414 PROCEEDINGS OF THE IEEE, VOL. 63, NO. 3, MARCH 1975

Theory of Optimum Economic Growth and Energy Policy

Abrmcr-The present paper gives a concise but selfcontPiwd b a t - merit of Optimum economic growth theory and then reformulate it study the p o k y implications of the present situation of limited energy resources of various forms. The optimum energy poky so obtained is justifiiMe in terms of common sense, and economic principles, but is at variance with current eneqy poky.

INTRODUCTION s OME BASIC results of economic growth theory and optimum growth theory are presented in Section 11. The notion of a production function, its various properties,

and manifestations due to technological progress are discussed. The equation for capital accumulation is derived. Optimum distribution of investment and consumption are then deter- mined from a utility function criterion.

In Section 111, optimum growth under resource constraint is studied. It is assumed that technologies for developing various forms of energy resources are at different stages of maturity and the resources may or may not be depletable. The de- pletable resources yield diminishing returns with the same mining effort. Energy resources with yet undeveloped tech- nology are represented by Arrow's learning by doing produc- tion function [ 1 1.

Significant features of the optimum policy based on the preceding assumptions are the following:

free market for private and industrial consumption of energy resources; tax on depletable resources at either the production or sales level; research and investment incentives for types of energy resources with immature technology.

11. THEORY OF OPTIMUM ECONOMIC GROWTH A . Investment and Consumption

Economic growth theory aims at studying the long range growth of an economy. Its central issues are production and distribution. Production is expressed by a production func- tion F , which gives the quantity of produced goods in terms of capital and labor used in the production process. The total produced goods are then distributed into consumption and investment or capital goods. For simplicity, only aggregated totals of consumption goods and capital goods are represented

K=I (2.1)

F ( K , L , t ) = I + C . (2.2)

In (2.1) and (2.2), I , K , L , t , and C denote net annual invest-

work was supported by the National Science Foundation under Grant Manuscript received June 14, 1974; revised October 16, 1974. This

The author k with the Department of Electrical Sciences, State Univer- GK-160 17.

sity of New York, Stony Brook, N.Y. 11790.

ment, productive capital, labor population, time, and annual consumption, respectively. Equation (2.1) means that the total worth of productive capital is increased by net annual investment, which is the resulting figure after capital deprecia- tion from usage, antiquity, etc. having been deducted from gross annual investment. Equation (2.2) represents that the annually produced goods and services F ( K , L , t ) are either consumed or reinvested into productive capital. The produc- tion function F ( K , L , t ) gives the total amount of goods p r e duced annually as a function of capital and labor employed and also as a function of time to take into consideration the effects of technological progress.

Because the study is aimed at long range growth, full employ- ment at all times is assumed. In (2.2), total available capital K and total labor force L are used as arguments in the produc- tion function F. However, this assumption is not essential. Basically the same results are obtained by assuming L to be a constant fraction of the labor force.

B. The Production Function and Technology Change [2 ] The production function F ( K , L , t ) is meaningful only in the

domain P, defined by K 2 0, L 2 0. Generally it is assumed to the following properties:

F ( K , L , t ) 2 0 in P for all t ; The function F has second partial derivatives with respect t o all its arguments; F ( 0 , 0, t ) = 0; Marginal products are nonnegative F , = aF/aK 2 0 ; F l =

F is homogeneous of degree one in K and L , F ( X , b 5 )

t ) = h F ( K , L , t ) ; F is strictly quasi-concave in K and L :

aFlaL 2 0;

For any given constant C1 and 0 < X < 1, if F ( K , L , t ) 2 C1 and F(K ', L' , t ) 2 C 1 then

F(hK + (1 - X)K', hL + (1 - h)L' , t ) 2 C1.

