Efficient and Optimal Utilizationof Capital Services

By GuiLLERMO A. CALVO*

In recent times there has been renewedinterest in the factors determining capac-ity utilization in connection with the tradecycle.' However, little attention has beenpaid to the question of the desirability ofdifferent rates of capacity utilization. Tnthe present paper I discuss this problemin the context of the neoclassical one-sec-tor model when capital depreciates as afunction of the intensity at which it isoperated.

I. The Model

Let us call:

iiLt= capital services per unit of timeavailable at t (proportional to capi-tal stock)

5t= capital services per unit of timeutilized at t

Li = labor services per unit of time avail-able at t

iVt = labor services per unit of time usedat t

Ft = output at tCt = consumption at t5t=rate of depreciation per unit of

available capital services per unitof time at t

We assume:

DLt(1) = M, exogenously given, n>0 Lo>O

(2) Y = F{S, N) for all 5 > 0, .V > 0

* Assistant professor, Columbia University. Thepresent work was completed while I was visiting profes-sor at the Centro de Investigacioiies para el Desarrollo{C.I.D.}. Bogota, Colombia, during 1970.

• See Robert Lucas, and Paul Taubman and MauriceWilkinson.

(3)(a) F linear homogeneous, and con-tinuous for all S > 0, N > 0

(b) F(0, A-) ^ 0

(c) F{S, 0) = 0

dF(d)

dS> 0

dF(e) > 0 for all S, N > 0

dN

(f)

dF(h) Urn = X if iV > 0

5—0 dS

(4) 0 < wt < 1

(5) 0 < N, < L,

(6) DK, = V, - Ct - 5,Ku Ct > 0

Furthermore,

(7) 5 - g{w) for nU 0 < m < 1

where

(8)(a) g{m) is twice-differentiable at every

0 < m < 1

(b) g'(m) > 0 for all 0 < m < 1

(c) g"{m) > 0 f o r a U O < m < 1

(d) lim g{m) = «

Restricting m to belong to the closedunit interval is equivalent to saying thatcapital services used are smaller than or

181

182 THE AMERICAN ECONOMIC REVIEW MARCH 1975

equal to its availability and that they can-not be negative; m can be given at leasttwo rather different economic interpreta-tions:

a) It can be a measure of the speedat which machines are operated per unit oftime with constant labor services used perunit of time; or

b) it can measure the share of timeper unit of time at which machines areoperated, given a capital-labor servicesratio. In other words, it may provide amodel for the choice of the number ofshifts, when the ratio of factor services isthe same for every shift.

The latter is perhaps better understoodif we define iV(X', mK/N) as the flow oflabor services required to operate capitalat full capacity, i.e., when S=K, withcapital-labor services ratio equal to mK/N.Clearly, then,

K mK

N

and, hence.mN = N

Therefore, recalling (3a), output whentotal capital services are K and the factorservices ratio equals mK/N becomes

where in view of our previous definitions,F(K, N) is the full capacity output whenthe factor services ratio equals mK/N andthe total supply of capital services is K.So m can be conceived as the share of themaximum output obtained when machinesare operated m parts of a unit of time perunit of time with a given factor servicesratio. In what follows, m will be referredto as the utilization ratio.

The rate of depreciation 5 is by (8) apositive strictly convex function of m;positivity of 5 is a natural assumption tomake, whereas convexity is assumed forthe sake of analytical convenience; thecondition lim^^i g(m) = oo is innocuousand it will be used to guarantee thatefficient m's are less than unity; however,there may be some justification for it ifwe interpret m= 1 as the case where ma-chines are operated at a maximum speedor without stopping for repairs (recallinterpretations (a) and (b) above).

II. Efficient and CompetitiveUtilization Ratios

By (3) and (5) we can assure that it isefficient̂ to employ all the available laborservices;^ thus Nt= Z-t for every t.

Let us define

Kt= — = ratio of available capital services

*• to available labor services at time t

So we can write, recalling (3),

(9) yt =/(mtAt)

where f{x) = F(.v, 1)

Moreover, by (3),*

(10)(a) f(x) is continuous for all i > 0

(b) /(O) = 0

(c) fix) > 0

(d) fix) < 0 for all :r > 0

* Aa usual we call efficient any feasible per capita con-sumption path such that there is no other feasible percapita path yielding larger consumption (for a non-null set of points of the real line).

' Notice that this would not necessarily be the case ifisoquants were L-shaped.

' TjaJUng Koopmans, p. 236, has shown that if equa-tions (3a-g) hold, there is some x such that/(3:)<«a: forall Ji:>f. The latter together with (10c and d) readilyimplies (lOf).

