Optimal government policy regarding a previously illegal commodity.

• Author: Caputo, Michael R.

1. Introduction

   The existing research on the taxation of economic goods and services has focused entirely on those commodities that are legally traded within the economy (for a survey, see Sandmo [9]). Even alternative approaches to the analysis of taxation, as in Barzel [1], have dealt exclusively with legal goods. What all such research ignores is a class of goods or services with perhaps an unsavory but important history: goods or services that are exchanged outside of legally sanctioned markets. Many illegal commodities, such as drugs, prostitution, and gambling, have extensive and well-organized "black markets" designed to facilitate exchange and overcome the complex problems facing entrepreneurs working outside the law. Simultaneously, considerable resources are allocated by the U.S. Congress and the executive branch to constrain, if not completely eliminate, black market trade. Nowhere is the battle between black market forces and government efforts at prohibition more apparent then in the "war on drugs." A host of hotly-contested issues exist concerning the nature and extent of drug use, the dynamics of drug production and distribution, the cost of illegal drug use and control, and the effects of drug prohibition on the justice system. What is not questioned, however, is that the market is sizable and that Americans spend billions of dollars a year for illegal drugs.(1)

   Development of a rational policy response to drug use must consider the consequences of the policies, strategies, and resources that are applied to controlling the market for drugs in the United States. Within the government's drug policy tool kit is the powerful option to choose alternative legal rules and tactics. The possibility of changing the formal legal constraints within the drug market has important implications for government expenditures and the allocation of resources. Furthermore, a change in legal constraints leads to changes in transaction and production costs, directly affecting the feasibility and profitability of operating within the black market for drugs.

   One critical issue for research is to examine and clarify the dynamics associated with a change in the primary legal constraint underlying U.S. drug policy: a change from a regime of complete prohibition to a regime of regulation or legalization.(2) A shift in the law means that the government must make numerous decisions concerning the optimal structure of an emerging and evolving market for legally sanctioned drugs. What strategies should the government employ to price or tax the commodity? How does the government protect and enforce its newly established right to drug revenue from the power of existing black markets? How do optimal taxing and enforcement strategies change, moving between the short- and long-run?

   Considering such a sea-change in the legal status of drugs raises a wide array of normative social issues that, for the most part, are beyond the scope of this paper. Instead, the approach taken here is to use positive economic analysis to model a number of short- and long-run taxation and enforcement strategies for a net benefit maximizing government existing in an environment of legalized drug use. Optimal control theory is used to model the government decision-making process, and to develop and discuss a number of implications for taxing and enforcement behavior. The approach developed in this paper, although motivated within the context of the drug market, is sufficiently general to be extended to any presently illegal good or service in which a non-trivial black market exists for the good or service.

2. Model Motivation

   Richardson [8] looked at the interaction between trade policy and drug legalization by modelling a two-tier system whereby a cartel supplies imperfectly competitive street dealers. A novel result from the model is that an endogenous level of violence emerges. Richardson [8] also looked briefly at the welfare consequences of alternative policies in this context. In contrast to Richardson [8], the model developed here focuses entirely on government policy and is dynamic, and considers both short-run and long-run government policies. Whereas Richardson [8] was unable to compare policy equilibria under a demand shift, the results presented here are specific in this regard (as they are for many of the comparative statics and dynamics results). For example, in the long-run, the optimal policy of the government in the face of rising total demand for the formally sanctioned product is to increase the tax rate on the legal good and increase the enforcement rate against the black market good, and let some of the increase in demand be absorbed by the black market.

   Examining the question of whether zero-tolerance drug policies inhibit or stimulate illicit drug consumption, Caulkins [5] showed that under plausible conditions, zero-tolerance policies may actually encourage users to consume more, not less, than they would if the punishment increased in proportion to the quantity possessed at the time of arrest. Caulkins' [5] model also suggested which drug policies will minimize consumption for a given drug user. Careful consideration was then given to the political feasibility of the policies, although all results pertain to the short run.