The equality holds only if K = K' and L = L'. Conditions l), 3), 4) mean that production cannot be

negative, nothing is produced if no capital, and no labor are employed, and production does not decrease if more capital or more labor is used. Condition 2) is a smoothness condition which makes good sense and narrows down the class of func- tions from which F is selected to fit experimental data. Condi- tion 5) means that if capital and labor employed are increased proportionally, the produced good is also increased by the same ratio. It is valid for an industry or an aggregate of indus- tries which has grown beyond its optimal size. The capital and labor employed are then broken up into productive units of the optimum size. More capital and labor employed results simply into proportionately more productive units and hence 5).

CHANG: ECONOMIC GROWTH AND ENERGY POLICY 41 5

The condition 4) follows from 5) . As one way of deploying hK + (1 - A)K’ and AL + (1 - X) L‘ is t o divide them into two groups, the total production of the two groups is the least one can do with the total capital and total labor. Therefore,

F(AK + (1 - h)K‘ , AL + (1 - A)L‘, t ) 2 F ( A K , A L , t )

+ F(( 1 - A)K’, (1 - h)L’) .

From 5 ) , the right-hand side is equal to

A F ( K , L ) + ( l - h ) F ( K ’ , L ’ ) > A C + ( l - X ) C = C

and 6 ) follows.

neoclassic production function. When all six conditions are valid, the function F is called a

One special form of neoclassic production function is

F ( K , L ) = X + (1 - a ) L - p ] -I”. (2.3)

For the enterprise to realize maximum profit, the following marginal relations must hold:

and

Equations (2.4) and (2.5) give wage W and rental R as marginal yields of labor and capital respectively. Let ij and k be defined as

* w K R L

a=- k = - - .

These are the wage to rental and capitalization per worker ratios. The elusticity of substitution 0“is defined as

It expresses the percentage increase in capital to replace labor in response to a wage increase relative to rental.

Evaluating W and R from (2.3) to (2.5) gives

Taking logarithm and differentiate gives

Equation (2.8) shows that the production function of the form (2.3) has constant elasticity of substitution (CES). The function satisfies the conditions 1) to 6 ) if its parameters are in the proper ranges: y > 0,O < a < 1, and - 1 < p < 00. In the limit of p -+ 0, it is reduced to the Cobb-Douglas form:

F ( K , L ) = y K a L’-OL. (2.9)

In the preceding discussion, we have not considered the effects of technological progress, or the parameter t . Econo- mists account for technological progress in the following way. In (2.3) and (2.9) K and L are replaced by effective capital K,, and effective labor Le , respectively. Many ways of representing technological progress have been proposed, and we shall

mention the following: 1 ) labor augmenting (Harrod neutral)

L e = @ ( t ) L

K , = K (2.10)

where @( t ) is an increasing function of t ; 2) capital augmenting (Solow neutral)

L , = L

K e = @ ( t ) K (2.1 1)

3) learning by doing

L , = K ~ L

K , = K .

The first two types are more suitable for well-developed industrial society where the progress is near saturation and depends almost entirely on general scientific and technological advance. The third type is more suitable for representing a new industrial society or a new industry where each new installation embodies some significant improvement over the preceding one, and the progress depends on how frequently new investments are made. Multiplying L by K a , where a > 0, means that with each additional investment, the overall effec- tiveness of labor is increased. The labor augmenting factor increases with K instead of t. The technological improvement or learning is a consequence of doing rather than that of general scientific advance. In studying the recent trends of economic growth, econo-

mists noted that 1) capital investment grows almost propor- tionately with total production, and rental remains essentially unchanged; 2) real wage is on the rise, and labor’s share in- creases faster than either total production or capital’s share.

The preceding trends can be easily explained by labor augmenting technological progress with constant K, /L , ratio.

