Some Initial Explorations of Interregional Linkages for Econometric Models

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SOME INITIAL EXPLORATIONS OF INTERREGIONAL LINKAGES FOR ECONOMETRIC MODELS

s.

L. Green* and M. R. Perryman**

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Abstract. The present paper offers some initial findings in an ongoing effort to isolate significant interregional patterns in economic activity. These preliminary explorations are designed to provide a partial contribution toward an ultimate situation in which an effective structure for multi-regional or "bottom up" econometric models may be promulgated. Specifically, the analysis concentrates upon a simple approach to the construction of a framework for analyzing relative cost factors as a determinant of regional employment performance. In particular, this approach is much less restrictive in its implementation than is the case with other recent efforts of a comparable nature. Initially, some basic employment functions are examined. This discussion includes the derivation of the relevant relation ships and a presentation of some simulation results for the Texas economy. The investigation then proceeds to a formal exposition of a flexible methodology for capturing interregional cost patterns. In addition to the determination of standard employment results from wellknown expressions for the aggregate production function, the pertinent equations are extended to incorporate such factors as a partial adjustment to equilibrium mechanism and the presence of technical progress. The essential purposes of this endeavor are (1) to suggest some tentative lines along which fruitful subsequent analyses of interregional linkages of various aspects of economic activity may be pursued and (2) to initiate a procedure for the specification of multi-regional econometric models which include a significant degree of geographic interrelationship. Keywords. Economics; identification; large-scale systems; models; statistics. INTRODUCT ION

smaller areas, has long been advocated, its actual use to date has been extremely limited. 1 Even in cases where multiregional models have been developed and applied, their implementation has usually been restricted to the simple aggregation of an essentially independent set of sub-models. A more meaningful approach is, of course, to permit interaction among the relevant areas in an economically and empirically meaningful fashion. The current analysis suggests some lines along which such an effort might be conducted. Given a very feasible strain in subsequent data base development, this approach may be fruitfully adopted on a widespread basis. Given the preliminary and exploratory nature of these findings, attention is

This paper presents some initial results regarding the interregional linkages of regional econometr ic models. While the "bottom-up" approach to econometric modeling, i .e., the construction of systems reflecting national or large regional patterns via individual structures representing *The author s are, re spectively, Graduate Fellow in ECGnomics, Brown University and Herman Brown Professor of Economics and Director of the Center for the Advancement of Economic Analysis, Baylor University. Professor Perryman also serves as Director of Baylor University Forecasting Service and the Texas Econometric Model Project . Appreciation is expressed to the staff of the Center for manus cript preparation and other support service s. Financial assistance from the Hankamer School of Business, Baylor University and the Brown Foundation i s als o gratefully acknowl edged .

1Discussions of var ious aspects of current multiregional modeling are given in F. Gerard Adams and Norman J . Glickman, eds., Mod eling the Multiregional Economic System, Le xin gton: Lex ington Books, 1980. 26 1

26 2

S.L . Gre en and M. R. Perryman

confined to the employment sector. As an initial point of departure, Section 11 presents a basic overview of the labor demand expressions from the Texas Econometric Model. This model, which was recently completed under the direction of one of the authors, is designed in a manner which is conducive to ultimate integration into a "bottom up" structure. Some relevant simulation results are included in this section. The analysis of employment in Texas is followed by a thorough discussion of an approach to interregional linkages. In particular, relative cost factors between Texas and the United States are examined as a key determinant of the flow of employment opportunities. To illustrate the generalization potential of these results and to conform with prior empirical evidence, alternative production technologies are assumed for the state and national economies. Moreover, the methodology could easily be extended (i.e., disaggregated) to encompass the relative cost information for two (or more) subregions. Once this exposition is completed, a concluding section summarizes the study and discusses relevant issues with respect to the multi-regional data base.

