Introduction

    Not only illicit drug damage physical and mental health of its users, but it also cause higher levels of crime in the society. Large volumes of researches
<footnote>Maden (1992), Edgar and O'Donnel (1998), and Gossop (1998)
</footnote>
 about the relationship between drug misusage and crime have all observed significant positive correlation. For example, Benneett (2000) reviewed that within his sample, as the measure for drug misusage increases, the measure of involvements in crime raises.
    Some
<footnote>See Collins (1985), Otero-Lopez (1994), Bennett (1991).
</footnote>
 pointed out this positive relationship is nature, because crime can be seen as the psychopharmacological side effect of drug use. Yet, most commonly it has been argued that their positive correlation is from the fact illegal drugs are expensive. As a result, majority of additives need to commit crime to raise sufficient fund. Turnbull, McSweeney, Webster, Edmunds and Hough (2000) found that within their sample of 126 drug users more than 80% finance their drug habit through criminal activities such as shoplift, burglary, theft, fraud and drug offends.
    Not only drug misusages reduce social welfare by imposing damages on victims of crimes, but also raising criminal justice system's expenditures on deal with them. Among all drug users, problem drug users have caused almost all (about 99%) economic and social costs. They have been commonly defined as "users of any age whose drug use is no longer controlled or undertaking for recreational purposes and where drugs have become a essential element of the individual's live"
<footnote>'The economic and social costs of class A drug use in England and Wales, 2000', Godfrey, Eaton, McDougall and Culyer, (2002), Home Office Research, Development and Statistics Directorate.
</footnote>
.
    In this essay, I am going to explore available policies that could be adopted by government to reduce drug related problems. As it is the problem drug users cause most of the damage on society, my essay will focus on this particular group. In the next section two simple models will be provided to get some interesting insight into this problem. It will be followed by short conclusion and discussion about future studies on this topic.

The model

    In this section I will try to formalise the relationship between drug misusage and crime into a constrained maximising problem. In the first subsection, I shall set up the basic model. Next In the following subsections, a static one period model and a simple two-period dynamic optimisation model will be discussed.

Setting up the model

Government budget constraint
    The total expenditure available for government to reduce drug related crime is exogenously given, denoted by "B". This amount is determined by factors such as current GDP level, the seriousness of drug problems as well as ideology of government in power. There are two actions government can take to reduce the level of drug misusage and crimes it caused.
    Let "H" denotes the total monetary value of help and treatment programs provided for problem drug users. Recent researches such as Turnbull, McSweeney, Webster, Edmunds and Hough (2000) and Bennett (2000) reviewed that provision of these programs can reduce problem drug users' dependence on illegal drugs, which can reduce total number of problem drug users. Because spending on these programs reduce drug misuage, in this essay "H" will be called government expenditure on drug reduction. Let the total numbers of problem drug users denoted by ′I(H)′, then I_{H} is negative.
    Such spending include programs like counselling (both individual and group), relapse prevention, physical examination, methadone detox, methadone maintenance (for some countries like Switzerland, Netherlands and Canada heroin maintenance programs has been trialed as part of their treatment), employment and housing assistance.
    Government can tack drug related crime by adopting tougher law and enforcement, which is represented by higher level of "θ" . In UK it normally takes the form of "target policing". It involves strategies like "buy and bust" operations, raids, sweeps and incovenience policing. Let the total costs of arrest, convict, police detention, court appearances and prison stay of a drug related criminal are denoted by C(θ,I(H)). It represents government direct spending on crime reduction.
    Carrying out these tougher measures require more efforts from police and other law and justice departments. In other words, higher "θ" means more resources will be needed, which shows C_{θ} is positive. I have mentioned in early section, problem drug users are far more like to commit in crime than general public. Thereby it is reasonable to assume that larger numbers of problem users in the society would lead to higher volumes of crime. This would raise total costs of dealing them, hence C_{I} is positive as well.
    From above discussion, we know government budget constraint can be written as:
    B=H+C(θ,I(H))
   