C. Optimum Growth Theory for a One Sector Model (31, [4 ] Both dynamic programming and maximum principle have

been used for optimum growth studies. Dynamic pro- gramming gives some additional conditions and interpretations which are useful. It will be followed in the subsequent study. From (2.1), (2.2), and (2.10), the equation for accumulation of capital can be written as

K = F ( K , L e * ) - C (2.12)

where

L = ,toeH (2.13)

representing an exponential rate of population growth, and L o , A, 7 are constants. The utility function to be maximized is

(2.14)

where the felicity function U ( C / L ) is of either of the following forms:

U ( x ) = A x ’ - O , 0 < a< 1 (2.15a)

U ( x ) = A log x . (2.15b)

Equation (2.15a) and (2.1 5b) represent decreasing marginal

416 PROCEEDINGS OF THE IEEE, MARCH 1975

return in felicity as personal consumption is increased. It is expected as one meets one's most urgent needs f i t . In the present theory the labor population L is assumed to be a constant fraction of the total population P. Per capita con- sumption C/P is proportional to the per worker consumption C / L . The integral J represents upto a proportionality constant the sum total felicity for the entire population with some discount on future returns. The future discount ratep can be explajned by the uncertainty in looking into the future, and that we do not care about posteriority as much as we do the present, and any number of other reasons. Let L e = Le" be defined as the number of effective workers. Let k, and c denote the &pita1 and consumption per effective worker, respectively. Equations (2.12) and (2.13) become

& = f ( k ) - c - yk (2.16)

J = L o exp [-(a- 7)tl U ( c ) d t + C 1 (2.17) 1" where C1 = 0 for (2.15a), and (I = 1 for (2.15b). The constants w and 7 are defrned as

0 = p + U ? (2.18)

r = n + r . (2.19)

Maximizing J is the same as maximizing the integral le:

Je = im exp [-(a- r)tl U(c) d t . (2.20)

Equations (2.16) and (2.20) give the state equation and utility function on per effective worker basis. Since k is the only state variable, the maximum value of J , is a function of the initial value of k alone. Let V ( k 0 , t o ) be defined as

f"

with the initial condition that

k ( t o ) = ko.

By a change of variable t f = t - t o , (2.21) becomes

(2.22)

m

V ( k 0 , t o ) = m a x i exp [-(a - r)(t' - t o ) ] U ( c ) dt' c ( f )

= V W O , 0) exp [-(a - 7 ) r o I . (2.23)

From dynamic programming, if a path is optimal from t to O D 7 it must be optimal from t + A t to m. Therefore,

V(k, t ) = max {exp [-(a - r) t ] U ( c ) A t + V ( k + A k , t + A t ) } C

= max {exp [-(a - 7)l U ( c ) A t + V(k, t ) C

+ - (f(k) - c - 7 k ) k A t av ak

- ( w - 7 ) V ( k , t)dt + o ( A t 2 ) } . (2.24)

Subtracting V ( k , t ) from both sides of (2.24) dividing the r e

sulting expression by At, and then let A t -+ 0 gives

o = max {exp -(a - r)t] ~ ( c ) C

av + - (f(k) - c - 7k) - (O - 7 ) V} . (2.25)

The value of c which maximizes the right-hand side of (2.25) is the optimal c corresponding to k:

ak

Canceling out exp [ - (w - r)tl from both sides of (2.26) gives

(2.27)

Note that (2.27) is to hold for all values of t . Similarly, (2.25) becomes

Let u(k) denote V ( k , O ) . Equations (2.27) and (2.28) give simultaneous relations between u', k, and i?

u'(k) = U'(E') (2.29)

(O - 7 ) ~ ( k ) = U(E') + U'(c^) [ f ( k ) - i? - 7kl . (2.30)

Differentiating (2.30) with respect to k gives

(2.3 1)

Substituting (2.29) into (2.31) gives I! A

(') [ f ( k ) - E' - 7k] - + f ' ( k ) - w = 0. (2.32) dc"

U'(E') dk

From (2.1 51, U"(E')/U'(E') = -o/E', and (2.32) becomes

U - - [ f(k) - E' - 7k] - + f ( k ) - w = 0. (2.33)

dE' E' d k

Let k* denote the value of k which satisfies

f ' ( k * ) = o (2.34)

c* =f(k*) - 7k*. (2.35)

Then c ( t ) = c* and k ( t ) = k* satisfies both the state equation (2.16) and the necessary condition for optimality (2.33). The path c*, k* is indeed optimal if w - 7> 0. It represents a steady rate of economic growth with investment proportional to income (total production) and real wage increasing with productivity.

