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The commodity flow gravity model
Regional and Urban Economics 4 (1974) 69-75. © Nor~h-Holland Pul~ishing Company
e
THE COMMODITY FLOW GRAVITY MODEL A Theoretical reassessme~t John H. NIEDERCORN* University of Southern California, Los Angeles, U..~I.A.
Josef D. MOOREHEAD* State of California, Department of Transportation, Sacram¢ "J, California, U.S.A. Received September 1973 A generalized gravity model is shown to be the proper tool for describing commodity flows in space generated by a revenue maximizing monopolistic firm. Hartwick's finding that the slope and intercept parameters of the logarithmic transformation of the commodity flow gravity equation are positively ,~orrelated is confirmed and a theoretical explanation of this phenomenon is provided.
1. Introduction
In a recent article in this journal, John M. Hartwick (1972) has demonstrated that the simple commodity flow gravity model i~ a relatively poor predictor of flows generated in a transportation cost minimizing system. He also came to the somewhat unexpected conclusion that when the gravity equation is transformed into double logarithmic form and the parameters estimalled using regression analysis, the estimated values of the slope and intercept of ll:hefunction seem to be related to each other in a positive linear manner. ~ Both of Hartwick's conclusions appear to be correct. However, this article will prove that given two reasonable speci~cations of the average revenue function over space, a generalized form of the gravity model iis still the appropriate tool for describing commodity flows generated by a revenue maximizing monopolistic firm~ Furthermore, within the context of this model, it will be e~:sy to identify the causal factors underlying Hartwick's unexpected findings about th~ slope and intercept ef the transformed gravity equation. The commodity flow gravity model presented in this paper was developed by direct analogy to the utility maximizing model of Niedercorn and Bechdolt (1969). *The authors would like to thank Professor Peter Gordon of the University of Southern California who read an earlier draft of this article and made several helpful suggestions for its improvement. ~Hartwick (1972) appears ~o have made a minor error and consequently misinterpreted his results. In eq. (2.9), the sign preceding ~ should 1": negative. As a result of this mistake he concludes that p and 0c are negatively correlate-" ,~hen, in fact, J and (-0c) are negatively correlated. The latter relationship implies that ,8 and ~ are indeed vos;tiivelycorrelated.
J.H. Niedercorn, J.D. Moorehead, The commodity f l o w gravity model
70
2. The utility maximization model In its most general form the utility maximization model for the individual consumer requires the maximizaldon of the objective function kU i
c
=
,
"
(a "~ o ) ,
O)
(0 < ~ < O.
(2)
subject to the constraint n
.M, = r Z (,/~)(~Zi~). j=l
where k~i a
n
t5 C kTij
aP~
kM ! r
d~ {g
total utility of individual k at origin i, interacting with persons or things at all destinations per unit time, a positive constant ef proportionality, --~ total number of origins and destinations, = population of destinationj, = a positive constant representing scale effects, --~- number of trips taken by individual k from origin i to destination j, per unit of time, = number of people or t+~ings at destination j, with whom individual k would like to interact, .per unit time,
m
utility of individual k at origin i, interacting with people or things at destination j (per number of persons or things with whom individual k would like to interact), : = total amount "f money indlvi'dual k, located at origin i, is willing to spend on travel, per unit time.. a positive constant of proportionality, = distance between origin i and destination j, and ---- a constant such that 0 < ~ < 1, insuring a concave travel cost function.
Assuming a utility per potential interactor function of the form
f
aP;'
(0 < b < 1),
(3.)
and substituting it into eq. (1), the former equation i~ maximized subject to the constraint embodied in eq. (2), using the method of Lagrangian multipliers. The followirg gravity equation is obtained" n
~
=" tM,{P~I[ '~. [P~/di: I(1 -b)]}(1/r(l-b)l'dij) "/(l-b), j=l
( j - 1, 2 , . . . n).
