A test for inflation-induced redistributions

Engineering Costs and Production Economics,

115

I5 (1988) 11S-1 20

Elsevier Science Publishers B.V., Amsterdam -

Printed in Hungary

A TEST FOR ~~FLAT~O~-~~DU~ED RED~STR~BUTIO~S Paul G. Reinhardt York University.

The intuitive notion that inflation redistributes income because some prices are ahead of others during inflation has stubbornly defied direct statistical analysis. The issue has attracted particular attention in connection with the wage-lag hypothesis. It holds that workers lose real income because wages do not rise as rapidly, with respect to time, as prices for the same discrete increases. Despite long-standing efforts to test the assertion* it is still far from clear what the data show. This paper presents an alternative test for adjustment-induced redistributions and applies it to recent German wage-price movements, It is suggested that German wages have lagged behind prices in their adjustments that brought about rising real wages over the period 1964198 1. It will be convenient to discuss the definitional problems against the background of previous tests. I DEFINITION OF PROBLEM

Before attempting to separate real and non-real changes in the data the basic questions considered in this paper are set out. * The tindigs of W. C. Mitchell (1908), A. E. Hansen (1925), C. Bresciani-Turroni (1937) E. J. Hamilton (1936, 1942, 1952) and R. H. Bbatia (1962) have supported the hypothesis, while those of R. A. Kessel and A. A. Alchian (1960) and T. F. Cargill (1969) have cast doubts on it. In the controversy between monetarists and non-monetarists over the causes of recent intlation, the question of a wage-lag has re-surfaced as a central issue. While D. E. W. Laidler (1976) and M. H. Parkin and M. Sumner (1972) find evidence in the U.K., of a wage-lag, A. Jones (1972), Sir John Hicks (1974) and P. Wiles (1973) offer a wage-lead explanation for the same period.

Downsview,

Canada

What we want to test for is the sign of the netredistributions, between two points in time, that result when wages and prices do not change at equiproportionate rates at every point in time. This raises first the question of how real income changes accrue with time when there is a time relation between wages and prices. Since economic theory is silent on this question we will explore the intuitive notion that income changes may depend on both the duration and the magnitude of the departures of the real wage rate from a horizontal linear time path. Secondly, there is a problem in that the time process under study can be observed only by periodic readings p(tJ and w(tJ of the time functions of prices and wage rates, p(t) and w(t), respectively, at ti,i=O,

1.. .n. The variations

in the ratio -w(t) PM

conG~uow~yincur

the redistributions we are interested in. Apart from a measurement problem, therefore, there is the question of how a continuously accruing quantity can be imputed from periodic w(4) w(t) readings Pfr3

Of

pft)

’

The linear periodic estimates of p(t,) and w(ti) available from time series data pi and wi, respectively, creates a separate statistical problem because p(t) and w(t)may be non-linear. If time series have a low reporting frequency, they average nonlinear observations across long time intervals. %r; can these data be used to provide estimates on -

PiI

at the midpoint ti of these time intervals? However serious these influences may be we cannot avoid them by the simplifying assumptions of non-dynamic models. It appears that the failure by

116

previous investigators to deal with these problems explicitly may have caused a number of difficulties for their tests. T. F. Cargill (1969) studies the co-movements of wi and pi over time and finds phasial shifts between them. Unfortunately, there is no analysis of how the sign of the phasial statistic relates to that of the redistributions we are after. His conclusions about the wage-lag hypothesis are therefore difficult to accept. W. C. Mitchell (1908), A. H. Hanser (1925), E. J. Hamilton (1936, 1942) and D. E. W. Laidler (1976) interpret the changes in the series F i=O. . . n as evidence of a wage-lag for particular itime periods. The implicit assumption is that real income changes are proportionate to the discrete changes, between successive periodic averages, of the real wage rate. But this is the same as assuming that the length of time it takes for changes in the real wage rate to develop, and to be offset, has no effect on the accrual of real income. As will be shown below, the assumption, in fact, makes redistributions, independent of the relative speed of wage-price adjustments, and thereby rejects, from the outset, that which is to be tested. In an effort to avoid this problem we will treat time explicitly. For convenience, we wili relate changes in the worker’s cost-of-living to changes in the ratio r(t) = 3.

