should we wait to ‘grow out of’ unemployment? the implications of a neoclassical calibration...

THE ECONOMIC RECORD. VOL. 74. NO. 215. JUNE IWH. Ih?-6Y

Should We Wait to 'Grow Out of' Unemployment? The Implications of a Neoclassical

Calibration Analysis WILLIAM COLEMAN" Eiwnomics Depurtmenr. University of Tusmuniu.

Hohurt. Tusmuniu

The puper irses u Solon~-Rumse~ ,qron'th model to evulrrute the efiectiveness of growth us u remcclv j i)r irnemploymetit in the fuce of u n u p ~ minimum. Plairsihle iulihrution of the model srr,qsqests tliut the eliniinution of 5 per ceni iinrmployment by the process of c~~pitcrl uccirniirlurion may t u k uhout 20 years.

I Introdirctioti The classic solution for unemployment is a reduc-

tion in real wages. The wisdom of this classic solution has been the subject of disagreement, and alternative remedies have been advanced. One such alternative is economic growth.

This paper uses a Solow-Ramsey growth model to evaluate economic growth as a remedy for unemployment. The evaluation is both favourable and unfavourable to economic growth as a remedy for unemployment.

The evaluation is favourable in that it reinforces the conclusion of the earlier literature (Sgro 1980, Sgro and Takayama I98 I, McDonald 1984) that capital accumulation will eventually eradicate any quantity of unemployment caused by a wage minimum (as long as the minimum is not 'too high').' Although this earlier literature assumed that the propensity to save was exogenous, the

* I am indebted to the comments of Hugh Sibly. Ian Mcdonald, and two anonymous referees. ' Sgro and Takayama (198 I ) analyze the impact of a wage minimum in a two-sector growth model with exogenous propensities to save. The impact of a minimum in a one-sector model with an exogenous propensity to save is reviewed in McDonald (1984). A body of literature with a kinship to the present paper is the study of dual-sector models of developing economies. where one sector's wage is fixed (e.g. Dixit 1968). A recent example of this literature is Robertson (1997).

present analysis shows that the effectiveness of capital accumulation as a remedy for unemployment is not sensitive to that assumption, and remains true in a Solow-Ramsey model where the propensity to save is determined by utility maximization of forward-looking decision makers.

However, the evaluation is unfavourable to economic growth as a remedy for unemployment in that it concludes that society may need to wait a long time for the growth process alone to solve the unemployment problem. Plausible calibration of the model suggests that it may take about 20 years to eliminate by capital accumulation alone an unemployment rate of 5 per cent.

Further. the analysis is unfavourable to growth as a remedy for unemployment in that the alternative policy of reducing wages will, in the long run, increase the wage rate above that of the economy with a wage minimum. Thus even if the only policy goal is high wage rates the policy of eliminating wage minima has something to recommend itself.

In summary. the present analysis suggests that although economic growth is a feasible remedy for unemployment. it is also inferior to the classic solution of wage reductions.

I! A Model of an Economy nith a Wage Minimum

The paper analyzes a model which combines the aggregative productive relations of Solow's

I62

c 1998. The Economic Societv of Australia. ISSN 0013-07-49

1998 SHOULD WE WAIT TO 'GROW OUT O F UNEMPLOYMENT'! I63

growth model (Solow 1956) with a Ramsey style explanation of consumption (Ramsey 1927).

Output is produced by way of a constant returns to scale aggregate production function.

YIL = y(KIL) ( 1 )

Y = output

y' > 0, y" < 0

K = capital, L = employment

Capital grows according to difference between output and consumption.

(1) dKIdt = !(KIL)L - C

C = consumption.

Consumption could be assumed to be a fixed fraction of output. as previous investigators of the wage minimum in a growing economy (Spro and Takayama 1981. Macdonald 1984) have sup- posed.' However. this paper will assume (in the style of Ramsey 1927) that every individual decides consumption by maximizing an identical utility function, which is a function of consumption from the current period unto infinity,

( 3 ) Utility maximization implies that the growth rate in consumption of every individual is proportional to the excess of the rate of profit over the para- metrical rate of time preference. 6

u = I;, r a u(C) dl.

dCICdt = 8 Lv'(KIL)-Sj (4)

8 = elasticity of intertemporal substitution

The total supply of labour. N. is constant. We choose units so that N = I .