Equation (2.33) can be rewritten as

(2.36)

Equation (2.36) can be used for numerical integration of c(k), except that the right-hand side of (2.36) is indeterminate

CHANG: ECONOMIC GROWTH AND ENERGY POLICY 411

C

3.0- 1 I I I I for the system described by (2.26) and (2.17) is 0'1, 7'3% H = exp [-(a - y ) t ] V ( c ) + q {f(k) - c - yk) . (2.39)

2.5- *GOLOEN RULE 1

Comparing (2.25) and (2.39) gives

a v a v a t q = a k H = - - - .

(2.40)

I I 1 1 I I I I 0 2 4 6 8 1 0 1 2 W G

k-

Fig. 1. Optimum consumption c versus capital investment k. The mows indicate the direction of trajectory 8s f increases.

Fig. 2. Capital accumulation (upper curves) and growth of personal income (lower curves). Curves are computed at population grawth of 2 percent annually and technological prolpesr of 3 percent (solid curves) and 1 percent (broken curves).

at the reference point c*, k*. To overcome this difficulty, differentiate (2.33) with respect t o k and set k = k*:

+f"(k*)= 0 (2.37)

dc 1

Equation (2.36) is integrated with the initial slope given by (2.38). The results are illustrated in Figs. 1 and 2. Fig. 1 gives the optimum value of c at each level of investment k. Fig. 2 shows the effect of technological progress on per capita con- sumption and accumulation of capitaL In a society with planned economy, the optimum annual

consumption and investment can be realized by decree. While this is not possible in American society, they can nevertheless be closely approximated with the aid of tax and fiscal policy. Arrow and Kurz call this problem the controllability problem and discuss it at length in [3] . (The term controllability as used here is somewhat different in meaning from its usage in control theory.) We shall postpone the discussion to Section 111 when we discuss the more difficult problem of energy

In economic literature, maximum principle is usually used instead by dynamic programming. The Hamiltonian function

policy.

In Section 111, maximum principle is used to determine the optimum energy policy, and relations of the form (2.40) are used to interpret the results.

III. OPTIMUM GROWTH WITH RESOURCE CONSTRAINT The optimum growth theory presented in the preceding

section is defective in a very fundamental way as the avail- ability of natural resources is very much a factor of production which has not been represented. It is assumed in fact that there are unlimited natural resources, capital is accumulated surplus product of labor, and production is a function of capital and labor only. The theory does not imply that natural resources are free for the taking, but does imply that any amount of natural resources can be obtained by a proportional amount of productive efforts without any limitation on total quantity or on the rate of expenditure.

A. System Equations The problem of energy resource constraint can be formulated

as

F ( K o , L o , R , t ) = Ii + e ( R o ) + L C i = N

i=o (3.1)

R = C R i - Lr i = N

i= 0 (3.3)

L = b L

L = C Li i = N

i = O

e-p* LU(c , r , b ) d t = max. (3.8)

It is assumed that there are N different types of energy r e sources Ri , i = 1, 2, * - , N. Each type R i has total deposits (known or unknown) D i , annual rate of consumption Ri , capital equipment K i , and labor force Li for its production. Note that Li includes only laborers working directly on pro- duction of R p The equipment K i is lumped together into a general goods item which includes everything else except Ri , i = 1, 2, * , N, and L O , K O represent the labor and capital, respectively, for production of general goods. The energy resources Ri , i = 1, 2, * . , N, are assumed to be exchangeable in the production of general goods and in direct consumption and the total energy resource for such use is denoted by R .

If two resources are not exchangeable in their usage, they can be represented by different letter symbols in the general

41 8 PROCEEDINGS OF THE IEEE, MARCH 1975

Equations (3.6) and (3.7) represent the growth of labor population in terms of net birth rate b , and division of labor into general production and production of various types of energy resources.