(4)
J.H. Niedercorn, J.D. Moorehead, The commodity flow graw!ty model
71
It should be noted tbat a logarithmic utility function of the form
f(kT, j~ = \aP~]
ln(~,~)
(5)
yields the special case of eq. (4) in which b = 0. These two equations generalize the earlier results of Niedercorn and Bechdolt, who worked only "~vltha linear transportation cost function, by incorporating a concave power cost function as en integral part of the mo:lel. When a = 1, this model reduces ~o the earlier one.' As noted by Niedercor, i and Bechdolt (1969, pp. 278-279), these results can easily be aggregated acros~ individuals by areal unit to yield results that are capable of making reasonably accurate forecasts of travel behavior.
3. The ~evenue maximization commodity flow model Now consider the case of the business firm which has a sp~ tial monopoly in the region defined -_~er the n different origin and destinaticn points. 3 In the short run such a firm can be connived as maximizing total reven~ue subject to a constraint embodying the assumptions that the firm's total transportation cost budget and total output are fixed. Therefore, the firm faces the problem of how to allocate its given total product over n different markets in ~uch a way that total revenue is maximized for a given fixed level of total transportation cost. Under these assumptions, the kth firm in area i can be conceived as maximizing the total revenue function
.R, = . Z * p /tqtj~
(6)
j=l
subject to the constraint n
kB, = r E
(o < • <" 1).
(7)
./=1
In these equations, kR~
= total revenue accruing to the kth firm in area i,
~qu
= amount of product sold by firm k in area i and shipped to customers in area j,
g Aq,A ~ - ~ ) -- revenue earned per potential customer in area j, 2For a more detailed discussion of this and other generalizations of the Niedercorn-. Bechdolt model, see Moorehead (1971). 3Several variants of the model described in this ~ction are presented ia Mo,t)rehead (1971).
72
~Bi
J.H. Niedercorn, J.D. Moorehead, The commodityflow gravity model
= the total fixed short run transportation cost budget of firm k located in area i. This amount can be conceived as the product of the fixed given level of total output times a fixed target level of transportation costs per unit of product given by historical experience.
It is aow clear that eqs. (6) and (7) are exactly analogous to (1) and (2). Instead of trying to maximize utility subject to a budget constraint, the firm attempts to maximize total revenue subject to a budget constraint. Forming the augmented obje,~tive function, n
kR~
/¢¢/~ ' ~ k"
n
X P~g'"-~"~' + 2 [ r ~ t(d,~)~qij) --kB,], j__, \ , r j /
(8)
_
taking partial derivatives with respect to the kqU and 2, and setting them equal to zero, yields a system ofn + 1 simultaneous equations. Ira revenue per potential customer function of the form
g(kqiflaP~) = (~qu/aPf) a,
(0 < fl < 1),
(9)
is assumed, and its first derivatives with respect to kqU are taken, these derivatives can then be substituted into the above n + 1 equations to obtain the solution"
t,q.,.~ = kB i {Pf ~ ~=l [P;Idi,""
- ')] }( l l rt' - #)"di,)flt ' - '), (j = l, 2 , . . . n).
(10)
Again, it should *~.enoted that if a logarithmic revenue per potential customer function is assumcd of the form
g(kqu/aP~) = ln(kq~j/aP~) ,
(11)
the special case of eq. (10) in which fl = 0 is obtained as the solution. It might bc interesting to check whether the revenue per potential customer functions assumed in eqs. (9) and (11) make economic sense. In other words, do they yield negatively sloped average revenue functions ? Taking each of these equations, and multiplying them by aP~ to obtain total revenue per area A and dividing by kqil yields the respective average revenue functions:
kAR~ = (aPSe]kqs:)l -p
(12)
i,ARlt = a_P~ln (o,qu/aP~) "
(l 3)
and :~qlj
It is easy to observe that so long as t:' < 1, eq. (12) will be negatively sloped, and that eq. (13) will aiways be negatively sloped. From eqs. (10) and (12), two things immediately become clear. First, as distance from the point of manufacture increases, the monopolistic firm sells smaller and smaller amounts per potential customer. Second, as distance
J.H. Niedercorn, J.D. Moorehead, The commodity flow gravity model
73
increases, the firm sells at a higher and h~gher price. Thus, the firm ma:~dmizes revenue by operating c~n the more elastic part of the demand curve of near customers and the more inelastic part of the curve of distant customer:~. This result conforms exactly to the results one ~vould expect from the theory of third degree price discrimination. In principle, it is possible to aggregate eq. (10) across all firms in a given indm~try located in area i. 4 Then, an equation similar to eq. (10) could conceiv. ably be used to predict aggregate commodity flows between areas. However, to do so would be more risky than it, the case of individuals' ~ravel for two rea:sons. First, different types of commodities, unlike individuals, may be subject to different demand and transportation cost functions. In other woros, the values of fl, r and 0t may differ for different commodities. If the valuers of fl or 0cdiffer, a simple method of aggregation is i~possible. Second, aggregation will wc,rk if and only if every firm is essenti~,lly a monopolist and either cannot or wil~ not attempt to take over other firms' markets. On the other hand, since the model is short run in nature and a good deal o f habit and inertia tend to persist in the business world, once calibrated, it may give good forecasts for relatively ,'i~hor~. periods of time, if the commodities over which it is aggregated have similar demand and transportation cost functions.
