彈力性・半彈力性の基本法則と新物価指数算式
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概要
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Suppose that two sets of real positive C^∞ -functions {y(t)} and {z(t} of the real variable t defined in the interval I[t_1, t_2] are given; and that the latter functions {z(t)} are, for their part, real positive functions of y(t) defined in the interval I'[y',y''] of y(t). We assume that all these functions {y(t)} and {z(t)} allow a one-to-one continuous mapping f_t to be operated on themselves, and that the mapping satisfies the following conditions: Fundamental Laws [I]:[numerical formula]Then, we can prove that the mapping f_t is represented by (d log)/(d log t)・We shall therefore designate "(y/t)=(d log y)/(d log t)" as tne "Elasticity of y with respect to t," and the above "Law [I]" as the "Fundamental Law of Elasticity." Similarly, the following law [II characterizes the mapping "(d log)/(dt)":[numerical formula]We shall therefore designate "[y/t]= (dlog y)/dt" as the "Semi-elasticity of y with respect to t" and the above "Law [II]" as the "Fundamental Law of Semi-elasticity." Now let v_t, p_t^s and q_t^s be, respectively, the total traded value, the price of the s-th commodity, and the quantity of the s-th commodity in the year "t", then we have : (1) v_t =Σ^^n__<s=1>p_t^sq_t^s Take the semi-elasticity with respect to t of both members of (1), and we have: (2)[(v_t)/t]=Σα(s,t)[(v_t^s)/t] where v_t^s=p_t^sq_t^s and α the weight α(s,t)=v_t^s/v_t=p_t^sq_t^s/Σ^^n__<s=1>P_t^sq_t^s Integrating (2) with respect to "t" from "k" to "l", we have: (3) log V_<kl> =log P_<kl>+log Q_<kl>+R where V_<kl>=(v_l)/(v_k) log P_<kl>=Σα(s,l)logp_t^s-Σα(s,k)logp_k^s log Q_<kl>=Σα(s,l)log q_l^s-Σα(s,k)logq_k^s R=Σ^^n__<s=1> Σ^^m__<i=1>{(α(s,t_i)-α(s,t_<i+i)logv^sξ_i} We then construct the following index numbers of prices P_<kl> and of quantities Q_<kl>; logP_<kl>=1/2 {log P_<kl>+log V_<kl>-log Q_<kl>} logQ_<kl>=1/2 {log Q_<kl>+log V_<kl>-log P_<kl>} It is easy to see that these index numbers satisfy the following five tests: [I] The circular test. [II]The factor test. [III]The continuity test. [IVI] The adjustment of the "Fallacy of the Beauty Contest". [V] The grouping test.
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