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Show APPENDIX F GEOMETRIC AND ALGEBRAIC PROOFS OF EXCISE TAXES INTRODUCTION " Excise taxes" can be classified as two types. One type is the " unit tax" or the " straight levy." Under this type, each unit of a taxed commodity or base is taxed a specific amount. A second type, the " ad valorem tax" is calculated as a percentage of the selling price of the taxed item. Most excise tax theory concerns itself with the effects of excise taxes upon monopolies, oligopolies, monopolistic competitors, and pure competition. Unless otherwise stated, all discussion of theory presented herein is in terms of partial- equilibrium analysis, since very little has or can be said about effects of excise taxes under dynamic- equilibrium. Three publications ( Bishop, 1968; Musgrave, 1959; Taubman, 1965) develop the theory of excise taxes in various industries and discuss special cases found in the theory. The discussion which follows is based largely on these works. The economic analysis presented utilizes simplifying functions for ease of understanding. The use of linear functions facilitates not only the mathematical presentation of the theory but also an accompanying graphic analysis. A more rigorous analysis is presented in the articles just cited and can be referred to if the linear assumption is inappropriate. Given a linear demand function of the form p = a+ bx and a supply function of the form s = c+ dx and the market equilibrium clearing condition s = p, it can be shown that if a unit tax of t is levied, the corresponding change in price ( A p) is given by the expression: AP = b - ( d) t If d> o and b< o as is the case in Figure F- l, price will go up less than the tax levied ( pi- p0< t). This is accounted for by the upward sloping supply curve. As price rises, the quantity taken goes down and part of the tax increase is absorbed in lower production costs. At this stage it is important to note several things: 1) if the industry is one of constant costs ( d = o) then price goes up by the amount of the tax no matter what the demand conditions; 2) change in price is dependent only upon the slopes of the demand and supply functions and not their intercepts; and 3) the ratio of the price increase to the unit tax is equal to the ratio of the elasticity of demand over the sum of the elasticity of demand and supply. In mathematical notation: Ag = Pi- Po e t t e+ T? In the diagrammatic analysis, Figure F- l, initial equilibrium is at M0 with output of Q0 and price of P0. A unit tax of BD is imposed and the new intersection of price net- of- tax and supply is at Mj. Q^ units are produced and sold at a price of Pj. Figure F- l. Competitive Industry. 89 |
