Optimal government policy regarding a previously illegal commodity.
Ostrom, Brian J.
I. 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.
II. 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.
III. 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], r, [Theta], w, [x.sub.0])
[element of] intA, where [Mathematical Expression Omitted] is compact
and convex, and [Beta] is constant over time.
(A.3) [there exists] an optimal solution to (P) denoted by ([Tau](t;
[Beta]), u(t; [Beta]), x(t; [Beta]), [Lambda](t; [Beta])), where
[Lambda](t; [Beta]) is the optimal path of the current value shadow
price of illegal sales.
(A.4) The optimal triplet ([Tau](t; [Beta]), u(t; [Beta]), x(t;
[Beta])) [element of] intU, where [Mathematical Expression Omitted] is
compact and convex.
(A.5) The Hessian determinant of the current value Hamiltonian with
respect to ([Tau], u) given by (3c) below, is nonzero along the optimal
path.
Assumption (A.1) imposes a standard smoothness assumption on the
market demand function, social benefit and cost functions, and
enforcement production function. It also asserts that the market demand
function is downward sloping in the legal market price and that the
marginal product of the variable input used in enforcement is positive
and declining. Furthermore, the demand function shift parameter affects
the total demand function by shifting it parallel to the right. Hence an
increase in [Epsilon] can be thought of as a change in the income of
consumers or a change in the tastes of individuals in favor of more
consumption of the good. The social benefit function is a strictly
decreasing and strongly concave function of illegal sales, since it is
defined as the benefits resulting from the decrease in criminal activity
associated with the purchase of the legal good. In other words, as
illegal sales fall, social benefits rise because, say, the externalities associated with making illegal purchases (i.e., supporting illegal
suppliers) likewise fall. Clearly, if illegal sales are zero, such
benefits are at a maximum [Mathematical Expression Omitted], while if
all sales of the good came from the black market, i.e., x = q, then the
aforementioned benefits are zero. In addition, the marginal social
benefits decrease as the benefit function shift parameter increases. In
contrast, since the social cost function captures such externalities as
drug treatment costs and the lower productivity of drug users, it is a
function of the total demand for the good. That is, social costs are
dependent on the total consumption of the good, not whether it was
purchased from legal or illegal sources, unlike the benefit function.
The social cost function is a strictly increasing and strongly convex
function of total sales, and the marginal social cost function increases
with an increase in the cost function parameter. Finally, social costs
are zero if total use of the good is zero.
Assumption (A.2) requires that the parameters of the problem lie in
the interior of their compact and convex set. Therefore the analysis
avoids the mathematical details associated with differentiating with
respect to parameters when they are at the boundary of their set. The
parameters of the problem are furthermore assumed to be constant
throughout the planning horizon, i.e., the government has static
expectations and perfect foresight with respect to the parameters.
Assumption (A.3) asserts that (P) has a solution, hence allowing an
economic analysis of the problem to be carried out. Assumption (A.4)
requires that the optimal path be bounded and nonzero. The nonzero
assumption about the optimal path makes economic sense. For example, it
is hard to believe that the government would not tax the legal good
(i.e., [Tau](t) = 0) even for a moment, since then it would earn zero
tax revenue from the good but still incur social and enforcement costs.
The assumption that black market sales are positive reflects the belief
that a current well established black market would not totally vanish
under competition from a legal domestic industry, and that some
enforcement of the law would always take place. Finally, assumption
(A.5) allows the use of the Implicit Function Theorem in solving the
Maximum Principle set of necessary conditions.
Because f(p(t); [Epsilon]) is the total demand function for the good,
illegal sales x(t) must be subtracted from total demand in order to
arrive at the residual demand for the legally produced good, f(p(t);
[Epsilon]) - x(t). Thus instantaneous tax revenue generated from legal
sales is the product of the residual demand and the unit tax rate of the
legalized good, [Tau](t)[f(p(t); [Epsilon]) - x(t)]. In order to arrive
at net instantaneous benefits, social benefits B(x(t); [Delta]) are
added and enforcement costs [Phi](u(t); w) and social costs C(f(p(t);
[Epsilon]); [Theta]) are subtracted from gross instantaneous tax
revenue. The government is asserted to have a positive rate of time
preference for net benefits, so the integrand of (P) is the present
discounted value of instantaneous net benefits for the government at
time t. Thus the optimal value function of the government, V([Beta]), is
the maximum present discounted value of net benefits from its optimal
taxing and enforcement policy.