social costs of drug related crime
   
    The costs of drug related crime is measured by function below,
    V(π(θ,I(H)),I(H),θ)
    which measures the total monetary lost from drug related crime directly and other social and personal lost such as costs of premature death due to drug related crimes. Recent research showed almost all problem drug users need to finance their addiction through crime, thus it is reasonable to assume as number of problem users increases, social costs of crime would raise, in other words V_{I}>0
<footnote>This postive relationship has been estiablised in numbers of studies, such as Hearnden (2000), Brand and Price (2000), Turnbull, McSweeney, Webster, Edmunds and Hough (2000), Godfrey, Eaton, McDougall and Culyer (2002) about UK and Belenko (1999) about US.
</footnote>
.
    What is more, studies such as Sherman (1990), Hope (1994) in US and Dorn (1992) in UK reviewed spending on law and enforcement makes criminal activities harder to take place. Tough law and enforcement would act as efficient deterrence for drug offends and other illegal activities related to drug misusage, so V_{θ}<0.
    Let us assume illegal profits from drug trade increase the expected benefit from engaging in activities such as drug trafficing, producing as well as dealing. They could either lead to higher volume of drug offends or the seriousness of these offends. Either way, it will cause higher level of social costs. This means costs from drug related crime is positively correlated with profits from drug trade (π) and V_{π}>0.
    The level of profits are positively correlated with toughness of law and enforcement that is measured by θ, and total numbers of problem drug users I(H), which means π_{θ}>0,π_{I}>0.
    This can be explained by demand D(I(H),P)and supply S(θ,P) of illicit drug. Demand is positively related with total numbers of drug users and negatively with market price as in other goods markets. By the definition of problem drug users I have quoted in earlier section, these users are dependent on drug psychologically as well as physically. Drugs are necessary for them hence whose demands are inelastic to price.
    I mentioned before, tougher law and enforcement makes it more difficult to conduct drug related crime, which includes illegal drug trades. Thus, higher value of θ reduces the level of drug supplied. Similar to other goods market supply of drug is positively correlated with market price.
    When total numbers of drug users "I" falls due to exogenous shock, such as increasing spending on drug reduction (H). Demand curve for drug would be shifted inwards for any price levels. Given other things been equal, it would leads to a fall in market equilibrium price and quantity, thereby lower profits. This shows π_{I} is positive.(see figure 1)
   
 [demand for drug.bmp]
 [Changes in government spendings on drug reduction.]

    On the other hand, if supply shifts left-toward due to exogenous shocks such as raise in θ, given other things been equal, this would cause a raise in market equilibrium price but fall in total quantity demanded. As I have mentioned earlier, problem drug users' demand for drug is inelastic to price, this kind of shock would raise profits from drug trade. This shows π_{θ} is positive. (see figure 2)
   
 [supply for drug.bmp]
 [ Changes in toughness of law and enforcement.]

Static model

    Total social costs should equal the sum of total direct costs from drug related crimes and spending of government to prevent these crimes. They can be written as:
   
SC=V(π(θ,I(H)),I(H),θ)+C(θ,I(H))+H
    Therefore, total welfare in the society, which is the objective function of this problem can be written as
<footnote>In this essay, I have assumed problem drug users do not get utility from drug consumption becuase drugs are essential for them. Also to aviod contravercy I have assumed that profits from drug related crimes are not counted as part of social welfare.
</footnote>
:
    SW=-SC
    The problem of choosing optimal government policies to maximise total social welfare can be represented by constrained maximisation problem below,
    maxSW=-[V(π(θ,I(H)),I(H),θ)+C(θ,I(H))+H]
    st.C(θ,I(H))+H=B
    Even though in reality the government can only choose the level of spending on crime-reduction C(θ,I(H)). Then depends on factors such as costs of making arrest, appear in court, etc. ( factors I have mentioned in earlier section) the toughness of law and enforcement θ will be determined implicitly. To simplify our analysis, let us assume government can choose the level of θ directly. Government can also choose its optimal level of spending on drug reduction H explicitly. To simplify this problem further, let us assume social welfare function I introduce above is twice differentiable with respect to both arguments, and the second order conditions satisfy requirements for welfare maximisation
<footnote>A sufficient condition for a local maxima is that if at the solustion of the FOC equation (1) through (3), the determinant of the bordered Hessian is postive, then the solustion maximise SW subject to government constraint. For simplicity, let us assume that the bordered Hessian of our problem has postive determinat, or in other words, our solustion maximise total social welfare.
</footnote>
.
    The solution of above problem can be found by maximising the Lagrangian below with respect to government's policy choices
<footnote>Becuase this objective function SW is real-valued and continuous and the constraint equation is compact, from Weierstrass theorem we know that optima exist for this objective function over its constraint. Also as none of the partial dericatives of the constraint equation are strictly zero, we know that the crtical value from the Lagrangain coincide with our objective function.
</footnote>
,
    maxL=-{[V(π(θ,I(H)),I(H),θ]+C(θ,I(H))+H}+λ[B-C(θ,I(H))-H(I)]
   