If the wage was flexible. then the labour market would be in equilibrium, L = I , and consequently. (2) and (4) could be rewritten as.

dcitir = B L V ' ( K ) - ~ I c ( 5 )

dKldt = y (K) - C (6)

(5) and (6) constitute a familiar dynamic system, which can be represented by way of a familiar phase d iagnm with a saddle path indicating the solution (e.g. Romer 1996, p.51). If the initial quantity of capital is less than its steady state mag-

Sgro and Takayama assumed a 'classical savings' hypothesis in which there are different propensities for profits and wages.

nitude. then capital and consumption rise towards their steady state values, K* and C*. Since the real wage, w, equals the marginal product of labour, which is a positive function of capital, the real wage will also rise towards its own steady state value, w*.

But what if the wage is inflexible downwards? Suppose that in period zero, some point in time before the steady state is reached, a minimum real wage is imposed, such that,

wn, = minimum wage w,~, = full employment wage at period zero n'* = steady state wage.

The immediate response to the introduction of such a minimum is a reduction in employment, and the appearance of a quantity of unemployment. The question this paper seeks to resolve is, 'How would this quantity of unemployment change over time?' To answer that question the equation\ of motion. (5) and (6). must be rcvised to allow for thc wage minimum.

I f we assume that the unemployed workers are not constrained from borrowing against their future incomcs. the structure of the consumption optimization problem remains the same as when there was no unemployment. So consumption growth continues to bc governed by the same equation, (5). Notice. however. that the wage minimum paruneterizes the rate of return on capital, y ' ( K ) , which in turn parmetcrizes the growth rate of consumption. at a value we denote ,q. So ( 5 ) may be rewritten.

wfe< wm < w*

(7)

dCldt = ,sc (8)

g = growth rate of consumption under a binding wage minimum.

With respect to the equation for capital growth. (6). it should also be revised to allow for the parameterization of the average productivity of capital (at value we denote 4). which follows from the parameterization of the wage. Thus,

(9) dKIdt = q K - C

q = average productivity of capital under a binding wage minimum.

(8) and (9) constitute the economy's equations of motions with a wage minimum.

Figure I depicts the dynamics of C and K, (8) and (9). under the wage minimum. The zero

164 ECONOMIC RECORD JUNE

investment locus is initially linear owing to the linear relationship between capital and output under a binding wage minimum. The linear locus extends until the quantity of capital which is sufficient to employ the whole labour supply at the minimum wage. Kfr. The locus then assumes a familiar curvature, reflecting the diminishing marginal product of capital once full employment h a s been reached.

Of all the paths which may be depicted in Figure I , there is a unique path which yields at K = K,* an amount of consumption equal to saddle path quantity of consumption under full employment. This unique path is the path the economy follows (see Figure 2).3 Thus capital grows until, at some time T, it has reached a magnitude. K such that ful l employment is reached. Tki minimum real wage is now equal to the full employment real wage. The path of the economy henceforth can be described by the flexible wage system, equations (5) and (6).

In summary. on the assumption that the minimum wage is smaller than the steady state wage, a reduction in the wage is not required for the elimination of unemployment? Rather, the accumulation of capital will eventually eliminate unemployment without any reduction in wages. This is not to say that capital accumulation is a desirable remedy for unemployment. I t has a crit- ical demerit: it may take a very long period of time for capital accumulation to eliminate unemployment. This is demonstrated in the next sec- tions of the paper.

Ill A Calihrarion of rhe Time Required ro Crow 0111 of 5 Per cent Unemployment

The Appendix demonstrates that an approximation of the time required to eliminate unemployment by capital accumulation. T. may be derived in terms of, CJ =the elasticity of technical substitution 9 = the elasticity of intertemporal substitution x = the profit share q = the average productivity of capital

Every other path will end (within a finite period of time) with zero consumption. This is inconsistent with utility maximization. ' If w,, > w* then the economy cannot 'grow out of unemployment. The capital stock will not rise; it will fall. at a constant exponential rate. forever. Employment will fall at the same constant exponential rate. See Coleman (1997) and Sgro (1981).