Equation (3.8) is similar to (2.14) except that the felicity function U now depends on three variables, per capita con- sumption c, per capita consumption of energy resources r , and birth rate b. The dependence of U on c and r are expected to be of the form (2.15). U should be quite flat for a reasonable range of b , but decreases if b is required to be either too large or too small.

I(-

Fig. 3. Learning-bydoing function for various energy resourceg

production function F and felicity function U. The optimiza- tion procedure and results are essentially the same as what follows.

In (3.1) to (3.8), F, I, e, and c are in units of some standard goods per unit time, for instance, real dollar yearly. K is in units of the same standard goods. R , Ri, and r are in units of some common, exchangeable measure of the various types of resources per unit time, for instance megajoules per year. Di, i = 1, 2, * . * , N, are then in units of megajoules. The functions @i(Di) are dimensionless.

Equation (3.1) means that the total production of general goods is channeled into net investment IO for the general goods productive capital K O , investments I i for the capitals Ki, expenditure e ( R o ) for exchanging energy resource R 0 from some external source, and consumption LC. Equation (3.2) represents that capital of various types are accumulated from corresponding net investments. Equation (3.3) gives an accounting of R as the sum of various resources R i (including R o ) with direct consumption of energy Lr deducted.

In equation (3.4), Gi(Ki , g(Ki) Li, t ) represents the mining effort for resource R i with capital Ki and labor Li. The func- tion g(Ki) is an increasing function of Ki and represents the learning by doing factor. @i(Di) decreases as Di is decreased. It represents decreasing returns as lower grade mines are used when a resource Ri is being depleted. Equation (3.5) gives the relation between Di and Ri.

To understand the meanings of these terms, some typical resources will be used as illustrations.

I ) Oil Wells: The techniques are well developed: g ( K 1 ) = 1. @ I (Dl ) represents decreasing returns as D l is reduced. At first only the gushers are used. Then the industry extends to deeper wells and marginal fields.

2) Off-Shore Drilling and Shale Oil: New techniques are being developed, and g ( K 2 ) increases with more investment K 2 , representing the improving technology. However, its deposit is finite and @ 2 ( D z ) is a decreasing function as Dz decreases.

3) Solar Energy: New techniques are being developed but are very far from commercially applicable on a large scale. The resource is unlimited, and @3(D3) = 1.

4) Nuclear Fusion: A workable technique for peaceful use of nuclear fushion energy has not been developed. However, it is promising. The resource is virtually unlimited if developed, and @ 4 ( 0 4 ) = 1.

Fig. 3 gives the typical g ( K ) versus K curves of various energy sources. Note that (4a) represents nuclear fushion which is eventually successful, and (4b) represents nuclear fushion if proved to be impossible for peaceful use.

B. Optimization In the system equations (3.1) to (3.7), (3.2), ( 3 . 9 , and (3.6)

are state equations; Ki, i = 0, 1, 2, * , N, Di, i = 1, 2, - * * , N, and L are state variables; and the other equations are con- straints which reduce the number of independent control variables. Without loss of generality, the following are taken ascontrol: Ii, i = 0, 1, 2, . * * ) N; Li, i = 1, 2, . * * , N; R o , r , and b. The variables L O ; Ri, i = 1, 2, . . e , N; R and care regarded as dependent functions of the state and control variables as expressed by (3.7), (3.4), (3.3), and (3.1), respectively.

Pontryagin's maximum principle will be used to obtain necessary conditions for optimization. However, it is under- stood from dynamic programming that the adjoint variables p and q are simply partial derivatives of the maximum return function with respect to the corresponding state variables. The Hamiltonian function is

i = N i = N

Tbe adjoint differential equations are obtained from H :

(3.10)

aH aKi

q i = - -- - (-e-"' U,FR + p i )

where C ~ K and G ~ L denote the partial derivatives of the func- tion Gi with respect to its first and second arguments, respectively,

(3.12)

The control variables maximize the Hamiltonian:

(3.14)

-- aH - aLi

- e pt U, [ - F L ~ + FR CiL gi(Ki) ai] - pi G ~ L g i ( K i ) ai= 0

aH - 0

= e-pf U, [FR - e'(Ro)] = 0

(3.15)

(3.16)

CHANG: ECONOMIC GROWTH AND ENERGY POLICY 419

aH - = e -Pr [ U, - U,FR ] = 0 ar

aH ab - = e - P ‘ L U b + q L L =o .

