General micro behavior and optimal macro space-time planning

Regional Science and Urban Economics S (1975) 285-323. 0 North-Holland

GENERAL AUCRO IEIEHAVIORAND OPTI1MAL MACRO SPACE-TIME PLANNING” Walter BARD

and Panagis LIOSSATOS

Depurtment of Regional Science and Department of Peace Science, UniLtersitj oJ Penwyk~ania, Philadelphia, Peim, 49174, U.S.A.

1. Introduction This paper aims to extend the traditional general interregional model of micro behaving units to: (a) incorporate dynamic aspects associated with investment and space-time development; (b) introduce government operations into the frat#lework; and (c) formulate a macro model useful in the context of a central planning agency which is consistent Jvith optimizing behavior on the part of individuals and organizations. In section 2 the static interregional general equilibrium model is extended to incorporate investment in production and planning by firms and individuals over a time period. The effects of balance of payments constraints are examined for a set of discrete regions. In section 3 the model is extended to cover government operations with respect to both the production of goods and their dstribution among constituents. Thereby a limited amount of regional planni.rg is introouced. In section 4 the central planning function is exar lined with respect to a set of discrete regions where the central plan concerns only those variables a:.ssociated with. micro behaving units. We introduce an Ombudsman’ to insure that the plan’s allocation of goods to consumers is consistent with their optimizing behavior. In section 5 the central planning model is formulated in continuous sIbace. This extension then permits us in section 6 to design a more operational, central planning model - in essence a macro space-time development model which is consistent with optimizing micro behavior. In section 7 the macro central planning model is extended to cover commodity flows generated by development force,s and possibilities. This permits the *Research supported by l%.tional Science Foundation grant t 751018. *An Ombudsman is typically conceived to be an independent and impartial arbiter between govcmment and the individual. Our Ombudsman performs this role by maximizing his payoff function.

rt:placement of the more limited concept of mtirket equilibrium with new concepts more effective for a centrzi planning agency concerned with the dynamics of development. This also suggests ways of internalizing externalities ;n !o the maximizing behavior of micro ui?its. In section 8, some concluding remarks are made. Throughout the paper we alre primaril:; interested in dcveIoping the forma!if;m far the fresh insights that it can provide, and in exzwining the re:nsonableness of the conditions for optimality. In effect we assume that soiu?ons exist for the various optimization .I)roblems posed i:a this paper. In a later manuscript, the technical problems of existence, uniqueness: and stability should be thoroughly explored. 2. The general interregional model extended to incorporate investmerit As fully detailed elsewhere [Isard (1969, ch. 1i)], we assume there exist & commodities [/z = 1, . . ., l, where 1-2 represents tcatisport inputs (e.g., in ton-miles), I- I represents money, and 2 represents labor], U regions (J, K, L = A U) each containing m consumers (i =T 1, . . ., m), n producers 0’ = 1, . .‘., $, and f traders (J= 1, . . .,j’). E ac h consumer i, J desires to maxiLtGze his utility over the relevant planning horizon (time period) t = 0 to f = t,, wilere at time t his utility (ordinal) per unit of time is given by

where ei = bid?) and is the purchase per lrnit of time of good h at time t, and where t representing time in eq. (1) indicates that tastes may charige with time. Correspoklding ,:s the set of’ I commodities in each region J, there is a set of price functions, ~3; = pi(t), 0 5 t s t,, where pi(t) is the price of good h in region J at time t. For any given set of prices pi(t), pi@, . . ., pf(t), 0 5 t 5 z,, at the market ot’ his region, i, J is subject to a budget constraint pertr .aing to the entire planning period! 2

(2) ‘Strictly speaking, we should add the terms

+ c a/L max 10, QL] +IL brfJL max [O,n/l

i. L

9

where rlJ”- and fiiJJL are i, J’s fractional share of the profit fl,’ of th, :irm_i, J over the entire planning period, and of the gair 3 from trade IIYL of the shipper f, L over the entire planning period, respectively. Since the optimal solution implies zero profit and gains from trade over the planning period we have not included these terms in 42); in effect we assume that positive profits realized in any umt of time during the planning period are held as ‘undistributed profits’ subject to ~0 interest rate. While this it: clearly not rhe best assumption that can be set down, it makes it possiblf 3r u; to avoid extremely complex notation Also, as with our other manuwipts we tic it convenial to assume a zero discount (upc~unt) on utility, and correspondingly a BYO interest rate on money (fluid capital). This per nits a simpler notation and avoids -in conceptual difficulties when both planning and ec inomic theory are fused as in this papee. The reader can adopt a different convention if he so desires.

W. Isard and P. Liossatos, Extension of micro behat-ior model

287

where gii = gii(t) represents the stock per unit of time of good iz which i, J holds ‘initially’ at time t and sells (supplies’) to the market at that time. Each producer j, J desires to maximize his profits over the planning horizori taking prices as given and subject to the transf’orrn;t.Ganfunction,3

(bi3 (Y J159 ...,y~;K:j,...,K~;t)=o, I

where

yij

,I$(z) represents an outp!lt of commodity h per unit of time when positive, and an input when negative; where K,Ji = K&(t) represents the stock of good k embodied as capital at time t; and where f indicates that the transformation function may change over time. At any point of time t, j, J’s profit (excess of current reveres over current expenditures) per unit of time is given bY Z$ = 1 Ly

~{(y;li-X{j)

l

(4)

Jl

However, over the planning horizon, j, J may realize gains (losses) by having accumlrolated (used up) capital at f = t, relative to initial endowment at t = 0. Hence, total profit over the planning period is defined as

(3 and total profit per unit of time may be viewed as

Each trader (,exporter),f, J desires to maximiri: the gains from trade over the planning horiz;sn, taking prices as given and subject to the condiiions that any 14’ 0 f good h to any region L must be non-negative. Per unit of time export shf his total gains from trade G! is

where the gain from trade per unit of commodity h at time t is

3Alternatively we might write K,‘; 0 = 0, #,‘(Y%,r m*Yc,; where K,’ = K,J(KJII, * . ., Kcclj) represents capital stock as a whole. Or we could consider c?f one, each a function of (KJIJ, severat general types of capital stcxk KJlr, l

l

KJjJ,,

. . ., KJ,J).

.

.

.

instead

W. Isard and P. Liossatos, Extension of micro behaoior model

288

Here oh is the ideal weight per unit of commodity h, and dJ”L the distance betweenJandL.” We find it convenient to think in terms of a fictitious market participant in each region whose behavior simulates the operations of a pureiy competitive market. He sets prices taking the demand and supplies of the various behaving units as given. 5 Per unit of time his payoff at time t is given by

cPi&

(9)

h

Here 4 is the excess demand for h at the market in region Jand is defined by

where’ JdL SW

=

s;;L+c5;-

2

c h’,L+J

o,dJ4Ls;tT;L.

Here l$,- 2 is the Kronecker delta which takes the value of unity when h = Z-2, and zero otherwise. In this system we assume each behaving unit maximizes his payoff. Each household consumer i, J acts to

(12) subject to (2). We thus form the Lagrangian (13)

to yield the necessary (first-order) condirions

where 2; may be viewed as the margin-cLIutility of income, which in this mar!-.i is independent of time, which simply reflects that the budget constraint !‘+athe individual pertains to the entire planning period. Each prcducer i, J acts to maximize total profits [defined by (5)] with respect ZGJ*&and K{j subject to (3). Accordingly we form the Lagrangian

“See lsard et a!. (1969, pp. 228-534 and 704-705) for further discussion of these conventions. 5For example, see Lange ( 1938). ‘isad et a!. /1964, pp. 542-543).

CV. lsard end P. Lkwatos, Exrension

ofmicro behavior model

289

where $ fs defined by (4) to yield the necessary (first-order) conditions’

Given that $5 is an implicit function, (16) and (17) yield

w our basic investment principle. See Isa rd and Lkssatos Each shipperf, Jacts to

(1975b). 8

(19) subject to $7” 2 0.

(20)

The first-order conditions are9

W The market participant in each region J acts to max sz pJhzidt ,

(22)

{i > subject to pJ # 0.

p; 2 0;

(23)

The first-order conditions are’ ’

(24) ‘Eq. (16) comes from setting C?LFjb/ZjJ~J = 0;

and (17) from the Euler-Lagrangian &P,JpKJ

RJ =

equation

dldt (i?YjJ/?kJ,,.i).