FIGURE 1

The Dyiuniics of C und K Unilrr u Wage Minimum

I C

I I

k = [K*-K,r] /K, i ,= the proportionate gap between steady state capital and the quantity capital which would fully employ the labour supply at the wage minimum.

ii = initial unemployment rate5 By assigning values to these parameters we can

quantify T. This section quantifies T for a range of plausible values of the six relevant parameters.6

The unemployment rate is defined here i ~ . the ratio of unemployed to employed, rather than the rdtio of unemployed to the employed plus unemployed.

For small values of unem loyment. e.g. one quarter of one per cent. such that PU P * I +uT, the value of 7 can be reasonably approximated as Follows;

U T -

y[ I -upc(T)[ I +u l I If, for example, the unemployment rate is 0.25 of 1 per cent ( u = 0.0025) and q = 0.25 and upc(T) = 0.90, the expression says it would take a bit over two months to eliminate unemployment of one quarter of I per cent. It is not difficult to make sense of the expression. We know that if the propensity to save, s. is constant then the growth rate is sq . If the economy is growing at sq it would obviously take ulsq of time to eliminate an unemployment rate of u. But Ulsq is essentially the expression above. So the expression can be understood as an approximation which works for stretches of time brief enough such that the propensity to save is approx-

I9Y8 SHOULD WE WAIT TO ‘GROW OUT OF’ UNEMPLOYMENT? 165

FIGURE 2

The Path lo Full Empluymeni

I I I

and a dependent value of u. Table 2 indicates how much unemployment would be eliminated over five years by economic growth for a variety of parameter values.

Table 2 indicates that the quantity of unemployment which will be eliminated within a five- year period may be substantial or insignificant, depending on the choice of parameters. Even within the range of parameters to which we have restricted ourselves, this quantity of unemployment might be as small as 0.9 per cent, or as large as 7.9 per cent.

TABLE I The Yeurs Required fur Growih 10 Elrnroiuie

5 per cent Unenrploymeni n = 0.3. 4 = 0.25.

e = 0.2, (0.5)

k 0 0.5 I

K’

The parameterization assumes that R (the protit share) = 0.3, y (the average productivity of capital) = 0.25, and u (the uncmployment rate) = 0.05 (i.e. 5 per cent). We allow o (the elasticity of technical substitution) to be either 0.5 or I . We allow 8 (the elasticity of intertemporal substitution) to be either 0.2 or 0.5.’ We allow k to be either 0.5, 0.25 or 0.125. Table I provides values for T for all combinations of these selected values.

Table I indicates that time required for the process of capital accumulation to eliminate 5 per cent unemployment is sensitive to the choice of parameters. Even within the range of parameters we have restricted ourselves to, the time required to eliminate unemployment may be as long 3s 68.2 years or as brief as 2.9 y e a .

Instead of calculating how much time is required to eliminate a given quantity of unemployment, one could use the model to calculate how much unemployment would be eliminated in a given quantity of time, say five years. We can use the preceding analysis to answer that question simply by making a parameter of T (say, T = 5).

imately constant. An expression of a similar sort is pro- vided by Sgro (1980. p.83).

The magnitude of the elasticity of intertemporal substitution is generally thought to be less than I . Some authors (e.g. Hall 1988) have judged it to be closer to zero than 1.

0.12s 33.2 68.2 (11.5) (15.4)

0 . 2 5 13.5 29.8 (4.9) (9.8)

0.5 8.7 22.4 (2.9) (5 .9 )

TABLE 2 The Rote of Unempluynrenr Elrnrinured by G r o W h

over 5 years K = 0.3. q = 0.25.

e = 0.2. (0.5)

k U 0.5 I

0.125

0.25

0.5

1.5 0.9 (2.9) (1.8) 2.6 I .3

(5.2) (3.2) 3.6 1 . 1

(7.9) (4.9)

Tables 1 and 2 reveal that we need to be more precise about parameter values if we are to obtain a better than vague idea of the time required to eliminate unemployment by growth alone. Is there any piece of information from which we may infer a more precise choice of parameters? One such piece of information would be an observation of a variable which is uniquely determined by the set

I66 ECONOMIC RECORD JUNE

of parameters we are concerned with: 8, a. and k. One such variable is investment, expressed as a percentage of national product. In the model this magnitude is uniquely determined by 8, Q. and k. Table 3 indicates investment as a percentage national product, for each of the combinations 8, (3, and k used in Tables 1 and 2.