Finally, the state equation of the model implies that for a given
enforcement rate, illegal sales will increase (or fall less quickly)
whenever the legal market price rises relative to the illegal price. In
other words, for a given enforcement rate, consumers view the legal and
illegal goods as perfect substitutes in which the price of the legal to
illegal good is the important determinant of demand for either class of
the good. A unit rise in the enforcement rate is assumed, without loss
of generality, to reduce illegal sales by one unit, ceteris paribus. The
increase in the enforcement rate may be viewed as a higher rate of
seizure of the illegal good coming from outside the home country, or it
may be thought of as an increase in the seizure rate of the illegal
domestically produced good. Unlike standard limit pricing models,
however, illegal sales may fall even if the legal market price exceeds
the illegal price so long as the enforcement rate is set high enough.
Hence the government has three methods of reducing illegal consumption
at its disposal: (i) reduce the unit tax rate on the legally produced
good, (ii) increase the enforcement rate, or (iii) employ some
combination of the above two policies.
IV. Government Decision Making in the Short Run
Define the current value Hamiltonian as
H([Tau], u, x, [Lambda]; [Delta], [Epsilon], [Gamma], k, [Rho],
[Theta], w) [equivalent to] [Tau](f([Rho] + [Tau]; [Epsilon]) - x) +
B(x; [Delta]) - [Phi](u; w)
- C(f([Rho] + [Tau]; [Epsilon]); [Theta]) + [Lambda][k([Rho] + [Tau]
- [Gamma]) - u]. (1)
The necessary conditions are, by Theorem 3.12 of Seierstand and
Sydsaeter [10, 234]:
[H.sub.[Tau]] [equivalent to] [Tau][f.sub.p]([Rho] + [Tau];
[Epsilon]) + f([Rho] + [Tau]; [Epsilon]) - x - [C.sub.q](f([Rho] +
[Tau]; [Epsilon]); [Theta])[f.sub.p]([Rho] + [Tau]; [Epsilon]) +
[Lambda]k = 0 (2a)
[H.sub.u] [equivalent to] -[[Phi].sub.u](u; w) - [Lambda] = 0 (2b)
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
The necessary conditions also require that the Hessian matrix of H
with respect to ([Tau], u) is negative semidefinite along the optimal
path, that is
[H.sub.[Tau][Tau]] [equivalent to] [Tau][f.sub.pp]([Rho] + [Tau];
[Epsilon]) + 2[f.sub.p]([Rho] + [Tau]; [Epsilon]) - [C.sub.q](f([Rho] +
[Tau]; [Epsilon]); [Theta]) [f.sub.pp]([Rho] + [Tau]; [Epsilon])
- [[[f.sub.p]([Rho] + [Tau]; [Epsilon])].sup.2][C.sub.qq](f([Rho] +
[Tau]; [Epsilon]); [Theta]) [less than or equal to] 0 (3a)
[H.sub.uu] [equivalent to] -[[Phi].sub.uu](u; w) [less than or equal
to] 0 (3b)
[Mathematical Expression Omitted].
By assumption (A.5) [H.sub.[Tau][Tau]][H.sub.uu] [greater than] 0,
which from equation (3a) implies that [H.sub.[Tau][Tau]] [less than] 0,
since [H.sub.uu] [equivalent to] -[[Phi].sub.uu](u; w) [less than] 0
holds by assumption (A.1) and the definition of [Phi]. Notice that
equations (2a) and (2b) are not simultaneous equations in ([Tau], u).
Because [H.sub.[Tau][Tau]] [less than] 0 and [H.sub.uu] [less than] 0
hold along the optimal path, the Implicit Function Theorem can be used
to solve equation (2a) for
[Mathematical Expression Omitted],
while equation (2b) can be solved for
[Mathematical Expression Omitted].
These solutions, are, respectively, the short-run unit tax rate
function and the short-run enforcement rate function. Equation (4a) is
the optimal value of the unit tax rate for a given value of the current
value shadow price of illegal sales, illegal sales, the demand shift
parameter, the response coefficient, the net price of the legal good,
and the social cost function shift parameter. Equation (4b) is the
optimal value of the enforcement rate for a given value of the current
value shadow price of illegal sales and the variable input price.
The necessary conditions have a straightforward economic
interpretation. For example, equation (2a) asserts that the government
should set the unit tax rate on the legal good so as to equate the total
marginal social benefit of the tax, which is comprised of the marginal
tax revenue from legal sales, [Tau][f.sub.p](p; [Epsilon]) + f(p;
[Epsilon]) - x, and marginal social benefit of the tax -[C.sub.q](q;
[Theta])[f.sub.p](p; [Epsilon]), to the response weighted current value
shadow cost of illegal sales, -k[Lambda]. The current value shadow price
of illegal sales, [Lambda], is negative, since a rise in illegal sales,
ceteris paribus, implies a reduction in the demand for the legally
produced good, and therefore a reduction in the present value of net
benefits for the government. More formally, rearrangement of equation
(2b) yields [Lambda] = -[[Phi].sub.u](u; w) [less than] 0, since
[[Phi].sub.u] [greater than] 0 by assumption (A.1) and the definition of
[Phi]. Equation (2b) asserts that the enforcement rate should be set so
as to equate the marginal cost of enforcement, [[Phi].sub.u](u; w), with
the current value shadow cost of illegal sales, -[Lambda].
