(SOCIAL) COST-BENEFIT ANALYSIS IN A NUTSHELL

(SOCIAL) COST-BENEFIT ANALYSIS IN A NUTSHELL RUFUS POLLOCK EMMANUEL COLLEGE, UNIVERSITY OF CAMBRIDGE 1. Introducton Cost-beneft analyss s a process for evaluatng the merts of a partcular project or course of acton n a systematc and rgorous way. Socal cost-beneft analyss refers to cases where the project has a broad mpact across socety and, as such, s usually carred out by the government. Whle the cost and benefts may relate to goods and servces that have a smple and transparent measure n a convenent unt (e.g. ther prce n money), ths s frequently not so, especally n the socal case. It should therefore be emphaszed that the costs and benefts consdered by (socal) cost-beneft analyss are not lmted to easly quantfable changes n materal goods, but should be construed n ther wdest sense, measurng changes n ndvdual utlty and total socal welfare (though economsts frequently express those measures n money-metrc terms). In ts essence cost-beneft analyss s extremely, ndeed trvally, smple: evaluate costs C and benefts B for the project under consderaton and proceed wth t f, and only f, benefts match or exeed the costs. That s: }{{} B }{{} C Benefts Costs So what makes thngs more complex? There are a varety of factors: Benefts and costs may accrue to dfferent sets of people. If ths s so we need some way to aggregate and compare dfferent benefts and costs across people. Benefts and costs may occur at dfferent ponts n tme. In ths case we need to compare the value of outcomes at dfferent ponts n tme.? Emal: rufus [at] rufus [dot] pollock [dot] org. Ths paper s lcensed under Creatve Commons attrbuton (by) lcense v3. 1

2 RUFUS POLLOCK EMMANUEL COLLEGE, UNIVERSITY OF CAMBRIDGE Benefts and costs may relate to dfferent types of goods and t may be dffcult to compare ther relatve values. Ths usually occurs when one of the goods does not have an obvous and agreed upon prce. For example, we may be spendng standard captal goods today n order to obtan envronmental benefts tomorrow. Benefts and costs may be uncertan. Benefts and costs may be dffcult to calculate and, as a result, there may be wdely dfferng vews about ther szes. One mght thnk ths could be subsumed under uncertanty, however the two ponts are rather dfferent: two people agreeng that an outcome follows some probablty dstrbuton s dfferent from them argung about ts mean and varance. Usually, n real-world cases the domnant ssue s usually the last one: the basc job of calculatng estmates for the project s costs and benefts. Ths especally true n the socal case where the projects under consderaton may nvolve costs and benefts that very dffcult to quantfy what s the beneft of the natonal securty derved from mltary spendng, how large are the benefts from educaton, etc etc. Necessarly ths quantfcaton only makes sense on a case-by-case bass. Here we are concerned wth general prncples and we therefore focus only on the precedng four tems and look at how they can be ncorporated nto the analyss n a general way. 2. The Basc Model We are consderng a project wth known (though perhaps uncertan) beneft B and cost C. Our task s to decde whether t s worthwhle. As already dscussed, f B and C were denomnated n exactly the same terms (.e. the same good, at the same tme) for a sngle person and wth no uncertanty thngs would be straghtforward: check whether B C. However, ths s unlkely to be the case so we wll need to do more work. All of ths work, n ts essence nvolves convertng benefts and costs nto values that can be easly compared. Equvalently we need to have benefts and costs denomnated n terms of some standard good or measure of value. We shall term ths good or measure of value the numerare. In theory, ths numerare could be anythng: apples, years of lfe, acres of ranforest etc. However, gven that many (though by no means all) goods are already denomnated n

