persons withdrawing from addiction is given by summarizing over individuals with different ages and numbers of years of addiction remaining:


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1 COST BENEFIT ANALYSIS OF NARCOTIC ADDICTION TREATMENT PROGRAMS with Specil Reference to Age Irving Leveson,l New York City Plnning Commission Introduction Efforts to del with consequences of poverty, rce, nd other socil problems hve resulted in prolifertion of tretment nd rehbilittion progrms for criminls, ddicts, nd the mentlly ill. These hve included skills trining, professionl nd nonprofessionl psychotherpy, drug therpy, nd incrcertion nd surveillnce. Mny such efforts re experimentl or pilot projects. To dte there hs been little in the wy of systemtic efforts to evlute, either publicly or privtely, the success of these progrms. The mount of informtion in the public domin is growing, nd public gencies providing funds re beginning to insist tht evlutions tke plce. To ttin comprbility mong such evlutions requires greement on methodologies of generl pplicbility. A "simplified" methodology which permits comprisons mong divergent ddiction rehbilittion nd tretment methods is developed nd illustrted here. It is believed tht the techniques hve pplicbility to mny other progrms. A criticl feture of rehbilittion efforts is tht they tend to del with very different groups. We would like to hve cost benefit informtion for ech tretment pplied to ech popultion type. In lieu of this, we find it necessry to ssume ech tretment to be most pproprite to prticulr popultion type. We exmine costs nd benefits for ech group nd then djust for differences between groups in ese of tretment nd in the vlue of benefits from successful tretment. In relted pproch, the efficient choice of client group for given progrm is considered. There hs been gret concern over the growth in the drug problem mong youth nd its fr reching implictions for institutions such s schools nd the militry. There hs been gret concern over the lck of consensus on ny tretment method to effectively del with the young. At the sme time, there hs been prolifertion of tretment progrms ttempting to del with the young which require ssessment. These efforts re criticlly bound up with discovery of the bsic cuses of ddiction. Rther thn develop more generl methodology for compring progrms deling with divergent popultion groups, we focus here on the role of ge in the success in tretment progrms. ssistnce. Becuse of finite life expectncy nd the possibility tht mny persons "mture out" of ddiction s they grow older, the number of yers of ddiction remining is greter for younger ddicts. The number of remining yers of ddiction is represented by R. The benefits of reduction of mn yer of ddiction (bi) re represented s function of ge (i) s3well s the number of yers (j) of ddiction remining.3 Future benefits re discounted by n pproprite rte (r). The full expression for the benefits of N = E Ni persons withdrwing from ddiction is given by summrizing over individuls with different ges nd numbers of yers of ddiction remining: B = E E N11 (1) i=1 j=1 (l+r)i Eqution 1 cn be gretly simplified. First, we ssume tht the number of yers of ddiction remining is solely function of ge. Therefore we cn write Nj B = E (2) j=1 (l+r)j Now we exmine the ssumptions implicitly required to use simpler mesures of success thn the full expression in Eqution 2. Most studies ignore differences mong ge groups in benefits per ddict successfully treted. The effectiveness mesure becomes E = E N. (l+r)j where B = be. The most commonly used mesure of success is the proportion () of persons dmitted to tretment (A) who successfully withdrw. The reduction in the number of ddicts is N = A. If we compred different progrms treting the sme types of people, bl would equl b2 nd would equl t,, so tht the probbility of successful withdf'wl would be sufficient to guge success. Otherwise, use of this mesure is equivlent to ssumming tht (3) The Underlying Cost  Benefit Model The benefit to society of hving one less ddict depends on the number of yers tht n ddict would hve remined ddicted without E = 1 j=1 (l+& (4) 327