Wbserve that profits flJJ = 0 if tij is characterized by const-ant returns to scale. (t) = 0, the partial of GfJ with respect to 91f a stationary value should occur at s,,~~‘+~ s’ k!J J+L must not be positive if its value is to be a maximum. Should the stationary value occur > 0, the partial must be zero. at shf J-rL loSee previous footnote. Note also from (18) that z,,*’is a function not only of the phJ(t) but also ofbhJ(t) and &J
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The interregiot,a! economy just described can be shown to contain as many independent relations as unknowns, there being W- 1 independent pkes.‘” Additionally we wish to be ;!i>leto encompass multi-region situations where there are two or more different currencies among the several regions and where we require balance of payment .. for each region. Hitherto all prices in any region J were stated as values in terms of a general unspecified unit of the multiregion system. Now WLmust define commodity I- 1 as a commodity universally designated as money, and which in the light of history we can conveniently specify as g4d, or some other ‘currency’ of limited qua:ltiiy. It immediately follows that one unit of money (or gold) defined in standard physical terms will have different prices in the different regions, specifically, p;' . 19 . . ., py_ I. Also, it is necessary, as wil! be discussed beiow, to take the price ofgold in one region, say region U, as arbitrarily set, that is to serve as a numeraire. It then follows tha’ the ratio

PL 35

vllue/unit of gold in J Z--W*+a rtlre/unitof gold in C7’

J=A,...,U-1,

represents the gold exchange rate which specifies the amount of geld in W which exchanges for one unit of gold in J. Suppose also that in region U we define the currency unit to be equivalent to one unit of gold, and that we do so for the currency of every other region. Then we can restate the previous ratio as p:_,=

PI”-!

value/unit of Jcurrency value/unit of Ucurrency

J=

A,...,U-1,

’

which represents the cuvrelzcy exchange rate, and tells us how much U currency exchanges for one unit of J currency- Note that by our convention the gold exchange rate and the currency excha;lge rate are the same; so tie can speak generally of the exohange rate. It also follows that P:_l = value/unit of gold-=in J Pk-I

value/unit of gold in L

value/unit of J currency value/unit of L currency

is the exchange rate specifying how muf:h gold in L (L currency) exchanges for one unit of gold in J(Jcurrency). With these conventions the price of good 11in region r in terrm of cumency of J becmes pilpf-,, that is it specifies the amount of J currency per unit of 11. We al ;o must redefine gains of trade for shipperfin region .I. To him, the relavant price in a region L to which he consi&rs exporting a good is pk/pf_ l x pk_ 1 /pf_ 1 1See lsard et

al. (1969,ch.

11).

W. hard and B. Liossato~, Exterlsion of micro belwior model

or p&:_ I, that is the price CCthe god in the receiving region in (his own) currency. The price he pays (in J currency) on his region’s the good is p#_, . The cost of the transport inputs he purchases region is pf_ 2t’pp;‘_ I (hcodJ+L’ J. So to him gains of trade per unit of currency, -.C-*L

z

Zh

_

Pk

Pi-1

w!i!i_ __‘k(whdJ’L). PL

Pi-

_.C

*1

terms of J market for in his own h arc, in J

05)

1

Observe from eqs. (8) and (25) that J-+L

=

?h

p;_iz;4L,

(26)

where T;-‘~ is in terms of some number of the general unspecified units; or general rdue units. KLJWwe wish to introduce a balance of payments constraint for each region of the system. We define region J’s gains from trade &” in terms of its own currency as -J = Q

(27) KfJ

where the first term on the right-hand side is total value oi exports in terms oi J’s currency and thus represents a demand for J’s currency on the world currency market, and v here the A+.,~& term is total value of imports in terms cf J’s currency and thus represents a supply of J’s currency to the world currency market. If we multiply both sides of (27; bye:+, we crbtain

e

(28) LSJ

LfJ

a definition of regionJ’s gains from trade in terms of general value units. For the enrtirc planning period we wish to maximize system gains from trade G per unit of time, max

j: Gdt,

f J-ct

-PI,/

where

G= CG i, .f,J

>

-J4L > subject to Shs - 0 and eq. (28) where we take the QJ as given constan’;;, J T=A, . . .c,U-i $ Sinc.e Qlursystem is closed,

292

W. lsard and P. Liossatos, Exrsttsion ofmicro bchauhr modei

and Q” is determined once the constants QJ, J = A, At the end of this anal;!sis we can set Q-’ = 0 for aI1.J. We form the Lagrar: @an

. . *, U-

1,

are

specified.

A; discussed above, there are only U- 1 independent constraints in (28). However in eq. (31) we sum over all J. So we take some ~9, say ~7~= 0, Differentiating (31) with respect to $TL yields the first-order conditions

What is the significance of HJ? Suppose that corresponding to a specified set of Q’, we determine an optimal solution satisfying (32) and (33). At the optimal solution. G = 2’; and hence we obta,in %Gj2QJ = -d.

(34)

Thus I+~, which is a pure number, indicates howI, in terms of general value units, the optimal gains of trade for the system changes as the constant QJ !also in general valrle units) is allowed to change. More concretely, since Q1/p;_, = QJ9 we may divide both numerator and denominator of the LIiS of (34) b) p;_ 1 to obtai3

which gives the system gains of trade (when cxprcssed in J’s currency) from a change m the eJ (also expressed in terms of J’s currency). The full significance of hJ, hotvever, is best seen from examining what happens when region J trades with region U, the region in which the price of gold is arbitrarily set; and for u-hich we have taken I?’ = CL(Also region #IIserves as our general reference region with n-hich al: regions .I trade.) Shipments of gold from J are in effect associated with such trade, and specifically we take s;a$ > 0, for some f and J = A, . . .+ U- I. Bearing in mind th;lt transport cost on gold may be taken to he zero (as when central banks clear balances among thems&es,). we have from (8) and (33)

W. hrd

and P. Liussatos,Exteursion of micro I~elxnviormodel

293

But. as already discussed, i#’ = 0. Hence from cq. (37) we obtain

=‘:-l_,

tar”

PL

J=A

’

)

.

(1

.,

u-1.

W

Thus 1~’is directly related to the exchange rate of’currency J with the standard currency of region U. So (32) can be rewritten

To interpret (39) divide both sides byp/_

1

to obtain (40)

or, using (26),

Now from the discussion of (25), ?i’” is in terms of J’s currency, and so is phLlpl_f. Thus the gains from trade eq. (25) are to be adjusted by a term expressed in terms ofJ’s currency since each ratio within the parenthesis is a pure number. When no balance of trade constraint is binding, uq = 0, for all J, and the second term of the LI-IS of (41) drops out. From (41) we 2.1~0see that for $iL : I 0, TX-‘= can be positive but only if [(pf- r/p,“-rf - (J$-_I ly,“-,)} is negative ; and ii’” can be negative, but only if ((p~_,/p~_ 1)- (p,“_ 1 /pf-,>> is positive. It :;hus is ciear that the expression in parenthesis reAects the strengths of J”s and L’s currency relative to each of her. If J’s currency is stronger than L’s? that is exchanges for more dollars (when the dollar is the standard world currency) per unit of L’s, then there is a ‘gain’ from currency exchange operations which offsets the ‘loss’ from $“* < 0. If J’s currency is weaker than L’s, that is trades for fewer dollars per unit than L’s, then there is a ‘loss’ from currency exchange operations which must he of&et by the “gain’ ?i’” > 0. Extension to cover limited regioriralplanning and government production and distribution

3 .*

We now wish to introduce government production and distribution. For notationtif convenience we assume that there is one and only one government in each region.’ 2 The government is assumed to operate both within and without *The extension to different numbers and kinds of government units is easily done. See fsard et al. (1969, chs. 13, 14,14A). l

‘*

W, hard and P. Liossaros, Extension ofmicro beharkw model

293

the market system. When it purchases inputs, it does so at its region’s market; and to obtain the wherewithal for these purchases it levies taxes on constituents in its region. When it handles its outputs, it does not self them at the market but distributes them directly to consumers (and in this connection we assume that what is distributed for consumption by one individual cannot be distributed for consumption by a second individual). In this latter respect it operates outside the market. Since it is concerned primarily with constituent welfare, it must estimate the utility (individual welfare) function of each of it: constituents and somehow aggregate them into an index of social welfare which is to be maximized. Each i,J with his tax base TBf defined by the government of his region confronts ;i tax rate rf, 0 5 r: < 1, and receives a set of government goods and services &‘,+&+, . . ., +‘ij’ all determined by the government. His utility and budget cons’traint must be redefined as

(42) and (43) respectively, where 1Nedefine TBJ as

The first-order conditions for his maximizing utility are given by (14). Using eqs. (14) and (43), we obtain bii, the amount of good lzpurchased by !, Jas bi!i

=:

bii(pi,

. . .,p;;

r’; )$+, . . .,f$+;

t).