TABLE 3 Net Ili\-esmrrnt us u Perivnrugr of Ner Nurionul

Prodircr x = 0.3. 4 = 0.25.

0 = 0.2. (0.5)

of of

x CT 0.5 I

0. I25

0.2s

0.5

0.5 0.3 (1.3) (0.7)

1.1 0.7 (3.4) ( I .h)

I .6 0.9 ( 5 . 8 ) (2 .7 )

In conjunction with an observation on investment Table 3 can point to an appropriate calibration. If , for example, the investment share was observed to be 0.5 per cent one could conclude that 0 = 0.2. cs = 0.5. and k = 0.125 was an admissible calibration. but thai (for example). 0 = 0.5, Q = 0.5, and k = 0.5 was not.

In order to make use of Australian data on investment for this calibration procedure we should allow for the fact that Australia is experiencing labour supply growth. The relevant meas- urement of Australian investment is therefore the increase in capital per unit of labour supply, as a percentage of national product per unit of labour s u p ~ l y . ~ Between l979/80 and 1995/96 the annual increase in capital per unit of labour supply in Australia averaged out at 1.2 per cent of net national

Coleman (1996) reworks the model to allow for growth in the labour supply, by modelling labour supply growth in the 'expanding household' fashion (see Blan- chard and Fisher 1989. p.38). The quantifications of the time required to grow out of 5 per cent unemployment for given values of 0. o and k are nearly the same as those in Table 2. Further, the increase in capital per unit of labour supply (expressed as percentage of product per unit of labour supply) for given values of 0, o and I; are very nearly the same as investment shares reported in Table 3.

product per unit of labour supp~y? This suggests we should be looking at those parameter combinations which yield in Table 3 entries of around 1.2. The two entries in Table 3 which are closest to 1.2 are 1.3 and 1 . 1 Inspection of the corresponding cells in Table 1 yields magnitudes of the time required to grow out of 5 per cent unemployment of 12.5 and 13.5 years.'" Thus, if the model analyzed in this paper is to be taken as a guide to events, it may take about 10-15 years to eliminate by economic growth 5 per cent unemployment. Such a length of time may seem unac- ceptably tardy to many.

Further, the estimate of 10-15 years is probably a significant under-estimate of the time required to grow out of 5 per cent unemployment. This is because the investment shares in Table 3 are those implied by the model under the assumption of a binding. permanent and perfectly rigid wage minimum. Therefore our calibration of parameters on the basis of Austrulian investment data between IY7Y/80 and 1995/96 is valid only i f the Australian economy in those years had been subject to a binding, permanent and perfectly rigid wage minimum. In other words our parameter calibration is valid only i f the Australian labour market in those years was such that the unemployment rate. no matter how large, would have had no impact on wage rates. in either the short run or the long run. Such a characterization con- siderably over-estimates the degree of rigidity in the Australian labour market. In order to gauge the limit of the misestimate caused by this over- estimate of wage rigidity, we will repeat the parameter calibration procedure under the polar opposite assumption of complete wage flexibility. As the first step, Table 4 presents the investment shares implied by the model under a regime of complete wage flexibility.

Sources: Capital = 'Ner Capital Stock at 1989190 prices: Equipment and Non-Dwelling Construction' (ABS Table 5221-5). Labour supply = 'Labour Force' (ABS Table 6202-1). Net Naiional Product = 'Gross National hoduct at 1989/90 prices (ABS Table 5206- I ) - 'Consumption of Fixed Capital at 1989/90 prices: Equipment and Non-Dwelling Construction' (ABS Table 522 1-5).

l o There are three other entries in Table 3 which are in the region of 1.2: 1.6. 1.6 and 0.9. Inspection of the corresponding cells in Table I yields magnitudes of the time required to grow out of 5 per cent unemployment of 8.7. 9.8. and 22.4 years. respectively.