expressed for some arbitrarily chosen industrial category i Yi = AKi ClLi B

2See especially Robert M. Solow, "The Production Function and the Theory of Capital," Review of Econoimc Studies 23 (1955-1956) and Joan Robinson, "The Production Function and the Theory of Capital," Review of Economic Studies 21 (1953-1954). 3Charles W. Cobb and Paul H. Douglas, "A Theory of Production," American Economic Review 18 (1928). 4A discussion of the estimation of these functions is given in M. Ray Perryman, "The Supply Side of Regional Econometric Models," Modeling and Simulation 11 (1980). See also Perryman, 'A Test of the Functional Form of Production Relationships in a Regional Econometric Model," presented to the 1980 Conference of the Western Economic Association, San Deigo, California, June, 1980.

output (gross product originating) in category i capital input in category i labor input in category i parameters of the production process.

where Yi

The labor demand function for each category may then be easily derived by assuming a desire for cost minimization in production subject to the constraint defined in (1). The application of a standard constrained minimization procedure to this Neoclassical function yields the following: 1

Li

= [Yi/A)(PKi/PLi)a JT-"S

(2)

where PKi = price of capital in category i PLi = price of labor in category i . As an initial point of departure, it is useful to assume constant returns to scale in the output process, i.e., that a + B = 1. In this case, the demand function reduces to

THE EMPLOYMENT SECTOR SPECIFICATIONS OF A REGIONAL ECONOMETRIC MODEL As is the case with any aggregate modeling effort which seeks to address supply-side (i . e., production) phenomena, the present study abstracts from the issues of the socalled "Cambridge controversy"2 and assumes the existence of aggregate production relationships. Specifically, the major sectors of the Texas economy are said to be characterized by Cobb-Douglas technology.3 This relatively simple specification seems reasonable in that, in a detailed consideration of twenty of the state's industries, Perryman found that Texas generally conforms to the Cobb-Douglas formulation. 4 In its most basic formulation, this function may be

(1)

1

Li = A(PKi/PLi) a • Yi

(3)

or, in logarithmic form, 1

lnLi = lnA + aln(PKi/PLi) + lnYi.

(4)

This relationship may then be expressed in a generalized stochastic form as

where Aj

parameters of the equation, j = 0, 1, 2

a random disturbance term. It is expected a priori that AO < 0 and Al < 1, whiTe A2 is constrained to be one. Equation (5) is, of course, linear in its parameters and hence, may be empirically determined by a restricted least squares estimator. If the constant returns to scale assumption is removed, the general form of the relationship remains intact, but A2 becomes an unrestricted parameter behavior. In the case of increasing returns to scale, A2 = (1/(a+6» will be less than one, while A2 > 1 in the decreasing returns case.

o<

A more meaningful labor demand function may be generated by accounting for the likely existence of technological change over the sample period. A simple and straightforward means of achieving this goal is to introduce the notion of Hicks' neutral technical progress into the relationship.5 The 5See , for example, J.R. Hicks, Capital and Growth, London: Oxford University Press (1965 ).

Some In i t ial Expl or a tions o f Int e rr eg i ona l Li nka ge s

263

resulting augmented production function mav be written as where Lit

Yi = [AKi aLi 8]e Yt, where y t

(6 )

a technical progress parameter time.

Solving for the labor demand function via a cost minimization process yields 1

a

(10)

observed employment in category i at time t Lit* optimal employment in category at time t in the absence of transactions costs and information problems ~ = the parameter indicating the degree of partial adjustment, i.e.,O

_y

Li = (Yi/A)~(PKi/PLi) ~e (~)t. (7) In logarithmic form, this relation becomes

<~< 1.

In the case of a simple Cobb-Douglas production technology, the optimal quantity of labor is obtained from the derived demand function described at the outset of this section i.e., 1

Lit* = [(Yi/A)(PKi/PLi) a]aTS 1

( 11)

- 1'

a+81n(Yi) + a+8 t.

