Demographic stipulations and earnings in drews Institutionalized Divvy Economy

DEMOGRAPHIC STIPULATIONS AND EARNINGS IN DREWS INSTITUTIONALIZED DIVVY ECONOMY

ARIE P. SCHINNAR

University of Pennsylvania

1. INTRODUCTION

Interrelations between population and economic variables have been the subject of numerous studies (Easterlin [17], Kuznetz [22], Leibenstein [23], Pitchford [28], and United Nations, [37, chap. 14], [38]). However, only recently have these interactions been explored systematically using components of national economic accounts (UN [36]) and national demographic accounts; Schinnar [31]. Stone [34], Morishima [27], Samuelson [29], and Schinnar [33] have attempted to describe the complete realm of relations among direct labor requirements of an economy, its input-output coefficients, and the subsistence requirements of a population. Fox [18, chap. 10] focused on the relations between the composition of the labor force (see also Cooper and Schinnar [12]) and the structure of interindustry transactions, whereas Artle [2] was principally concerned with the relations between the latter and patterns of household consumption.

Noteworthy among these works is Richard Stone's analysis which also extends to modeling the interdependencies between the relative size of various occupational groups and their respective shares in the distribution of income. This extension is of particular relevance to developing countries which must often consider mechanisms to effect desired realignments in the population distribution so as to meet specific economic plans. Wages inevitably are an important instrumentality in such mechanisms, and are related to "income policy" where shares of the national wealth are distributed among various demographic groupings, subject to social and political constraints.

The present paper considers one such income policy modeling possibility. It involves a version of Dantzig's [13] variant x of an Institutionalized Divvy Economy set forth by W. P. Drews [15]. ~ In these terms we demonstrate that for every stipu-

An early draft of this paper was included in Schinnar [31]. Subsequent revisions and extensions were supported in part by the National Science Foundation Grant No. SOC76-15876. The developments in this paper were inspired by the author's collaboration on related research with Professors A. Charnes and W. W. Cooper whom the author gratefully acknowledges for research guidance and helpful suggestions.

x Charnes and Cooper [9] have also completely characterized the above economy by a nonlinear extremal principle which was interpreted as a minimization of an economic potential function. See also Charnes, Littlechild, and Rousseau [1 II for a numerical example with interpretations for an index of standard of living.

2 With help from T. C. Koopmans and others; see Drews [16].

59

60 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION, VOLUME FORTY-ONE

lated distribution of income, there exists, under certain conditions, a distribution of population groups and wages for which the cost of consumer goods is in exact balance with the revenue (income) of each population group. The underlying as- sumption in this model, emanating from Drews "d iwy type economy" model, is that through changes in relative wages, public expenditures, and tax policies, a government can alter the proportions of monetary flows received by each population group, thereby bringing about a realignment in the proportions of populations associated with various work activities. Optimality and resulting equilibrating concepts are preserved but there are no (and need not be) assumptions as to competi- tion in the economy. 3 Indeed, it is one purpose of this analysis to replace the latter with a mixed system in which policy and planning conditions of a governmental- social variety can be accommodated.

In Section 2 we formulate an ecodemographic 4 accounting variant of Drews Institutionalized Divvy Economy. In Section 3 we prove the existence of an income- stipulated equilibrium in such a model. Section 4 is devoted to an exposition of the implications of various demographic stipulations for wage rates in the economy. We conclude in Section 5 with an aggregate form of the model for use in manpower planning.

2 . C O N C E P T U A L F R A M E W O R K

In this section we develop an ecodemographic variant of Drews Institutionalized Divvy Economy by reformulating the model in Schinnar [32] in terms of Dantzig [13]. To initiate the analysis, consider a Leontief [24] type economy for trans- forming resources into consumer goods. An open model is used, as is best suited for planning purposes, where the final demand vector, Y, of consumers and government is related to the level of industrial activities needed to sustain its components via a Leontief matrix (I-A) of interindustrial activities. Formally,

(I-A) X = r (n Xn) (n • (n xl), (1)

where X is the vector of industrial activity levels related to Y. Definitions and notations are now introduced consistent with those used in

Schinnar [32]. Let Y= Yg+ Yh be a decomposition of the final demand vector where Yg, n x l,

denotes the final demand associated with government purchases and yh, n• the final demand generated directly by household consumption.

p=(m • l) population vector whose components are classified by demographic groups, for example, age cohort, skill, or activity. A typical element

For a review of various approaches to general equilibrium theory and basic references see, for example, Chap. 9 in Intriligator [20].