The production activity of the government in region J is constrained by a transformation function

where {f’

represents the production per unit of time of program e, e = 1, * . ., i;; f{represents the input per unit of time of commodity h, h = I, . . ., I, all of which inpu.t is purchased at the market at J; and K&, represents the amount of commodity h embodied in its capital stock.13 For && to be positive the government must purchase commodity h at the market in J. convenience, we assume zero depreciation rate. The reader can easily introduce depreciationinto the model. See Isard and Liossatos (f975a). * 3F~t

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In addition, the government’s operations are s&ject to a budget constraint over the planning horizon. Assuming that al; its tax revenues come from C-X tr1.xrate r: applied to the taxable base TBf of each constituent, we have

where the first term in the in&grand represen!.s tax receipts at time t per unit of time; the second, government expenditures on current account; and the third, government expenditures on investment account. Once government goods and services are produced, they must be distributed to all i, J. (For convenience, we assume that no government goods and services are distributed to firms, traders and fictitious participants.) Each i, J k’eceives a share yfi of the goods and services associated with any particular program c’, such that

In determining its allocation, J+

Pei

=

l&f:+?

(49)

to any consumer i,J, the government is concerned with maximizing its contribution to the welfare of its constituents. We therefore assume it estimates the cardinal utility for each i, Jas

and somehow constructs an appropriate welfare function over the estimated utility functions of all its constituents,f 4 WJ(t)

=

WJ[u”ll@),

. . ., iii(t);

t],

(51)

setting the marginal social welfare GVJ/a$ from increase in utility of an;,&’ constituent i, 9 as positive. Taking the bii as given and bearing in mind thr: definition (49), it maximizes the region’s social welfare over the planning horizor:

s’,lWJ(t)dt, l STE,~le are of cc’ucse numerous diffkxlties in constructing such a welfare function and much debate in the field oi welfare economics on how this should be done. “8e wish to avoid being involved in such debate, and therefore adopt sim,plistic approaches. The reader is free to adopt others. No? : a!so that we assume that utility is additive over time and later, over space.

296

W. lsard and P. Liossatos, Extension of micro behasior model

with respect to r{ yii, fi+,+‘L- and K;ih, subject to (43, (46). (47) and (48) and appropriate initial conditions. Forming the Llqrangian

+ pJII/J+ pJ 9r; = CZ’J 1

we obtain the necessary first-order conditionc

(56) _pJ.!.$

=

#Pi 9

h

(58) We now interpret these first-order conditions. Since xi is the social marginal welfare of income to i, J, as estimated by the government, eq. (54) equates this marginal social welfare for all i in the region J through difherential taxation a consequence of a taxation scheme consistent with optimality. Eq. (55) requires that the marginal social welfare of government goods and services from any program e be the same for all i in J; and the government is able to achieve this condition by varying the yfi*(Both,{:” and e: are independent of i in J). This equation also defines
W’. Zvard and P. Liossatos, Extension yr micro behavior model

297

or, equivalently,

where BP;;‘- is a small change in+:- and A&+ is the corresponding small change required by (16) when all other variables in (46) are held constant. Recalling that # = 2: = marginal social welfare (utility) of a dollar of income to each and every i, J, and Cat pJhis the price in terms of dollars of a unit of input of Ii, we interpret pJpi as the cost (in terms of social welfare forgone) per unit of iz used as an input. Thus the RHS of (60) corresponds to the cost of the additional input - A{‘-, that is marginal cost in terms of social welfare; and the LHS of (60) corresponds tc the social welfare value of the additional government output dpeJ+, or marginal revenue in terms of social welfare. Thus (60), and also (59) to which it is equivalent, is a marginal revenue = marginal cost condition. By dividing both sides of (59) by ph, we can interFret (59) as requiring that the last dollar spent on each input on any program yield the same marginal welfare, namely pJ, which is also equal to A:, the marginal social welfare (utility) of a dollar of income to each i in 9. Dividing (58) by (56) yields

(61) Multiplying both sides by dd and integrating from t to t,, we obtain

which is our basic government investment principle wherein the cost of a unit of capital (marginal social welfare foregone at time tj must be equal to the value of the marginal product of the unit of capital summed over all points of time from t to tr plus the scrap value of this unit at t, .’ ’ At this point we can fully embed government behavior regard,ing production, taxation and distribution within a general equilibrium framework. Assume all producers, shippers and fictitious market participants behave as described in section 2, and that consumers behave as indicated in this section and section 2. All we need to do then is to change cq. (IO) IO rcatl

15See Ismi and Liossatos (1975b).

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298

and to consolid8atethe discussion of this section and section 2 into one statement. t

4. A central plranningm~Iel (micro) with many discrete regions In this section we develop a model having in mind the needs aall operations of a central planning commission. We also shall lay the groundwork for a fusion of macro and micro models in sections 6 and 7. Assume that, in addition to all the behaving units, commodities, and regions of sections 2 and 3, there exists a central planning commission motivated to maximize the social welfare of the individuals over the entire system of U regions. It takes system welfare to be a function of regional welfares,

w=

W( WA, w”,

. . ., WV; t),

WV

where IIY-’= It

W"(trf

, . . ., ii;;

t) ,

” (69

J = A, . . ., U.

aims to max 1: Wdt,

with respect to all variables: 6$, rij, KiJ, s&‘~, t$,,&+,+~-, I&,, y$, subject to constraints (3), (20), (46) and (48), appropriate initial conditions and also

J = A, . . ., U, J = A,. . ., U.

(67) (68)

In the above, hii refers to the allocation (delivery) of good lt to consumer i, J,’ 6 and rii refers to a direct tax on the ‘initial stock’ of good /Zheld by i, J at time t. It specifically is not concerned with and does not consider a budget constraint for each i, J. We form the Lagrangian

-

c

f,L#J

*6This aliocation is distinct from the b’,,, which refers to a purchase decision by consumer i, 3.

K

hard L_ndP, Liossatos, Extensbn

of micro lehucior model

299

are costate variables.’ 7 The first-order conditions

U@)

(71) (72) (73) (74)

and recalling that fzJ = ~$$’ is an argument in th,e utilty iunction uiJ, we have, keeping (64) and (65) in mind,

By eq. (70), we set that the central planning commission should allocate goods for consumption among the i in any region J so that the marginal social welfare (utility) of the last unit consumed by each i in J is the same, atid equal to the costate variable phJ. As discussed in isard and Liossatos (1975b), pi is the imputed price of a unit of capital stock of good h. It also serves as the price of good /z in J whether that good is consumed, from which therefore marginal social welfare from serving consu.mer i in J is Gbtained, or whether invested, in which case marginal social welfare from consumption by i in J is foregone. ’ ;‘A11costare variables except VhJ have already been used in the prcvlous sections. While there is a one-to-one correspondence between their use there and here, the specific magnitudes and dimensionsassociated with them will be different.