(8)

This expression may then be given in an estimable stochastic manner as

Expression (10) may be written in terms of Lit, i.e., as a demand relationship, in logarithmic form as (12 )

lnLi = AO + A11n(PKi/PLi) + A21nYi + A3t + ( i,

(9 )

where, as previously noted, AO < 0 and o < Al < 1. The parameter A2 is constrained to one in the constant returns case and, of course, unrestricted in the nonconstant case. The final parameter, A3, is equivalent to - y under constant returns and (- y/ a+8) otherwise. Hence, assuming technical progress does occur over time, A3 is expected to be negative . An additional extension of the basic approach to the specification of the labor function is to incorporate partial or gradual adjustment into the analysis. Such a structure, which is embodied in numerous existing econometric models, tends to greatly enhance the predictive power within the sectoral equations by accounting for the responsiveness of employment levels to changes in price and output conditions. 6 In a theoretical setting characterized by the perfect information and absence of transactions costs, it is expected that the labor sector would adjust fully and immediately to variations in market situations. The "real world", however, is characterized by longterm contractual commitments, established capital stock and capacity constraints, substantial training and relocation costs, and countless other intervening factors. Hence, a gradual adjustment model seems to provide a reasonable descriptive mechanism for short-run employment reaction. A simple functional relationship for any industrial catgory i is given by 6The gradual adjustment structure discussed herein is a highly simplified version of that discussed in Frank J. Hahn, "Economic Equilibrium and Transaction Costs," Econometrica 39 (1971).

Substituting equation (11) into this function under the constant returns assumptions, i.e., where ex + 8 = 1, gives

~ lnYi

+

(L- ~ )lnLit_1.

(13)

In the non-constant returns case, this relationship becomes

~

a+81nYi +

(1- ~ )lnLit_1.

(14)

As general stochastic relationships, both (13) and (14) are expressed as follows: lnLit = AO + A11n(PKi/PLi) + A21nYi + A31nL it_1 + ( i.

(15)

The two cases differ, however, in that constant returns to scale requires that A2 + A3 = 0, while non-constant returns imposes no such restrictions. Finally, it may well be desirable to formulate employment functions which incorporate both long-term (technical progress) and short-term (gradual adjustment) factors. The simplest methodology to achieve this goal is to define the optimal labor position, say Lit**, from a production relationship which includes technological advancement. In the previously discussed Cobb-Douglas expression with Hicks' neutral technical progress, the ideal derived demand adjustment is, in the case of constant returns,

264

S.L. Gr een and M.R. Perryman

due to the relaxation of two highly restr ictive assumptions which are critical to the Massachusetts model.

1 In(-) + aln(PKi/PL i) +

1nLit **

A

1.0lnYi - yt.

(16)

Substituting this equation into (12) yields lnLit = 'l'ln(1/A) + 'l'aln(PKi/PL i) + 'l'lnYi - 'l'yt + (1- 'l' )lnLit_1 . (17) Similarly the optimal labor function in the more general non-constant returns case is given by

with the resulting sectoral employment expression being

'l'a - ( a+S)t + (1- 'l' )lnLit_1,

(19)

In both cases, the empirical form is lnLit

= AO +

A1ln (PKi/ PLi) + A2lnY i +

A3t + A4lnLit_1 + £i.

(20)

Note that Aj < 0 in both cases and that, given constant returns, A2 + A4 = 1. In the non-constant returns to scale framework, this restriction is not required. It is also expected that A3 will be negative in both expressions. Some explorations of interregional employment functions are now provided. INTERREGIONAL FLOWS IN RESPONSE TO ECONOMIC ACTIVITY: SOME INITIAL RESULTS