2 s well s ignoring ge differences in benefits per mn yer verted. It guges success by the number of ddicts successfully withdrwing without regrd to l'ngth of time they would hve been ddicted. In doing so, it overemphsizes very immedite gins. Reducing the verge number of ddicts over number of yers requires mesuring success by the number of mn yers of ddiction verted, or Assume tht progrms with lrger hve lower. If cross progrms is proportionl to, the rtio of the cost effectiveness rtios is the sme s the rtio of the costs per person treted. If vries more thn, the costs per person treted would hve to be lower in the progrms with the higher k for them to compre fvorbly with the others. If cu is smll reltive to or is lrge M = N. This cn be derived from Eqution 3 s ct A M cs (9) M = N. R. j=1 where r = o nd B = b M. This mesure is prticulrly simple to use. However, it overemphsizes benefits which re deferred in time. If we wish to tke into ccount tht we vlue benefits tody more thn those which will not be received for mny yers, it is necessry to discount future benefits by n pproprite discount rte (r). The mesure which does this is the discounted number of mn yers of ddiction verted, designted s E s defined in Eqution 3. (5) Next we consider cost effectiveness. Where is used, discounted could esily be substituted. Let individuls going through progrm either be treted successfully t cost or unsuccessfully t cost cu. The verge cost (ct) of treting n ddict in progrm hving proportion successfully treted of is ct = cs + (1) cu The cost of treting A ddicts is A ct, so tht the cost per successfully treted is A A et (6). Designting the cost per suc  cessfully treted ddict s c, c = + cu (7) As the proportion successfully treted increses, cost per success flls becuse the costs of reltively fewer unsuccessful cses re included. The rtio of costs of ddicts dmitted to tretment to mn yers of ddiction verted (cost effectiveness rtio) is ct A cs + (1 ) cu. M (8) Empiricl Evidence on Age Differences We hve exmined the evidence on differences mong ge groups in the benefits of successful tretment nd the chnces for successful tretment. The nlysis indictes () tht differences in the prevlence of opite use mong ge groups reflects some sort of mturtion (nd/or mortlity) since without mturtion they imply implusible ptterns in the number of users over time, (b) derives the verge number of yers of ddiction remining by ethnic group bsed on ge ptterns under lterntive ssumptions bout pst rtes of growth of ddiction, nd (c) compres the higher retention rtes of the New York City Methdone Progrm for the youngest ddicts with the lower derived mturtion rtes nd suggests tht the young my do the sme or worse if "nturl" mturtion ccounts for prt of the dropping out of older persons. (This mteril is vilble upon request.) A Cursory Look t Mjor Progrms Virtully ny progrm will py in terms of benefits to the nonddict popultion. Even if we incrcerte persons t cost of over $8,000 per yer for severl yers, if crime of $10,000 to $20,000 per person per yer is verted,6 the progrm cn be justified. However, this clcultion does not consider tht, even with high pyoff, funds for ddiction services my be limited, nd it does not tke into ccount the undesirbility of involuntry incrcertion. While it my be cler tht the pyoff to ddiction progrms in generl is very high, there is still mjor issue s to wht proportion of the ddict popultion cn be delt with by ech pproch. As of the end of 1969, there were bout 11,000 New York City ddicts in tretment progrms. In ddition, there were bout 5,000 in pretril detention7 nd others serving prison terms or on prole. Nerly ll tretment funding comes from the New York Stte Nrcotic Addiction Control Commission which hs current operting budget of $38 million per yer nd $200 million cpitl construction progrm. The three 328
3 lrgest progrms for the tretment of drug ddicts in New York City re the progrm of the New York Stte Nrcotic Addiction Control Commission, the Methdone Mintennce Progrm, nd the Phoenix House Progrm of New York City's Addiction Services Agency. The Stte progrm combines voluntry or involuntry incrcertion with therpeutic nd rehbilittion services for up to five yers, followed by n ftercre phse involving surveillnce. The progrm, depending strongly on court nd prentl coercion, hs suffered bscondnce rtes of over one third per yer nd high rtes of recidivism. However, it hs been successful in moving persons from n institutionl setting into ftercre. Furthermore, it is the only one of the three lrgest progrms to hve given primry ttention to younger ddicts. One third of the persons in the progrm re under ge 20 nd two thirds re under ge 25. The vilbility of rehbilittion services under the progrm ppers to be incresing. At the end of 1969 there were 5,000 ddicts in the progrm, with lrge expnsion plnned s new fcilities being constructed become vilble. The Methdone Progrm