W. Isard aud P. Liossatos, Extension of micro behavior modei

300

From (71) a.nd (72) which are formally identical with ( 16) and (17), respectively, we derive (18), our basic investment principle, which then relates the costate variable 1~::to the value of marginal product of capital. Eqs. (73) are the conditions of optimality for trade as discussed in section 2. Dividing eq. (74) by (75) yields

Multiplying both sides of (79) by dt, integrating, and using definition (49), gives

our basic prinsiple for government investment. Note that the imputed price of government capital vi(t) is approyriately related to the social welfare value of its marginal product. DivP- lg eq. (7t;) by (75) and using definition (49), yields .----

z:

S-W

which states that the social welfare of the marginal product of any input I2 must be the same over all programs e and specifically equal to vi, tb.e imputed price of lt as government capital. This condition parallels (59), a marginal revenue = marginal cost type of condition. Eq. (77) Aates to taxation of i, J. It staltes that each i, J should be taxed up to the pokt where the social welfare value of the lkst unit of good h taxed away from him - which value is pi: and is the same for all i in J- is equal to the imputed price (the social welfare value) of that good as government capi:A. Finally, e-q. (78) relates to the distribution of government goods associated with any program e. It states that these goods should be distributed in such a way that the marginal social welfare stemming from the consumption of the last unit that i, J receives is the same for all i in J arid specificall,y eyu21 %o <;!/f+, atl average social value of the goods associated w,ith a unit of program 4. Observe that the model of this section concerning micro behavior identifies optimal magnitudes for variables associated with individual behaving units. For each producer j- in J, whose transformation function 4: is known to both the central planner knd himself,’ a conditions (71) and (72) are also conditions f’ r his maximizing profits as defined by (5) whenever the central planner lBThis is a very strong assumpiiort. Additionally, we assume here and eisewhere in this pap3 ftio cost in obtaining, pr0c :.sslng, and transmitting information.

W. hard and P. Liossatos, Extension of micro behat:ior model

301

announces p:(t) as market prices. For each shipper f in 9 whose shipment must always be non-nega.tive, conditions (73) are also conditions for his maximizing gains fi*om trade as defined by (7) whenever the central planner announct
where qii(t) are functions that are variables which he controls and changes, taking the 6ii and b{i as given. Accordingly his optimality conditions are

(83) which are interpreted below. In our redefined system, the centr;: planner is sens’tive to the Ombudsman and incorporate> the q$ in his estimate 2: of uf. 111particular we assume that

i 9Eqs. (682 rep laze (47). B

W. Isard and P. Liossatus, Extension ofmicro behavior model

302

FurtLcr we take

where the assumed cardinal utility function fi&has the usual properties, namely, positive marginal utility and diminishing marginal utility. Moreover we assume additive -welfarefunctions, namely

and

Note fro,% eq. (85) that for fixed prices, ShJiis a decreasing function .Lifq[imSO The central planner now identifies his optimal path by assuming the 3s given functions. Accordingly, all magnitudes he computes are functions of these functions. In particular,

and

Individual behaving units make their computations functions, For example, i, J finds b!i

=

bhJi(.ms, pi(t),

l

m , ; t). l

taking

p;f(t)

as

given

(9W

On account of @Q-(90), eqs. (85) determine t-he functions qhJi(t); and in turn P,‘(Z),Giji(,l),bhJi(t) ancl all other quantities can be specified. Operationally we may imagine the process to take place in stages. At the Start (i;Ii ‘= 1, for all t, h, i, and J. The central planner announces the b,J,(t), pi(t) and other elements of his plan at the end of stage I. The consumer i, J is then able to calculate his time rate of income xhP{(l)ghJi/t) and know his budget constraint. Given his preference structure, which we assume can be represented by an ordinal utility function z&t), he chooses and announces at the end of stage II the set Of b{i(t), that is the consumption plan over the time period which he most prefers. The Ombudsman then compares the 6,5iand bnJiand varies the d!* He increases L$ -.KLcneverE;ii>&decreases yii whenever bii
of positive and diminishing

W. lsard and P. Liossatos, Extension

ofmicro behavior model

303

During stage 1V the central planner does a completely new calculation of his plan in accordance with the new Gf (associated with the announced values of the q,“,,, which he now takes as relevant. Since S{i is a decreasing function of (I:- 21 the new & which he announces at the end of stage IV are presumably c&er to the b:*than the 6li which he announced at the end of stage II. He also announces a new set of J& to which each i, J responds with a new set of bhJi,and which then leads the Ombudsman to announce a new set of q&. It is reasonable to expect, although we cannot rigorously prove this, that the set of actions and reactions leads to the convergence of (ii,-bhJi) to zero after a series of rounds.2 2 This outcome then yields a central plan which allocates goods for consumption to i, J; goods for use as inputs and capital investment to producer j, J; goods for use as inputs to shipper$, J; goods for use as inputs to government J; and which also requires outputs from producer j, J, shipper f, J and government J equal to the corresponding magnitsldes which these behaving units would choose were each to function as a self-interested, independent maximizing unit confronted with a set of prices which clear the market. Thus the central plan comes to have the identical magnitudes as the interregional ocneral micro behaving system. _ I ‘I The central plamiag model (Qlicro) in continuous space Assume a space economy consisting of a set of N regions, N being some positive integer, ordered along a straight line (the x-axis).23 This; set occupies the interval [0, B](B > 0). Consider a partition of this interval {x’, x2, . . ., p+11 i, where x1 = 0 c x2 < . . . -C xNhl = B. Then we take [x’, x2] to be the first region (region A), [x2, .x3] to be the second region (region B): . . ., 2‘Strictly speaking, a w/(?bJ,,,=

&&/i)b”,,i=

~),,,~‘/q~,,

s,

which with (70) yields

J’ VA1

:= qJ,,#hJ.

DifTerentiating with respect to qJhr yields

v,f(abJ~~~~qJ,,~) = pkJfqJR1(ijphJ/~qJh,). can take i3phJ/2qJhI as negligible; that is, we can assume that i, J’s alioc::Ltion Qx.u-chases) has an insignificant eff’t in the determination of the costate (imputed price) variable. So it must be that ?bJhr/&fhrc 0, since ~7;~~~” < 0 Eut

we

by our diminishing returns assumption. 22An analysis of the convergence properties would be quite involved and would involve a full treatment of different institrltionai ;trr~ngem~nts. Such is beyond the scope of this paper. 23Strictly speaking, the points (regions) in the sequence to bc considered do not uccd to be on a straight line, We only require that there is a definite ordering of them and that llhc di:>tancc between any ttvo points is the sum of the distances between the intermediate pC:s of neighboring points.

304

W. lsard and P. Liclssatus, Extension of micro behavior model

[2, .xJ+l] to be Jth region (region d;, . . +, [xN, d”+‘] to be the Nth region (region v). We let dx’ E ?+l-~‘, J = A, B. . ., N. If max,dx’ 6 J?, then, approximatdy, we can look upon the discre,,+e set of regions as a continuum of regions occupying the interval [O, S]; tack region becoming _a point in [0,81]. It is this continuum which we have in mind when we a%w da? + 0 for all J. Consider any magnitude Q:(t) where a refers to the a-type of behaving unit and J refers to the region [xJ, x+’+l]. We posit that Q:(t) + 0 as Ati -4 0 in such a way that Qi(t)/dxc’ approaches a def!nite limit which we designate Q&q I) where x replaces x’. Q&, f) is then the local density of [email protected] the region [J?, ti+l] there exist nf of a-type behaving units. We posit that as dx’ -+ Of,ni + 0 in such a way that &lx’ approaches a definite limit which we riesignate n&x, t): the density of the a-type population. Further we can take the ratio Q&Y, i)/n,(.u, Z) to represent the amount of Q per a-type behaving unit. ‘bViththese conventions, we now may treat b,i(x, t), ghi(x, t),$@, t), J+&J), I<,,,-@,i); jz(x, t),/<(~, t), &o(x, t), as relevant densities. If we assume rt&, t) = 1, then Q&, 8) numerically gives the amount of Q per x-type behaving unit, such that the numerical values of bhi(X,I), ghi(X.9l), . . . may be considered to pertain to one of these behaving units. By the convention already established it is seen that the magnitude ~$7~ corresponds to shipment from [x$ J?] to [xL, ,++l]. Thus we posit as AX’, Ati + 0, s;l,-”+ 0, in such a way that siiL/Ax6A* approaches a definite limit which we designate shf(x, x’, t). Here x’ = xL and x = X! In view of the above discussion, we transform z&j into Ui(x, t) and take Ui(X,1) to be the utility density of consumers of the ith type at (x, t), The numerical magnitude of uI(x, t) may be said to pertain to a single individual of the ith type if the density of iti! type consumers at (x, t) is unity. As u;(t) depends on bii and other magnitudes of i, J, SO we assume that the utility density Ui(X,S) becomes a function of the corresponding densities bhi(X,t), that is .