4 The term "eco-demographic" is introduced in Schinnar [32] to describe simultaneous economic and demographic considerations.

SCHINNAR : DEMOGRAPHIC STIPULATIONS AND EARNINGS 61

Q=

With

(I-A)X---- Y, + Qp

where ~ p ~ > 0 andpj>0 . j = l

Turning now to wage cost and related income considerations, we let W ~

p~ (i= 1 . . . . , m) denotes the portion of the total population which is a member of a particular demographic group; for example, of a particular age cohort and engaged in a particular occupational or other activity (such as attending school, household activities, etc.). (n x m) consumption matrix where the element Q~j denotes the rate of per capita consumption of the ith produced (i= 1 . . . . . n) by thejth demographic (consumer) group (j----1 . . . . . m). Q~. denotes the consumption vector associated with the jth demographic (consumer) group. these definitions, we can rewrite equation (1) as follows:

(2)

(/x k) vector of wage rates and associated costs per unit labor of type i employed (in various industries). The labor is provided by the economically active portion of the population vector p--that is, by k of its m demo-

k graphic groups. ~ w~>0 and w~>0.

E---- (k x n) matrix of employment patterns reflecting the labor force partici- pation rates and providing a mapping from various industries to the demographic distribution. A typical element E~ reflects the proportion of the labor force which is a member of demographic group i and required by industry j. Use is made also of the notation E ~, E~ to denote the ith row and jth column of E, respectively.

A0 = (n X n) diagonal matrix of labor coefficients with A0~. denoting the amount of labor required per unit of activity in sector j when i=j, and A0~s.=O for all i4=j.

g = (lx n) vector of consumer good prices. V= (k x l) vector of nonwage income. S = (lx m) vector of saving and taxation.

To interpret what we have to this point, observe that E*AoX represents the total amount of labor coming from the ith demographic (cohort-occupational) group of the labor force distribution, and that this group is employed at a wage rate w~. For if the total money flow (income) to the ith demographic (cohort-occupational) group is stipulated to be proportional to some given positive constant ?~, we would have for i= 1 . . . . . k

w,E'AoX + V, = ~', (3)

where ~?-~= 1, ~-~>__0, and ~, a scalar, denotes the total "money" that flows in the economy.

Noting that Q~. is a characteristic consumption vector of the j th demographic group of the population distribution p, we express the cost of this typical bill of goods,


for the j th demographic group, as follows:

gQ.~pj + S j = ~6~. (4)

j = 1 . . . . . m, where g ;>0, ~ g j = 1, denotes the proportion of expenditures stipulated

for consumption by the j th demographic (consumer) group. We now draw on the Dantzig [13] formulation by assuming, with Dantzig,

that each industry in the economy is under an equilibrium of receipts, that is, g(I-

A)/~', and expenditures, that is, wEAof (+r f ( , where ~ r = (lx n) vector of per activity resource cost and capital investment.

This equilibrium then implies that

wEAo 4- r = g(I-A)

o r

(s)

g = wEAo(I-A) -1 + r(I-A) -~ . (6)

Next, assuming that (I-A) -~ exists, we can use equation (2) to substitute for X in equation (3) and equation (6) to substitute for g in equation (4). Then letting

H = EAo( I -A) - IQ (7) (kxm)

c - E A o ( X - A ) - ' Y g (8) (k x I)

D =-- r ( I -A ) - IQ (9) ( l •

we finally obtain

~Hp + ~G 4- V = @ (10) (k X l)

wHfl + Dff 4- S = ~0 (11) ( l xm)

where ff and p are diagonal matrices with components of w and p as diagonal elements, respectively. G is interpreted here as the distribution of employment opportunities (measured in suitable units of labor) generated by government expenditures. D designates a vector of physical resources and investment per capita (in various demographic groups). H is a matrix with elements H~j denoting the amount of labor-resource of type i required per consumer (member of a demographic group) of type j.