I wwn SHOULD WE WAIT TO 'GROW OUT OF' UNEMPLOYMENT'! I67

I U 0.5 1

0. I15

0.25

0 .S

To implement the repeat calibration we proceed as before. Since the change in capital pe r unit of labour supply has been between 1.2 per cent of product per unit of labour supply. we seek those cells in Table 4 with entries close to 1.3. There are 3 such entries (1.1. 1.3. and 1 . 1 1. Inspection of the corresponding cells in Table I yields magnitude of the time required to eliminate 5 per cent unemployment of 33.2. 25.4, and 29.8 years respectively. Thus, i f Australia had completely tlexible wages in recent years. her investment experience in these years suggests that i f a binding wage minimum was now imposed, causing 5 per cent unemployment. the process of capital accumulation would take about 25 to 35 years to eliminate it.

To draw together the previous paragraphs: i f we allow that wages in Australia in recent years have been neither perfectly rigid nor perfectly flexible, then Australian investment experience in recent years, in conjunction with the paper's model, suggests that the period of time required to eliminate 5 per cent unemployment by growth alone would be greater than 10-15 years and shorter than 25-35 years. ,Perhaps 20 years is the best round figure estimate.

One other consideration makes for a still more pessimistic conclusion. Australia is not only experiencing population growth. It is also experiencing technical progress. Coleman ( 1997) shows that Table 1's quantifications of the time required to grow out of 5 per cent unemployment for a given 8. u and k are only slightly affected by labour augmenting technical progress as long as we (plausibly) assume that the binding wage minimum grows at the rate of technical progress. But if we allow for labour-

augmenting technical progress, the appropriate measure of investment in our calibration procedure is the increase in capital per unit of 'effective' labour supply, expressed as a per centage of product per unit of effective labour supply. In recent years this magnitude appears to have been very low indeed in Australia, and possibly negative. A rate of labour-augmenting technical progress of only 0.5 per cent would imply that between 1978/79 and 1995/96 the increase in capital per uni t of 'effective' labour supply was less than 0.3 per cent of product per unit of effective labour supply. Thus, even if we were to use Table 3 as our guide to calibration (the Table most favourable to growth as a remedy) i t would appear to take more than 68 years to grow out of 5 per cent unemployment,

To summarize, plausible calibration of a Solow-Ramsey growth model suggests that the elimination of unemployment by economic growth is likely to be unappeallingly slow.

The unwisdom of waiting for unemployment to be remedied by growth is reinforced by the modesty of the reduction in wages which would be required. according to the model. to eliminate unemployment immediately. I f the elasticity of technical substitution is 0.5. i t would require only a 3 per cent reduction in the wage rates to eliminate 5 per cent unemployment immediately. If the elasticity of technical substitution is 1.0. only a 1.5 per cent reduction in the wage rates would be required.I'

IV A Wu'ye Minimum us Wage Reducing It is worth noting one other cost in tolerating

a wage minimum, and relying on economic growth to solve unemployment: the toleration of a wage minimum will, paradoxically, be actually detrimental to the cause of high wages. This can be demonstrated as follows. Regardless of whether the wage is fixed or flexible, the actual wage will exceed w, only when K exceeds Kip. The wage minimum economy, owing to i ts lower investment, will accumulate a quantity of capital equal to Kfe only by a later date than the wage flexible economy.I2 Consequently, the wage in the wage minimum economy will exceed w, only at a later date than in the wage flexible economy. Thus, enforcing higher woges now comes at the cost of delaying even higher

I I

ECONOMIC RECORD J U N E I68

W J , ~ P . T later. Figure 3 compares the wage profile an economy which has a wage minimum intro- duced at period zero with the wage profile of an economy without a minimum.

FIGURE 3

A Wage Mitriniimi Entails Lower Wugcs In the Lotif Rlol

wage

I mthwt I wage minimum I

predicated upon the acceptability of a suite of familiar neoclassical assumptions, and the wisdom of judgements of certain parameters.

APPENDI x

This Appendix derives an expression for T . In the fixed wage model.

dCld1 = gc ( A l )

( A 2 ) dKIdt = 4 K -C.

( A l ) and (A?) imply,

I time 0 1

V Conihrsiotis The pJper is concerned with the suggestion that

economic growth may bc sufficient to eliminate unemployment, without any need for wagc reductions.