(91) Similar considerations pertain to the transformation functions 4; and #J. Corresponding to the Ui(X,t), i = 1, . . ., m, we have a social welfare density at (x, t), namely '~~(X,tj=~(...Ui(X,t)...;X,t),

i=

l,...:i?Z.

(92)

System welfare W becomes in general a functional

ITowever, we find it convenient to employ the simple functional (941 ,

W. Isard and P. Liossatus, Extension of micro behavior model

305

With these conventions the central planning problem is max Jz IV(t) db, with respect to the full set of relevant variables subject to C 6ai(x, t) + F J>&X, xf7 t)

dx’

i

+c j)hf(x’,x,

dx’,

t)

+ C R,j(X, i

11= 1, s .

t)

\ (96 I’

,, I,

f

=O,

9j(...Yhj(X,t)...;...Khj(X,t),...;X,t)

j=

l,...,n, (97)

#(. . .ta,-(x, t) . . .; . . .f$yx, t) . . .; . . . &6-(X, t) . . .; x, t) = 0, (98)

%&~ t) -f%% t) =

c r&, i

tlgh~(x,t) ,

(9%

ww

l-C Yei(X9 t, = 09 i

shf(x,x',t)

2

f = l,...,

0,

j-;

h = I,.

l

l

Y0’

(W

and appropriate initial and boundary conditions. Form the Lagrangian density 9 =

qx, t)+C Ph(& t)

-rhi)ghl+C

$- p(-x,t)$ +

.Yhj+JB i

h

c l’h(-x, h

rhighi

1 Of

-Kh,+,f?[ 1

sh/Cx’9

x9 t.l dx’

W. hard arsd P. Liossa tos, Extensiort of micrB21 behacior model

306

wherep,, i$, /I, v,,,and 5, aresostate variables. The first-order conditions are

W-W

yhj

*- = -,&

:

(104)

ww s,,/(.., x”,

t) :

t,,(X, x”,

t)

:;.;

rh(x, XI,

0;

x‘,

t)s,f(“,

t)

=

0,

(106)

where r&i x1, j) = p&x’, I)-p&,

f) --PI- 2(x, t)o,lx’--xl

9

(106’) K kG:

P aK,,= -’

(107)

h',

(108)

-.

h

%i :

l

P

(109)

-p&hi

aw

Yei

-{+

Q!;

1’&ki = O; or

f

e

=

c,;

or

2-K ---

vh

=

(110)

Ph,

auf=:-5,

Euf iy;

.

+ fe

l

(111)

‘I% interpretation of eqs. (103)--(111) MI identical, respectively, with those for eqs (70)-(79) excer t that we replace rei$on J with location x ,and magnitudes associated with region J with densitie; associated with location x. We also refer to ix-type behaving units instead of the a behaving unit, although we are free t-o set the density of the or-type behaving units at any location x as unity so that the densities at location x may be viewed numerically as magnitudes relating to a sin$e or-type behaving unit at location x. With regard to the operations of the Ombudsman and the central planner’s responses, eqs. (82) and (84)-(87) must be replaced respectively with (112)

aY(x,t, = C cj, i p(t) = jB i?+, t) dx. 0

(116)

By this procedure, wr2 extend to continuous space (a cs.4rltinuum of regions along a Pine) our notion of the compatibility and ‘identity” of a central plan and an interregional general micro behaving system. This extension, however, does not provide any additional insights as such. These will come in later sections when we introduce still more new concepts to make it possible to use calculus and differential equations more effectively in our framework, 3 hereby we will also be able to exploit more effectively parallels with physical systems. 6. A macro central pbrmhg model We now Gsh to developa macro model foracentral planner whosemagnitudes are simple aggregates of magnitudes in the micro model of section 5. Specifically, we define

(121) In connection with eq. (119), we define q, as a weighted average tax rate such that

Now we wish to introduce another concept into the model, namely the flow U&Y, t), of good h through location x at time t per unit of time. To do so,

308

consider the discrete model. There, when exports are in the positive x-direction the flow v,‘(.?, t) of good it through the point x’ may be viewed as consisting of all exports from all originating regions up to and including region J- 1 to all terminating regions extendhg from J up to and including N (or U). That is. J-l

IV

c c

iY;(_r’, t) =

F=l

~,F’~(tj.

J = 1, . . ., N,

(123)

M=J

where (124) and where we define for our closed system,

When exports are in the negative x-direction the flow Ui (x”, P.) of good ir through the point xb may be viewed as consisting of all exports from all originating regions from N down to and including region J to all termina.ting regions from J- 1 down to and including region 1. That is, J-l

u&Y, t) - -

N

c c

F=l

J = 1,. . ., N,

sy(t),

M=J

(126b

where U” (Q, t)

=

U;(B,t)

=

0.

(127)

Formal& the net flow through 2 is

w9

v,(xJ, 0 = &Xx’, O+ v,(x”, 0,

but we shall show below that if one of the terms on the RHS of (128) is nonzero, the other must be zero. If, follows from (123) and (126) that we must have U,*(x’, :) Z 0

and

V,(X’,

t)

s

(129)

0.

From (123), (126) and (128) we obtain, in passing to the continuum,

&(x9 0 =

(s,(x’, x”, r:.=,j~o_

t) -

sh(x”, s’,

t))

dx’ dx” .

(130)

.4dditionally, we wish to specify the net exports (imports) associated with location x (region J). In the discrete model, when U,,(xJ, t) = v,‘(x”, t) the net exports of good h from region J is naturally defined as the difference between the flOW &(ti+‘, t) passing through x’+’ (whiich is the flow leaving region J)

of micro

W. Isard und P. Liossatos, Extension

behavior model

309

and the flow U&r’, t) passing through X’ (which is the flow entering region Jj. t), the net exports of good h is naturally defined as When &(xJI t) = U&c’, the difference between the flows CT,&‘”I. 1) passing through xJ+l (which is the flow entering region Jj and the flow U&T’, tj passing through xJ which is the flow leaving region J. Consequently, net exports of region J is

U/)(x-,

o- CG(xJ, 1) = c sy(r)-

c

KfJ

LfJ

sf’J(t).

(131)

‘gg-+.‘(t),

(132)

Observe also that cTk+(xI+l, tj-

W&J, 1) =

f M=J+

and u,(.uJ+‘,

sy”(t; 1

d

F=l

t)- s/,-(.8, t) = Jo,(xJ, t)]- Iu;(xJ+l, t)l

Ji’p(t)-

=

+ g-J(t). M=4+1

,

f=l

We obtain (13 1) by observing that by the definitions

V,(xJf I, t) = ‘F’

5

($-+M

Fe1 M=J+l

(I 33)

(126) and

(123),

28)

(I

-s”‘F)+ AI=d+l

By these same definitions (i3Sj F=l

M=J+

F=l

1

So by cancellation

The RHS of (i36) is identical with that of (131). Also, from (136). derive (132) and (133). Dividing (13 1) by dsJ and doing some trivial operations, we have ,J -+L

rr,(xJ+I, t)- U&xJ,t) Ax-” Letting

%I

=

AsA, . . ., AxJ -+ 0. we obtain

1-f

c AxJ J

Ax'

we can

K-J

AxL_

‘r

%

K$s,oxJ A,?'

dXK. (137)

IV. hard and P. Liussatos, Extension ofmicro

310

behavior model

where sh(x, x, t) = 0. Note that (138) is directly derivable from (130) by differentiating with respect to X. With flows and net exports defined, we now can specify transport requirements. To do so, we first change an arbitrary convention used in section 2. Instead of assuming that transport inputs required by a shipment is obtained by the exporter at his region’s market (or by the importer, as is sometimes done in other models), we adopt an equally good convention, and in one sense a more logical and symmetrical convention for a continuous space model. We a.ssume that in making a shipment from J to L over the distance xL-xJ, the exporter f (or society) obtains the required transport inputs at all regions J,J+ I,. . ., L- l,, where at each region K he buys just the amount of transport inputs to move the shipment through that region over the distance d_& If we had adopted such a convention in section 2, the gains from trades definition of (8) would be

r; p;_2A.&h, JS .‘;
-J+L =

for

JCL,

(139)

rh

for

L<~5JdbAdpw,,

J > L,

where the equilibrium condit’on continues to be

f;i’L =
-J+L

r,,

.I+L

s,,f

= 0.