    From the first order conditions, we know that :
    ((∂L)/(∂H))=-(V_{π}π_{I}I_{H}+V_{I}I_{H_{I}}+C_{I}I_{H}+1)-λ(C_{I}I_{H}+1)=0   (1)
    ((∂L)/(∂θ))=-(V_{π}π_{θ}+V_{θ}+C_{θ})-λC_{θ}=0   (2)
    and
    ((∂L∂)/(∂λ))=B-C(θ,I(H))-H(I)=0   (3)
    We can simplify above expression by divide equation (1) through (2). The reduced conditions for social welfare maximum are as below:
    ((V_{π}π_{I}I_{H}+V_{I}I_{H}+C_{I}I_{H}+1)/(V_{π}π_{θ}+V_{θ}+C_{θ}))=((C_{I}I_{H}+1)/(C_{θ})) 
    and B=C(θ,I(H))-H(I)
   
    The left-hand-side of first condition equals ((∂SW/∂H)/(∂SW/∂θ)), which is the marginal rate of substitution(MRS) between spending on drug reduction and toughness of law and enforcement. Its the ratio between marginal improvement of social welfare as a result of one unit increases in spending on supports for problem drug users and one unit increase in toughness of law and enforcement. This measures at given level of (H,θ) how much tougher law and enforcement is required to remain social welfare constant, if one unit less is spent on help and treatment program by government. The right-hand-side of above equation measures the slop of government constraint, in other words, it is the measure of relative prices of proving higher level of H and θ. The above equations say, government's optimal policies (H^{∗},θ^{∗}) should be chosen where MRS equals their relative prices and government budget holds at the chosen level.
    After deriving the conditions for optimal government policy, it would be in our interest to find out how changes in levels of θ, H and B would affect total social welfare. From partial derivatives of the objective function (SW), we can see, for given level of total government spending B and spending on drug reduction H, changes in toughness of law and enforcement by one unit would change total social welfare by the amount below:
    ((∂SW)/(∂θ))=-(V_{π}π_{θ}+V_{θ}+C_{θ})
    where V_{π}π_{θ}>0 measures the marginal costs of crime to society as a result of the marginal increase in profitability of drug trade due to higher θ. This is indirect costs to the society because, when law and enforcement become tougher as I showed early, inelastic demand for drug would lead to higher profits. This would encourage more drug offends hence higher social costs.
    V_{θ}<0 represents the marginal reduction in social costs as a direct result of raising level of θ. It measures the marginal effectiveness of tough law and enforcement as the direct deterrence of crime.
    As it is costly to impose and enforce tougher laws on drug related crimes, C_{θ}>0. It measures the direct marginal reduction of total social welfare as a result of increasing θ.
    Sign of above equation is determined by the relative strength of these three effects. Thus, general conclusion about the effect of tougher law and enforcement on the welfare of a society cannot be derived. Yet, it is clear that tougher law and enforcement would only benefit a society when the benefits outweights the costs (direct and indirect) it caused.
    Keep the level of government total spending B, and the toughness of law and enforcement θ constant, if government spend one more unit on drug reduction, its effect on social welfare is measured below:
    ((∂SW)/(∂H))=-(V_{π}π_{I}I_{H}+V_{I}I_{H}+C_{I}I_{H}+1)
    where V_{π}π_{I}I_{H}<0 measures the marginal reduction of social costs from drug related crime as a result of decreases in marginal profitability of drug trade from the marginal reduction in total numbers of drug users. This raises social wellbeing because higher support H would reduce total drug users, which reduces the profitability of drug trade and incentives to commit drug offends. This would in turn reduce crime and social costs.
    V_{I}I_{H}<0 indicates the direct fall in costs of drug-related crime as a result of marginal reduction in numbers of drug users.
    C_{I}I_{H}<0 measures savings of government spending on crime reduction as a result of marginal fall in total numbers of problem drug users as spending on drug reduction increase. The last term represents the only marginal social cost higher H caused (it equals 1).
    Similar to the effects of tougher law and enforcement on social welfare, the effects of higher spending on drug reduction on social welfare cannot be determined in general. when deciding if increasing spending would be beneficial for a society, government need to consider above four effects carefully.
    From the Lagrangian above, we can derive the value of λ. It is the Lagrangian multiplier, measures marginal increase of total social welfare as government relax its budget by one unit. In other words, if λ>0, it shows as government increase its total expenditure by one unit, total social welfare would increase by λ. However, if the value of Lagrangian multiplier is negative, then increase total government expenditure could reduce total social welfare, which would be inefficient from social point of view. This suggests higher spending by government would not always increase social welfare. Total expenditure ought to only be increased when λ is positive.

