The Ramsey rule revisited.

• Author: Yang, Chin W.

1. Introduction

   In pursuit of an optimum tax scheme in terms of minimizing welfare loss, economists have been searching for a better solution since the seminal contribution of Ramsey [17]. The literature is voluminous: they include the works by Hotelling [9], Hicks [8], Harberger [7], Dimond and Mirrlees [2], Dixit [3; 4], and Sandmo [19]. The Ramsey optimum tax rule, that is, the percentage reduction in quantity demanded of each commodity be the same was interpreted by Kahn [11] as the inverse elasticity rule. The inverse elasticity rule states that the optimum tax rates and price elasticities of demand should be inversely related. The merit of the inverse elasticity rule is that the optimum tax rate does not require the explicit knowledge on prices and quantities of the taxed commodities. To evaluate policy implications, one needs only to examine the published price elasticities of demand to arrive at optimum tax rates. However, the inverse elasticity rule normally applies only to a constant cost case in which average cost (hence marginal cost) is horizontal [10; 18]. In a competitive market, while the constant cost case is plausible, it assumes away the producer's surplus. In this paper, we prove that the inverse elasticity rule with the ad valorem tax on the gross demand price is equivalent to that with the supply ad valorem tax developed by Ramsey.(1) However, the inverse elasticity formula with the unit tax as shown in this paper is different. Furthermore, we propose a sufficient condition for optimum tax ratio with a positively sloped supply schedule. In addition, we explore a sufficient condition for an optimum equiproportional tax across all commodities. Last, we employ some empirical estimates of price elasticities of both demand and supply to calculate the optimum tax ratios.

2. Inverse Elasticity Rule with Increasing Costs

   Given a set of linear demand and supply functions an ad valorem tax v based on the gross demand price [P.sub.G] is equivalent to pivoting down the demand curve by v[P.sub.G] [14]. In Figure 1, the excess burden is measured by the well-known welfare triangle whose size is determined by the tax rate [P.sub.G] - [P.sub.N] = v[P.sub.G] (difference between gross and net demand prices) and loss in output [Delta]Q. We first measure the excess burden via the following price elasticities of demand ([Alpha]) and supply ([Beta])

   [Alpha] = ([Delta]Q/Q*)(P*/([P.sub.G] - P*)) [is less than] 0 (1)

   [Beta] = ([Delta]Q/Q*)(P*/([P.sub.N] - P*)) [is greater than] 0 (2)

   where P* and Q* are pretax equilibrium price and quantity. Hence it follows immediately from (1) and (2) that

   [P.sub.G] - P* = ([Delta]Q/Q*)(P*/[Alpha]) (3)

   [P.sub.N] - P* = ([Delta]Q/Q*)(P*/[Beta]) (4)

   [P.sub.G] - [P.sub.N] = (([Delta]QP*)/Q*)((1/[Alpha]) - (1/[Beta])) (5.A)

   or

   v[P.sub.G] = ([Delta]QP*/Q*)([Beta] - [Alpha]/[Alpha][Beta]) (5.B)

   where v is the demand ad valorem tax rate. Solving for [Delta]Q from (5.B) yields

   [Delta]Q = (Q*[Alpha][Beta]v[P.sub.G])/(P*([Beta] - [Alpha])). (6)

   The size of the excess burden W is therefore

   [Mathematical Expression Omitted]

   For a group of commodities (i) whose income effect is negligible such that the compensated demand curve is a good approximation for the ordinary demand curve and (ii) that are insignificantly related, i.e., commodities are not close substitutes or complements,(2) the Ramsey rule can be cast into the following minimization problem:

   [Mathematical Expression Omitted]

   subject to [summation over i] [v.sub.i][P.sub.iG][Q.sub.i] = R (9)

   where subscript i denotes the ith commodity and R denotes the tax revenue constraint. The Lagrangian equation and its corresponding first-order conditions are:

   L = [Phi] + [Lambda](R - [summation over i] [v.sub.i][P.sub.iG][Q.sub.i]) (10)

   [Mathematical Expression Omitted]

   [Mathematical Expression Omitted]

   and from (11) and (12) we have

   [Lambda] = (

   .sub.i] - [[Alpha].sub.i])[Q.sub.i]) (13)

   [Lambda] = (

   .sub.j] - [[Alpha].sub.j])[Q.sub.j]). (14)

   Note that the changes in both prices and quantities must be small enough for the elasticity formulas to be valid. Besides, such a small change is needed to free us from the theoretical difficulty of income effect and interrelatedness between a pair of commodities. As...