4. Reconciliation with Hartwick's results It is easy to see why the commodity flow gravity model does not yield accurate solutions to the Hitchcock-Koopmans transportation problem. The former assumes revenue maximization by monopolistic firms; the objective of the latter is only to minimize transportation costs between two sets of points with fixed total quantities supplied at each point and fixed total quantities demanded at each point. Thus, these two different analytical techniques are addressed to the solution of two entirely different kinds o f problems; each has its place. The explanation o f Hartwick's second result is as yel by no me~.ns obvious. If natural logarithms are taken of both sides of eq. (10), it is clear that lc(kB~) would be analogous to Hartwick's fl and 0~/(1-/5) to his ~. His results show that his/~ and (-0c) are negatively correla~.ed, or what is the same thing, fl and ~ are positively correlated. In order to corroborate Hartwick's results it will be sufficient to show that ln(,B~) and 0c/(1-fl) can be expected ~o be positively correlated, s However, this can only be c~one within a long run framework. 4]~his process assumes that each firm in the industry has a monopoly with respect tO its product, even though the products may be similar, such as different alloys of steel. Shipments of product could be measured either by weight or by volume. 5In actuality, this is not exactly what Hartwick found. The base to which his slope parameter applies is apparently rd~j. In the derivation presented here, the base is closely related, but differs from this amnunt by a cc~sta~t factor. : ". ;ertheless, the ~roof given in this at~'.icle implies Hartwick's result because the existence oi a constant factor cannot influence any correlation that may exist between the two variables.
d.H. Niedercorn, d.D. Moorehead, The commodity flow gravity model
74
Assume that in the long run the kth firm at origin i desires to maximize its prcfits and has constant average costs of production, K. Then, total profits for this firm per unit time can be written as
........... kfl, = a j ~ 1P~g(,'~j)c[kqu'~_ Kj= xhqu - ri= xZ(di~l)~,q,J)•
(14)
Taking partial derivatives with respect to the kqil, and setting them equal to zero yields n equations in n unknowns. Again, taking the first derivatives of eq. (9), substituting them into these n equations, and solving, yields
#tl(1-tJ)(aP~) (j = 1, 2 , . . . n). (15) k~lrl = (K+ rd~),l(a-p) , Assuming the logarithmic revenue function given in eq. (ll), taking its first derivatives: substituting them into the above mentioned n equations, and solving, yields aef _
(j = I, 2 , . . . n).
(16)
Given the logarithmic and power revenue functions, total long run transportation costs of the firm are represented by the following equations:
kB~ = ra
II
Z
(d~])e~
(17)
j= l (K + rd~) ' and n
•
c
kB~ = ra~t/(l_#),.., (d,~)Pj j (K+rdl~) 1/°'#)"
(18)
It is now necessary to show that InG,B3 and ot/(1 - p ) are positively correlated. It is reasonable to assume that the structure of consumer preferences described by the average revenue function remains con qant; therefore, it is only necessary to show that lnb,B3 and ot are positively correlated. In order to demonstrate this correlation it will be sufficient to aseertzin that the first derivative of InG,B3 with respect to ot is positive. Denoting the summation factor in eq. (11) by Y, and the summation factor in eq. (18) by Z, it is easy to show that don ~B~) _ K ~ P:fd,9 In dq dot - "Yj=~t ( - ' ~ r ~ > O, so long as all d u > 1, and don kB/') 1 " c ' =-'~" Z Pj(d,])In dot "-" l - t
du(K+rd,7)
(19)
J.H. Niedercorn, J.D. Moorehead, The commodityflow gravity model
75
Therefore, if K > [fir/(,
-
-
~]d~
(21)
for every destination, then don
Br)/du > O.