Two-period dynamic model

    The model in last section has been extremely simplified. Especially it ignores all possible dynamic effects of how current policy choices might affect future social welfare . Empirical studies established that total numbers of drug users in current period is positively correlated with past values. This is because, higher numbers of problem drug user in first period might provide negative influence on young people, which could encourage more people to develop addictions and raise the total number of problem drug users in the second period. In this section, I am going to take this kind of dynamic relationship explicitly into account by extend earlier model into a simple two-period setting. I shall derive the conditions for government optimal policy choices again, then compare them with these conditions I have derived early.

Setting the model
    There is a society only exist for two periods (period 1 and 2), let us assume there is no discounts. A subscript is added to each function I have specified in section 2.1 to denote which period it represents. Let all function take the same form as before, except the function for total numbers of drug user in the second period. It takes the form I₂(H₂,I₁(H₁)). It shows the total number of drug user in second period is not only depended on total spending on drug reduction in that period, but also depended on the total numbers of drug users in first period. And I_{2I₁}>0 to take the possible positive dynamic relationship observed in empirical studies into account.
    In this setting, government facing a dynamic choice of their optimal policies to maximise the total welfare in these two periods subject to its budget constraints. This problem can be written as below constrained maximisation problem,
    maxTSW=-[

V₁(π₁(θ₁,I₁(H₁)),I₁(H₁),θ₁)+C₁(θ₁,I₁(H₁))+
H₁+V₂(π₂(θ_{2,}I₂(H₂,I₁(H₁)),I₂(H₂,I₁(H₁))+
C(θ₂,I₂(H_{2,}I₁(H₁))+H₂
]
    st.C₁(θ₁,I₁(H₁))+H₁+S=B₁
    C₂(θ₂,I₂(H₂,I₁(H₁)))+H₂-S=B₂
    where S represents saving (or dissaving) in the first period. This problem can be again solved by maximising Lagrangian equation below:
    maxΓ=TSW+λ₁[B₁-C₁(θ₁,I₁(H₁))-H₁-S]+λ₂[B₂-C₂(θ₂,I₂(H₂,I₁(H₁)))-H₂+S]
    The first order conditions with respect to H₁ and H₂ can be written as below:
    ((∂Γ)/(∂H₁))=-[