uses the drug Methdone, itself ddicting but providing no "high," to block the effects of opites. After few weeks of withdrwl, the drug is dministered orlly on n outptient bsis for n indefinite period of time. There re bout 2,000 ddicts currently under tretment, nd doubling of efforts is under wy. This progrm tends to concentrte on older ddicts s indicted by comprisons with the Nrcotics Register for September 1969: Methdone Progrm Nrcotics Register Less thn 25 10% 33% nd over The Methdone Progrm is the only one of the three to mke public regulr evlution reports. The evidence of high retention rtes, improvement in employment experiences, nd reduced criminlity hve led to belief tht this is the most effective method currently vilble. The Phoenix House progrm relies on encounter techniques involving confronttion of ddicts with ech other nd with ex ddicts. After two yers of institutionliztion, process of grdul re entry into the community begins. Phoenix House hs generlly served bout thousnd persons, but only smll frction hve re entered the community through the progrm. Scnt informtion is vilble with which to compre the effectiveness of mjor progrm lterntives, even using simplified frmework devised to minimize dt requirements. success rte for persons on drug mintennce for 36 months. Lck of success in tht progrm reflects both voluntry nd involuntry termintion stemming from fctors such s criminl behvior nd lcoholism. The Stte progrm reported tht fter its first 21 months of opertion, 44 percent of the 1,893 persons returned to the community hd not resumed drug use. Thirty percent were returned to rehbilittion center fter it ws discovered tht they were reusing nrcotics, nd 26 percent could no longer be locted (wrrnts were issued ginst them).8 In view of the fct tht the progrm hd only 188 persons on ftercre fter its first yer nd 836 fter 21 months, the verge length of time in the community would hve to hve been substntilly less thn one yer. In view of the short time, the progrm's experience is discourging. A study of roughly similr progrm, tht of the Cliforni Rehbilittion Center t Coron, found tht 35 percent were in good stnding (or hd been removed from the progrm while in good stnding) one yer fter relese. After three yers the figure hd fllen 19 percent.9 There is nothing in the experience of the New York Stte progrm to dte to indicte it is doing ny better thn this. If we tke seriously the public mndte to prepre ddicts for entry into the community, the fct tht only hndful of Phoenix House residents hve ctully done so fter their period of institutionliztion must be treted s if the proportion delt with successfully is negligible. This progrm could only be considered successful if society were willing to plce high vlue on withdrwl in setting of permnent institutionliztion. We re now in position to compre progrms on the ssumption tht they continue to tret the sme types of people s they now tret with the sme degree of success. If we compre the benefits of the Methdone progrm designted by subscript 1 reltive to the Stte progrm designted by subscript 2, we hve B1 b1 B2 b2 02 t2 (10) The number of persons dmitted to ech progrm is ssumed to be the sme. Assuming the Cliforni experience to be pplicble to New York Stte, equls bout.7/.2 or 3 1/2. The men ges nd corresponding yers of ddiction remining for the three mjor progrms re s follows: The Methdone progrm reported bout.7 329
4 Approximte Yers Age Remining Nrcotic Addiction Commission Addiction Services Agency Methdone Mintennce 32 8 The ge distribution of the Methdone progrm implies tht the cost of the progrm must be bout 2/3 of the others to compenste for the difference in yers of ddiction remining, if ll hd eqully high success rtes. If the yers of ddiction remining were discounted, this difference would be modified somewht. The vlue to society of treting the kinds of persons treted by the Stte progrm my be higher thn those treted with Methdone. There is some evidence tht older ddicts commit fewer crimes. Younger ddicts cn be expected to remin on drugs longer, with greter dverse effects on helth.. Furthermore, tretment of younger ddicts my hve greter benefits of preventing other persons from being induced to use drugs. However, there would hve to be rtio of bl to b2 s low s bout 3/7 in order for the Stte progrm to hve eqully lrge benefits. When one considers costs, the problem of progrm choice becomes more difficult. In the pst, the Stte progrm required over $8,000 in operting costs lone per mn yer of incrcertion, nd currently