The major concept to be employed in the analysis of interregional activity is that of the economic base of a region. The base is the portion of local output which is produced for export to other areas. The remaining output, of course, is utilized within the region. Traditionally, studies of the economic base have sought to categorize various industries as either basic, i.e., export oriented, or non-basic~., local serving. Many methods have been utilized for this purpose, one of which is merely ar-bitrary characterization. In an emp i rical model of the Susquehanna River tlasin, for example, several agricultural industr i es at the two digit level, mining, and manufacturing are treated as basic, with all other economic activity being viewed as non-basic. 8 Two widely adopted approaches are based on comparisons of regional and national employment allocations to various sectors. These techniques, known as location quotients and minimum requirements, are not arbitrary, but are highly restrictive in that each industry is assigned exclusively to a local or export role. In reality, of course, most production categories may be expected to serve both basic and non-basic functions. Such a choice-theoretic procedure is developed within the Massachusetts model, and the employment sector des cribed herein is structured in an analytically similar fashion . The basic methodology embodied in the Massachusetts relationships involves the estimation of regional employee ratios of an input-output nature for all pairs of industries within the economy. Similar ratios are calculated between regional employment and final demand for each resulting ratios are then multiplied by an appropriate value (employment or demand) and then summed in order to obtain the estimates for local-serving employment by category. Each of these relationships requires the calculation of a "regional purchase coefficient" for the relevant good. The model thus assumes the availability of a complete input-output table for all industrial areas within the state. In the absence of this information, a more simple method within the same spirit may be derived.

Significant advances in the modeling of responses to interregional price changes have recently been embodied in the econometric model of Massachusetts.? The employment sector of that system permits factor substitution, movements engendered by shifts in comparative locational advantage, and relationships among the various industries within the region. The present approach incorporates several essential elements of this structure within an alternative setting. Specifically, the framework discussed herein is more generally adaptable

The basic structure of employment functions found in the Texas model are, of course, discussed in Section II. It should be noted that these equations are expressed in terms of total labor demand within each sector, i.e~ distinction is made between export and local activity. In order to approach this

?See George I. Treyz, Ann F. Friedlander, and Benjamin H. Stevens, "The Employment Sector of a Regional Pol i cy Simulation Model," Review of Economics and Statistics 62 (1980).

8See H.R. Hamilton, S.E . Goldstone, J.W. Milliman, A.L. Pugh Ill, E.B. Roberts, and A. Zellner, Systems Simulation for Regional Analysis: An Application to River Basin Planning, Cambridge: M.I.T. Press (1969).

Some Initi a l Expl o rations of Interregion a l Linkages

problem given existing data constraints, it is initially plausible to relax some of the more restrictive assumptions of the Massachusetts model. Specifically, this structure is based on the hypotheses that all production functions are of the Cobb-Douglas form, exhibit constant returns to scale, and are identical for the region and the nation. This assertion is critical to the remainder of the model in that the derivation of the equations to be estimated involves several ratios of regional to national variables. The simple functional forms and geographical homogeneity permit the cancellation of several unobservable output concepts and, hence, greatly facilitate the empirical implementation of the resulting equations. In fact, the final non-basic employment relationships utilize only relative input prices as explanatory variables. While extremely convenient, these constraints on the nature of the production process are difficult to defend from a theoretical standpoint and are in conflict with recent emp i rical evidence. With respect to Texas and the United States, for example, it was recently demonstrated that Texas tends to exhibit Cobb-Douglas technology, while the nation as a whole conforms to a constant elast i city of substitution (CES) specification. Moreover, the constant returns assumption also seems overly restrictive. 9 Th i s regimented structure is dropped within the present explanatory model, as the regional and the national economies are assumed to exhibit, respectively, Cobb-Douglas and CES production functions with non - constant returns to scale. As previously noted, the primary goal of the delineation of basic and non-basic employment lies in the need to examine the effective response to changes in interregional cost patterns. Because of data limitations, the labor sector in its present form simplifies the empirical analysis of the Mas sachusetts mode 1 in that (1) it abstracts from long and complex lag structures, (2) it limits non-manufacturing production functions of two inputs, and (3) it does not utilize the general equilibrium framework of input-output vis-a-vis local serving employment. While this approach is somewhat unique, it nevertheless permits the explicit quantification of the impact of changes in relative cost patterns. It also has the advantage of not being reliant upon what would be, at best, only a crude approximation of the proper labor deviation. The model is developed via the basic Cobb-Douglas non-constant returns function, but can be easily extended to account for multiple inputs, technical progress, gradual gSee Perryman, "Functional Form." The problems in using the Cobb-Douglas form at the national level are discussed in Dale W. Jorgenson, Laurits P. Christensen, and Lawrence J. Lau, "Transcendental Logarithmic Production Frontiers," Review of Economics and Statist i cs 55 (1973).