The short-run comparative statics of (P) are derived by substituting
the short-run unit tax rate and enforcement rate functions, equations
(4a) and (4b), into the necessary conditions from which they were
derived, equations (2a) and (2b), and then differentiating with respect
to the parameter of interest using the multivariate chain rule. Such a
recipe yields
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
where all terms are evaluated at [Mathematical Expression Omitted].
It is important to point out that in the short-run functions, illegal
sales and the current value shadow price of illegal sales are parametric to the government, that is, they are taken as given in the short run,
but not the long run. Moreover, the short-run comparative statics for
perturbations in ([Epsilon], [Rho], [Theta], x, w) are qualitatively
identical to the comparative statics of a static net benefit maximizing
dominant firm, as can be verified by setting [Lambda] [equivalent to] 0
in equations (2a) and (2b) and computing the comparative statics. Thus
the short-run policy responses of the government ignore the effects that
the policy response to the market perturbation may have on future
illegal sales, and hence on the government's long-run market share
and present value of net benefits, in contrast to the long-run policy
responses of the government to be discussed later. This point is
important to keep in mind when interpreting the short-run and long-run
comparative statics, as well as the differences between the short-run
and long-run policy responses.
V. Government Policy in the Short Run
The results in equation (5) have a meaningful economic
interpretation. For example, equation (5a) says that if the loss in the
present value of net benefits from increased illegal sales is smaller
(i.e., [Lambda] [less than] 0 becomes a smaller negative number), then,
in the short run, the government's optimal policy is to raise the
tax rate and reduce the enforcement rate, both of which lead to a loss
in the short-run market share of the government. Illegal sales do not
change in the short-run because they are parametric. Therefore, the rise
in the tax rate and subsequent fall in the quantity demanded of the
legal good are entirely responsible for the fall in the
government's market share.
Equation (5b) says that a rise in illegal sales, which lowers the
present discounted net benefits of the government, results in the
government lowering the tax rate in an effort to capture back some of
its sales and deter consumption of the illegal good. The enforcement
rate, however, remains unchanged in the short run when illegal sales
rise. Notice that in the short run, the policy instrument used by the
government to mitigate the rise in illegal sales is the tax rate (it
falls) rather than the enforcement rate. That is, in the short run,
rather than prosecute and jail the users, suppliers, and distributors of
the illegal good more intensely, it is optimal to simply lower the tax
rate so as to entice users to substitute towards the relatively cheaper
legal product. This result contrasts sharply with the U.S.
government's current drug control policy and practice.
The economic content of equation (5c) is that an increase in the
total demand for the good (both legal and illegal), say from a change in
tastes, will result in the government taking advantage of the positive
demand shock by raising the tax rate. Because illegal sales are given,
such a policy is optimal, at least in the short run. In contrast, the
short-run enforcement rate is unaffected by the demand shock since
illegal sales are fixed, again pointing out that the tax rate is the
best policy instrument for the government to use in the short run.
Now turn to equation (5d), where the short-run comparative statics of
the illegal suppliers' response coefficient are displayed. One way
to interpret an increase in k is that the ability of the black market
suppliers to respond to the tax policy of the government is enhanced or
more efficient. Given this interpretation, the optimal short-run policy
of the government due to an increase in k is to lower the unit tax rate
but not to change the enforcement rate. The lower tax rate combined with
the parametric illegal sales in the short-run results in the rise in the
government's market share. It is again seen that the tax rate is
the optimal policy instrument to adjust in the short run.
Equation (5f) asserts that as the marginal social cost of the
good's use rises, the optimal short-run response by the government
is to raise the unit tax rate but leave the enforcement rate unchanged.
The increase in the tax rate reduces total demand for the good, and thus
mitigates some of the increase in marginal social costs. Again the tax
rate is the policy instrument of choice by the government in the short
run.
Finally, we point out an important policy conclusion that equation
(5) does not reveal. Careful inspection of equation (4) indicates that
the short-run unit tax rate function and enforcement rate function are
independent of the government's discount rate, marginal social
benefit shift parameter, and the price of the illegal good. Hence, in
the short run, the optimal policy for the government is to do nothing
when either the discount rate, marginal social benefit or price of the
illegal good change. In the short run, therefore, it is not always
optimal for the government to change its policy in the face of changing
market conditions.