employs nerly one person for every ddict in residentil or ftercre ctivities. Completion of the progrm probbly requires n verge of bout $15,000 per person. For comprble cost, the Methdone progrm could provide detoxifiction nd drug mintennce for more thn decde. After tht time, it my be possible for mturtion effects to be substituted for drug mintennce. While there my not be lrge difference between the Stte nd the Methdone progrms in the cost per completed progrm, those progrms within the lowest proportion of persons treted successfully will tend to hve the lowest cost per person dmitted to tretment. This occurs becuse the cost per person treted (ct) is n verge of the cost per person treted successfully (cs) nd the cost of persons who do not successfully complete the progrm, some of whom hve prtil tretment (cu). (See Eqution 6.) Higher costs per person dmitted to tretment in more successful progrm will tend to offset prt of the gins from more successful tretment. The lter stges in tretment t which persons drop out the lrger will cu be reltive to cs, nd the less the cost per person dmitted to tretment will vry with the probbility of success. When the cost per dmission is considered, the weight of evidence is probbly still hevily in fvor of Methdone, since in the Stte progrm there is probbly only modest difference in cost between those who successfully withdrw nd those who do not. However, this bers closer scrutiny, since success my depend on the mount of rehbilittion provided nd the Stte hs been expnding the mount of rel rehbilittion service provided. Furthermore, some deduction of welfre of the ddict must be mde for the involuntry nture of the Stte progrm nd the dependence on Methdone. On the surfce, the Addiction Services Agency progrm is by fr the chepest, estimted by the gency to cost $7,500 for the entire 2 1/4 yer process. While in theory it could hve n dded benefit beyond the time of mturtion out of drug use becuse it is directed to improvement in mny spects of socil djustment, this does not occur without re entry.10 Choice of Age Group within Progrm Selecting ge groups for tretment so s to mximize the number of mn yers of ddiction verted is equivlent to minimizing the number of ddicts. If the success of tretment were the sme t ech ge, we would concentrte on the youngest ddicts who hve the gretest number of yers of ddiction remining. On the other hnd, if the number of yers of ddiction remining were the sme t ech ge nd success of tretment were greter for older persons, we would concentrte on the oldest ddicts. (We re defining success net of ny norml mturtion tendencies.) Here we consider the optiml ge when vritions in both yers of ddiction remining nd success of tretment re tken into ccount. The number of mn yers of ddiction verted for progrm is given by M=Ak. (11) For simplicity, we ssume tht the verge number of yers of ddiction remining in progrm cn be represented s proportion (p) of the difference between some fixed ge ( *) nd the verge ge () of persons being treted = p (* ). (12) Thus if * is ge 40 nd p =.6, ddicts ge 20 would be expected to remin ddicted for nother 12 yers on the verge. Those who were still ddicted t ge 30 would be expected to remin ddicted for nother 6 yers. We further ssume tht the rte of success in withdrwl from ddiction cn be represented by liner function of ge where t > O. = s + t, (13) Combining these equtions we hve M = A s p ( * ) + A t p ( * ). (14) 330
5 Multiplying through nd differentiting with respect to ge ddiction will be prevented for lrger number of yers when the young withdrw, rther thn relying on the more visible success rte lone. d = At *p Asp 2Atp. (15) Finl Comments The second derivtive is d2m d 2 =  2 A t p (16) which, since A, t nd p re positive, is negtive indicting mximum. Setting the first derivtive equl to zero nd solving, we derive the optiml level of = * s 2 2t (17) One inference tht cn be drwn from this result is tht the effect of ge differences in the success rte on the optiml ge is importnt. If, for exmple, s were .3 nd t were.03, indicting success rte of.4 t ge 20, nd.7 t ge 30, the term would dd five yers to the optiml ge. If s is positive, however, the optiml ge is reduced. For exmple, if s =.2 nd t =.02, indicting success of.6 t ge 20 nd.8 t ge 30, the optiml ge would be reduced by five yers. With the functionl form we hve chosen for, p does not enter into the result since it does not chnge the terms of trde between ge groups. Unfortuntely, we hve little informtion with which to determine *. Our