Wage theory and growth theory

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PAPER NO. 1521

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Wage Theory and Growth Theory

Hans Br ems

10V

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

BEBR

FACULTY WORKING PAPER NO. 1521

College of Commerce and Business Administration

University of Illinois at Urbana- Champaign

December 1988

Wage Theory and Growth Theory

Hans Brems , Professor
Department of Economics

Digitized by the Internet Archive

in 2011 with funding from

University of Illinois Urbana-Champaign

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1103 South Douglas Avenue Urbana, Illinois 61801

WAGE THEORY AND GROWTH THEORY

HANS BREMS

Abstract
According to the "natural" rate hypothesis in the short run, by
accepting a "natural" rate of less than full employment, labor can
have a real wage rate higher than under full employment. To see that
hypothesis in a long-run perspective the paper solves a neoclassical
growth model for capital stock., output, and factor prices and finds
that in the long run, by accepting a "natural" rate of less than full
employment, labor can have a real wage rate no higher than under full
employment: levels of capital stock and output are correspondingly
lower. Nobody benefits.

Under profit maximization, pure competition, and a given capital
stock, demand for labor is simply labor's marginal-productivity curve.
As a result, in the short run, by accepting a "natural" rate of less
than full employment, labor can have a real wage rate higher than
under full employment.

-2-

But only In Che short run may capital stock be considered given.
The purpose of the paper is to examine how much of such a short-run
wage-employment tradeoff will survive once capital stock has become a
variable. We shall solve a neoclassical growth model for its capital
stock, output, and factor prices and examine the sensitivities of such
solutions to a "natural" rate of less than full employment. The model
is this.

I. THE MODEL

1. Variables

C = physical consumption

g = proportionate rate of growth of variable v

I = physical investment

< = physical marginal productivity of capital stock

L = labor employed

P = price of goods and services

r = nominal rate of interest

p = real rate of interest

-3-

S = physical capital stock

w = money wage rate

X 5 physical output

Y = money national income

2. Parameters

a E multiplicative factor of production function
a, 8 = exponents of production function
c = propensity to consume
F = available labor force

X = "natural" fraction of available labor force employed
M E supply of money

V = velocity of money

3 . National Income

Money national income defined as the aggregate earnings arising
from current production is identically equal to national product
defined as the market value of physical output:

Y = PX (1)

-4-
4. Production Function

We must be careful with our aggregation and begin at the firm
level. Let the inputs of an individual firm be labor L and physical
capital stock S and its physical output be X. Then let a Cobb-Douglas
production function be common to all firms:

X = aLaS8 (2)

where 0 < a < 1, 0 < 8 < 1, a + 8 = 1> and a is what growth measurement
[Maddison (1987: 658)] calls "joint factor productivity."

5. Demand for Labor

Demand for labor is a short-run commitment to be determined by
maximization of profits. Here the firm may consider its physical
capital stock. S a constant and ignore the effect of investment I upon
it. Maximizing its gross profits PX - wL with respect to employment
L, the firm will then hire labor until the last man costs as much as
he contributes, and under pure competition the real wage rate will
then equal the physical marginal productivity of labor:

-5-

w 3X -

- = — = aaLa " 1S6 (3)

P 8L

Since a + S = 1, a - 1 = - B, so raise to power -1/B, rearrange,
and write firm demand for labor

-1/B

P

w
L = (aa)1/B(-) S (4)

On the right-hand side of (4) everything except S is common to all
firms. Factor out all such common factors and sum (4) over firms.
Then S becomes aggregate physical capital stock and L aggregate demand
for labor.

6. Supply of Labor

Current labor-market literature, e.g., Lindbeck and Snower (1986)
and Blanchard and Summers (1988) distinguish between "insiders," who
are employed hence decision-making, and "outsiders," who are unemployed
hence disenfranchised. Facing our short-run demand (4) the decision-
making insiders can, in the short run, have a higher real wage rate by

accepting less employment. Let them accept the fraction X employed of
available labor force, where 0 < X <_ 1. In other words, if L > XF
insiders will insist on a higher real wage rate. If

L = XF (5)

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

Consider the fraction X a parameter, then (5) will be a solution
for employment corresponding to Friedman's (1968: 8) "natural" rate
1 - X of unemployment. The fraction X would reflect institutional
dimensions of the labor market such as union density. Cross-country
measurement of movements in employment and union densities is repro-
duced in Appendix I and found to be in good accordance with our inter-
pretation of labor supply (5).