VI. Government Decision Making in the Long Run
We now turn towards an examination of the government's optimal
decision making in the long run. One obvious reason for investigating
the long-run policy of the government is to compare and contrast its
behavior with that in the short run. The most important reason, however,
is the degree of realism the long-run analysis provides. Specifically,
recall that in the short run, black market sales and the current value
shadow price of illegal sales are held fixed (or are parametric) when
the market parameters [Alpha] [equivalent to] ([Delta], [Epsilon],
[Gamma], k, [Rho], r, [Theta], w) change. In a short period of time such
an implicit assumption is tenable, but it is clearly not in the long
run, for black market suppliers are sure to change their behavior if,
say, the demand for the good rises. Thus the long-run analysis formally
allows illegal sales and the current value shadow price of illegal sales
to adjust to changes in market parameters.
The necessary conditions (equation (2)) can be reduced to the
following pair of ordinary differential equations and boundary
conditions upon using the short-run unit tax rate function and short-run
enforcement rate function from equations (4a) and (4b):
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
x(0) = [x.sub.0] (6c)
lim x(t) = [x.sup.*]([Alpha]) where t [approaches] +[infinity]. (6d)
The steady state or long run of the model is defined as [Mathematical
Expression Omitted]. In the steady state, the necessary conditions (6a)
and (6b) reduce to
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
Assuming that a point ([[Lambda].sub.+], [x.sub.+], [[Alpha].sub.+])
exists that satisfies (7), which is assumed explicitly in the statement
of (P), then a locally unique solution to (7) exists, denoted by
[Lambda] = [[Lambda].sup.*]([Alpha]) (8a)
x = [x.sup.*]([Alpha]). (8b)
The long-run functions ([[Lambda].sup.*], [x.sup.*]) allow for
adjustment in the current value shadow price of illegal sales and
illegal sales when the market parameters change, for they depend
formally on such parameters [Alpha]. This is how the endogeneity of
([Lambda], x) takes place in the long run. Given the existence of the
steady state solution (8), the long-run unit tax rate function and
enforcement rate function can be defined from equation (4) as
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
Equation (9) makes explicit the difference between the short-run and
long-run policy functions. The short-run policy functions in equation
(4) hold ([Lambda], x) fixed when the parameters [Alpha] change. In
equation (9), however, ([Lambda], x) are evaluated at their long-run
values ([[Lambda].sup.*] ([Alpha]), [x.sup.*]([Alpha])), which adjust to
changes in [Alpha], as the equation formally points out. Thus equation
(9) defines the long-run policy functions as the short-run policy
functions when ([Lambda], x) adjust to changes in [Alpha] according to the long-run functions ([[Lambda].sup.*] ([Alpha]), [x.sup.*]
([Alpha])).
In the Appendix, where the technical details are presented, the
steady state or long-run equilibrium is shown to exhibit local
saddlepoint stability. A geometric interpretation of the local
saddlepoint stability is given by the following proposition, whose proof
can also be found in the Appendix. The slopes of the isoclines are given
in the last equation of the proof. The phase diagram in x[Lambda]-space
is depicted in Figures 1 and 2.
ISOCLINE PROPOSITION. The steady state of (P) is a local saddlepoint
if and only if the slope of the [Mathematical Expression Omitted]
isocline is greater than the slope of the [Mathematical Expression
Omitted] isocline, evaluated at the steady state.
The steady state or long-run comparative statics are found by
substituting equation (8) into equation (7) to create identities in
[Alpha], differentiating with respect to the parameter of interest using
the multivariate chain rule, and then solving the resulting linear
system via Cramer's rule. This process yields
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted], [Mathematical Expression Omitted],
[Mathematical Expression Omitted],
[Mathematical Expression Omitted]
[Mathematical Expression Omitted], [Mathematical Expression Omitted],
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]
where the derivatives are evaluated at the steady state given by
equations (8) and (9). The steady state comparative statics for
([[Tau].sup.*]([Alpha]), [u.sup.*]([Alpha])) were derived from the
identities (9a) and (9b). Note also that the inequalities in parentheses follow from sufficient condition (a) or (b) of the Appendix.
VII. Government Policy in the Long Run
First consider an increase in [Delta] as given by equation (10a),
which represents a decrease in marginal social benefits. The optimal
long-run policy of the government is to increase the unit tax on the
legal good and the enforcement rate, a response initiated by the now
higher opportunity cost of illegal sales (i.e., [Lambda] [less than] 0
is more negative). The policy has the result of reducing the total
quantity demanded and illegal sales, thus offsetting, in part, the fall
in marginal social benefits. By way of contrast, recall that in the
short-run, the optimal policy of the government in the face of declining
marginal social benefits was to do nothing.
Recall that an increase in [Epsilon], given by equation (10b),
represents an increase in the total demand for the good, resulting from,
say, a change in the income of consumers or a change in tastes in favor
of more consumption. In the long run, the optimal policy by the
government is to raise the enforcement rate and the tax rate, which is
again a response to the increased opportunity cost of illegal sales. In
general, illegal sales may rise or fall, for the increased tax rate
decreases total demand, but encourages substitution towards illegal
purchases, while the higher enforcement rate acts to deter illegal
sales. Under sufficient condition (a) of the Appendix, illegal sales
rise. Recall that in the short-run (see equation (5c)) the demand shock
resulted in the government raising the tax rate and leaving the
enforcement rate unchanged, which makes economic sense because illegal
sales are fixed in the short run.