(3.17)

(3.18)

Equation (3.14) shows that all qi’s are the same and can be written simply as q :

q i = q = e - ’ r U , - , i = O , 1 ,2 ;* . ,N . (3.19) A

From (3.10) and (3.19)

4 d 4 d t

- - = p - - 1 n U c = F ~ , . (3.20)

From (3.1 1) and (3.20)

From (3.12) and (3.19)

(3.22)

From (3.15)

From (3.16) and (3.17)

FR = e ’ ( R o ) (3.24)

U r = U,F,. (3.25)

From (3.18) and (3.13)

qL = -e-’‘ U, (3.26)

As in Section 11, an optimum utility function V of all the state variables is defined as

V ( K , D , L , t ) = max e-’‘ L U ( c , r , b ) d t ’ . (3.28) LW Equation (3.1 9) gives

(3.29)

The exponential factor e-pr on the right hand side of (3.29) arises because the value of the utility function has been dis- counted from t = 0 as a starting point. If the discounting is referenced from the time t , V becomes

Y ( K , D, L , t ) = max l- e exp [ -p( t ’ - t ) l LU(c , r , b ) dt’ (3.30)

and (3.29) becomes

(3.3 1)

C. Interpretations of Mathematical Results In American society, the control variables are not directly

under the command of any person or government. However, the variables can be influenced by tax, fiscal policy, laws and education. The underlying assumption is that each individual tends to maximize his felicity function with the limited financial means at his disposal. There are individual variations to be sure, but the statistical mean values are almost com- pletely predictable and controllable. To keep within the con- fines of economics, only tax and fiscal policy implications of the mathematical results will now be discussed:

I ) Free Market Condition for Energy Resources: In (3.25), U, and U, are marginal felicities of energy resources and con- sumption goods. The ratio UJU, represents the price at which users are willing to pay for energy resources in terms of con- sumption goods. FR is the additional units of consumption goods that can be produced if an additional unit of energy resource is used in production. Equation (3.25) means that price of oil to consumer should equal the price of additional goods produced if the same amount of oil is used in pro- duction. In other words, neither the consumer, not the oil consuming industries should be subsidized. They should be charged the same price.

2) Import and Export Duties on Energy Resources: Equa- tion (3.24) means that the usual import duties should be applied equally to energy resources. Because price rises on increased demand and lowers on increased supply, the marginal cost e ’ (Ro) is higher than the average import price e ( R o ) / R o . FR is the domestic price and should be equal to e’ (Ro) accord- ing to (3.24).

As the price of imported and domestic oil must be equal in a free market, the difference e ’ ( R o ) - e ( R o ) / R o represents the import duty to be levied. Under export situation, the marginal revenue e’(R0) is lower than e ( R o ) / R o , and the difference is equal to an export duty.

3) Equal Marginal Util i ty of All Investments: Equation (3.31) shows that marginal utility of investment should be equal to the marginal felicity of consumption for all types of investments. However, as the utility function is difficult to calculate, this result has on€y conceptual value, and is difficult to apply.

4) Capital Rental and Interest Rate: Equation (3.20) gives the relation between capital rental F K ~ and interest rate. In an economy with rising income, U, is expected to decrease due to increasing c. A unit of goods consumed today would yield a utility Uc( t ) . The same goods with interest at A t later will yield an utility

exp ( i A t ) U c ( t + A t ) exp ( - p a t )

where i is the interest rate. Under rule of free substitution, this expression is equal to U,:

exp ( i A t ) Uc( t + A t ) exp ( - p a t ) = Uc( t ) . (3.32)

Solving i from (3.32) and letting A t --* 0 gives

Equations (3.33) and (3.20) give

(3.33)

(3.34)

The four preceding relations are the ones which normally exist

420 PROCEEDINGS OF THE IEEE, MARCH 1975

TABLE I OFTIMUM ENERGY POLICY

Technology Research and Depletion Multiplier Production Investment

Energy Source D @(Dl g(K) TaX Incentive

Well Oil ftnite / 1 yes no Shale and Offshore Oil finite / / yes Yes Natural gas finite / 1 Yes no S o h energy OD 1 / no Yes Fusion energy OD 1 / no yes

under free-market conditions, and it is significant that these relations are not changed by resource constraint.