7 . The Wage-Employment Tradeoff

The real wage rate insiders will be happy with, given their natural
rate X of employment, is found by inserting (5) into (3):

-7-

w

- = aa(XF)"0S6 (6)

What is the implied slope of the Phillips curve? As long as the
ratio w/P satisfies (6) the levels of the money wage rate w and price
P can be anything: the Phillips curve is vertical. Where labor can-
not negotiate real but only money wage rates, short contract periods
will have to do, and a temporarily finite slope of the Phillips curve
is possible until successive rounds of collective bargaining have
restored levels of the money wage rate w and price P satisfying our
wage-employment tradeoff (6). Cross-country measurement of movements
in employment and real wage rates is reproduced in Appendix II and
found to be in good accordance with our wage-employment tradeoff (6).

8. Physical Output

Write the firm production function (2) as

L a
X = a(-) S (7)

S

-8-

and the firm demand for labor (4) as the factor proportion

T -1/6

L l/fi W

- = (aa)1/0(-) (4)

S P

Insert (4) into (7), then on the right-hand side of (7) everything
except S is common to all firms. Factor out all such common factors
and sura (7) over firms. Then S becomes aggregate physical capital
stock, and X aggregate physical output. We already know that the fac-
tor proportion (4) holds for the firm as well as for the economy at
large. Read it for the economy at large, multiply out in (7), and
arrive at a production function of the form (2) now holding for the
economy at large. Into such an aggregated (2) insert (5) and write
physical output:

X = a(XF)aSB (8)

9. Desired Capital Stock and Investment

Desired capital stock and investment are long-run commitments to
be determined by maximization of present net worth. Here the firm can

-9-

no longer consider Its physical capital stock. S a constant or ignore
the effect of investment 1 upon it.

We begin by defining the rate of growth of a variable v as the
derivative of its logarithm with respect to time:

dlog v

gv -z e- (9)

dt

To find the capital stock desired by the firm define physical
marginal productivity of capital stock as

3X X

< = — = aBLaSB = 6 - (10)

3S S

Firms were purely competitive; then price P of output is beyond
their control. At time t, then, marginal value productivity of capi-
tal stock is <(t)P(t).

Let there be a market in which money may be placed or borrowed at
the stationary nominal rate of interest r. Let that rate be applied
when discounting future cash flows. As seem from the present time x,

-10-

then, marginal value productivity of capital stock, is <(t)P(t)e
Define present gross worth of another physical unit of capital stock
as the present worth of all future marginal value productivities over
its entire useful life:

k(-r) e / <(t)P(t)e r(t T)dt

Let firms expect physical marginal productivity of capital stock
to be growing at the stationary rate g :

g (t - t)
<( t) = <(r)e <

and price of output to be growing at the stationary rate g

e ( t - t)
P(t) = P(r)egP

Insert these, define

p = r - (gK + gp) (11)

-11-

and write Che integral as

<(x) = / <(T)P(x)e p(t T)dt

Neither <(t) nor P(t) is a function of t hence may be taken out-
side the integral sign. Our g , g , and r were all said to be sta-
tionary; hence the coefficient p of t is stationary, too. Assume
p > 0. As a result find the integral to be

k = <P/p

Find present net worth of another physical unit of capital stock
as its gross worth minus its price:

n = k - P = (</p - 1)P

Capital stock desired by the firm is the size of stock at which
the present net worth of another physical unit of capital stock would
be zero:

< = P

-12-

Insert (10) and find capital stock desired by the firm

S = BX/p (12)

Define investment desired by the firm

I = gsS = BgsX/p (13)

What is g ? Let it be correctly foreseen by firms that because
a + B = 1 the economy at large will have the solution (20), to be
found presently, and let that solution be common to all firms.

What is p? In its definition (11) let it be correctly foreseen by
firms that because a + B = 1 the economy at large will have the solu-
tion (23), to be found presently, and let that solution be common to
all firms. Historically the marginal productivity < of capital has
indeed remained stationary. In that case (11) simply collapses into
the real rate of interest, common to all firms.

On the right-hand sides of (12) and (13), then, everything except
X is common to all firms. Factor out all such common factors and sum
(12) and (13) over firms. Then X becomes aggregate physical output
and (12) and (13) aggregate desired capital stock, and investment,
respectively.

-13-
10. Consumption; Equilibrium; Money

Let the aggregate consumption function be

C = cX (14)

where 0 < c < 1.

Aggregate equilibrium requires aggregate supply to equal aggregate
demand:

X = C + I (15)

To determine the rate of inflation we must, first, define the
velocity of money as the number of times per year a stock of money
transacts money national income:

Y = MV (16)

and, second, consider the money supply M and its velocity V to be
parameters growing at the rates g^ and g , respectively.