265

adjustment, and more complex production techno 1og i es,1 0 Equation (5), i.e., the Cobb-Douglas labor demand function with no restrictions on returns to scale, may be expressed with minor rearrangement of terms as

In implicit form, this relationship becomes simply Li = Li (PK i> PLi, Yi )

(22 )

and, thus, its total differential is given by aL i

aL i

dLi = aPKi • dPKi + 1PLi aLi dPLi + 3yi

(23)

dYi'

Expanding this expression by computation of the partial derivatives from (21) yields

-s

1

dLi = (l/A) a+S[( a/ a+S)(PKi/PLi) a+S(l/PLi) -S dPKi - ( a/ a+ S )(PKi/PLi)~(PK i /PLi2)

1 1-a- S dPLi + (a+S)(Yi)"""U'f"B • dYi]'

(24)

Regional output that is produced with the aid of the labor input may now be usefully classified according to its ultimate purpose, i.e., basic goods for export or nonbasic goods for local utilization. For any industrial category i, this relation is merely Yi = YLi + YXi, where YLi YXi

(25 )

output of category i sold to local customers output of category i exported to customers outside the region.

Applying the total differential as a linear operator to (35) provides an equat ion for changes in output by destination, i.e., dYi

= dYLi

+ dYXi.

(26)

Thus, the change in regional employment in a particular industry which is attributable to fluctuations in export demand is essentially the portion of equation (24) that reflects 10The most complex of these problems is, perhaps, the estimation of complex multiple input production functions. A straightforward approach to this difficulty with respect to the CES function is given in V. Karuppan Chetty, "On Measuring the Nearness of Near-Moneys," American Economic Review 59 (1969).

S.L. Gr een and M.R. Pe rr yman

266

dYLi. The resulting equation may be stated as follows: (32 ) where dYj where dLXi

change in labor demand attributable to movements in export demand for production in category i. dPI

Notice that empirical estimates of the parametric terms in (27), i.e., l/( a+S), l/A, and (l- a- S)/( a+S), may be directly obtained from the logarithmic expressions discussed in Section 11. Hence, the only remaining unknown factor is dYXi. A framework for its determination is presently discussed. From equation (26), it is immediately apparent that dYXi

dYi - dYLi,

( 28)

change in regional output of good j, j=l, 2, ... , P being an index of the regional outputs of industries which utilize the production of category i as an input change in personal income in the region, a primary determinant of final local demand change in national output in category i, a proxy for changes in overall export demand.

Because this expression is in differenced form, there is no intercept paramater to be estimated, i.e., 80 is constrained to be O. Given the above equation or a comparable empirical function, dYLi may be defined as simply the local component of the explanatory relationship. In the present case, the resulting variable is

which is equivalent to dYLi dYXi = dYi - (dY i )

P

. dYi·

(28a)

NOW, defining the term 4> = (dY Li / dY i ) ,

(29)

it becomes possible to restate (28a) in the form dYXi = dYi - 4> . dYi,

dYLi = E 6.' dYj + Bp+1 • dPI . j=l J

This estimate, combined with the observed dYLi provides a plausible value for and, hence, a solution for (41). Following the construction of dYXi, it becomes possible to test a relationship of the form

(30)

or, more simply, dYXi = (1-4»

dYXi • dYi'

(31)