(22)
More specifically, if
< xl(x+ra
s)
for every j, then don ~Br)/da > 0.
(24)
Clearly, the expression on the fight hand side of inequality (23) approaches unity either if r approaches zero or d o does likewise. It will generally not fall much below that level so long as rd~ is small relative to K. Therefore, it appears that the derivative given in eq. (20) will be positive unless [I gets close to one. Finally, one would not expect [] to approach unity in any case, because eq. (12) indicates that under such conditions quantity demanded approaches zero for all nonnegative values of average revenue. In other words, the demand curve tends to approach the vertical axis as a limit for all non-negative values of average revenue.
5. Conclusion This paper has shown how a short run commodity tlow gravity model based on revenue maximization can be derived in a manner analogous to the case of utility maximization by the individual traveler. A long run profit maximization model is also presented. The implications of these two models are then discussed. It is concluded that the gravity model best describes the pattern of shipments of a price discriminating spatial monopoly. In addition, Hartwick's finding that the slope and intercept parameters are positively correlated was confirmed and a theoretical explanation of this phenomenon was provided.
References Hartwick, J.M., 1972, The gravity hypothesis, and transportation cost minimization, Regional and Urban Economics 2, 297-3t~7. Moorehead, J.D., 1971, A generalized economic derivation of the gravity law of spatial interaction, unpublished Ph.D. Dissertation (University of Southern California). Niedercorn, J. and B. Bechdolt, 1969, An econov,:tic derivation of the gravity law of spatial interaction, Journal of Regional Science 9, n~. 2.
e
THE COMMODITY FLOW GRAVITY MODEL A Theoretical reassessme~t John H. NIEDERCORN* University of Southern California, Los Angeles, U..~I.A.
Josef D. MOOREHEAD* State of California, Department of Transportation, Sacram¢ "J, California, U.S.A. Received September 1973 A generalized gravity model is shown to be the proper tool for describing commodity flows in space generated by a revenue maximizing monopolistic firm. Hartwick's finding that the slope and intercept parameters of the logarithmic transformation of the commodity flow gravity equation are positively ,~orrelated is confirmed and a theoretical explanation of this phenomenon is provided.
1. Introduction
In a recent article in this journal, John M. Hartwick (1972) has demonstrated that the simple commodity flow gravity model i~ a relatively poor predictor of flows generated in a transportation cost minimizing system. He also came to the somewhat unexpected conclusion that when the gravity equation is transformed into double logarithmic form and the parameters estimalled using regression analysis, the estimated values of the slope and intercept of ll:hefunction seem to be related to each other in a positive linear manner. ~ Both of Hartwick's conclusions appear to be correct. However, this article will prove that given two reasonable speci~cations of the average revenue function over space, a generalized form of the gravity model iis still the appropriate tool for describing commodity flows generated by a revenue maximizing monopolistic firm~ Furthermore, within the context of this model, it will be e~:sy to identify the causal factors underlying Hartwick's unexpected findings about th~ slope and intercept ef the transformed gravity equation. The commodity flow gravity model presented in this paper was developed by direct analogy to the utility maximizing model of Niedercorn and Bechdolt (1969). *The authors would like to thank Professor Peter Gordon of the University of Southern California who read an earlier draft of this article and made several helpful suggestions for its improvement. ~Hartwick (1972) appears ~o have made a minor error and consequently misinterpreted his results. In eq. (2.9), the sign preceding ~ should 1": negative. As a result of this mistake he concludes that p and 0c are negatively correlate-" ,~hen, in fact, J and (-0c) are negatively correlated. The latter relationship implies that ,8 and ~ are indeed vos;tiivelycorrelated.