We define p(t) and w(t) as the purchase price

of a specified commodity basket consumed, and the wage for a specified type of labor received, respectively, per unit of time, at time r. We define the cost-of-living as the function R(t) of time that measures the number of hours the worker has to spend on his job to maintain consumption of his commodity basket. With time passing continuously, R(t) acrues at the rate r(t) at time t. Since r(t) is the ratio of two money flows, i.e. cost of a given consumption basket per period wages per period it denotes the fraction of the time spent on the job to maintain consumption. The total cost, in hours, worked for this consumption over the period test 5 t,, , at time t, , for an initial R(t,) = 0, is thus R(t,)= 7 r(t) dt .

To see how R(t,) can be imputed from periodic readings of r(r), assume r(t), and its derivatives r’(t), r”(r), . . . r”(t), are observed n + 1 times over the time intervalte~t~ttn,atpointst,, t, ,... ti ,... t,,ti-ti + 1 = i time unit apart. It can be shown that the integral can be represented by the sum of the Taylor ri+I expansions of j r(t) dt at ti so that we obtain ti (1)

R(&) = i=n-

1

izo r(ti)+ $ iTglr’(tJ+ $j iYE1 ?+“(tj) +

=

I

...

0

Relation (1) shows how R(t,) depends on periodic observations of the function r(t). In testing for changes in R(t) therefore, we will need, in addition to the averages of observations, a statistic that reflects the effect of the time shape of r(t). II THE BIAS IN LINEAR COST-OF-LIVING STATISTICS WHEN PRICES CHANGE AT DISPROPORTIONATE RATES

In re-examining the proportionality assumption of previous investigators we can see that it is the n-l r(ti) , and changes, same as equating R(t,) and c

The subscripts k and k+ 1 refer to successive periods of equal length and the sums of r(ti) are taken, separately, over each of the successive sets of n + 1 readings. Only then can, in period k, the average

n-1 1k rr, = ?+tti) [a:

be proportionate

n

equal @I=------~

to

the

average

cost

of

to R,(t,),

or

consumption,

Rdt,)

48--o Equation (1) shows that, representing R(t,) by n-1 r(ti) assumes the effect on R(t,) of the time shape c of r(t), or of the relative speeds of wage and price adjustments, 1

2!

n-1

c

r’(tJ+

1

ji

1-l

c

r”(ti)+

. . . =o.

117

The magnitude per unit of time of the resulting error, e - f, is shown in Figure 1. Let r(ti) be the reading to represent r(f) for the period ti 5 t 5 ti + 1. It is clear n-l

that

c r(t,) adds increments resulting from suc-

cessive’ periodic slope reading r(Ci), whereas R(t,) correctly takes r(t) as the rate of accrual of R(t) at every point t. Generally, therefore, p and r, the slopes m-1 of the rays from the origin to R(t,) and 5 r(t& respectively, will differ. The larger the effect of relative adjustment speeds on the cost-ofliving, the greater the differences p-r, the greater the bias of linear estimates. It seems, therefore, that any direct analysis of series ri, without imputing pi, precludes, from the outset, that we can detect any redistributive effects, however large they may be.

as the unknown points t,, , d time units apart. ft means r(t) describes cycles around the line r,(t). Whatever the relative magnitude of periodically observed changes in wages and prices, the wage-lag hypothesis is that R(t,)>

R(r,) or that f r(r) dt - R(r,) > 0

:(I

Substituting the Taylor expansion (1) the wage-lag hypothesis can be written in the form

+ . . . -R(t,)>O

(2)

III DEFINITION OF WAGE-LAG

If we knew the wage lag were completed within f. 5 t 5 t, the redistributions in (2) would be free of the effect of real changes. In the absence of this knowledge however, we face the problem of eliminating the redistributions from (2) that result from the arbitrary choice of to and t, .