In addition, the definitional eq. (1 I) would need to be replaced by

c *yJ-+L c s;-~+S;- 2 c qJxJ h

1 $-+” +

s

LfJ

L#J

h’

FSJ M>J

c F
s,“?‘~ . > ww

In view of definitions (123) and

c

L#J

J+L =

sh

z LfJ

(I

26), eq. (141) becames

S,“-L+d;-z

1 OhdXJ[~;(XJ+l, h’

t)--~&&8,

t)].

Also, note that if we we,re to use definition (141) in the central planning model of section 4, the relevant definition of gains from trade would be given by (139) where thep;i are imputed prices. Going to the continuous space economy and letting

(i33)

IV. lsard and Y. Liossatos, Extemion

ofmicro hehasior model

311

expressed in units of ton-miles (transport inputs) per mile per unit of 11,(142) becomes

where from

(I

23) and (126), respectively, we have

W;(x, t) = j;

j”,sh(x’, XI’, 8) dx’ dx"

,

WW

and u;(x,

t)

=

-I”,

sff s&v”,

x’,

t)

dx’ dx”,

(146)

and where (128) becomes

wx,0

=

u;(x,

t&-(x, t) .

t) +

(147)

With HIS. (117)-(120) and (122), (138) and (143) we now may rewrite constraints (96) as Ch(X,t)+&(x,

t)

(1 -

=

Qtx’,f))%#(X, t)+

Yh(X,

t) -

6:(x, t)

;,-(.x, t)--s;-” x r3~~(Uhs;(X:, U&,

-

t)-

2)).

h'

Assume the central planner’s transformation function l

l

.

Yh(& t) . . . ; . . . &(?c, t) . . . ; 3, tj = 0

W-W

to be related to the transformation functions 4i which we take to be characterized by constant returns to scale. Thle relationship is such that fur any K, = & Khjand Yh = zj.Ybj,

aK\l

and ZKhj

-=i!

for allj.

(150)

r,

and

Relations (150) imply that always thle marginal productivity of capital is the same for ah j and the central planner, and 1iLcwise for the r(larginal rate oi‘ substitutio.ra between capital and any other input.

312

W. hard and P. Lbssatos, Krtension

ofmicro

beharkw model

The central planner constructs a welfare density function,

f(C1,

‘

.

.,

Gj,‘, - * l ,j:;

(W

x, z),

where C,, ;bndPe+ are de5ned by (117) and (121). We assume he does this by summing cstimatcd individual utility functions as in (115). When the central planner is motivated to optimize, such a procedure then implies

that is, good h and program e are allocated among the i so as to equalize estimated marginal utilities over all i at (x, tj 24 Taking system welfare as W(t)

==

I: f

uw

dx,

subject to (14fJj,(149), (98), and

~&!I 0 -f i-(x, 0 = rdx, 0%(x, 0, the appropriate restatement of (99), and

UZ(x,z) 2

W6)

4x& 0 5 0,

0;

and appropriate initial and boundary condit.ions. AccordingFy, we form the Lagrangian density Aif = j’+~p,~x, h

(I -r,)9,+

t)

Yh--

6:- iii -S:-2

[

h

24F~t a Connat tre:e:tmentof aylgregation problems, sectGreen (1964).

$ tih.(Uc- Vi)

P. hard and P. Liouatos,

Ext.wsiott of micro behavior mode?

313

The first-order conditions are

(159

(161) (162) (163)

(164)

(165) (166) Cond.ition (158) requires that the marginal social welfare from consumption of good II [which by (152) is also the same as the estimated marginal utility of good k to all i] must be equal to the costate variable ph, the imputed price of a unit of capital stock of h. 25 Conditions (159) and (160) are formally identical with (16) and (17), and together yield the basic investment principle applicable to the central planner, 6’Ye

= yj,,, PiI-i’K,,

g,h-

l)...)

1.

(167)

Condrtions (Ml) and (162) cm best be understo4 if we refer to definition (139) where we consider shipments front J to J+ I, and L-pm J to J- 1, respec25Where the central planner’s behavior is taken to be directly sensitive to ,the allocatilm of goods tc :ach individual household, he might replace C&C, t) in comtraint (148) w .th XlIbhi(x,t)t9h(x,t) with &ghl(.v,t); definition (151) with (92), and thuscondition tlS8) with (103).

IV. bard an,;l P. Liossatos, Extension

314

ofmicro behavior model

tively, and which thereby give, respectively, J-J+1

=

Jti Ph

-&-p~-2dxJwh,

J-+J-1 =h

=

pi-’

-p;i

Th

-p:_ 2Ax.‘o,, .

We observe that Fh - 01,/~~1 __L =( 0 is the continuous version of $+J+l s 0, and &,+~~p~+ 2 0 is the continuous version of ~i-l~-r :i 0. Accordingly, (161) and (‘162)state, respecti’vely, that gains from trade at x’ from shipments to the points x+ dx and x- Ax (for Ax >, 0) is zero. Also using (168) and (169) we may define J+J+l

iin-i AxJ+O

,;;yo

?h -

%%

t)

=

=

‘;(-‘7

t,

=

$h(x,

1) -

a/#,-

2(x:,

(170)

t),

A-r’

J+JTh &,

1 -$h(x,

t)+~&-.

,(x,

t)).

c

We then may rewrite (I 61) and (162) as

?,+(X,t)

5 0;

TJX, f) =< 0;

$(x, t)&yx, t) = 0,

(172)

c=Jx, 1)&-(x,

(173)

t)

=

0.

observe from (161) and (162) that if U’(x, ?) > 0, U;(X, t) = 0; and if z&-(x, 2) < 0, v,‘(x, t) = 0. Conditions (163~(166) are to be interpreted as (107~(110) except that reference to the individual household in (164) i.s suppressed; but bearing in mind (152) we see that the individual household can be reintroduced in the interpretation. This completes the general discussion of the macro model. However, we are also interested in restructuring the central planner’s macro model so as to be consistent with independent maximizing behavior of each of many micro) units. First, as in section 4, while the choice of ch(& t) and the corresponding ahocations 6hi(x, t) maximizes estimated individual utilities 6:(x, t) given the strncture of the macro model, it does not follow that the true utilities Ui(X, t) are maximized for price-taking consumers when ph(& 2) are viewed as prioes. Further it does not follow that C,,(x, t) = xi b,i(.k, t) when all consumers are allowed to act independently to maximize utir:iy when the ph(& t) are announced as prices. So, as in section 4, we must introduce an Ombudsman in order to develop a macro model responsive to maximizing behavior of

W. lsard ard P. Liossatos, Extension of micro behavior model

314

individual households. However, note that meeting the central planner’s firstorder conditions regarding production does imply the first-order condition:; for maximizing n+), the profit density of thejth type producers at X, where

-

C PhCx9 OIIKhjCx, O) h

(174)

l

It is also insured that xi yhj = Yh and Cj Khj = &. This follows from tht: fact that the first-order conditions for the jth type producers at X,j =z 1 . ., 17, together with (150) imply that Cj yhj and Cj Khj satisfy the centra.1 p;a;lner’s first-order conditions on production. Moreover, meeting the central planner’s first-order coaditic. IX regarding export activity does imply the first-order conditions (zero gains from tradejt for all types of shippers. From (170) and (17 1) we obtain by integrating

f/2h(X’, t) -fIh(X,

r&, x1, t)

=

1

t) -

s:’a/#,-

&(X’, t) -ph(X, t) - 3‘1r ~hj+-

#:

2(x’,

t)

t)

d.y, for

x c x’,

for

x > x’,

d.%

which, as expected, is the continuous version of (I 39); and from (172) and

(I

73)

(1761 Conversely, the validity of (176) for al! pa&s of x and x’ implies the first parts of (172) and (173). However it does not follow that the mere summation of shipments of all f will yield the central planner’s &,(_y,t). We thus require an Ombudsman type of procedure. One such procedure would entail an Ombudsman at each location9 who permits exports or imports up to the level of &(x, t). Once this level is reached, he permits no further cvports or imports. Since this or an equi*jalent activity is a trivial operation we ..i\?not formalize it, in order to maintain simplicity. TO insure that the central planner’5 C&y, t) = xi b,i(tiY,t) we must redefine our conceptual framework to include an Ombudsman and make specific assumptions about welfare functions. The Ombudsman is conceived to ha?Je the payoff function

beharior mudei IV. hard arrd P. Liossatos, Extension ofm.krt*

3116

the macrcb str u cture of our revised framework the welfare function ltakes the particular f’c.rz~ lrr

f

=

+G,(. .