J.H. Niedercorn, J.D. Moorehead, The commodity f l o w gravity model
70
2. The utility maximization model In its most general form the utility maximization model for the individual consumer requires the maximizaldon of the objective function kU i
c
=
,
"
(a "~ o ) ,
O)
(0 < ~ < O.
(2)
subject to the constraint n
.M, = r Z (,/~)(~Zi~). j=l
where k~i a
n
t5 C kTij
aP~
kM ! r
d~ {g
total utility of individual k at origin i, interacting with persons or things at all destinations per unit time, a positive constant ef proportionality, --~ total number of origins and destinations, = population of destinationj, = a positive constant representing scale effects, --~- number of trips taken by individual k from origin i to destination j, per unit of time, = number of people or t+~ings at destination j, with whom individual k would like to interact, .per unit time,
m
utility of individual k at origin i, interacting with people or things at destination j (per number of persons or things with whom individual k would like to interact), : = total amount "f money indlvi'dual k, located at origin i, is willing to spend on travel, per unit time.. a positive constant of proportionality, = distance between origin i and destination j, and ---- a constant such that 0 < ~ < 1, insuring a concave travel cost function.
Assuming a utility per potential interactor function of the form
f
aP;'
(0 < b < 1),
(3.)
and substituting it into eq. (1), the former equation i~ maximized subject to the constraint embodied in eq. (2), using the method of Lagrangian multipliers. The followirg gravity equation is obtained" n
~
=" tM,{P~I[ '~. [P~/di: I(1 -b)]}(1/r(l-b)l'dij) "/(l-b), j=l
( j - 1, 2 , . . . n).
(4)
J.H. Niedercorn, J.D. Moorehead, The commodity flow graw!ty model
71
It should be noted tbat a logarithmic utility function of the form
f(kT, j~ = \aP~]
ln(~,~)
(5)
yields the special case of eq. (4) in which b = 0. These two equations generalize the earlier results of Niedercorn and Bechdolt, who worked only "~vltha linear transportation cost function, by incorporating a concave power cost function as en integral part of the mo:lel. When a = 1, this model reduces ~o the earlier one.' As noted by Niedercor, i and Bechdolt (1969, pp. 278-279), these results can easily be aggregated acros~ individuals by areal unit to yield results that are capable of making reasonably accurate forecasts of travel behavior.
3. The ~evenue maximization commodity flow model Now consider the case of the business firm which has a sp~ tial monopoly in the region defined -_~er the n different origin and destinaticn points. 3 In the short run such a firm can be connived as maximizing total reven~ue subject to a constraint embodying the assumptions that the firm's total transportation cost budget and total output are fixed. Therefore, the firm faces the problem of how to allocate its given total product over n different markets in ~uch a way that total revenue is maximized for a given fixed level of total transportation cost. Under these assumptions, the kth firm in area i can be conceived as maximizing the total revenue function
.R, = . Z * p /tqtj~
(6)
j=l
subject to the constraint n
kB, = r E
(o < • <" 1).
(7)
./=1
In these equations, kR~
= total revenue accruing to the kth firm in area i,
~qu
= amount of product sold by firm k in area i and shipped to customers in area j,
g Aq,A ~ - ~ ) -- revenue earned per potential customer in area j, 2For a more detailed discussion of this and other generalizations of the Niedercorn-. Bechdolt model, see Moorehead (1971). 3Several variants of the model described in this ~ction are presented ia Mo,t)rehead (1971).
72
~Bi
J.H. Niedercorn, J.D. Moorehead, The commodityflow gravity model
= the total fixed short run transportation cost budget of firm k located in area i. This amount can be conceived as the product of the fixed given level of total output times a fixed target level of transportation costs per unit of product given by historical experience.