We assume that no disturbances ather than real, or those resulting from variations in the relative speed of adjustments in prices and wages, affect R(t,).

IV THE EFFECT OF THE TERMINAL POINTS ON THE LAG

r&o) ; r&) (1, _ to), Let R(t,) be the cost 17(t,) = -~----

that would have been incurred had the real ratio r,(t) of perfectly synthronized wage and price adjustments prevailed at every point in time. That is, we assume in this case what static analysis assumes generally, that consumption costs are independent of the time shape of the price wage ratio. In restricting the assumption to the relation of R(t) and real r,(t) we have a basis for separating R(t) as the real component, from the actually accruing, consumption cost R(t). We assume variations in the relative speed to develop over time intervals of variable length t. Any

The problem of the dependency of statistical averages of real wages on the particular location of the reporting periods was pointed out by R. A. Kessel and A. A. Alchian (1960). By switching the base, for purposes of comparison, from a year of an exceptionally high, to one with an exceptionally low real wage, they turned a period of allegedly low real wages, obtained by W. C. Mitchell (1908) and E. J. Hamilton (1936, 1942) into a period of relatively high imputed real wages. Clearly, the success in testing (2) depends entirely on the ability to render the estimators independent of this effect. In analyzing the problem, we note that in a dynamic setting there exists a unique r(t) at time

given variation approaches zero,* as dt+d < k(t,, * For example, a price rise is i~~sin~y

-to). Differing adjustment speeds will, thus periodically, generate r,(t)

wage-

price adjustments because we want to investigate, exclusively, the effect of variations

r(L) = r,(t)

compensated for by

increasing wages. Our assumptions ~r~lude,uncompleted

price changes.

in the speed of adjustments

for &WI

wage-

118 t only. Any representation

of r(t) by the average 1 n-l _ r(ti) is an estimate of r(t) at the midpoint n c to + t” Its statistical error does not pertain to the 2 . likely deviation from some timeless “true” ratio but only from E rq II-1

[

1

In summary, the suggested test takes the net effect $ B’(t) on the wdrkers’ cost-of-living of real changes in r(t) as given. It tries to answer the question of whether or not the time shape of how these real changes are implemented increases the cost-ofliving, if what we observe goes on forever.

. The same holds for any

change

f(t). c If t, were known we could place the midpoint of to + t” = t,, . Because of the our observations at 2 symmetry involved, we would have

/d pict;)

[c 1 n-l

E

/R(t)

r’(tJ

- r&, + t.) (t. - to) =o

What we would test for wotild simply be the effect of the time shape of r(t) on R(t,) in as much as it differs from a linear r,(t).

1-t

to

tl

Figure 1 A Diagrammatic Comparison of the Dynamic and Static Consumption

V THE SUGGESTED

Costs, S and F

TEST

Even though t, is unknown we can provide for the to+& symmetry around by assuming the process 2 that generates the wage lag R(t&-R(t.) goes on indefinitely. Any estimate obtained from periodic observations during the randcmly drawn interval to 5 t 5 t, will be located such that

It can be shown that the hypothesis E(L,,,) > 0 is tested for by merely obtaining the average of the positive and negative second differences, d2ri, separately, over n - 1 successive readings,* A, and D, (for average acceleration and deceleration), respectively. The test of (2) then is to see if A,+B,>o.

(3)

Accordingly, wage-price adjustments increase, or decrease the cost-of-consumption R(t,) as, relatively, prices accelerate more, and decelerate less, rapidly than wages. Intuitively, this amounts to for any given real change E determining whether the positive curvatures in r(t) extend over a shorter time interval than the negative assumption we test (2) by analysing ones. We are thus looking for a succession of 1 n-l inverted U’s in the r(tJ of data that can be expected r”‘(ti)+ . . . =E(L,,,,)>O. to preserve the time shape of the adjustments. r”(4)+ r -t_=O

and

.c

E

1

In terms of Figure 1, we are thus testing for the significance of the sign observed in the difference e-r.