VOL. 65 NO. 1 CALVO: CAPITAL SERVICES 183

(e) lim/'(:c) = «>I—0

(f) lim \j{x) - Ka;] = - »i-O

Due to the fact that we are in an essen-tially one-good economy, static efficiencysuffices to determine the value of Wt.Clearly, we will choose m so as to

(11) max [/MO - g{m)h]l>m>0

By (8) and (10) there exists a unique in-terior solution. It satisfies the followingfirst-order necessary condition when &i>0.

(12) ) = g'im)

In other words, efficient utilization ofcapital at a given point in time t calls forequating the marginal productivity,f imkt), to the marginal cost of capitalservices at t, g'im). Figure 1 shows howthe efficient level of m—indicated bym*{ki)—is determined. Notice that sincef"<O,f will shift to the left in Figure 1as kt is increased; hence,

(13)dm*ik)

dk< 0

Thus, the efficient utilization ratio de-creases as the ratio of the available capitalto labor services k is increased. To put itdifferently, the richer is this idealizedeconomy in terms of available capitalservices per capita, the smaller will be the(efficient) share of total available capitalservices utilized.

By (8) and (12) it readily follows

(14) lim m*ik) = 1

On the other hand, in a competitivesituation the (instantaneous) real rate ofinterest r should in equilibrium be equalto the net marginal productivity of thecapital stock, i.e.,

-m'flt.)

FIGURE 1

Moreover, it is clear that profit maximiza-tion implies mt=m*ik). Hence, by (12)and (14), we get

By implicit differentiation of (16) weobtain

(17)1

> 0

i.e., the larger is the rate of interest thelarger will also be the utilization ratio.

An interesting implication of our as-sumptions is that by (16), the utilizationratio can be determined as soon as u isknown, and that calculations can be per-formed by using function g exclusively.

Finally, differentiating (12) totally withrespect to k yields

(18)dimk)

dk

dm*ik)— - -

dk

and since dm*{k)/dk<0, we get

d{mk)

(15) - g(mt) =(19)

dk> 0

184 THE AMERICAN ECONOMIC REVIEW MARCH 197S

which means that the ratio of capitalservices used to the labor services used, mk,is positively correlated to the correspond-ing ratio of available factor services k.Notice that this is obtained despite thefact that the efficient utilization ratiom*{k) is negatively correlated with k(recall (13)).

III. Optimal Utilization Ratios

In view of (l)-(7), the notation intro-duced in Section II, and assuming effi-ciency, one can write without loss ofgenerality,

(20) Dkt = fim^kO - [gimt)-\-n]kt - c^

= h{kt) - Ct

where

(21) kik) ^f(m*ik)k) - [gim*ik)) + n]k

Clearly, fik)-nk>fimk)-[5-\-n]k, andhence, by (lOf),

(22) lim k{k) = - 00

On the other hand, recalling (12), for allwe have

(23) k'ik)=fim*ik)k)[m*'ik)k-\-m*ik)]

-g'im*ik))m*'ik)k-g{tn*ik))-n

=fim*ik)k)m*{k) - gim*ik)) - n

= g'{m*ik))m*{k)-gim*ik))-n

hence, by (13) and (23), for all A>0,

(24) h"{k) = g"im*{k))m*ik)m*'ik) < 0

For any given 0 < m < l , (10) can beused to show that there is some ^>0 suchthat for all 0 < -fe < A

(25) fimk) - [gim) -f n]k > 0

while

(26) fimk) - [g(m) -j- n]k = 0 for * - 0

Hence, since m*ik) solves (11), then (22),(24), and (25) imply the existence ofsome k>0 such that

FiGUKE 2

(27) hik) > 0 if and only if 0 < k < k

Furthermore, by (8b) and (26),

(28) hiO) = 0

Figure 2 depicts a function satisfying (24),(27), and (28).

We are now ready to prove the first cen-tral result of this section.

PROPOSITION 1: There exists one andonly one golden rule capital-labor ratio, kg.The corresponding unique golden rule utiliza-tion ratio mg satisfies

(29) g'im)m. - gim) = n

PROOF:From (20), golden rule consumption per

capita Cg satisfies

(30) = max hik) =*>o

By (24), (27), and (28), k exists and isunique. Also hikg)>0 (see Figure 2).Clearly, by (23) and (30),

VOL. 65 NO. 1 CALVO: CAPITAL SERVICES 185

(31)

- nBy (8), g'im)m—gim) is a strictly increas-ing function of m, 0 < m < l ; hence, thereis a unique m for which (31) holds. Exis-tence of Mg follows from the fact thatmg=m*ike) for which existence has beenproved above.