Now consider the effects of an increase in the illegal price as given
by equation (10c). Under sufficient condition (b) of the Appendix, the
higher illegal price lowers the opportunity cost of illegal sales, which
allows the government to respond by decreasing the enforcement rate as
buyers substitute away from the illegal good. Such substitution allows
the government to in turn raise the tax rate. In contrast, recall that
in the short run, changes in the illegal price had no effect on the
government's policy.
An increase in the government's discount rate, as given by
equation (10f), can be thought of as increased impatience on its part.
The increased impatience results in a lower opportunity cost of illegal
sales, and as a result, the government's long-run policy is to
lower the tax rate, which causes a rise in total sales of the good.
Because the enforcement rate also falls, illegal sales increase in the
steady state. By way of contrast, recall that in the short run, the
government's optimal policy was to do nothing when the discount
rate rose, thus pointing out the sharp differences that can occur when
different time frames are used for policy.
An increase in marginal social costs, as seen in equation (10g),
causes an increase in the opportunity cost of illegal sales. As a
result, the optimal long-run policy by the government is to raise the
tax rate to reduce total sales of the good, and raise the enforcement
rate to reduce illegal sales. In general, however, illegal sales may
rise or fall, and in fact they rise under sufficient condition (a) of
the Appendix, even though the enforcement rate is higher in the new
long-run equilibrium. This is not paradoxical though, since social costs
depend on total demand. The increased tax rate is effective at reducing
total demand, and hence offsets some of the increase in marginal social
damages that has occurred.
A rise in the marginal cost of enforcement (see equation (10h))
results in the government lowering the enforcement rate in the long run.
Even though the government's long-run optimal policy is to reduce
the tax rate, and thus increase total demand for the good, it is not
enough to entice consumers to substitute towards the legal product, and
as a result, long-run illegal sales rise. In the short run (see equation
(5g)), the government did not change the tax rate when the marginal cost
of enforcement rose, it only lowered the enforcement rate.
Finally, note that the long-run values of the tax rate and
enforcement rate are unaffected by the initial size of the black market.
Hence the long-run policy of the government is the same regardless of
the size of the black market when the legalized market is first
established.
VIII. Policy Dynamics
In the final section of the paper we turn to the dynamics of the
government's optimal policy. One reason for looking into the
dynamics of policies is completeness: we have looked at the short-run
and long-run optimal government policies, so it makes sense to complete
the picture by looking at the transition from one long-run policy
position to another. A second and more important reason is that by
considering only the long-run policy, one gets a distorted view of the
policy dynamics. For example, we showed that in the long run the optimal
policy by the government to a decrease in marginal social benefits is to
raise the tax rate. One could then infer from this long-run result that
the transition path from one long-run equilibrium to another involves
gradually raising the tax. Such intuition, however, is wrong. We will
show that the optimal transitional dynamics dictate a lowering of the
tax rate initially, and then a reversal so as to raise the tax rate
above its old long-run equilibrium value until its new higher long-run
equilibrium value is reached.
The local comparative dynamics of governmental policy can be found by
solving the linearized (at the steady state) version of equations (6a)
and (6b) by standard methods, using the boundary conditions equations
(6c) and (6d) to find the specific solution, and then differentiating
the solution with respect to the parameter of interest [2]. There exists
a more geometrically appealing approach, however. It is only necessary
to determine how the [Mathematical Expression Omitted] and [Mathematical
Expression Omitted] isoclines shift in response to changes in [Alpha]
[equivalent to] ([Delta], [Epsilon], [Gamma], k, [Rho], r, [Theta], w),
and combine this information with the long-run comparative statics to
determine the local comparative dynamics. The latter approach is
followed here and is presented in a series of phase diagrams.
Before moving on to the comparative dynamics, though, it helps to
define the optimal paths of the tax rate and enforcement rate. The
optimal paths of the policy variables are derived by substituting the
optimal paths of illegal sales and the current value shadow price of
illegal sales into the short-run tax rate and enforcement rate
functions, that is, by substituting the solution of equation (6), (x(t;
[Beta]), [Lambda](t; [Beta])), into equations (4a) and (4b):
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
Differentiation of equation (11) with respect to the parameter of
interest, evaluated at t = 0, gives the impact effect of that parameter
change on the optimal path, a procedure which will be used below. Note
that at t = 0 illegal sales are fixed at their long-run value, so that a
change in a parameter does not affect the value of illegal sales. This
is also verified by inspection of Figures 3 through 6 at the initial
moment the parameter is changed.