Questions of existence remain, which will be addressed in the subsequent section, In that section we follow Dantzig [13] and utilize the Brouwer fixed-point theorem for this purpose. This will then be extended in ways that will generalize the results to a wider variety of situations that are likely to be encountered in national development planning.

The notation ~" denotes a diagonal matrix construct from the component of the vector X.

SCHINNAR: DEMOGRAPHIC STIPULATIONS AND EARNINGS 63

3. INCOME STIPULATED ACCOUNTING BALANCE

Now assume a sociopolitical process or context which stipulates and fixes, a priori, the proportions of monetary flows. That is, let ~',, the proportion of total monetary inflows (income) received by the ith demographic (cohort-occupational) group, and ~., the proportion of total monetary outflows (expenditures) generated by the j th demographic (consumer) group, reflect a status (with respect to income and expenditures). These we shall assume can be meaningfully stipulated (if such solutions exist) in a variety of ways. In particular, we assume that a political- economic process can be devised to secure the desired distributions ~- and t~.

In our case the demographic groups act as the relevant consumers and owners of labor resources, and in this section we assume they are free to adjust their relative wages and group sizes in response to such income policies, and that they will do so in a way which produces an accounting balance (10)-(11). We also require that V, Yg, S, r be stipulated exogenously, but a parametric variation of these values can be effected to study or set government policies in order to obtain the desired balance of incomes and expenditures.

Our first problem is to show whether there exist such wage rates, w, and a demographic distribution, p, for which the accounting balance stipulated by equations (10) and (11) holds. We shall first need to rewrite equations (10) and (11) in a form where w and p are expressed as relative distributions 2 and #, respectively. To this end we let

~, = w, / (Z w,) (12)

tzj = p / ( ~ p j) (13) J

1 a ---- [(Z w,)(Z pj)] (14)

i 5

G, = G,/( Z p~) (15) J

Dj = D / ( Z w,). (16) i

Next, letting

2=_(l• vector with 2i (i=1 . . . . . k) for its components; ~ - - ( m x l ) v e c t o r w i t h ~ j ( j = l . . . . . m)fori tscomponents; J=--diag (21, 22 . . . . . 2~); p_=diag (#1, #~ . . . . . #~);

so that equations (10)-(11) can be equivalently expressed as

,~H/z + ~G + a V = a~ r (17)

(18)


where 2 ~ A and # ~ U, and A and U are the unit simplexes, that is, the convex hull defined over the unit vector:

k A ---- {2: Y], 2, = 1, 2, >_ 0} (19)

i = l

U -- {#: ~ / ~ j = 1, #j > 0}. (20) ~'=1

The following theorem is a variant of Dantzig's [13] main result? THEOREM 1: Given H, 7, 6, G, V, D, S>O, ~'~----1, ~6i-~1 and H#+G>O, then

i

there exist 2 c A, l~ ~ U, and a scalar a>O such that equations (17)-(18) hold, where A, U are as defined in equations (19)-(20).

Proof: First solve for 2 in equation (17) as follows:

,~(H# q- G) ---- a(~r -- V) (21)

or, equivalently,

2(Ht z + G) = a(~?" -- V) r, (22)

that is, [J(H~+d)]r=2(H/~+tT) where the notation ( . ) denotes a diagonal matrix with the components of the vector H ~ + G > 0 as diagonal elements. Hence, ( . ) - 1 exists and

2 -- tr(~ r -- V)r(H# + G) -a. (23)

Similarly, we solve for/~:

# = (2H + b)-xa(~6 -- S) r. (24)

Next, substituting equation (23) in (24) gives

/z = <(~r -- v)r<Ht z + G> -IH + D/a>-I(~ 6 -- S) r. (25)

Finally, since U is a nonempty convex and compact set and equation (25) is a continuous function of U into itself, by virtue of H ~ + d > 0 , so that, by the Brouwer fixed-point theorem, there exists at least one point /z ~ U which is mapped by equation (25) into itself, r

The above development assures us that for every set of incomes, expenditures, and tax stipulations there exists, at least formally, a demographic distribution and a set of relative wages such that the fulfillment of an accounting balance is preserved.