Remark: Equation (16) above and Proposi-tion 1 imply that, as usual, the (instan-taneous) real rate of interest in the goldenrule path equals the rate of growth oflabor. In fact, given the latter and thedepreciation function, one can use (16) todetermine m^. The golden rule value of theutilization ratio becomes, under the pres-ent assumptions, completely independentof the production function.

The present model can easily be suitedto study optimal utilization ratios in anoptimal growth plan. Consider the fol-lowing problem:

'L'"^(32) max

subject to Dki=hikf)

*t > 0, Ct > 0

ko given

> 0

for every t

where « is assumed to be strictly concaveand increasing, and to satisfy lim^^o uic)= — =0. P'unctions associated with thesolution of (32) will be called optimal. Letus define k{a) as the value of k such that(see Figure 2)

h{<j) exists and it is unique. Existence isshown in the Appendix; uniqueness, onthe other hand, follows from (24).

PROPOSITION 2: For any initial capital-labor ratio k>k[i>0,^ there exists a unique

* Recall the definition of k in (27).

Optimal path of k's; kt converges monotoni-cally to kia). The associated mt=m*(^t)converges monotonicaUy to m*ik{(r)).

PROOF:Existence and monotonic convergence

of k will not be proved here since it in-volves a trivial application of results ob-tained by Koopmans, for example.

Monotonicity and convergence of mtfollows directly from the properties of theoptimal path of ^'s and (13).

Remark: Since by (13) there is an inverserelationship between m and k, we havethat if in an optimal plan ko<kia-)[ko> k{(T)], then m.o> w*(i(ff)) [nto < m*(/̂ (o-))];therefore optimal ki increases (decreases)and optimal Wt decreases (increases) overtime. By (23) and i33) we have

(34) c = h' = g'im*ikia))m*ikia))

- gim*iHc))) - n

But, by a similar argument as in the pre-vious Remark, one can show that there isone and only one m for which (34) holdstrue. The asymptotic value of m is againwholly independent of the productionfunction.

In view of (16) and the results of thissection it would be interesting to fmd outabout the family of functions s(m) satis-fying (8) such that

(35) z'{m)m — zim) = g'(m)m — g(m)

for every 0 < w < 1

Tt can easily be shown that every functionbelonging to that family can be expressedas

(36) s(m) =g{7n)-\-bm, for some b such that

— bm<g(m), for all 0 < w < l

Therefore, golden rule and optimal utiliza-tion ratios are not going to be affected bychanging the depreciation function fromg to z (satisfying (8)) as long as z'im)

186 THE AMERICAN ECONOMIC REVIEW

— g'{m) = b-constant for ail 0 < m < l ,(A8)

or equivalently,

MARCH 1975

lim = 1

APPENDIX

LEMMA: Let a->0; then there is some k>0such that h'ik) = a. (A8')

PROOF: „ ^ ^ n .V, •Taking into account (22), (24), and (27), ^ence, for every e>0, there is some

it is sufficient to show

d In gim)lim = 1m-*i dm

(Al) lim h'ik) = 00

0<m(e)<l, such that

d In gim)(A9)

dmBy (14) and (23), the latter is equivalent to

(A2) lim [g'im)m — gim)] = oo

We will show that (A2) holds given that gsatisfies (8). First notice that

(A3)dm

lg'im)m-gim)] = g"

for all 1 > m > m(e)

Therefore,

(AlO) In gim) < In githit))

-{• i\-\-e)im - 7h{t))

for all 1 > m > fhie)

which is equivalent to

Suppose now that (A2) is false. Then by (All) g(m

(A4) lim [g'(m)m - gim)] = B,

exp

for all 1 > m >

Hence,

B a real number (A12) gil)<g{rh{e)) exp

Mojeover, by (A3)

(A5) B> - g(0)

Due to C8c and d) there is some m < l suchthat

(A6)(a) gim) > 0

(b) gim) is increasing for all 1 > w > m

Thus (A4) is equivalent to

[ g'im)m "I^^7— ' 1

gim) J= B

The latter implies, recalling (8d), (A5), and(A6),

contradicting (8d). Therefore (A2) holdsand the Lemma follows.

REFERENCES

T. C. Koopmans, "On the Concept of OptimalEconomic Growth," in The Econometric Ap-proach to Development Planning, Amster-dam 1965.

R. E. Lucas, Jr., "Capacity, Overtime and Em-pirical Production Functions," Amer. Econ.Rev. Proc, May 1970, 60, 23-27.

P. Taubman and M. Wilkinson, "User Cost,Capital Utilization and Investment Theory,"Int. Econ. Rev., June 1970, 11, 209-15.