It is aow clear that eqs. (6) and (7) are exactly analogous to (1) and (2). Instead of trying to maximize utility subject to a budget constraint, the firm attempts to maximize total revenue subject to a budget constraint. Forming the augmented obje,~tive function, n
kR~
/¢¢/~ ' ~ k"
n
X P~g'"-~"~' + 2 [ r ~ t(d,~)~qij) --kB,], j__, \ , r j /
(8)
_
taking partial derivatives with respect to the kqU and 2, and setting them equal to zero, yields a system ofn + 1 simultaneous equations. Ira revenue per potential customer function of the form
g(kqiflaP~) = (~qu/aPf) a,
(0 < fl < 1),
(9)
is assumed, and its first derivatives with respect to kqU are taken, these derivatives can then be substituted into the above n + 1 equations to obtain the solution"
t,q.,.~ = kB i {Pf ~ ~=l [P;Idi,""
- ')] }( l l rt' - #)"di,)flt ' - '), (j = l, 2 , . . . n).
(10)
Again, it should *~.enoted that if a logarithmic revenue per potential customer function is assumcd of the form
g(kqu/aP~) = ln(kq~j/aP~) ,
(11)
the special case of eq. (10) in which fl = 0 is obtained as the solution. It might bc interesting to check whether the revenue per potential customer functions assumed in eqs. (9) and (11) make economic sense. In other words, do they yield negatively sloped average revenue functions ? Taking each of these equations, and multiplying them by aP~ to obtain total revenue per area A and dividing by kqil yields the respective average revenue functions:
kAR~ = (aPSe]kqs:)l -p
(12)
i,ARlt = a_P~ln (o,qu/aP~) "
(l 3)
and :~qlj
It is easy to observe that so long as t:' < 1, eq. (12) will be negatively sloped, and that eq. (13) will aiways be negatively sloped. From eqs. (10) and (12), two things immediately become clear. First, as distance from the point of manufacture increases, the monopolistic firm sells smaller and smaller amounts per potential customer. Second, as distance
J.H. Niedercorn, J.D. Moorehead, The commodity flow gravity model
73
increases, the firm sells at a higher and h~gher price. Thus, the firm ma:~dmizes revenue by operating c~n the more elastic part of the demand curve of near customers and the more inelastic part of the curve of distant customer:~. This result conforms exactly to the results one ~vould expect from the theory of third degree price discrimination. In principle, it is possible to aggregate eq. (10) across all firms in a given indm~try located in area i. 4 Then, an equation similar to eq. (10) could conceiv. ably be used to predict aggregate commodity flows between areas. However, to do so would be more risky than it, the case of individuals' ~ravel for two rea:sons. First, different types of commodities, unlike individuals, may be subject to different demand and transportation cost functions. In other woros, the values of fl, r and 0t may differ for different commodities. If the valuers of fl or 0cdiffer, a simple method of aggregation is i~possible. Second, aggregation will wc,rk if and only if every firm is essenti~,lly a monopolist and either cannot or wil~ not attempt to take over other firms' markets. On the other hand, since the model is short run in nature and a good deal o f habit and inertia tend to persist in the business world, once calibrated, it may give good forecasts for relatively ,'i~hor~. periods of time, if the commodities over which it is aggregated have similar demand and transportation cost functions.
4. Reconciliation with Hartwick's results It is easy to see why the commodity flow gravity model does not yield accurate solutions to the Hitchcock-Koopmans transportation problem. The former assumes revenue maximization by monopolistic firms; the objective of the latter is only to minimize transportation costs between two sets of points with fixed total quantities supplied at each point and fixed total quantities demanded at each point. Thus, these two different analytical techniques are addressed to the solution of two entirely different kinds o f problems; each has its place. The explanation o f Hartwick's second result is as yel by no me~.ns obvious. If natural logarithms are taken of both sides of eq. (10), it is clear that lc(kB~) would be analogous to Hartwick's fl and 0~/(1-/5) to his ~. His results show that his/~ and (-0c) are negatively correla~.ed, or what is the same thing, fl and ~ are positively correlated. In order to corroborate Hartwick's results it will be sufficient to show that ln(,B~) and 0c/(1-fl) can be expected ~o be positively correlated, s However, this can only be c~one within a long run framework. 4]~his process assumes that each firm in the industry has a monopoly with respect tO its product, even though the products may be similar, such as different alloys of steel. Shipments of product could be measured either by weight or by volume. 5In actuality, this is not exactly what Hartwick found. The base to which his slope parameter applies is apparently rd~j. In the derivation presented here, the base is closely related, but differs from this amnunt by a cc~sta~t factor. : ". ;ertheless, the ~roof given in this at~'.icle implies Hartwick's result because the existence oi a constant factor cannot influence any correlation that may exist between the two variables.
