引力モデルの一般化とその成立機構

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Many types of gravity models have been proposed. However, they are generally expressed by the expression which is similar o the expression showing the law of universal gravitation: W_<ij>=Г<ij>(M_iM_j)/L_<ij> (i) where W_<ij> is the quantity of a social phenomena which is found between the i th zone and the j th zone, Г<ij> is a coefficient or a coefficient proper to i, j, or i and j, M_i and M_j are the index of the magnitude of the effect of the i th zone and the j th zone on the W_<ij> respectively, L_<ij> in the index of the distance between the i th zone and the j th zone. But, some other generalized forms of the gravity models are already proposed to express the characteristics of the models. Department of Scientific and Industrial Research, Road Research Laboratory in England showed the generalized form by the formula: W_<ij>=kM_iM_jf(D_<ij>) (ii) where k is a parameter, and f(D_<ij>) is a function of D_<ij> which is the distance from the i th zone to the j th zone, or the time or cost of travelling between these zones. The function f(D_<ij>) is called "deterrent function". Morton Schneider, Walter G. Hansen, and Alan M. Voorhees proposed the generalized forms of gravity models, too. In this paper, the present writer tries to show the generalized form of gravity models expressed by the expression (iii) to represent the characteristics of the models and to classify the models. W_<ij>=K^αK_<i.^β>K._<j^γ>K_<ij^δ>A_<i.^κ>A._j^λL_<ij>^<-1> (iii) where K is a coefficient, K_i., K._j and K_<ij> are the coefficients proper to the j th zone, and the i th and the j th zones, respectively, α, β, γ, and δ are the parameters whose values are 0 or 1 and α+β+γ+δ is 0 or 1, and k and λ are also parameters whose values are 0 or 1, A_i., and A._j are the index of the magnitude of the effect of the i th zone and the j th zone on the W_<ij>, respectively. When we express the gravity model generally by the expression (iii), we can classify the many types of the gravity models proposed up to this time into 4 types of model. (I) Type ακλ: α, k, λ=1, β, γ, δ=0 (W_<ij>=KA_i.A._jL_<ij>^<-1>) (II) Type γκ:γ, κ=1, α, β, δ, λ=0 (W_<ij>=K._jA_i.L_<ij>^<-1>) (III) Type γλ:γ, λ=1, α, β, δ, κ=0 (IV) Type δκλ:δ, δ, λ=1, α, β, γ=0 (W_<ij>=K_<ij>A_i.A._jL_<ij>^<-1>) Prof. W. Isard explained the mechanism of the existence of his gravity model of trips. The present writer tried to find the explanations of the mechanism of the existence of the generalized gravity model and a special gravity model of interzonal flow of goods expressed by the expression: W_<ij>=κ(W_i.W._j)/W..D_<ij>^α where W_<ij> is the interzonal flow of goods between the i th zone and the j th zone, W_i. is Σ^^n__<j=1>W_<ij>, W._j is Σ^^n__<i=1> W_<ij> and W.. is Σ^^n__<i=1>W_i. or Σ^^n__<i=1>W._j, D_<ij> is the distance between the i th zone and the j th zone (n is the total number of zones), and κ and α are parameters. The expression (iv) is also found at a stage in the process of the explanation of the existence of the Isard's gravity model, which is found by Prof. Isard. The explanations which are tried in this paper are based on the explanation by Prof. Isard. Firstly, we assume that when the index of distance L_<ij> or the distance D_<ij> can be neglected, the values of the W_<ij> in the formula (iii) or (iv) W_<ij'> are given by the expression: W_<ij'>=F)Ai.^κ, A._j^λ) (v) W_<ij>'-(W_i.W._j)/W.. (v') And secondly, we assume that when the index of distance L_<ij> or distance D_<ij> can not be negrected, the ratio W_<ij>/W_<ij'> is determined by the function g(L_<ij>^<-1>) or κD_<ij>^<-α>. Using this assumption, we can get the expression: (W_<ij>)/(W_<ij'>)=g(L_<ij>^<-1>) (vi) or (W_<ij>)/(W_<ij'>)=κD_<ij>^<-α> (vi') Using the expression (v) and (vi), we can get the expression: W_<ij>=F(A_i.^κ, A._j^λ)g(L_<ij>^<-1>) (vii) This is the general form of the gravity model - the formula (iii) would be expressed by the expression (vii). And we can derive the expression: W_<ij>=κ(W_i.W._j)/(W..D_<ij>^α) (vii') from the expression (vii') which is equal to the expression (iv). It will be able to say that we can get many types of gravity models by changing the forms of the functions f(A_i.κ, A._j^λ), and g(L_<ij>^<-1>).
流通経済大学の論文

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鈴木啓祐
流通経済大学

引力モデルの一般化とその成立機構

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