First consider the policy implications of the dynamics of Figures 1
and 2. To do so, differentiate equation (11) with respect to t:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
Inspection of Figure 1 reveals that if [x.sub.0] [greater than]
[x.sup.*]([Alpha]), then illegal sales and the current value shadow
price of illegal sales decrease monotonically over time to the steady
state, i.e., [Mathematical Expression Omitted] and [Mathematical
Expression Omitted]. It then follows from equation (12b) that if initial
illegal sales are larger than their steady state level, the
government's optimal policy is to allow the enforcement rate to
rise monotonically over time in an effort to reduce the relatively large
initial illegal sales to their smaller long-run level. If, however,
initial illegal sales are smaller than their steady state value, then
the government's optimal policy is to reduce the enforcement rate
monotonically over time until the higher steady state level of illegal
sales is achieved. Note that in general no such monotonicity property
can be claimed for the tax rate policy for Figure 1.
In contrast, Figure 2 shows something quite different is possible.
Here, if initial illegal sales exceed their steady state level, then
[Mathematical Expression Omitted] and [Mathematical Expression Omitted]
hold on the approach to the steady state. It therefore follows from
inspection of equation (12) that the government's optimal policy is
to raise the unit tax rate over time to reduce total demand for the good
and thus illegal sales to their lower long-run level, while at the same
time reducing the enforcement rate. So the dynamics of Figure 2, which
hold under sufficient condition (b) of the Appendix, reveal monotonic policies for both of the government's instruments.
First consider the comparative dynamics of the marginal social
benefit shock displayed in Figure 3. By equation (7) and the Implicit
Function Theorem, the [Mathematical Expression Omitted] isocline shifts
down since [Mathematical Expression Omitted], but the [Mathematical
Expression Omitted] isocline is unaffected because [Mathematical
Expression Omitted]. The initial effect of the marginal social benefit
decrease is to decrease the current value shadow price of illegal sales
(i.e., the opportunity cost of illegal sales increases). Since net
social benefits are inversely related to illegal sales, and because the
opportunity cost of illegal sales is now higher, the immediate policy by
the government is to lower the tax rate and raise the enforcement rate.
This can be verified by differentiating equation (11) with respect to
[Delta] and evaluating at t = 0:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
Such an immediate response by the government has the desired effect
of lowering illegal sales, which then decrease monotonically to their
new lower steady state level. The enforcement rate does remain higher in
the new steady state helping to reduce illegal sales, but the government
must eventually reverse the initial decline in the tax rate so that it
too can reach its new higher steady state level. Even though such an
optimal policy by the government does successfully achieve its desired
end of reducing illegal sales so as to partially offset the decline in
marginal social benefits, the present value of net benefits is still
reduced, for by the Dynamic Envelope Theorem of Caputo [3] or LaFrance
and Barnay [7]
[Delta]V([Beta])/[Delta][Delta] [equivalent to] [integral of]
[B.sub.[Delta]](x(t; [Beta]); [Delta])[e.sup.-rt]dt between limits
+[infinity] and 0 [less than] 0.
Moreover, in this instance an intertemporal policy reversal is
optimal, pointing out that an examination of the short-run and long-run
policy strategies is insufficient for a full understanding of the policy
implications of the model. In particular, the long-run policy of raising
the tax rate in the face of falling marginal social benefits overlooks
the immediate reduction in the tax rate called for in an optimal
intertemporal policy.
Consider the comparative dynamics of the demand shock under
sufficient condition (a) of the Appendix displayed in Figure 4. From the
Implicit Function Theorem and equation (7), the [Mathematical Expression
Omitted] isocline shifts down because [Mathematical Expression Omitted],
as does the [Mathematical Expression Omitted] isocline since
[Mathematical Expression Omitted]. Because the steady state value of the
current value shadow price is lower due to the demand shock, the
isoclines must shift down in such a way so as to lower its value. The
demand shock initially depresses the current value shadow price of black
market sales, that is, raises the opportunity cost of illegal sales. As
a result, the initial policy of the government is to raise the
enforcement rate and to raise or lower the tax rate, as can be verified
by differentiating equation (11) with respect to [Epsilon], and
evaluating at t = 0:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
The initial rise in the enforcement rate is an attempt by the
government to keep the increased demand for the good out of the black
market. In the long run, however, the policy is not completely
successful at reducing black market sales since they rise monotonically
to their new higher steady state level. As a result, the government
finds it optimal to increase the tax rate in the long-run in an effort
to take advantage of the increased demand. In addition, the government
will increase the enforcement rate so as to keep at least some of the
increased demand out of the black market. The optimal response by the
government may or may not result in an increase in their present value
of net benefits, for by the Dynamic Envelope Theorem,
[Mathematical Expression Omitted].