;y

.,rce+, . *0; x, t)*

(151)

(178)

In (177), vi > 0 and 17:< 0. It therefore follows from (158) that Cn is a decreasing function cf qh. 26 ( See the discussion in section 4.) Thus, as in section 4, we can visualize a process where during stage I, the central planner takes qh = 1 and at the end of the stage announces the C& t) and ph(& t) for ali (x, t). Each consumer i at a2 x, t is then able to calculate the time rate of his income x1, Ph(X,tlshi(x, t) and know the budget constraint. Given his preference structure, represented by an ordinal utility density u~(x, t), he chooses and announces at the end of stage II the set of bhi(X, t),that is the consumption plan over the time period which he most prefers. The Ombudsman then compares the C&c, t) and x1 bhi(X, 2) and varies the q,,(x. t) which he announces at the end of stage III. The central planner then does a completely new calculation of his plan. And so forth, as in section 4. 7. Interregional micro behavior in conjunction with an extended macro central planning model

The macro framework of the preceding section can be extended to cover a second law of motion governing the u,, variable. For detailed discussion of this second law, see Isard and Liossatos (1975a and 1975b). In particular, the central planner’s macro model can be reformulated as max jz JgBf(. . ., Ch, . . .; . . .,+zp

. . .,.

l

Y t)dxdt, )

(.173)

subject to (148), (I 49), (98), (129), and ir, = -Yhih-G(hLlh,

(180;

where r/, is defined by ( 147) and subject to appropriate initial and boundary conditions. To motivate the interpretation of eq. (180). we take development for a system [O, B] as j,” W,

tj dx,

(181)

26Until we can demonstrate otherwise, we assume that Ph varies only slowly with qh so that the ter’rRIQ&Y~~~[;lh~ is negligible in comparison withy,,, see footnote 21.

W. lsard and P. Liossatos, Extension oj’micru behatlim model

317

where D(x, t) is an index of development per unit kngth at (x, I). We let

with d&K,, gradient is

> 8, h = I, 2,

i(x,

t)

=

;g

. . .? 1.

It

therefore fol1ow.s that the dcwZopme~t

iJx,

(183)

t).

h

It is to be expected tha: the level (and this index) of development will differ among locations. Typically, such development falls off with distance from a major metropolis or growth pole. Whatever the case, we may presume that there exists a demand for development by those locations not at the highest existing level of development. Given the field characteristic of our model, it is both convenient and satisfactory to specify the demand for develo,nment at location x+ Ax (Ax > 0) as a demand for additional flow of goods from the neighboring location x to location x+ dx for deve!opment purposes at x+ As, assuming that the level of development at x is higher than at x+ Ax, that is that D(x, t) > D(x+ dx, t). If D(.x, t) < D(.u + Ax, t) then tlhere is demand for additional flow of goods from X+ Ax to x for development at A-.(The neighborto-neighbor interactions of our field approach of course relates this demand for additional flows from x to x -/-Ax to similar demands for all other pairs of neighbors.) in efl-ect then we can state for ir,,, the additional flow of h per unit of time, that for small

(184) Here, &,(x, t) can be associated with a change in U~(.x, tj, or in U&Y, t), or both.2 7 On the RHS of (184), the contribution of good lz (in the forti of K,,) to the development gradient at (x, t) is specified. This contribution can be clther positive or negative. From (I 84), we then have (I 85)

where j,, is a coefficient which converts the development gradient YC:/Zas gken by the RHS of (184) to an effective demand for additional &IW (per unit of’ time) of 11from s to x+ As for development at x-f- Ax So Uhe RI-% of ( IF?) z7We do not consider &+(A-, t j or i/,-(x, f) as such because there associate either with any given development gradient. C

is ~:a a. priori

retL:on

to

318

CF.kwd and P. tiossatos, Ektensiort ~4‘micro beharior ma&l

represents the ef$ective demand for additional flow for development. per unit of time which is then equated to the supply &(A-, t) of additional flow per unit of time Multiplying both the numerator and denominator of the RHS of (I 85) by Ax, and letting Ax + 0, we obtain

0,(x, if)_= -i

aK h

where I,, = Jim,,_,,&/dx. (186) becomes

U&x, t) = -

x Kh(x,0,

1ao

w9

h

Further if aD/Z& = &, a posiitive constant, then

Y&X? 0,

(W

where yh has the dimensions of (length/time)z or (velocity)2. In effect it measures the speed of response of flow to the capital gradient.2 8 Recall that underlying the demand for additional Bow for development is the difference in the marginal productivity of capital at the neighboring locations x and x+ AX, the marginal productivi,ty of capital being grc~ter at x + Ax when K(x, ‘) 3 K(x + AX, t). If we now add to (187) a term -a&, to represent the diminutio7, .?f flow on account of diverse ftictions (social, political, economical and physr,,: , where a, is a decay rate (involving resistance and inertial effects)2g associated with Q,+we obtain the additional equation of motion (I 80). Having developsd a rationale for (ISG\, we now return to the optimization problem of 0 79), and form the Lagrangiar

where Y is given by (157). The first-order conditions are the same except that we replace ( 160;. ( 16I) and (16.2). respectively, with

2RNote that (IS?) can be generalized :o reflect the effect of capital gradients in all h’, !I’ = 1 . ., 1, upon iih, in order to capture CYOSS effects such as the crow+productivity of capital ikestment. Doing so would yield

O&q t) = - 5 yr, ,k,(x,

t).

UW

29StricIly speaking, ak = J&J!,,where R,, is a friction coefficient and I,, is an inertia1 coefficient. See Tsard and Liossatos (1975h),

W. hard and P. Liossatos, Extension of micro behacior model

319

To interpret eqs. (191) and (192) we bear in mind that it can be shown that30

(193) That is q&, t) may be regarded as the imputed price of one unit of V, at (x, t) (whether Uh is positive or negative) from the standpoint of its contribution to social welfare density realized at location x from time t to t, . However, note that if q,,(x, t) > 0, then flows in the positive x-direction are positively valued, while flows in the negative x-direction are negatively valued.. If q,(x, t) < 0, then flows in the positive x-direction are negatively valued, while flows in the negative x-direction are positively valued. Also from our discussion of the derivation of (187) 2nd the interpretation of (180), qh which is the costate variable associated with (I 30) can be interpreted as the price of an additional unit of flow of What (x, rj for developrnznt at thie neighboring location x+ .4x. Given pk, the imputed price of capita?, qh equates the supply and demand for additiona flow at (x, t). It behaves as a price just asp,, behaves as a price of a unit of capital stock given qh. Also, from observing eqs. (191) and (192) it is clear that if Uz > 0, then V,- = 0; and if i?,; < 0, u; = 0. For all h for which r/h’ > 0, we multiply both sides of (‘191)by dr and integrate to obtain (193) Note that the terms in the integrand are the same as in thz gains of trade condition of (161” except that in the extended model of this section we must include in the gains from trade concept the loss (if any) in value per unit of Ul due to decay, that is - a&. Accordingly the integrand term on the RHS of (194) is the cumulative sum over all points of time from t to t, of the gains of trade from one unit of Ul. This plus the ‘scrap’ value of one unit of Ul at time t, must equal the value of one unit of Uz at time t. In one sense, (194) is an investmenttype principle for the stock of momentum associated with Uz. 3 ’ The discussion of eq. (194) now permits a direct interpretation or (191 j asa transport principle. Eq. (191) now states that, when Uz > 0, the difference in “OSee bard and Liossatos (1975b). 31LIb+ can also be viewed as a momentum concept in the senst’ that ii depicts the motion of goods. Thus when we consider I<,,as ihe density of goods at a point, and put aside the Paxk of fluidity of futed capital goods, u;1+ = &,Q where t’his a velocity. Since & is goods per unit length, &+ is momentum per unit of length, or momentum stock dencrity.

price 01' a good 12at any pair of neighboring locations must ecjual the transport coot of a unit of flow, ph~s the cost of its decay per unit of time minus the gain b).r plus the loss) in its price per unit of time. For all If for which i& c 0, we may proceed in similar fashion to obtain (104) except that -+a,,p, _ ,r! suhstitutcr; for --agl_ 2 in the integrand. Eq. (192) Grttnstates the transport p inciple for these t?.32 To interpret ey. f 190), combine it with (159) to yield

The d,ifT’erenccbetween (167), as interpreted in the previous section, and (195) is in the term r,,&,. This term accounts for a spatial gain (loss) associated with the additior>al units of flow, oh, generated by change in capital stock Kh via the induced change in &. We have aIready established that qh is the price of one unit of U’. But also, as is very clear when a formal Hamiltonian is used, qh may be viewed as the price of one unit of ri,, (just as pJI may be viewed as the price of one unit of &,J. Thus, when we consider for intuitive purposes the dilrlcretecase and two neighboring region; tit x and x+ Ax, with Ax very small, any change dK,(x, P) ilmpks a change- ,I&(& t)/dx in the slope of &(x, t). So by ( I Mlj and for ah == 0, AIj,t.i-. E) =: (y,,/Ax)Aii;,,

U9Q

ALj,/AK,

(197)

or = y,JAx.