(SOCIAL) COST-BENEFIT ANALYSIS IN A NUTSHELL 3 terms of money, t s often natural to use a numerare that s money-metrc. We also need to specfy money n whose hands (for example, 1 n the hands of someone on breadlne s lkely of dfferent value to 1 n the hands of a bllonare). For the purposes of socal cost beneft analyss a very natural numerare s (uncommtted) government funds, that s money the government has but s not yet allocated to any gven project. Ths s a natural numerare snce t s lkely to be government funds whch are used n payng for the project beng consdered. We wll also assume, to make our lves straghtforward, that, unless specfed otherwse, the cost of a gven project s exactly one unt of government funds today. Ths allows us to focus only on the benefts whch makes thngs smpler whle sacrfcng no real generalty. We begn, n the secton that follows, by focusng solely on the dstrbutonal ssues and gnorng temporal and uncertanty. We then ntroduce temporal consderatons and dscountng and conclude by dscussng uncertanty. 3. Dstrbutonal Concerns Let us suppose there are N people or groups. The beneft to group from the project under consderaton s b of ncome/consumpton. 1 Income/consumpton s construed broadly here to nclude not only normal traded goods such as food or dgtal musc players but also thngs lke securty, a stable bosphere etc. Indvduals exstng ncome/consumpton s denoted by x. Those operatng the project (the government) have a utltaran welfare functon: 2 W = u(x ) The change n welfare (assumng away any changes n behavour) arsng from the ndvdual benefts s: W = u(x + b ) u(x ) If the gans are small relatve to exstng ncome we may approxmate the change n ndvdual welfare usng the dervatve: u(x + b ) u(x ) = u (x )b. Defnng, w = u (x ) we then have a set of dstrbutonal weghts, that s weghtngs for ndvduals such that 1 We won t dstngush between ncome and consumpton here snce t wll not matter for our purposes. 2 We could avod any reference to ndvdual utlty here by postng a socal welfare functon defned drectly n terms of the outcomes beng affected be that money, educaton, securty etc.

4 RUFUS POLLOCK EMMANUEL COLLEGE, UNIVERSITY OF CAMBRIDGE the total (welfare) beneft s just the sum of the weghts tmes the ndvdual benefts: W = w b One last step remans: we need to convert utlty back nto our numerare (government funds) va multplcaton by some constant θ the overall beneft B wll then be θ W. To specfy ths constant we choose a benchmark project and then defne ts beneft B to exactly one unt of the numerare snce costs are also 1 ths mples ths project s just worthwhle. The standard approach s for the benchmark to be a project whch generates benefts equvalent to one unt equally dvded equally among all groups,.e. b = 1/N. 3 Ths mples that: θ w 1 N 1 θ = N w Note that f ncome were already equally dstrbuted so x = x and utlty (whch s only defned up to a constant) were normalzed so that that the margnal utlty of ncome at the reference ncome x were exactly one we would have w = w = 1 θ = 1. To summarze: N = Number of benefcares b = Benefts to group x = Income of group w = Weghts = Margnal Utlty of group W = Welfare beneft = w b θ = Converson factor from SW to Numerare B = Beneft = θ w b 3.1. Calculatng the Converson Factor θ. Wth a few assumptons on the form of the utlty functon and knowledge of the dstrbuton of ncome we can calculate an actual 3 Clearly the choce of the benchmark project s mportant. Why then choose ths project? The answer s that equally dstrbuted s what standard government projects lke provson of free educaton or free healthcare actually amount to. As such such ths equal dstrbuton project s defntely ncluded n the government portfolo and furthermore no worse project should be worthwhle because we could reallocate from that project to the equal dstrbuton and mprove well-beng.