curves of yers of ddiction remining suggest tht over the rnge of ges the curves could be pproximted by n * t bout 40, with different p's for ech ethnic group.11 In tht ge rnge, figure of ge 20 for á* would be vlid. 2 is very pproximte of course. If * This were s high s 25 nd function such 2 s = were pproximtely correct, we would come out with n optiml ge very close to the verge ge now being treted in the Methdone progrm. If, insted, s the progrm clims, success vries little with ge so tht function such s = t is pproprite, then to corroborte the ges now treted we would need n * of round 70 which is completely impossible. Even if * is s high s 50, s.2 nd t.02, then the Methdone progrm should be concentrting on persons s much s 10 yers younger thn they re now, even though the proportion successfully withdrwing would be lower if it did. This would tke into ccount the likelihood tht The results of this nlysis re illustrtive nd suggestive rther thn definitive. They tend to support the efficcy of the Methdone progrm reltive to the Stte progrm, if the gol is to minimize the number of ddicts. However, they rise serious questions bout the merits of the Methdone progrm's prctice of treting disproportionte number of older ddicts. Combining informtion on the number of yers of ddiction remining from successful withdrwl t ech ge with ssumptions bout success in chieving withdrwl t ech ge, suggests tht it my be possible for the Methdone progrm to increse its impct on the number of ddicts by the order of mgnitude of t lest one fourth, with no dditionl expenditures, by shifting emphsis in tretment to younger groups. Footnotes 1. Director of Reserch nd Director of Helth Plnning, Office of Comprehensive Plnning. The views expressed re those of the uthor nd need not represent those of the City Plnning Commission, the Community Renewl Progrm, or ny of their ffilites or consultnts. Beginnings on this study were mde while t the Rnd Corportion. The comments of Zili Amsel, Sidney Leveson nd Clrence Teng were most helpful. 2. The term ddict is used to imply users of hrd drugs whether or not physiologiclly or psychologiclly ddicted. 3. One study of ddicts with men ge 30.7 showed men ge of most rrests of See Bernrd Greenfield, "The Riverside Study  Ten Yers Lter." Pper prepred for the New York City Deprtment of Helth, Helth Reserch Trining Progrm, Summer If we mke the ssumption tht the rtio of the benefits per mn yer of ddiction verted to the discount function is n inverse function of ge then or bj (l+r) k B = E Nj j=i B = M. The benefits of reducing ddiction re proportionl to the number of mn yers of ddiction 331
6 verted nd inversely relted to the ge of those ssisted. 5. The model could esily be extended to llow for some positive benefit for nonsuccesses due to period of withdrwl prior to recidivism. 6. See Irving Leveson, "Drug Addiction: Some Evidence on Prevention nd Deterrence," pper presented t the meetings of the Econometric Society, Detroit, Michign, December 1970 nd John Holhn, The Economics of Drug Addiction in Wshington, D.C.: A Model for Estimtion of Costs nd Benefits of Tretment nd Rehbilittion, Report No. 33, District of Columbi Deprtment of Corrections, October Bsed on Bronx District Attorney Burton Robert's estimte tht bout four tenths of the persons in detention re ddicts. 8. New York Stte "Report of the Nrcotic Addiction Control Commission for the First Twenty One Month Period." The Commission did not relese comprble figures in its second nnul report. In the week of Jnury 2531, 1970, only 48 percent were known to be ginfully occupied. 10. The progrm voids constrints on the vilbility of professionl mnpower by using ex ddicts. However, retining ex ddicts in the progrm delys the time they will hve to re enter the community. The djustment process my be mde more difficult in the future if lrge numbers of persons must be bsorbed into the community in reltively short period of time. If the rtio of the number of ddicts to ex ddict employees is R, the rte of growth per two yer tretment period needed to retin S percent of persons treted s employees is S R. If the growth of the progrm were suddenly hlted, R of those in the progrm would be R ending tretment nd hve to find jobs. 11. At older ges the slopes re smller nd *'s higher. 9. John C. Krmer nd Richrd A. Bss, "Civil Commitment for Addicts: The Cliforni Progrm," Report of the Committee on Problems of Drug Dependency of the Ntionl Acdemy of Sciences nd the Ntionl Reserch Council,
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