If now Ail, leaves region x and enters region x-t- Ax, the spatial change in

value per urit of All;, (additional capital) is (A&,/AK,) q,,(x+Ax, t)-(AZ&,/ A&) q&, I) which by (197) equals y,,([q,,(x+ Ax, t)-q,,(x, t)J/Ax}. The limit of this t.erm as d.u -+ 0 is then Y,&~(x,t). This intuitive argument for interpreting yk&is not basically changed when Q,, > 0. Rigorously speaking, for our continuous model, it can be shown that (198)

where y,(.\-, t) is the spatial imputed price of one unit of y,&(x, t) from the $Iitncipoint of its contribution to social welfare per unit of time at all locations frtlm Ytc, B. From (1801, ~&‘&-, t) may be viewed as the flow of momentum of OWL!it through .v at time t, per unit of time. Therefore per unit of K,,(x, t), thl::rcit~e j(*units of the flow of momentum. For each unit of flow of’momentum ‘LHerc WTmust btxxrin mind that when CIA-is positively valued by society, q,, c 0 by (193).

W. Isard and P. Lsbssatos. ExteMon of micro behcwior model

321

through location x to its neighborin location there is a spatial gain of value of 4(x, tp= So per unit o;‘&(x, t), there is a y,,$spatial gain in value. [This roughly corresponds to the spatial gain from the resulting additional units of do,, flowing from region x to region x + Ax.]34 Having extended our macro model to incorporate a second law of motion, having interpreted the new costate variable q,,(x, t) as a price of one unit of flow (for development purposes) of good 1zat (x, t), and having revised our investment and transport principles accordingly, we now wish to examine the implications of this for micro behavior within the central planner’s macro model. With regard to consumers, the discussion of section 5 still pertains. With regard to producers, the definition of the profit density of the jth typ.9 producers at x’in eq. (174) must now be modified to read

That is the profit density of thejth type producers at x must now cover neighborhood elects, that is must now cover the cumulation over each point of time from 0 to f, of the spatial (neighbor-to-neighbor) gains (losses) from the additional units of flow of each good h generated by the stock of the good h held at X. Such neighborhood effects may be viewed as externalities to these producers. Thus we may imagine that the central planner provides a subsidy (exacts a payment or imposes a tax) of &yh per unit of K,ij whenever &, positive (negative). With regard to shippers the definrtion of gains from trade per unit k shipped of eq. (175) must now be modified to read, when x < x’, ‘th(x,x’, t) = ph(& t)-p&-,

t)-f”,’ I&-- a(% t)$+ c(hqh(%t, +ijJ t, t,] dX s 0.

(201:l

fi(x,

33This parallels the spatial gain of value t) when one unit of good tt flows. Thus strictly speaking, &x, t) are the gains from t.he transfer of a unit of momentum through a location x to its neighboring location. 34By multiplying both sides of (193) by dx and integrating, we have

which is the spatially imputed value of a unit of K,, (at x, t), and which must be equal to the cumulative sum of she ‘current ’ value of a unit of capital at all locations between x and B

plus a ‘spatial’ scrap value ai B.

Jr1 tJli?i papc’r wc have dcvelol~@din a step-by-step fashion a general model wk))i& ~ynthcsizc~ optimal macro behavior at a central planning level with m;~,imir.ing micro hchavior for a general eyuilibrium-type system of numerous *prir;c.ta&in$ rrnitri. After cxtcnding the standard general interregional equilibriratn frrtmewrprk to embody investment and consumption planning over time WC itttr&JCecJ the government both in the production and distribution of ~c~oJc ;rnd scrvic’cs. While we considered only one level of government, it IC ;rossihlc 111introduce many levels and forms of government.. [See Isard ( I’~fD‘~, Ch% I3 and n4j.l This in turn necessitated considerations of investment J~l;lImirl,~MC! the construction of an appropriate social welfare function. After rrli,lrtlrulrttilllr the st:rndrrrd model of a discrete se%of regions to embody con~WLWU~ sp~c (11c=ontinuum of regions), we were able to include another basic cfyllitmic-fitctar in the macro model. This factor was expressed through an sa&lific)n;~l~C~lJiI~iOlI relating the explicit time rate of change of commodity ~Iw\ wilh dif2’rtr<:ntir~J spittiirl development. In turn differential spatial develop~ttcnt ws rcktteck to sp:rM difTerentials in enpit:ln stock densities. It was hypoUlcG/ctJ th:rt tftt: f:tftcr ztr’r’associated with different marginal productivities (srGrll prrrlit dilkrrnti;rIs) which then provide the rationaIe for the time rate OI’Ch;tngs clf fluw. This step vields a macro mod4 kth a richer interplay between sptf i:ll rrtlri l~fll~~l~~~d processes. Each basic investment principle (equation) t;?r :t l~rc’;ltioncaplicitly invoives pains ;Ind losses from interactions with locations rrt its trciFJt~~~Ph~y;lirt. tM1 Mic’ transportation principle (equation) for any two h*+z.,tfionsinv~lvcs explicitly g;rins and iosses from effects on the investment J~r003~s Hf both itheSennd intervening locations.

W. hard arrd P. Liossatos, Ex!emiorr of micro bL$ar?iormodel

323

In the model resulting from the Con of macro vptimality and micro t;ehavior, irformation is passed down from. the cei;tral planning level which in turn conditions (constrains) the behavior of the many micro units. Tllejr behavior in aggregate in turn forces the central planner to change his assumptions on basic parameters. Through round-by-round iteration, with the assistance of the Ombudsman, consistency is achieved between the optimal values of the macro and micro variables. When only one law of motion is embodied in the macro model, the information passed down to micro behaving units relates to prices, taxes and allocations of government goods and services. When two laws of motion are embodied in the macro model, the additional information passed down to micro units covers subsidies and penalties. While the results for the ca3e involving two laws of motion are presented for only a continuum of regions, they can be developed for a discrete set. Likewise the balance of payments analysis of section 2 can be brought into the analysis of this case. Introduction of the second law of motion in combination with the first which presumably is an important step in making the macro model more realistic involves the explicit recognition of externalities. This step makes it possible to internalize these externalities in the revenues and costs of the micro behaving unit. Unfortunately the general nature of the macro model does not permit us to be specific about these externalities. Finally, all the formalism can be trivially extended to all and any radius from x = 0 in a two-dimensional space with the explicit assumption that there is zero interaction among radii, Additional development is possible which does incorporate interaction among radii, but such does involve major cc-mplexities.

n

~x’J,K.J., 1968, Application of control theory to economic growth in mathematics of the

drzision sciences, Part 2 (American Mathematical Society, Providence, R.I.). Green, H.A.J., 1964, Aggregation in economic analysis (Princeton University Press, Princeton, NJ.). Isard, W. et al., 1969, General theory: Social, political, economic and regional (MIT Press, Cambridge, Mass.). Isard, W. and P. Liossatos, 1975a, Parallels from physics for space-time development models, Part I, Regional Science and Urban Economics 5,540. Isard, W. and P. Liossatos, 1975b, Parallels from ph.kcs for space-time development mode]s, Part II, Interpretation and extensions of the basic model, Papers, Regional Science Association 34. Lange, O., 1938, On the economic theory of socialism (McGraw-Hill, New York).