   The current paper develops an optimal control model and derives short-run and long-run policy strategies for the government in the face of changing market conditions. The distinction between short-run and long-run policies is crucial, for policies which are successful in the short run are not necessarily the same policies which are optimal in the long run. The short-run versus long-run distinction will be emphasized throughout. We define a policy response as the optimal response by the government using the instruments under its control (the unit tax rate on the legal good and the enforcement rate against black market activity) to changes in the parameters of the environment. More precisely, an exhaustive qualitative analysis of the model is carried out via the methodology of comparative statics and dynamics. This gives a complete picture of the properties of the model, and forms the basis from which specific policy recommendations can be made. For example, in the short run, the government's optimal policy response to a rise in the black market price of the good or service is to do nothing. In the long run, however, the government's optimal response to a black market price increase is to increase the unit tax rate on the legal good and decrease the rate of enforcement against black market suppliers.

3. The Model and Assumptions

   The use of a dynamic model is dictated by the nature of the problem. For example, by reducing the enforcement rate against black market suppliers and/or raising the tax rate, the government is creating some incentives for black market producers to expand their production and sales, possibly reducing the government's present value of net benefits and future market share. The dependence of the future state of legal sales on the government's current policy is what dictates the use of a dynamic model. A variant of the Gaskins [6] limit pricing model is developed here to describe the government's optimal policy with regard to taxation of the legal good and enforcement effort expended toward the well established black market.

   A dynamic limit pricing model appears to be an appropriate form to model the dual legal/illegal nature that recently legalized goods or services take on. The government is the dominant firm, controlling the unit tax rate of the legal good and the enforcement intensity against the black market suppliers of the good. Black market suppliers, however, wield little market power once the government supplied good is legalized, and therefore are treated as a competitive fringe. The formal model is presented first, and is followed by an economic interpretation of its structure.

   The government is asserted to select the time path of the unit tax rate of the legal good [Tau](t) and the enforcement rate exerted against black market suppliers u(t) so as to solve the optimal control problem

   [Mathematical Expression Omitted]

   s.t. [Mathematical Expression Omitted]

   x(0) = [x.sub.0], [limits of x(t) as t approaches +[infinity] = [x.sup.*][Alpha]

   ([Tau](t), u(t), x(t)) [element of] U, (P)

   where

   p(t) [equivalent to] [Rho] + [Tau](t) is the market price of the legal good,

   V([Beta]) [equivalent to] maximum present discounted net benefit for the government,

   x(t) [equivalent to] illegal or black market sales,

   [x.sup.*]([Alpha]) [equivalent to] steady state illegal or black market sales,

   [Delta] [equivalent to] social benefit function shift parameter,

   [Epsilon] [equivalent to] demand shift parameter,

   [Gamma] [equivalent to] price of the illegal or black market good,

   k [equivalent to] illegal or black market suppliers' response coefficient,

   [Rho] [equivalent to] net price of the legal good,

   r [equivalent to] government's rate of discount,

   [Theta] [equivalent to] social cost function shift parameter,

   w [equivalent to] unit price of the variable input used in enforcement,

   [x.sub.0] [equivalent to] initial level of illegal or black market sales,

   q = f(p; [Epsilon]) [equivalent to] total (sum of legal and black market) demand,

   B(x; [Delta]) [equivalent to] social benefit function,

   C(q; [Theta]) [equivalent to] social cost function, and

   [Phi](u; w) [equivalent to] enforcement rate cost function of the government,

   which is defined as

   [Mathematical Expression Omitted],

   where v [greater than] 0 is the scalar variable input used for enforcement and F is the enforcement rate production function. The following assumptions are imposed on problem (P).

   (A.1) [Mathematical Expression Omitted],

   [Mathematical Expression Omitted],

   [Mathematical Expression Omitted],

   [Mathematical Expression Omitted].

   (A.2) [Beta] [equivalent to] ([Alpha], [x.sub.0]) [equivalent to] ([Delta], [Epsilon], [Gamma], k, [Rho]...