(SOCIAL) COST-BENEFIT ANALYSIS IN A NUTSHELL 5 fgure for θ. Assume that utlty takes CES form: u(x) = x1 α 1 1 α Thus, the weghts (equal to margnal utlty) are w = x α and hence θ 1 = E(x α ). At ths pont, t s useful to move to contnuous rather than dscrete varables so: 1 θ = w (x)df (x) = x α df (x) = E(x α ) Now, the dstrbuton of ncome x s (approxmately) log-normal LN(ν, σ) n whch case usng the formula for the moment generatng functon of the normal we have: E(x α ) = e αµ+ α2 σ 2 2 3.2. Example 1: Benefts n Proporton to Income. Suppose the project generates value V whch s then dstrbuted n proporton to ncome. Let λ be the rato of ndvdual beneft to ncome so b = λx. Now b = V λ = V/ x. Thus usng our formula from above the total beneft n terms of the numerare s: B = θ w b = θλ w x = θ NV 1 x w x N Usng a CES utlty functon so w x = x 1 α and usng expectatons we have: B = V E(x1 α ) E(x α )E(x) Usng a log-normal dstrbuton for ncome and the expresson for the MGF as before ths further reduces to B = e ασ2. For log-utlty, α = 1, and a reasonable estmate of σ s 0.47 (Newbery 2008) whch mples B = 0.8V. Thus each pound/euro/dollar dstrbuted generates a beneft n terms of the numerare of 0.8 and, f the project s to be worthwhle, t must have a drect yeld of at least 25% (= 1/0.8). 3.3. Dscountng. We now come to the tme factor: benefts of effort or expendture today may not be realsed untl tomorrow. In the sprt of keepng thngs smple let

6 RUFUS POLLOCK EMMANUEL COLLEGE, UNIVERSITY OF CAMBRIDGE us fx everythng about the problem except the temporal aspect. In partcular, gnore dstrbutonal ssues, uncertanty and varatons n the types of goods nvolved. Assume that by gvng up one unt of expendture today we gan V unts at tme T. Let expendture today be x and at tme T (n the absence of the project) x T. There s a utlty functon u whch converts expendture nto contemporaraneous utlty (.e. utlty n that perod). Let the numerare be present perod utlty normalzed so that the margnal utlty today equals 1,.e. u (x) = 1. Thus there are two major factors to take nto account. Frst, how to convert utlty from perod T to now. Humans tend to prefer thngs sooner rather than later. Hence, even wth all else equal, utlty today s preferred to utlty tomorrow. The measure of ths s termed the level of pure tme-preference and we wll denote t by ρ(t). Second, the reference stuaton tomorrow (n terms of resources, consumpton etc) may not be the same as today and margnal utlty from ganng or losng a unt wll dffer across tme, qute apart from tme-preference. Assumng changes n expendture are relatvely small we can approxmate utlty changes usng dervatves we have: C = Cost = u (x) = 1 b = Beneft at T = V u (x T ) B = Beneft n today s utlty = ρ(t )b So the project s worthwhle f: V ρ(t )u (x T ) 1 Ths mplctly defnes a dscount factor δ(t ) = ρ(t )u (x T ) wth the payoff of V at tme T valued at δ(t )V today. 3.4. Example 2: Dscount Rates and Clmate Change. Suppose we can spend resources (or equvalently forgo consumpton) today to mtgate the effects of clmate n the form of benefts relatve to the do-nothng scenaro, at some pont T n the future. Suppose, n the absence of ths project, the economy would grow at rate g per year so