5) Tax on Production on Depletable Resources: Under ideal condition, the wage of a worker is his marginal production for general goods FL, :

w = F L ~ , (3.35)

Assuming that workers for different products are exchangeable, the same wage is paid to a worker who marginally produces GiL@igi(Ki) units of ith resources. Therefore, the price at which a manufactuer can offer the ith resources is

(3.36)

According to (3.23), as FR represents the market price, pi /q is the tax which can be levied on the product. From another point of view, pi = a V/aDi represents the decrease in total utility if Di is decreased by one unit, and p i /q represents the amount of compensatory investment required to bring V to its original value. In all fairness, this compensatory investment should be charged to the firm which caused the depletion and offered the product for sale for its own profit.

An alternative form of tax is on excessive use of depletable resources, for instance, a horsepower tax on automobiles. Quite likely such a tax would serve the society better than direct tax on production or sale of gasoline. However, it is not possible to predict this from our theory unless a much more complicated model is used. 6) Investment Incentive: Equation (3.21) can be expressed

as marginal product of capital = net receipt from oil sales after tax X marginal oil production of capital. Let MO denote the marginal oil production of capital. From (3.2 1)

Mo = [ Gik + G i ~ g i ( K i ) L i ] Qi. (3.37)

In the preceding, oil is used in the sense of an unspecified energy resource. As already mentioned, gi(Ki) > 0 significantly only in a new industry. Suppose there are Nfirms engaged in the new industry of exploration and production of shale oil:

Li = L,j. Ir

The production function for the pth fm is

Equation (3.38) means that the technological progress factor g ( K i ) due to the total investment K i is shared by all the N firms. This may appear to be a strong assumption in view of

the existence of industrial secrets and patents. But in long range terms each competent firm must be very much in the know. Industrial secrets and patents may represents an edge of one fuin versus another for a short duration of one or two years, they rarely make a significant difference on the total productivity. Assuming constant K / L ratio at the beginning, a firm making an investment unilaterally would obtain a marginal increase in production:

aQ - = [ CjK + GjL g;(Kj) L,i] a+. (3.39) i

Comparing (3.39) and (3.37), the difference in return to investment must be financed by the government in some form of investment incentive:

x? aKpi

Mo - = G ~ L gi (Ki ) (Li - L,i) @i

Z G ~ L gi(Ki) Li Qi. (3.40)

It may appear odd that production tax and investment incen- tive are two components of the same optimum policy. Do they cancel each other? The fact is that firms do not use windfall profits to invest in some venture which may eventually benefit the society unless such ventures definitely benefit themselves. Even if they can be persuaded to do so, there is no reason to give these firms monopolistic power in developing future energy resources. A far better arrangement appears to be to create the conditions for independent free competitions in the areas: present oil production and development of future resources. 7) Government Organized or Sponsored Research: An un-

proved or undeveloped energy resource (solar or fusion) can be represented by the condition that

FL, > FR GiL Qigi(Ki) (3.41)

for all values of Li . Then Li = 0 and there is no marginal return to investment. However, a V/aKi # 0, and can be quite large. Such investments are usually called long range research projects. They have no prospect of immediate financial return, but they must be made because of (3.31). This gap can be filled by government organized or financed research.

8) Population Control: Equations (3.26) and (3.27) can be rewritten as:

av aL - = -ub (3.42)

ub = U,(FL - FR r - C ) + u+ (Jl- b ) ut,. (3.43)

If each additional person reduces the optimum return V , a V / a L is negative. A positive ub means that b is below its

PROCEEDINGS OF THE IEEE, VOL. 63, NO. 3, MARCH 1975 42 1

natural level, and some measure of population control is implied. Equation (3.43) gives the optimum path for imple menting such a policy.