-14-

II. SOLUTIONS

1. Convergence

The key to our solutions for growth rates and levels Is Solow's
(1956) convergence proof. We apply it as follows. Differentiate
aggregate physical output (8) with respect to time, consider our
natural rate X of employment a stationary parameter, and find

gX = ga + agF + e«S (17)

Insert (14) and the definitional part of (13) into (15), rearrange,
and write the rate of growth of physical capital stock as

gs = (1 - c)X/S (18)

Differentiate with respect to time, use (17) recalling that
a + 8 = 1, and express the proportionate rate of acceleration of
physical capital stock as

-15-

ggS = gX_ gS = a(Sa/a+ gF " gS} (19)

In (19) there are three possibilities: if gQ > g /a + gP, then

S ' sa'

ggs < °* If

gS = ga/ot + gF (20)

then g = 0. Finally, if gc < g /a + g,,, then g _ > 0. Conse-
gb b a r gb

quently, if greater than (20) g<, is falling; if equal to (20) gg is
stationary; and if less than (20) g is rising. Furthermore, g_
cannot alternate around (20), for differential equations trace con-
tinuous time paths, and as soon as a g -path touched (20) it would
have to stay there. Finally, g cannot converge to anything else than
(20), for if it did, by letting enough time elapse we could make the
left-hand side of (19) smaller than any arbitrarily assignable posi-
tive constant e, however small, without the same being possible for
the right-hand side. We conclude that gq must either equal g_/a + gp
from the outset or, if it does not, converge to that value.

Once such convergence has been established we may easily find the
corresponding values of other growth rates: insert (20) into (17),
recall that a + Q = 1, and find the long-run growth rate of physical
output

-16-

gx - gs (21)

Differentiate (6) with respect to time, use (20), and find the
long-run growth rate of the real wage rate

«w/P ■ ga/a (22)

Differentiate (10) with respect to time, use (21), and find the
long-run growth rate of the physical marginal productivity of capital
stock

g< = 0 (23)

As we recall from the definition (11), g was one part of the
definition of the real rate of interest. To solve for the other part,
insert (1) into (16), differentiate with respect to time, use (21),
and find the long-run rate of inflation

gp = §M + gv " gs (24)

where g stands for the solution (20)

-17-

We have found our natural rate X to be absent from all our long-
run growth rates (20) through (24). But might it be present in the
long-run levels at which our variables are growing? We shall see.

2 . Real Rate of Interest

To solve for the long-run level of the real rate of interest
insert (13) and (14) into (15), divide any nonzero X away, and find

Bgq

P - — (25)

1 - c

where g stands for our solution (20). Our solution (25) has no X in
it: the long-run real rate of interest is invariant with the natural
rate X of employment. Differentiating our solution (25) with respect
to time, we find it to be stationary — as we assumed in Sec. I, 9 above.

-18-
3 . Physical Capital Stock

To solve for the long-run level of physical capital stock insert
(8) and (25) into (12) and find

. l/o
1 - c

S = (a ) XF (26)

gS

where g„ stands for our solution (20). Our solution (26) does have X
in it: the long-run physical capital stock is in direct proportion to
the natural rate X of employment. Differentiating our solution (26)
with respect to time, we find it growing at the rate (20), invariant
with X — as it should.

4. The Real Wage Rate

To solve for the long-run level of the real wage rate insert (26)
into the short-run level (6) and find

-19-

B/o

- = aai/a ( ) (27)

P 2

where g stands for our solution (20). Our solution (27) has no X in
it: the long-run real wage rate is invariant with the natural rate X
of employment. Differentiating our solution (27) with respect to time,
we find it growing at the rate (22), invariant with X — as it should.

5. Physical Output

To solve for the long-run level of physical output insert (26)
into the short-run level (8) and find

. i 1 - c B/o
X = a ' ( ) XF (28)

gS

where g stands for our solution (20). Our solution (28) does have X
in it: the long-run physical output is in direct proportion to the
natural rate X of employment. Differentiating our solution (28) with
respect to time, we find it growing at the rate (21), invariant with
X — as it should.

-20-

III. CONCLUSION

We have found a stark, contrast between the short-run and the long-
run scope for wage policy. The simple mathematics of the contrast is
this.

In our real wage rate (6) neither a, a, 6, nor F is a function of
the natural rate X. Physical capital stock S may or may not be. In
general differentiate the natural logarithm of (6) with respect to X
and find the elasticity of the real wage rate with respect to the
natural rate X to be

31og (w/P) 31og S

S = - 8 + 6 — (29)

31ogeX 31ogeX

In the short run physical capital stock S can be considered a
constant depending on nothing:

Slog S

— = 0 (30)

31og X

-21-

Insert (30) into (29) and find the latter collapsing into - 8:
labor can have a 8 percent higher real wage rate by accepting a one
percent lower natural rate X of employment.

