The Effect of Inner Mobility of Shops on Tax Revenue

In this note we consider a simple model of a service sector. In this service sector there are two types of shops, each offering distinct services at differential prices with fixed cost possibly being also unequal. When the decrease of customers is experienced by one of these two types, not both, then having less profits, those shops belonging to the damaged type can shift to the other type, thus alleviating losses on the whole. This can also mitigate the effect on tax revenue. The typical sector which the authors have in mind is that of pubs and restaurants where two types are the 'first-rate (more expensive with more services)' and the 'popular' class. Based on this model, we conduct an elementary comparative statics analysis. Most results are natural consequences, which can be inferred without using equations. And yet we supply proofs based upon equations and inequalities. Our main result is that with migration between two groups, the sector may incur less loss against the decrease in customers, especially that in the first-rate class. This means the effect on tax revenue is also milder with shiftability. Our model in this note is quite restrictive because the size of customers is exogenously given and does not respond to price differentials. Moreover in the analysis of comparative statics, the number of shops is fixed with no entry and no bankruptcy. These constraints should be removed in a more general framework

Factor Price Equalization : Geometrical Conditions

This paper presents a geometrical approach to the univalence problem for a system of cost functions. We present a natural (almost tautological) extension of a geometrical theorem due to McKenzie: our sufficient condition is related to the non-separability of two cones formed by convex combinations of the rows of the Jacobian matrix. This means that the cones spanned by the rows of Jacobian matrix (i.e., production coefficients) do not move wildly so that the two cones corresponding to the two end points (i.e., factor price vectors) cannot be separated by the hyperplane orthogonal to the vector of changes in factor prices. Unlike most ofthe previous propositions, our condition can naturally include as a special case such linear systems as having a non-singular matrix. We also give an alternative condition employing the concept of monotone functions. Dual to the above result is one more condition, which is shown to be closely connected with Kuhn's WARP-like requirement when the given functions are concave as well as homogeneous of degree one

Univalence of Nonlinear Mappings : A Qualitative Approach

This note is a sequel to the previous one published in this journal (Vol. 30, No.1). In that article, we used one of mean value theorems to prove the univalence of a nonlinear mapping based on the qualitative regularity ofthe Jacobian matrix. The qualitative regularity is a property of a matrix whose regularity is shown to be valid by using only the sign patterns of mappings involved. In this note, we extend the result into a vector space over an integral domain. The vectors themselves are of n-tuples of elements in the integral domain. This integral domain is totally ordered, and some natural properties are assumed concerning this order. First two lennnata are given, and the first one is in fact a sort of mean value theorem for mappings from a direct product of discrete spaces into a discrete space, and utilizes mathematical induction. The second lemma depends on the fact that theory of matrices and determinants can be constructed also on a ring except for inverse matrix. Finally, our main proposition derives from the very integrity of a given domain. Another merit of the result is that the domain of a mapping need not be convex, and can be even a set oflattice points

Global Comparative Statics for Models with Hicksian Imperfect Stability

This note is to provide a bridge between traditional local analysis for comparative statics and its global counterpart. Many economists vaguely believe that it is possible to obtain a global result by applying consecutively a series of local results. This belief is not well founded in models where parameters enter in a not-so-simple way. An example is given to show that local analysis is after all local. In the proof of the first main theorem, a consecutive use of a well known local result is employed. Some necessary assumptions are explicitly stated. Then this theorem is applied to establish another main theorem in which a simple repetitive application of local analysis may break down because some required properties cease to hold. This two-stage approach seems to be useful in tackling with other types of equations. As an application of our theorems, a general equilibrium model with Hicksian imperfect stability is taken up. A comparative statics result due to Hicks is extended to the case of global changes. An interesting point to note is that when dealing with global comparative statics, the old system plays no explicit role. Only the new system matters together with the new and old equilibrium values. Understanding this point is important when we come to consider such real situations as involve technical changes in which new processes as well as new commodities turn up