4. DEMOGRAPHICALLY STIPULATED WAGE RATES

There are, of course, problems of empirical correspondences in this model. For instance, the above formal development assumes that consumers are free to adjust their group sizes and that they will do so in ways which are implied by the

8 Dantzig's theorem is a special case of the above development obtained when Y6, S, V, r=0 and H>0. Charnes and Cooper [9] relax the strictly positive H condition to H~0.

r See Kuhn [21] and Scarf [30].


stipulated income and expenditure proportions--that is, the ways in which these incomes and expenditures are "divvied" for the economy. In fact, age-cohort group sizes are fixed and it is doubtful that the related occupational groupings would behave in the manner indicated by Theorem 1. Indeed, from a pure planning perspective it may not even be desirable for them to do so. Economic development phms are usually associated with specific manpower requirements and other demographic characteristics, so that such distributional effects cannot be looked at in iso- lation from their production consequences. In this section we explore the implications of these demographic stipulations for the accounting balance equations (10) and (11).

We assume here that the distribution of p or ~ is determined a priori and in accordance with some development plan or economic projection. The problem, therefore, is that of finding a distribution of wages w so that the accounting balance of income stipulations is preserved.

To commence with the analysis, we reconsider equations (10)-(11) in an alter- native form:

w(Hp + G> + v , = ~r"

wHp + Dp + S = ~

or equivalently in matrix notation:

(10a)

(ll)

Now let

f<Hp + c>] N-- L flH r j "

Because Hp§ the diagonal matrix (Hp§ is nonsingular and N is of full- column rank. The system of linear equations (26) may not be consistent, however. If equation (26) is consistent then it has a unique solution (see Ben-Israel and Greville [4, pp. 189-190] or Bouillon and Odel [7]).

Since (Hp+G) is nonsingular,

w r = (,Hp %- G)-I(~T -- V), (27)

and equation (26) is consistent if and only if

flHr<Hp § a)- ' (~ r -- V) = ~ear --/3D T -- S r. (28)

Hence, givenp and r and V, equation (28) specifies what adjustments in consumption expenditures, ~, and saving and taxation policies, S (that is, an income redistribution pol[icy), are necessary in order to achieve the accounting balance (10)-(11). Alter- natively, we might restate equation (26) in goal programming form (as developed by Charnes and Cooper [8]), to solve for the necessary adjustments in income imputations (~r- V) in order to satisfy the distribution of expenditure stipulations, &

I<Hp+G>] , r e r - v ] pH T jw = L ~ , _ p o T _ s~ j . (26)


If equation (28) holds, then the interpretation of the solution of equation (26) is straightforward in terms of equation (27). First, note that:

Hp + G ---- EAo(I-A)-IQp + EAo(I-A) -~ Yg

= EAo(I-A)-~(Qp + Y~) = EAo(I-A) -~ Y

= EAoX ----- pw (29)

denoting a (k • 1) demographic distribution of the economically active portion of the labor force (that is, a subcomponent vector of p consisting of people actually employed in the economy). Hence, the condition p~,>0 of Theorem 1 implies that at least one worker in each occupational category is employed in the economy.

Next, by substituting equation (29) in (27), w h e r e / ~ denotes the diagonal matrix with the components ofp~ as diagonal elements, we have

w ~ = pw-l(~r - v ) , (30)

or that the wage per worker, wi, is defined as the earnings of a demographic group, ~6i-V~, divided by the total number of workers in that demographic (occupational) category. That is, w is a distribution of the mean wage per worker. 8

When equations (10)-(11) are not consistent, we may still obtain access to approximate solutions of equation (26) by means of generalized inverse methods. Using N + to represent the Moore-Penrose generalized inverse of N, because the matrix N is full-column rank:

N + = ( N r N ) - I N r = [Hfi'H r +pL]-~tH/3,/~w] (31)

with the associated norm-minimizing least-squares properties of

w r = [H• 2Hr +/5~]-1[H/~(~ :6r --/~Dr -- S r) q-/~w(~7 -- V)]. (32)

Note that the above solution involves no loss of generality because the Moore- Penrose generalized inverse always exists and is unique. If equations (10)-(11) are consistent, the solutions (27) and (32) will coincide.