Consider Figure 5, where the comparative dynamics of the illegal
price are displayed under sufficient condition (b) of the Appendix. By
(7) and the Implicit Function Theorem, the [Mathematical Expression
Omitted] isocline does not shift, since [Mathematical Expression
Omitted], while the [Mathematical Expression Omitted] isocline shifts up
because [Mathematical Expression Omitted]. The initial effect of the
increase in the illegal price is to increase the shadow price of illegal
sales. As a result, the initial policy by the government is to raise the
tax rate but lower the enforcement rate. The latter two results can be
verified by differentiating equation (11) with respect to [Gamma] and
evaluating at t = 0:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
The initial rise in the tax rate and fall in the enforcement rate
work together to initially reduce illegal sales, which then decline
monotonically from one steady state to the other. The tax rate remains
higher in the new steady state helping to reduce illegal sales by
reducing total demand, which allows the government to reduce the
enforcement rate to cut back on costs. This makes sense too, as the
higher level of the tax keeps total demand depressed, allowing the
government to lower the enforcement rate in order to increase its
present value of net benefits. The latter point is verified by applying
the Dynamic Envelope Theorem to (P)
[Delta]V([Beta])/[Delta][Gamma] [equivalent to] -k [integral of]
[e.sup.-rt] between limits +[infinity] and 0 [Gamma] (t; [Beta]) dt
[greater than] 0.
Finally, in Figure 6 the comparative dynamics of the discount rate
are presented. The [Mathematical Expression Omitted] isocline shifts up
because [Mathematical Expression Omitted], while the [Mathematical
Expression Omitted] isocline is unaffected since [Mathematical
Expression Omitted]. The increased discount rate initially results in a
lower opportunity cost of illegal sales. Because increased black market
sales now result in a smaller loss in the present value of net benefits,
the initial optimal policy for the government is to raise the tax rate
and lower the enforcement rate, as can be verified by differentiating
equation (11) with respect to r and evaluating at t = 0:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted].
Both of these immediate policy responses by the government result in
the ensuing rise in illegal sales. Observing the increase in illegal
sales, the government must eventually reverse its initial policy of
raising the tax rate and lower it below its original long-run level to
reach the new steady state. The tax rate reversal, however, does not
induce enough substitution towards the legal good, and as a result
illegal sales are higher in the new steady state. Again we see the
importance of the policy dynamics, for they indicate it is optimal for
the government to reverse its initial policy of lowering the tax rate, a
phase of the policy response missed entirely by long-run considerations
only.
IX. Conclusion
We identify four major conclusions coming out of our paper. The first
is that policies that are optimal in the short run are in general not
the same ones that are optimal in the long run. For example, in the
short run, the optimal response by the government to a rise in the
marginal cost of enforcement is to lower the enforcement rate but not
change the tax rate. In contrast, the optimal policy in the long-run is
for the government to reduce both the tax rate and enforcement rate.
Second, the comparative dynamics analysis has shown that
intertemporal policy reversals can be optimal. For instance, the
government's initial optimal policy response to a decrease in
marginal social benefits is to lower the tax rate and raise the
enforcement rate. In the long run, however, the government will increase
the tax rate above its original long-run equilibrium value, while the
enforcement rate will continue to remain above its original long-run
equilibrium value. This points out the importance of short-run and
long-run policy differences, as well as the transition from one long-run
position to another.
Third, the short-run comparative statics analysis shows that in most
instances, the tax rate is the optimal policy instrument for the
government to use in the short-run in response to changing market
conditions. Only in one instance, increased variable costs of
enforcement, is the enforcement rate the optimal short-run instrument to
use in response to market perturbations. Moreover, for some market
perturbations, it is optimal for the government to do nothing in the
short-run.
Finally, the long-run comparative statics analysis shows that in
response to every market perturbation, it is optimal for the government
to use both instruments under its control, the tax rate and the
enforcement rate. Moreover, sometimes, as in the case of a positive
demand shock, it is optimal to allow black market sales of the good to
increase at the expense of the government's market share.
Appendix
Because the functions M and N defined in equation (7) are locally
[C.sup.(1)] by assumptions (A.1), (A.5), and the Implicit Function
Theorem, the functions ([[Lambda].sup.*], [x.sup.*]) [element of]
[C.sup.(1)] [for every][Alpha] in some open ball about [[Alpha].sub.+],
provided the Jacobian determinant of equation (7), given by
[Mathematical Expression Omitted]
is nonzero at ([[Lambda].sub.+], [x.sub.+], [[Alpha].sub.+]). The
signs of the individual elements of J follow from the definition of the
functions M and N in equation (7), using equations (5a) and (5b). Note
that the terminal boundary condition [lim.sub.t[approaches]+[infinity]]
x(t) = [x.sup.*]([Alpha]) asserts the existence of equation (8), since
equation (8b) is, by definition, the terminal level of illegal sales for
(P). Sufficient conditions for the existence and differentiability of a
steady state solution are given by the conditions of the Implicit
Function Theorem used above.
