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Full text of "Wage theory and growth theory"

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JAN 4 2000 



NOVie 



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L162 




BEBR 

FACULTY WORKING 
PAPER NO. 1521 



IfnLr Ml 




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 



http://www.archive.org/details/wagetheorygrowth1521brem 



(217) 344-0171 
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 = aL a S 8 (2) 



where < a < 1, < 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 - 

- = — = aaL a " 1 S 6 (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 < 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)" S 6 (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) a S B (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 

g v -z e - (9) 

dt 



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



3X X 

< = — = aBL a S B = 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)e g P 



Insert these, define 



p = r - (g K + 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 = g s S = Bg s X/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 < 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 

g X = g a + ag F + 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 



g s = (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- 



g gS = g X _ g S = a( S a /a+ g F " g S } (19) 



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



S ' s a' 



g gs < °* If 



g S = g a /ot + g F (20) 



then g = 0. Finally, if g c < 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 g q 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- 



g x - g s (21) 



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



«w/P ■ g a /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< = (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 + g v " g s (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 



Bg q 

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) 

g S 



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 

- = aa i/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) 

g S 



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) 

31og e X 31og e X 



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



Slog S 

— = (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) 

31og e X 



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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