V_{1π}π_{1I₁}I_{1H₁}+V_{1I₁}I_{H₁}+C_{I₁}I_{1H₁}+1+V_{2π}π_{2I₂}I_{2I₁}I_{1H₁}+
V_{2I₂}I_{2I₁}I_{1H₁}+C_{2I₂}I_{2I₁}I_{1H₁}
]-λ₁(C_{1I₁}I_{1H₁}+1)-λ₂C_{2I₂}I_{2I₁}I_{1H₁}=0   (4)
    ((∂Γ)/(∂H₂))=-[V_{2π₂}π_{2I₂}I_{2H₂}+V_{2I₂}I_{2H₂}+C_{2I₂}I_{2H₂}+1]-λ₂(C_{2I₂}I_{2H₂}+1)=0   (5)
    The first order conditions with respect to θ_{1 }and θ₂ are showing below:
    ((∂Γ)/(∂θ₁))=-[V_{1π₁}π_{1θ₁}+V_{1θ₁}+C_{1θ₁}]-λ₁C_{1θ₁}=0   (6)
    ((∂Γ)/(∂θ₂))=-[V_{2π}π_{2θ₂}+V_{2θ₂}+C_{2θ₂}]-λ₂C_{2θ₂}=0   (7)
    Budget constraint need to be satisfied are
    ((∂Γ)/(∂λ₁))=[B₁-C₁(θ₁,I₁(H₁))-H₁-S]=0   (8)
    ((∂Γ)/(∂λ₂))=[B₂-C₂(θ₂,I₂(H₂,I₁(H₁)))-H₂+S]=0   (9)
   
    Similar to the static model, we can again simplify above conditions by divide equation (4) through (6), equation (5) through (7) and derive the 4 conditions below:
    MRS₁=((V_{1π}π_{1I₁}I_{1H₁}+V_{1I₁}I_{H₁}+C_{I₁}I_{1H₁}+1+V_{2π}π_{2I₂}I_{2I₁}I_{1H₁}+V_{2I₂}I_{2I₁}I_{1H₁}+C_{2I₂}I_{2I₁}I_{1H₁})/(V_{1π₁}π_{1θ₁}+V_{1θ₁}+C_{1θ₁}))
    =((λ₁(C_{1I₁}I_{1H₁}+1)+λ₂(C_{2I₂}I_{2I₁}I_{1H₁}))/(λ₁C_{1θ₁}))   (condition1)
   
    MRS₂=((V_{2π₂}π_{2I₂}I_{2H₂}+V_{2I₂}I_{2H₂}+C_{2I₂}I_{2H₂}+1)/(V_{2π}π_{2θ₂}+V_{2θ₂}+C_{2θ₂}))=((λ₂(C_{2I₂}I_{2H₂}+1))/(λ₂C_{2θ₂}))=((C_{2I₂}I_{2H₂}+1)/(C_{2θ₂}))   (condition2)
    and
    B₁=C₁(θ₁,I₁(H₁))-H₁-S   (condition3)
    B₂=C₂(θ₂,I₂(H₂,I₁(H₁)))-H₂+S   (condition4)
   