(SOCIAL) COST-BENEFIT ANALYSIS IN A NUTSHELL 7 consumpton at tme T s e gt tmes consumpton today. Suppose pure tme preference takes an exponental form so ρ(t ) = e ρt and we have CES utlty as before. Then: B = V e ρt e αgt = V e (ρ+αg)t Comparng ths wth a standard exponental dscount rate e δt gves an mpled dscount rate δ = ρ + αg. 4 3.5. Uncertanty. The models dscussed above nvolve no uncertanty: all relevant values, e.g. the project s payoff, future consumpton levels etc, are known wth complete certanty. Ths s clearly unrealstc and t s useful to be able to consder stuatons whch do nvolve (known) uncertanty. The natural approach here s smply to replace costs and benefts wth ther expected values. If those recevng benefts (or bearng costs) are rsk averse as s lkely uncertanty wll act to reduce benefts and ncrease costs (the certanty of 1m s worth more to most people than an evens chance of 0 or 2m). We llustrate by returnng to our clmate change example. 3.6. Example 3: Uncertanty and Clmate Change. Consder our prevous example regardng clmate change. We wll ncorporate uncertanty a lttle ndrectly here by assumng that drect costs and benefts reman certan but that we are uncertan as to the growth rate g n the absence of acton (.e. what the effects of clmate change wll f we do nothng). 5 Uncertanty n the growth rate wll affect our calculaton of benefts because t wll alter the level of consumpton, and hence margnal utlty, at whch benefts n the future are evaluated. Remember, the benefts of mtgaton are greater the more terrble the world when we of dong nothng (.e. n the case of unmtgated clmate change). Formally, suppose the growth rate g follows some probablty dstrbuton gven by G. Then: 4 Note that ths s, not concdentally, the same as the real nterest rate found n the Ramsey Kass Coopmans model. 5 An alternatve formulaton of the clmate change queston would be to change the default scenaro to full mtgaton and the acton scenaro beng do nothng. In ths framework the beneft would be ganed consumpton today whle the cost would be the reducton n consumpton n the future. Ths cost would then be drectly related to the growth rate n our man formulaton of ths problem.

8 RUFUS POLLOCK EMMANUEL COLLEGE, UNIVERSITY OF CAMBRIDGE C = E(u (x)) = 1 B = ρ(t )V E(u (e gt x)) E(u (e gt x)) s expected margnal utlty at tme T. Wth rsk averson ths wll be ncreasng n uncertanty (e.g. a ncrease n the varance of the growth rate that preserved the mean). To say more we need to specfy a functonal form for u and G. Take u as our prevous CES form and suppose growth s normally dstrbuted: g N(µ, σ 2 ). Then, recallng that the moment generatng functon for X, a normal random varable, s E(e tx ) = e tµ+ 1 2 t2 σ 2, we have that: E(u (e gt x)) = E(e αgt )E(x α ) = E(e αgt )E(u (x)) = e αµt + 1 2 α2 T 2 σ 2 Thus under uncertanty the dscount rate s ρ+αµ 1 2 α2 σ 2. Recall that under certanty t was ρ + αg. Takng the case where the growth rate under certanty, g, s equal to the expected growth rate under uncertanty µ, we see that the mpact of uncertanty (n the form of varance n the growth rate) acts to reduce the dscount rate. To llustrate, take as a benchmark case ρ = 0.5, α = 1.0, g/µ = 2.5 so the dscount rate s 3% n the case of certanty over the growth rate. Suppose however we aren t sure how bad (unmtgated) clmate change wll be so we replace our certan growth rate of 2.5% wth a normally dstrbuted one wth the same mean but a standard devaton (σ) of 2%. Ths would reduce the dscount rate to 1%. Moreover, a standard devaton greater than 2.5% would result n an negatve dscount rate n expectaton, consumpton n the future s more valuable than consumpton today (even though on average we expect consumpton to grow at 2.5% per year)! The crucal pont here s that uncertanty acts asymmetrcally because of dmnshng returns (equvalently rsk averson) n the utlty functon. Increasng uncertanty n the growth rate ncreases the chances both of beng n a great world where we are (relatvely) well-off and of beng n a terrble world where we are (relatvely) poor. In the former case our gan from extra consumpton n the future (the result of clmate change mtgaton n ths model) s reduced because we are already n a good stuaton but n the latter case the

(SOCIAL) COST-BENEFIT ANALYSIS IN A NUTSHELL 9 gans are greatly ncreased because we are so badly off. Moreover, thanks to dmnshng returns to consumpton the gans n the bad case greatly outwegh the reducton n benefts n the good case. As a result ncreasng uncertanty ncreases the value of each unt of beneft n the future and thus reduces the dscount rate.