We summarize the above development as follows:

THEOREM 2: Given p and all other components as in Theorem I, then i f p~o>O and equation (28) holds, a solution for w in equations (10)-(11) exists, it is unique and of the form given by equation (27). I f equation (28) does not hoM (that is, i f equations (10)-(11) are not wholly consistent), equation (32) provides a so- called norm-minimizing least-squares solution for w?

With the above development we can explore the effects of various income

8 Related interpretations can also be effected for the consistency condition (28). 0 See Albert [1] and Ijiri [19] for a related treatment of inconsistencies in such systems. See

also Bacharach [3] who addresses a special subclass of the problem, for which H is a square matrix.


distribution policies and demographic distributions for their wage rates and other consequences in the ecodemographic system.

$. A N AGGREGATED V A R I A N T OF THE MODEL

We now proceed to examine an aggregate version of the ecodemographic system characterized by Theorem 1 in order to elaborate on certain effects which might be needed for manpower planning purposes. We obtain access to this variant via a reclassification of the demographic distribution p. To do this we note first that implicit in our previous formulation are two different demographic classifications associated with the vectors p and w. The former characterizes the distribution of consumers which are distributed along m demographic categories (k economically active and m-k economically inactive), while the latter distribution is confined only to the demographic groups which are economically active. We now effect a new distribution for p by letting p~. denote the portion of population (economically active and inactive) which is sustained by the income of the./th demographic (such as occupational) group, j-----1 . . . . . k. Thus both p and w are now distributed in k demographic categories of the economically active population (classified, for example, by head of household) and each such group is tied together via the income of its economically active population subset? ~

As in the previous section, we assume that the distribution of p or /z is determined a priori. Consistent with this condition, we now restate the problem of finding a distribution of wages w so that it conforms to the accounting balance stipulated by

~r, = ~6j (33)

which is to hold for every i=j. That is, for each demographic grouping (i, j = 1 . . . . . k), total monetary inflows (income) equals total monetary outflows (expenditures).. Via the reclassification we have just effected, however, the income and expenditure groups for each i or j are the same, and hence this accounting balance holds by definition in this special version of the ecodemographic system. The problem now is to find the consequences associated with anyp that may be stipulated or predicted since we do not budget here, a priori, the proportion of the monetary flows for each population group.

We can assess the consequences of equation (33), as wanted, by equating equations (10) and (11) to obtain

~Hp 6- r~G 4- V = fiHTw r § T + S T (34)

where ~ d i a g (w, . . . , wk) and f i~d iag (p~ . . . . , pk). Factoring out w T on the left-hand side of equation (34), we have

~0 This involves development which extends beyond the ordinary theory of economic production and income imputation and extends to the use of transfer payment and the corresponding political, social, and perhaps even anthropological characteristics concerning who should earn or produce and how much they should get or consume. See, for example, Daugherty [14].


(Hp + G)w r + V = p H r w ~ + p D r + S r (35)

where ( . ) denotes the diagonal matrix whose elements are made up of the components of the vector Hp-k G. We now solve for w as follows:

w = (Dff + S -- Vr)[(Hp 4- G) -- HI~] -~ (36) = ( D p + s - V )B

where

or from equation (29)

B=_ < Hp + G> -- Hp (37)

B = pw - - H p . (38)

To underscore the above development we introduce

LEMMA 1" Let H>O, G>O and ~ G~>O, then i

(i) B is nonsingular (ii) i f H is irreducible, B-I~O.

Proof: Since Z G ~ O , B is diagonally dominant. Hence B is nonsingular. I f i

H is irreducible B is an "irreducibly diagonally dominant"n matrix with B , ~ 0 *~ and B~j<0 for all i4=.j. Then by Corollary 1 in Varg a [39, p. 85], B-*>0 .