Even though the steady state solution exists, is unique, and is
locally differentiable, the stability of the steady state must be
examined to determine if it can be reached starting from x(0) =
[x.sub.0] [not equal to] [x.sup.*] ([Alpha]). A linear approximation of
equations (6a) and (6b) using Taylor's Theorem, evaluated at the
steady state, leads to a coefficient matrix of the resulting linear
system that is identical to the steady state Jacobian matrix, whose
determinant is given by equations (1A). The local stability of the
steady state is determined by finding the eigenvalues of J. The
eigenvalues, [[Delta].sub.i], i = 1, 2 are found by solving [absolute
value of J - [Delta]I] = [[Delta].sup.2] - (trJ) [Delta] + [absolute
value of J] = 0. From the factorization ([Delta] -
[[Delta].sub.1])([Delta] - [[Delta].sub.2]) [equivalent to]
[[Delta].sup.2] - ([[Delta].sub.1] + [A.sub.2])[Delta] +
[[Delta].sub.1][[Delta].sub.2] = 0, it is seen that
[[Delta].sub.1] + [[Delta].sub.2] [equivalent to] trJ = r [greater
than] 0 (2A)
[[Delta].sub.1][[Delta].sub.2] [equivalent to] [absolute value of J]
(3A)
must hold. Since the sum of the eigenvalues is positive from equation
(2A), at least one eigenvalue must be positive, ruling out the
possibility that the steady state is locally asymptotically stable. Now
if [[Delta].sub.1][[Delta].sub.2] = [absolute value of J] [greater than]
0, then both eigenvalues are positive (or have positive real parts), so
no path could reach the steady state from x(0) = [x.sub.0] [not equal
to] [x.sup.*]([Alpha]), as all paths diverge from the steady state as t
[approaches] +[infinity] in this case. Thus [absolute value of J]
[greater than] 0 violates the terminal boundary condition
[lim.sub.t[approaches]+[infinity]] x(t) = [x.sup.*]([Alpha]), hence
[absolute value of J] [less than] 0 must hold. Thus by equation (3A) the
eigenvalues are of the opposite sign, or equivalently, the steady state
displays local saddlepoint stability.
Proof of Isocline Proposition. If the steady state is a saddlepoint,
then [absolute value of J] = [[Delta].sub.1][[Delta].sub.2] [less than]
0, hence
[Mathematical Expression Omitted],
where all terms are evaluated at the steady state. Since
[N.sub.[Lambda]] [greater than] 0 and [M.sub.[Lambda]] [greater than] 0,
the above becomes
[Mathematical Expression Omitted].
By the Implicit Function Theorem, the left hand side is the slope of
the [Mathematical Expression Omitted] isocline evaluated at the steady
state, while the right hand side is the slope of the [Mathematical
Expression Omitted] isocline evaluated at the steady state. Sufficiency
follows by reversing the above steps. Q.E.D.
Inspection of equation (1A) and use of equations (5a) and (5b)
reveals that the following are sufficient conditions for [absolute value
of J] [less than] 0 to hold:
(a) [Mathematical Expression Omitted]
(b) [Mathematical Expression Omitted],
where all terms are evaluated at the steady state. Sufficient
condition (b) implies that the [Mathematical Expression Omitted]
isocline slopes down in a neighborhood of the steady state.
An earlier version of the paper was presented at the 68th annual
Western Economic Association International conference at Lake Tahoe,
Nevada, June, 1993. We thank Carlos Ulibarri for helpful comments and an
anonymous referee for important insights that have resulted in an
improved product. We also thank Jeffery T. LaFrance for his
encouragement in pursuing this line of research. The implications and
conclusions drawn in this paper are those of the authors and do not
necessarily represent those of the University of California or the
National Center for State Counts. Giannini Foundation Paper Number 1122
(for identification purposes only).
1. Estimates range from $41 billion in 1990 by the Office of National
Drug Control Policy to more than $140 billion by the Select Committee on
Narcotics Abuse and Control. Caputo and Ostrom [4] estimated that
between $2.55 billion and $9.09 billion would be generated in tax
revenue if marijuana was sold as a regulated good in the U.S.
2. We offer no estimate on the likelihood of a major revision to U.S.
drug laws. However, the possibility gains credence when former U.S.
Surgeon General Joycelyn Elders argues that legalizing drugs would
markedly reduce the crime rate and that more studies on the impact of
legalized drugs are needed. Moreover, "The Criminal Justice Policy
Foundation agrees that a study of regulating licensing and taxing the
commerce in now-illegal drugs is urgently needed" [11].
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