While data on dYi are generally available, the determination of dYXi requires estimates of the local output coefficient 4> .11 Several approaches seem reasonable for the task. For example, a principal components analysis of subcategories of each industry might well produce a latent vector which provides appropriate weights for computing dYLi and, hence, 4> .12 More directly, a regression equation might be calculated and utilized to isolate the local market. As an initial point of departure, an equation of the following form seems to be a reasonable specification:

lIThe Massachusetts model assumes this term to be constant. 12An example of a comparable application of this nature is given in Perryman, J. Larry Lyon, Lawrence Felice, and Stephen Parker, "Community Outputs and Population Increase: An Empirical Test of the Growth Machine Model," American Journal of Sociology (1980) .

(33)

L L f(Cit/Cit-1) N N Cit/ Cit-1

(34)

L

where Cit

cost of local production in industry i during period t. cost of national production in industry i during period t.

This expression merely reflects the original hypothesis of this section, i.e., that export demand responds to changes in relative regions cost patterns. In a manner comparable to that utilized in the Massachusetts model, the equation may be explicitly written as dYXi

eW

L N L N [4> (Cit/Cit)/(Cit-1/Cit-l)]T e Eit •

(35)

Such an expression may, of course, be given in a readily estimable log linear form as

L

N

L

N

(36)

Some Initial Explorations of Interregional Linkages

The only task remaining is, of course, the derivation of the terms in the cost relationships, i.e., the computation of:

L N L N (Cit/ Cit)/(Cit-1/ Cit-1)· For expositional purposes, it is useful to rearrange this statement in the form

L

N

L

N

(Cit/ Cit)/(Cit-1/ Cit-1) =

L

L

N

N

(Cit/ Cit-1) • (Cit/ Cit-1)·

( 37)

with

~ <

0

U >

0 -1.

o< p ~

0

<

267

1

In this specification, ~ is an efficiency parameter, G is a distribution parameter, u is a returns-to-scale parameter, and is a substitution parameter. In the constant returns case, of course, u is constrained to be one. This function may be estimated in a straightforward linear fashion following a logarithmjc transformation and a 8aylor series expansion. 15 Given the empirical determination of ~, 0, - u, and p, the relevant cost equation may be derived as 16

For the regional economy of Texas, the cost function associated with the general two input Cobb-Douglas function for industry i is given by 13 p

1

_1_.

U 1+P

( ~u ) [ 0

PN Lit

S

( pL )a+S Lit (38 ) where, as in previous cases, the superscript "L" denotes the local or regional economy. Dividing this expression by its value lagged one period yields

N

PKit

l+ P

1+P

-

---L ~ - 1+Pl P

+ (1- 0 ) 1

(yN) . it

u

(41 )

Thus, the second term in (37), i.e., N N (Cit-1/Cit), becomes -p

1

[ (a1+P

pf>J

Kit-1

l+p

+

1 (1- 0 )TfP

a

(Pkit/P~it-1)~ S

(PLit/PLit-1~a+S

(39 )

This equation defines the first of the two multiplicative terms in equation (37). For the national economy, the assumption that individual industries follow aCES technology with non-constant returns to scale is adopted. Hence, the specification for some arbitrary category i is, by definition,14 Yit =

~[OK i t

_p

_p

+ (1- 0 ) Lit]

- u/p

(40)

13This function is derived in Hal R. Varian, Microeconomic Analysis, New York: W.W. Norton (1978). 14This form was originally suggested in Kenneth J. Arrow, Hollis 8. Chenery, 8.S. Minhas, and Solow, "Capital Labor Substitution and Economic Efficiency," Review of Economics and Statistics 43 (961) •

-p

1+p

PLit

-1

p

N 1+P

)1

N

N

. (Yit_/Yitl

u

(42 )

Substituting (42) and (39) into (37) gives the value of the dependent variable to be utilized in (36). Hence, the employment response to relative cost differentials across regions may be estimated. It should be noted that, relative to the Massachusetts model, the framework of the present system (1) can be implemented with substantially less data and (2) permits much more flexible function forms (i.e., any production technology for which a cost equation may be estimated).

