30)

31

3P ^ P

— = - acf(XF)'^S ' - < 0 (31)

31 H

3r ac(XF)^S^

— = < 0 (32)

31 b

So if the tax rate T is down, both price (25) and the rate of
interest (26) are up, but physical output (22) is unaffected.

The monetary-policy instrument is the money supply M. So take
the partial derivatives of (22), (25), and (26) with respect to M:

-If

3X
— = 0 (33)

9M

3P b

— = - > 0 (3A)

3M H

3r
— = 0 (35)

3M

So if the money supply is up, price (25) is up in proportion, but
physical output (22) and the rate of interest (26) are unaffected — -as
Hume (1752) had said they would be.

IV. SUMMARY i^T) CONCLUSION

A deraand-side equilibriura encompassed two markets, a goods market
and a money market, and had two equilibrating variables, physical out-
put and the rate of interest. There was no price mechanism: price
was a parameter. At that price industry would always produce a
physical output matching demand. We have seen such a demand-side

-19-

equilibrlum (9) as a partial one having neither enough mar'Kets nor
enough equilibrating variables.

A supply-side equilibrium adds the missing market, i.e., a labor
market. Here entrepreneurs demand labor and are facing diminishing
returns to it at a frozen physical capital stock. Consequently their
demand for labor (19) is a function of the real wage rate. Facing
such a demand function, unions choose the natural rate of employment
(20). The natural rate (20), in turn, determines a unique natural
supply of goods (22) .

Such a supply-side equilibrium also adds the missing equilibrating
variable, i.e., price. An unfrozen price will enable the natural
supply (22) to coincide with the demand-side equilibrium (9) and will
reverse both fiscal-policy and monetary-policy conclusions.

In a demand-side equilibrium larger government purchases or a tax
cut had raised physical output and the rate of interest: crowding-out
was incomplete. In a supply-side equilibrium larger government pur-
chases or a tax cut raise price and the rate of interest but leave
physical output unaffected: crowding-out is complete.

In a demand-side equilibrium a larger money supply raised physical
output and lowered the rate of interest: there was crowding-in. As
Hume had observed, in a supply-side equilibrium a larger money supply
raises price proportionately but leaves physical output and the rate
of interest unaffected: there is neither crowding-out nor crowding-in.

-20-

Our demand-side and supply-side equilibria were both static equi-
libria in the sense that capital stock S was assumed to be frozen--as
in early new classical macroeconomics.

The present paper will not go beyond such statics. But in closing
we do observe that in the form of derivatives with respect to time
inherent dynamics kept creeping into macroeconomics, at first only as'
afterthoughts. The Phillips curve took the derivative of the money
wage rate with respect to time. The augmented Phillips curve added
the derivative of price with respect to time. The government budget
constraint took the derivatives of the money and bond supplies with
respect to time. Growth theory went beyond the afterthoughts and took
the derivatives of all its variables with respect to time. Capital
stock was unfrozen, and its derivative with respect to time was the
definition of investment.

-21-

FOOTNOTES

Correspondence may be addressed to the author, Box 99 Commerce West,
1206 S. Sixth Street, Champaign, IL 61820.

1. Further documentation in Brems (1986: 19-2A).

2. Further documentation in Brems (1986: 33-37).

3. Hume, to be sure, knew neither unions nor insiders. But if
reflecting the equality sign of our 0 < X _£ 1 , i.e., full employment,
eighteenth-century institutions would still generate a unique natural
supply of goods (22).

4. This is true even in our absence of a real-balance effect--and
even more so in the presence of one.

For candid, hence helpful, criticism I am indebted to two anonymous
referees .

-22-

REFERENCES

Blanchard, Oliver J., and Summers, Lawrence H. , "Hysteresis and the
European Unemployment Problem," in Cross, Rod (ed.), Unemployment
Hysteresis and the Natural Rate Hypothesis, Oxford, 1988.

Brems, Hans, Pioneering Economic Theory, 1630-1980 — A Mathematical
Restatement , Baltimore, 1986.

Cantillon, Richard, Essai sur la nature du commerce en general,
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publisher: "A Londres, Chez F. Gyles, dans Holborn"; reprinted,
Boston, 1892; edited and translated into English by Henry Higgs ,
C.B., London, 1931; translated into German with an introduction
by F. A. Hayek, Jena, 1931; republished in French, Paris, 1952.

Friedman, Milton, "The Role of Monetary Policy," Amer. Econ. Rev.,
March 1968, l%_, 1-17.

Hansen, Alvin H. , Fiscal Policy and Business Cycles, New York, 1941.

■23-

Hume, David, "Of Interest" and "Of the Balance of Trade," Political

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-24-

Sargent, Thomas J., "Rational Expectations, the Real Rate of Interest,
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Sargent, Thomas J., and Wallace, Neil, "'Rational' Expectations, the
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Smith, Adam, An Inquiry into the Nature and Causes of the Wealth of
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distribution des richesses," Epheraerides du citoyen, Nov. 1769-
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P. D. Groenewegen, The Economics of A. R. J. Turgot, The Hague,
1977, 43-95.

-25-

Yarranton, Andrew, England's Improvement by Sea and Land. To Outdo
the Dutch without Fighting. To Pay Debts without Moneys. To Set
at Work all the Poor of England with the Growth of Our Own Lands
. . ., London, 1677; quoted from P. D. Dove, Account of Andrew
Yarranton, Edinburgh, 1854.

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