* See Reinhardt (1974)

119 VI DATA PROBLEMS

A2r, within, and

Ideally, we look for time series of the consumption price index and of the wage rate as they pertain to one representative worker in each wage group. The frequency of reporting would be large enough to record every change in r(t). That is, averages across time of prices and wages for that worker would not be used. Time averages conceal the changes Ar and

observations. Nor are averages, at a point in time, desirable as prices and wage rates of workers across wage groups. The reason is that we expect price and wage changes between different types of wage groups and between geographic locations to be poorly synchronized. The greater the aggregation of the data the poorer their ability to reveal information on the time shape.

TABLE

1

Wage-lag Hourly

between successive periods of

test on monthly

German

wages and retail prices,

cross wages of disaggregated

19641981*

wage groups Significance

Name and location

I. AN male workers. Efficiency Group

of wage group

I

levels indicating

degree of inconsistency

with the null hypothesis of unlagged and price adjustments (Sign “ + ” for d > 6, interpreted t = +I.87

Public utilities Hamburg Bremen Berlin

+ 2.43 + 1.64

Non-metallic

Hamburg Bremen

+ 1.40 + 2.09

Berlin

+2.00 + 1.12

Oil-refining Chemical

Machine

Breweries

+ 2.62

Hamburg Bremen Berlin

Building

Ship Building Printing

supplies

Hamburg Industry

+ 2.59 + .38 + 1.13

Hamburg Bremen Berlin

+ 1.83 + 3.40 + .32 + 2.26

Hamburg Bremen

Hamburg

+4.24

Bremen Berlin

+ 2.87

+ I.71 + 1.28

Hamburg Bremen

+ .93 - 1.44

Berlin II. AN female workers, Efficiency group Chemical Fibres North-Rhine-Westfalia Baden Wuertemberg Saw mills and wood processing

Glass Industry

Meat Processing

as wage-lag)

- .I7 - 1.51

Coal Mining North-Rhine-Westfalia Iron Ore Mining Lower Saxony building

wage

3 -

.95

+ 1.14

Rhinel.-Pfalz Baden-Wuert.

+ -

.05 .31

Saar

+ + + + +

2.06 2.98 .45 .20 1.50 I.41

Rhineland-Pfalz Baden-Wuertemberg Saar Baden-Wuertemberg Saar

* Source: Slatistisches Bundesamt Wiesbadn. in Industrie und Handel, I Arbeiterverdienste

Fachserie

M, Preise, Lohne, Wirtschaftsrechnungen,

Reihe 15, Arbeitnehmerverdienste

120 The historical data on wage rates and prices that have been investigated are almost exclusively in the form of annual averages. In addition, they represent high levels of aggregation. As a result test (3) cannot be applied to reexamine historical evidence. Current data hold out better prospects if they are very detailed and frequently reported. However, they continue to be biased against (3) to a degree that needs to be studied in each case against the time paths of ideally disaggregated and reported components. Therefore, the insigni~cant results on (3) that were obtained for UK series, for example, do not yet support negative conclusions. On the other hand, if significance levels do prevail against this bias they would appear to carry all the more weight.

VIII CONCLUSION

The present investigation of a large body of German wage price series suggests that the real incomes of workers would have been higher over the period 1964-1981 had wage increases been implemented at speeds equal to those of price increases for given real wage changes. REFERENCES Bhatia,

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DATA

monthly ratio ri = I$ was constructed by us:. ;d the same wage rate in th: three consecutive months of the quarter for which IV4is reported. Thirty-three wage rate series were randomly selected from the most disaggregated wage groups for the geographic classification that represented the smallest area. The t-values of the ri series were computed on the basis of annual quantities A,+& and on the assumption, ti=O. s& Table 1 lists the results. The t-values are generally, and sometimes strongly, in the positive direction. They suggest the remoteness of the probability that the observed t-values could have been generated by randomly accelerating and decelerating wage rate and price series. The likeliho~ that prices accelerated more than wage rates is thus very great.

u=s:.+

* Wagerateschangedrelatively figures are not expected to seriously

infrequently so that quarterly impair estimation.

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in

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