CONCLUSION Table I illustrates the policy implications of optimum growth

theory. The arrows // mean that @(D) reduces as D is re- duced and g(K) increases as K is increased.

Optimum growth theory provides a logical link between mathematical models of energy resources and objective func- tion on one hand, and optimum policy on the other. If the models and objective function are accurate and complete, then the conclusions are infallible. It is of interest that our conclu- sions are obtained from the model structure and qualitative de- scriptions of the functions without exactly specifying the functions. The conclusions are direct results of the nature

of the resources and are quite independent on estimates of total deposits at various economic feasibility level. However the quantitative aspects of the optimum policy do depend on these estimates, and can be determined from a numerical solution of the set of equations (3.1) to (3.8) and (3.19) to (3.27).

REFERENCES [ 1 1 K. J. Arrow, “The economic implications of learning by doing,”

I 2 1 E. Burmeister and A. R. Dobell, Mathematical Theories of Eco- Rev. Econ Studies, vol. 29, pp. 155-173, June 1962.

131 K. J. Arrow and M. Kurz, “Public investment, the rate of return, nomic Growth. New York: Macmillan, 1970.

and optimal fmcal policy.” Baltimore, Md.: Johns Hopkins Res, 1973 (published for Resources For the Future, Inc.).

141 R. Britto, “Some recent developments in the theory of economic growth: An interpretation,” J. Econ. Literature, vol. 11, no. 4, pp. 1343-1366, Dec. 1973.

Economic Theories and Empirical Models of location Choice and Land Use: A Survey

Abstmct-This is a survey of the theonticd md a n p W models of 1 ~ ~ c h o i c e d L n d u s e m c i t i e s . l l t e s u r v e y i s ~ c t e d f n r m a histaicdpaspectivemdthemodelsarecritiallyevrhutedintermsof their logicrl structrue. Special attention b given to the techid dif- ficultics mated d e n the -tion of the convexity of the produc- tion-isdmpped. Itisindicrtsddyfurlbaprogressinthis area requires a fundrmenbl rethinLiRg of the way m which we con- ceptualizethekhrriorofindividarlloationchoia.

I. PREFACE

0 UR OBJECTIVE is to present a unified and reasonably self-contained survey of the economics of locational choice and land use in urban areas. The existing knowl-

edge on location and land use is not satisfactorily integrated into central economic theory. In order to identify as precisely

work was supported by the National Science Foundation under Grant Manuscript received August 7, 1974; revised October 8, 1974. This

GK41603. R. Artle is with the Graduate School of Business and the Electronics

Research Laboratory, University of California, Berkeley, Calif. 94720. P. P. Varaiya is with the Department of Electrical Engineering and

Computer Science and the Electronics Research Laboratory, University of California, Berkeley, Calif. 94720.

as possible the nature and causes of this lack of integration we propose to evaluate location and land use theory from the perspective provided by the Arrow-Debreu framework of general equilibrium, and our rationale for choosing the Arrow- Debreu framework is as follows. If, on the one hand, such an evaluation places location theory within the main body of general equilibrium theory, then the researcher would be able to exploit the mathematical structure of the Arrow-Debreu framework. If, on the other hand, the evaluation reveals such intrinsic properties of the location and land use problems as to vitiate fundamental assumptions of the Arrow-Debreu theory, it could still serve a constructive purpose by its very identifica- tion of such properties.

This frame of reference imposes some severe limitations on the scope of the present study. Firstly, we assume that loca- tional choices are made in the context of a capitalist economy, where all resources, including land, are privately owned, and where all transactions occur in competitive markets. Secondly, governmental or, more generally, collective action is excluded. Hence, we do not consider the literature covering the location of such public facilities as hospitals and fiie stations nor the literature which is motivated by the efficiency and equity