15See , for example, Jan Kmenta, Elements of Econometrics, New York: Macmillan, 1971. 16Varian, Microeconomic Analysis.

268

S.L. Gree n an d M.R. Pe rryma n

CONCLUSION This paper has sought to present a simple initial exploration of a feasible means of integrating interregional phenomena into an econometric modeling effort. In particular, the incorporation of relative costs into an employment relationship is illustrated within the context of a general framework which permits substantial flexibility with respect to the functional form of the underlying production process. Given cont i nued development of data series to measure regional capital stocks costs along present lines, the implementation of this basic structure will be further facilitated in the future. 17 Beyond the immediate application to the labor sector of an individual regional economy, the present analysis seeks to suggest a method which may be utilized in a variety of settings as a basis for "linking" regional models into a truly interactive system. Through continued endeavors of this nature on the part of members of the modeling community, the ability to ultimately generate simultaneously determined "bottom up" models of national and large area economies wi ll be significantly enhanced. 17A method for deriv i ng these measures which seems somewhat adaptive to regional systems is given in Jorgensen and Mielco Nishimizu, "U . S. and Japanese Economic Growth, 1952-1974: An International Comparison," Economic Journal 88 (1978). See also the highly relevant discussion in Stephen Putman, An Empirical Model of Regional Growth, Philadelphia : Regional Science Research Institute, 1976. REFERENCES Adams, F.G. and N.J. Glickman. (1980). Modeling the Multiregional Economic System. Lexington Books, Lexington. Arrow, K.J., H.B. Chenery B.S. Minhas, and R.M. Solow. (1961). Capital labor substitution and economic efficiency. Review of Economics and Statistics, 43, -225-250. Chetty, V. K. (1969) . On measuring the nearness of near-moneys. American Economic Review, 59, 270-281. Cobb, C.W. and P. H. Douglas. (1928). A theory of production. American Economic Review, 18, 1139-1165. Hahn~ (T971). Economic equilibrium and transaction costs . Econometrica, 39, 417-440. -Hamilton, H.R. , S.E . Goldstone, J.R. Mill iman, A. L. Pugh Ill. B. Roberts, and A. Zellner. (1969). Systems Simulation for Regional Analysis: An Application to River Basin Planning. M.I.T. Press, Cambridge. Hicks, J.R. (1965). Capital and Growth. Oxford University Press, London. Jorgenson, D.W., L.P. Christensen, and L.J. Lau. (1973). Transcendental

logarithmic production frontiers. Review of Economics and Statistics, ~, 28-45. Jorgenson, D.W. and M. Nishimizu. (1978). U.S. and Japanese economic growth, 1952-1974 : an international comparison. Economic Journal, 88, 707-726. Kmenta, J. (1971). Elements of Econometrics. Macmillan, New York. Perryman, M.R. (1980a). The supply side of regional econometric models. Modeling and Simulation, 11, 1309-1316. Perryman, M.R. (1980b). A test of the functional form of production relationships in a regional econometric model. Presented to the 1980 Conference of the Western Economic Association. Perryman, M.R., J.L. Lyon, L. Felice, and S. Parker. (1980). Community outputs and population increase : an empirical test of the growth machine model. American Journal of Sociology , 86, 1387-1400. Putman, S. (1976). An Empirical Model of Regional Growth. Regional Science Research Institute, Philadelphia. Robinson, J. (1953). The production function and the theory of capital. Review of Economic Studies, 21, 81-106. Solow, R.M. (1955). ThelProduction function and the theory of capital. Review of Economic Studies, 23, 101-108. Treyz, G.I., A.F. Friedlander, and B.H. Stevens. (1980). The employment sector of a regional policy simulation model. Review of Economics and Stat i stics, 62, 63-73. -Varian, H.R. (1978). Microeconomic Analysis. W.W. Norton, New York.