d.H. Niedercorn, d.D. Moorehead, The commodity flow gravity model
74
Assume that in the long run the kth firm at origin i desires to maximize its prcfits and has constant average costs of production, K. Then, total profits for this firm per unit time can be written as
........... kfl, = a j ~ 1P~g(,'~j)c[kqu'~_ Kj= xhqu - ri= xZ(di~l)~,q,J)•
(14)
Taking partial derivatives with respect to the kqil, and setting them equal to zero yields n equations in n unknowns. Again, taking the first derivatives of eq. (9), substituting them into these n equations, and solving, yields
#tl(1-tJ)(aP~) (j = 1, 2 , . . . n). (15) k~lrl = (K+ rd~),l(a-p) , Assuming the logarithmic revenue function given in eq. (ll), taking its first derivatives: substituting them into the above mentioned n equations, and solving, yields aef _
(j = I, 2 , . . . n).
(16)
Given the logarithmic and power revenue functions, total long run transportation costs of the firm are represented by the following equations:
kB~ = ra
II
Z
(d~])e~
(17)
j= l (K + rd~) ' and n
•
c
kB~ = ra~t/(l_#),.., (d,~)Pj j (K+rdl~) 1/°'#)"
(18)
It is now necessary to show that InG,B3 and ot/(1 - p ) are positively correlated. It is reasonable to assume that the structure of consumer preferences described by the average revenue function remains con qant; therefore, it is only necessary to show that lnb,B3 and ot are positively correlated. In order to demonstrate this correlation it will be sufficient to aseertzin that the first derivative of InG,B3 with respect to ot is positive. Denoting the summation factor in eq. (11) by Y, and the summation factor in eq. (18) by Z, it is easy to show that don ~B~) _ K ~ P:fd,9 In dq dot - "Yj=~t ( - ' ~ r ~ > O, so long as all d u > 1, and don kB/') 1 " c ' =-'~" Z Pj(d,])In dot "-" l - t
du(K+rd,7)
(19)
J.H. Niedercorn, J.D. Moorehead, The commodityflow gravity model
75
Therefore, if K > [fir/(,
-
-
~]d~
(21)
for every destination, then don
Br)/du > O.
(22)
More specifically, if
< xl(x+ra
s)
for every j, then don ~Br)/da > 0.
(24)
Clearly, the expression on the fight hand side of inequality (23) approaches unity either if r approaches zero or d o does likewise. It will generally not fall much below that level so long as rd~ is small relative to K. Therefore, it appears that the derivative given in eq. (20) will be positive unless [I gets close to one. Finally, one would not expect [] to approach unity in any case, because eq. (12) indicates that under such conditions quantity demanded approaches zero for all nonnegative values of average revenue. In other words, the demand curve tends to approach the vertical axis as a limit for all non-negative values of average revenue.
5. Conclusion This paper has shown how a short run commodity tlow gravity model based on revenue maximization can be derived in a manner analogous to the case of utility maximization by the individual traveler. A long run profit maximization model is also presented. The implications of these two models are then discussed. It is concluded that the gravity model best describes the pattern of shipments of a price discriminating spatial monopoly. In addition, Hartwick's finding that the slope and intercept parameters are positively correlated was confirmed and a theoretical explanation of this phenomenon was provided.
References Hartwick, J.M., 1972, The gravity hypothesis, and transportation cost minimization, Regional and Urban Economics 2, 297-3t~7. Moorehead, J.D., 1971, A generalized economic derivation of the gravity law of spatial interaction, unpublished Ph.D. Dissertation (University of Southern California). Niedercorn, J. and B. Bechdolt, 1969, An econov,:tic derivation of the gravity law of spatial interaction, Journal of Regional Science 9, n~. 2.