Lemma 1 is fundamental for our development in this section. Note that we would usually have ~ G ~ > 0 as a very realistic stipulation since G=_EAo(I-A) -1Yo. In fact,

i

we have shown in Schinnar [33] that ~G~ (the portion of employment opportunities i

in unit-labor generated by government activities) accounted for approximately 45 percent of the jobs in the Hungarian ecodemographic system, aa We also observe here that H > 0 , as required by Dantzig [13], implies H_>0 and irreducible so that Lemma 1 is a relaxation of this condition; see also Charnes and Cooper [9].

With Lemma 1 at hand we can conclude this section as follows: THEOREM 3: Given H ~ O and irreducible, G>O, Y].G~>O andDf l+S> V r as defined

i

in equation (34), then there exists a unique w>O for maintaining the accounting balance of a population's income and expenditures for all p>O, ~p~>O; w is

i

given by equation (36). If, however, D ~ + S > V r then w>O. Proof follows immediately by virtue of Lemma 1. The inequality D p + S > V T implies that the total resource cost, investment, savings, and taxes for every demographic (for example, occupational) group must exceed or equal the nonwage income of that group. In general, both D f i § V T and H-irreducible can be relaxed and still produce a unique w20 .

,1 This term is taken from Varga [39, p. 27ff.]. *~ Note that the B~i are the components of p~>0. In case somep~=0 we can drop this popu-

lation category from the accounting framework since no wage will be required for it. *a In recent analyses, Esrafill Kasraie and Leslie Goldworm have shown that Y:. Gi constitutes

13 ~ and 32 ~o of the employment opportunities in Iran and the United States, respectively.

SCHINNAR.. DEMOGRAPHIC STIPULATIONS AND EARNINGS 69

It is of interest to recognize that given a desired demographic distribution, p, the government can exercise considerable discretionary control over the wage rates by mere manipulation of Yg (government expenditures), S (saving and taxation), and so on. This means that a considerable proportion of its planning can be ac- complished by this route. But this, of course, is only a logical consequence of our developments and will generally require, as we have already suggested, accompanying institutional arrangements and organizational competencies. In any case, before fully assessing the operational viability of the above (wage control) mechanism, further empirical work and calibration will be required.

6. CONCLUSIONS

In this paper we have developed a variant of Drews Institutionalized Divvy Economy with particular emphasis on the planning interactions between economic and demographic variables. We were also able to isolate several policy parameters so that the effects of various government policies on wages and demographic realignment may be examined.

The major objective which guided this development has been the need for a framework explicitly relating demographic variations with wage fluctuations. In Cooper and Schinnar [12] we examined one aspect of such a relationship; in particular, the response of the demand for cohort-occupational groups to wage variations. The present paper, however, considers circumstances where demographic manpower stipulations are an integral part of the economic development plan so that a set of wages must be computed in a manner which on the one hand maintains an accounting balance of income and expenditures for each demographic group whi,[e on the other hand satisfying other social criteria such as an income distribution stipulated by sociopolitical arrangements.

The general literature on the theory of income distribution is largely concerned with statistical modeling and interpretation of the skewness of the distribution of incomes. 14 More recently various models of a human capital variety have also been proposed; see Mincer [26]. The "national accounts" approach pioneered by Stone [34] and extended herein provides for a departure from "traditional" income models. However, before the fruitfulness of the present approach can be fully ascertained, much empirical work is needed.

For another variant of the income-policy model adapted for long range (dy- narnic) manpower planning and analysis, see Chapter 9 in Schinnar [31]. Related extensions need be also effected along the line of analysis in Schinnar [33] in order to determine the implication of invariant accounting regularities in the structure of the H matrix (see Section 2) for income and wage policy. Finally, for further gener- alizations of the model in this paper one might also consider the line of work proposed by Charnes and Cooper [10].

:L~ For a review of the literature see Blinder [6], Bjerke [5], Lydall [25], and Tinbergen [35].


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Demographic stipulations and earnings in drews Institutionalized Divvy Economy