    Condition 3 and 4 above stated that government total spending in these two periods must equal the total amount available B₁+B₂. These conditions slightly differ from government budget constraints under static model. Here government has the extra option of borrowing, which means under this dynamic setting government can spend more than in the static model. Yet, from condition 3 and 4 alone it is impossible to determine how government's spending behavior would differ from earlier model.
    Condition 2 takes very similar form as the condition we observed in static model. It says government should choose its level of spending in second period on drug reduction and toughness of law and enforcement in such a way the ratio of the marginal improvements of social welfare caused by one unit increase in these two areas equal the relative costs of providing them.
    The first condition is much more interesting while more complicated to explain, it states government optimal policies H₁,θ₁should be chosen to ensure that the ratio between the marginal benefits to society from one more unit of spending on drug reduction in first period and the marginal benefits from increasing the toughness of law and enforcement by one unit in first period, equals their relative costs in that period.
    There are two things we need to pay particular attentions to. First of all, comparing with left-hand-side of condition 1 above with the condition we observed under static setting. Here not only the marginal effects of higher spending on drug reduction in first period has been taken into account through the term V_{1π}π_{1I₁}I_{1H₁}+V_{1I₁}I_{H₁}+C_{I₁}I_{1H₁}+1, but also its marginal effects on second period social welfare through lower total numbers of problem drug users in second period represented by V_{2π}π_{2I₂}I_{2I₁}I_{1H₁}+V_{2I₂}I_{2I₁}I_{1H₁}+C_{2I₂}I_{2I₁}I_{1H₁}.
    This latter term is negative, means social costs from higher level of first period spending on drug reduction H₁ would be lower under dynamic than under static setting. Ceteris paribus, it shows total social welfare would be higher here. This provide extra incentive for government to spend more on drug reduction in first period. Because the extra linkage between higher first period spending on drug reduction leads to lower total numbers of problem drug users in first period that in turn would induce lower total numbers of problem drug users in the second period. Under current setting government's choice of extra spending in first period would be a rational choice.
    We also need to notice that on right-hand-side of condition 1 above, there is an extra term written as λ₂(C_{2I₂}I_{2I₁}I_{1H₁}). Here λ₂ again represents the marginal improvements of social welfare as a result of one unit increase in total government expenditure in second period B₂. It has been widely agreed that higher government expenditures would be desirable to reduce drug related problems. As I have explained in the earlier section, increase government spending will only benefit society when λ is positive. Thereby it is relatively uncontroversial to assume λ₂>0. Because I_{1H₁}<0,I_{2I₁}>0,C_{2I₂}>0,⇒C_{2I₂}I_{2I₁}I_{1H₁}<0.This means λ₂C_{2I₂}I_{2I₁}I_{1H₁}<0. It represents the effects of indirect reduction in second period costs on crime reduction as a result of increasing in first period spending on drug reduction. Given other things been equal, this negative term here ensures the net marginal costs of increasing spending on drug reduction would be less under dynamic (represented by λ₁(C_{1I₁}I_{1H₁}+1)+λ₂(C_{2I₂}I_{2I₁}I_{1H₁})) than under static setting (represented by λ(C_{I}I_{H}+1)). This also provides extra incentive for government to spend more on drug reduction in first period.
    These two points above shows, in a dynamic model government need to be more foresighted. When choosing its optimal policies not only effects of the same periods need to be considered, but also the extra linkages between different time spans.

Summary and future studies

    Recent studies reviewed that the positive correlation between the numbers of problem drug users and total numbers of drug related crimes is statistically significant. while there has been large volumes of empirical studies in this topic little attention has been paid to formulate their relationship into theoretical model. In this essay I have tried to formalise their positive relationship under both static and dynamic settings. Conditions under which government policies over the optimal toughness of law and enforcement and the optimal spending on drug reduction ought to be chosen has been derived with some brief discussion about comparative statics. Some interesting comparison about government spending on drug reduction in these two models has been carried out. From which we established that when dynamic relationships were taken into account, there is extra incentives for government to spend more on drug reduction than if this dynamic correlations was ignored.
    My study so far only scratched the surface of this problem, more detailed analysis about this matter is required. Personally, there are few things I feel are worth further attention. For instance, one of the interesting question has not been answered in this essay is given the extra option of borrowing from second period, whether government under dynamic setting would spend more (S<0) than under static model. (Even though we know there is extra incentive for government to spend more on drug reduction in first period, we still not sure whether the total spending by government would be higher under this dynamic setting.) To solve this question, more detailed comparison between conditions for optimal policy between static and dynamic model would be needed.
    Furthermore, empiricial studies need to be carried out to estimate the marginal rate of substitution and relative prices in a society. They would then be used to test if current government policies are social optimal. And if not, how can government improve total social welfare by alter its existing policies.
   

References:

    Becker. G. S, 'Crime and punishment: An Economic Approach', (1987), Journal of Political Economy.
    Bennett. T (2000),'Drugs and Crime: the results of the second developmental stage of the NEW-ADAM programme', Research, Development and Statistics Directorate, Home Office.
    Bramley-Harker. E, (2001), 'Sizing the UK market for illicit drugs', National Economic Research Associates.
    Godfrey. C, Eaton. G, McDougall. C and Culyer. A, (2002), 'The economic and soical costs of class A drug use in England and wales, 2000), Resarch, Development and Statistics Directorate, Home Office.
    Turnbull. P.J, McSweeney. T, Webster. R, Edmunds. M and Hough. M, (2000), 'Drug Treatment and Testing Orders: Final evaluation report', Resarch, Development and Statistics Directorate, Home Office.
    Ramsay. M, Baker. P, Goulden. C, Sharp. C and Sondhi. A, (2001), 'Drug misuse declared in 2000: results from the British Crime Survey',Resarch, Development and Statistics Directorate, Home Office.