The paper has argued that process of capital accumulation as a remedy for unemployment i s inferior to a wage reduction because the time required to eliminate unemployment by growth alone is probably large. Plausible assumptions imply that i t may take about 20 years to eliminate 5 per cent unemployment through growth. However, the value of the analysis remains

I? Supposing otherwise leads to a contradiction. If it did take the same length of time then C(0) must be lower in the wage minimum economy. in order to allow the accumulation of K' to occur in spite of the reduction in output. Therefore, i f the wage minimum economy did take the same time to accumulate KL,..,consumption must grow at a faster rate in the wage minimum economy, in order to reach the saddle path quantity of consumption associated with K,c. But consumption in the wage minimum economy must grow more slouiy. since the rate of profit is lower in the wage minimum economy. Thus supposing the accumulation of Kfc in the wage minimum economy takes the same length of time as in the wage flexible economy leads to a contradiction.

The roots of ( A 3 ) are g and 4. thus.

K ( t ) = A ! d' + A? 8'. (AJ)

(A4) implies K(O) = A I + A:, and K(0) = ,vAl + 4A2. Solving for A l and A:. and substituting back into (A41 yields.

(A5) K ( O = [ q K ( O ) - K'cO)l/lq-,~lAI cVr

+ I lc(O, -,gK(O)I/[q-,vIr\~ (4'.

But since K(0) = yK(0) - C(0). (AS) may be rewritten, CCO)

4-s K ( I ) = - [~,l" - @/'I + K ( 0 ) PI'. (Ah)

(A1 ) implies. C(t ) = C(O)r,V'.

Since 7 is the time required to grow out of unemployment,

C(0) = C(T)e-.V r. (A7) (A6) and (A7) imply.

Since the capitalflabour ratio is a parameter when the wage minimum is binding. the quantity of capital which will employ all the unemployed, Kfp. under a binding wage minimum. equals the initial quantity capital increased proportionately by the unemployment rate.

K/c = [I+u]K(O) (A9) (AS) into (A9) imply,

That is.

I wx SHOULD WE WAIT TO 'GROW OUT OF' UNEMPLOYMENT! 169

To solve (AIO) for T we need expressions for ,v and u p c f T ) . To obtain an approximation for ,g we expand ,g = 0 Lv'(K,,.) -61 around K*.

By taking advantage of the definitional truth. \"(K*) = - \ ' (K*) [ I -H*I/K*u. and letting y* and K* be the steady- \!ate values of 4 and n, we may rewrite ,v as.

( A l l )

To derive u p d T ) we turn to the model when the wage is tkxible (and conwquently L = I ) . since at I = 7 the economy is at fu l l employment. Lineariring ( 5 ) and ( 6 ) around the w a d y state values o f K and C yields;

,v 5 e yw*)IK,, - K+I .

,v = l e / ~ l n * ~ * l I -n*IIK*-KlrVK*.

tlK,tlt = \ ( K * ) -C + \ ' ( K * ) ( K - K * I ( A l ? )

c i ~ i t i I = e y w * ) l K - K * l c*. T h i s \?\tern has the characterihtic equation.

.,? - , v' ( K * ) + c* 8 y ' ( ~ * ) = 0 .

The r u m are.

. \ , = + L - J!" c* e11/2 > o ( A 1 3 )

.\, = I!' - \.I\? - -I\" c* ell/? < o

c' = B I l J I + B, (>I'

l l w quai ion fiir c' i \ .

+ ('* The tendency of ( ' to the steady-state requires B , = 0 . Thus.

( A I J ) C = B , 1,': ' + C*.

tlClilt = .s? B , v ' ' ~

Therefore. at I = T.

U h g ( A 12). c*e [ K - - K * I = .,, B , P. tt' - -

Therefore using (A13).

B, = z c*e \ * * [ K , ~ , - K * I / Lv'-t'Lv? - Jx" CL elIl.l'r, (A15)

By taking advantage of the definitional truths. v * * = -v?[ I --X*I/II* y* U = -!'I 1-7T*I/K*U.

(A15) mny be rewritten as,

(A16) and (A I4 ) imply.

(A8). (AY) and (A17) implicitly solve for 7 in terms of a. 8. n*. y*. IK/, -K*]/Kfr. and u. Since [Kl> -_ K * ] / K * is (trivially) expressible in terms of [K,t, -KS1/K and since the magnitudes of x* and y* are derivabri'from the magnitudes of I[. 6, and k , T can be solved in terms of k , n, 4. u. and i t .

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