By contrast, in the long run physical capital stock. S cannot be
considered a constant but is a variable to be solved for. When we
solved for it we found (26) whose elasticity with respect to X was

31og S

— = 1 (31)

31ogeX

Insert (31) into (29) and now find the latter collapsing into
-8+8=0: labor can have a no higher real wage rate by accepting a
lower natural rate X of employment.

In plain English the reason for the stark contrast is that in the
long run the levels (26) and (28) of capital stock and output simply
adjust to X and are correspondingly lower: the economy is impover-
ished, accumulates less capital stock and produces less output. Labor
does not benefit. Nobody benefits.

-22-

APPENDIX I. U.S. -EUROPEAN DIFFERENCES IN EMPLOYMENT AND UNION DENSITY

Friedman (1968: 9) never meant his natural rate to be "immutable
and unchangeable." Indeed, over the thirteen years 1973-1986 Freeman
(1988b: 294-295) found actual employment as a fraction of working-age
population declining steadily in OECD-Europe as a whole but largely
rising in the United States.

Among the institutional dimensions reflected by the natural rate
Friedman (1968: 9) mentioned union density. One would expect the
employment fraction and union density to be moving in opposite direc-
tions. Roughly speaking, so they did: over the fifteen years
1970-1985 Freeman (1988a: 69) found union density rising sharply in
Denmark, Finland, and Sweden; rising moderately in Australia, Canada,
France, Germany, Ireland, Italy, New Zealand, and Switzerland; rising
slightly in Norway and the United Kingdom; declining slightly in
Austria and the Netherlands; declining moderately in Japan and sharply
in the United States. In Sweden, however, the employment fraction and
union density moved in the same direction. Allowing for Sweden, Barro
(1988: 36) found the persistence of low employment to go with high
union density and large size of government but only in countries
lacking centralized bargaining.

-23-

APPENDIX II. U.S. -EUROPEAN DIFFERENCES IN WAGE-EMPLOYMENT TRADEOFF (6)

Differentiating our wage-employment tradeoff (6) with respect to
time would suggest increases in employment and the real wage rate to
be of opposite orders of magnitude, and so they were: over the
twenty-five years 1960-1985 Freeman (1988b: 296-297) indeed found
countries in most of OECD-Europe to have larger increases in their
real wage rates and smaller increases in their employment than had the
United States and Sweden. The pairing of the United States and Sweden
was also noticed by Ergas and Shafer (1987-1988).

Economists from Keynes (1936: 14) to Summers (1988) have insisted
that relative real wages do matter. Under decentralized bargaining,
wage restraint by an individual union may lower its relative real
wages. Centralized bargaining removes such fears. Sweden with her
centralized bargaining and very high union density did show more wage
restraint than countries with decentralized bargaining — as Freeman and
Ergas-Shafer found.

-24-

REFERENCES

Barro, Robert J., "The Persistence of Unemployment , " Amer. Econ. Rev. ,
May 1988, 78_, 32-37.

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: Blackwell,
1988.

Ergas , Henry, and Shafer, Jeffrey, "Cutting Unemployment Through
Labour-Market Flexibility," The OECD Observer, Dec. 1987-Jan.
1988, 19-21.

Freeman, Richard B., "Contraction and Expansion: The Divergence of
Private Sector and Public Sector Unionism in the United States,"
Journal of Economic Perspectives, Spring 1988, 2_, 63-88.

* , "Evaluating the European View that the United States Has

No Uneraployment Problem," Amer. Econ. Rev., May 1988, 78, 294-299.

-25-

Friedman, Milton, "The Role of Monetary Policy," Amer. Econ. Rev. ,
March 1968, _58_, 1-17.

Keynes, John Maynard, The General Theory of Employment, Interest, and
Money, London: Macmillan, 1936.

Lindbeck, Assar, and Snower, Dennis J., "Wage Setting, Unemployment,
and Insider-Outsider Relations," Amer. Econ. Rev. , May 1986, 76,
235-239.

Maddison, Angus, "Growth and Slowdown in Advanced Capitalist Economies
Techniques of Quantitative Assessment," J. Econ. Lit. , June 1987,
25_, 649-698.

Solow, R. M. , "A Contribution to the Theory of Economic Growth,"
Quart. J. Econ., Feb. 1956, 7£, 65-94.

Summers, Lawrence H. , "Relative Wages, Efficiency Wages, and Keynesian
Unemployment," Amer. Econ. Rev., May 1988, 78, 383-388.

-26-

Windmuller, John P., "Comparative Study of Methods and Practices,"
Collective Bargaining in Industrialised Market Economies, Geneva
I. L. 0., 1987, 3-158, especially p. 19.

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