A way of thinking about saving vs improving lives

Through­out this es­say I ig­nore flow through effects, non hu­man an­i­mals, and effects on the dis­tant fu­ture, even though I don’t dis­count those in my per­sonal al­tru­is­tic de­ci­sions. Feel free to ex­pand this anal­y­sis if you want to in­clude those fac­tors.

Ob­vi­ous dis­claimers: I think it’s a dumb idea to kill poor peo­ple. Also, well be­ing varies for rea­sons other than poverty.

TL; DR: If we’re choos­ing be­tween spend­ing money on sav­ing lives and re­duc­ing poverty, we need to con­sider how to com­pare the hap­piness cre­ated by sav­ing a life to the hap­piness cre­ated by in­creas­ing the con­sump­tion in a so­ciety. Differ­en­ti­at­ing an es­ti­mate of hap­piness as a func­tion of in­come lets you es­ti­mate the hap­piness cre­ated by in­creas­ing marginal con­sump­tion. If you as­sume hap­piness is log­a­r­ith­mic in in­come, then the dol­lar amount at which you’re in­differ­ent be­tween sav­ing a life and in­creas­ing to­tal con­sump­tion that much in­creases as in­come * log(in­come). My ex­tremely ten­ta­tive num­bers sug­gest that AMF is a much bet­ter char­ity for he­do­nic util­i­tar­i­ans than GiveDirectly.

Thanks to Claire Za­bel, Marie La, Daniel Filan, and oth­ers who helped me with this.

Some­times it’s use­ful to be able to put a value on a hu­man life. To use a crass metaphor, these days hu­man lives go for about $3400 if you want to buy them from AMF. That’s the sel­ler’s price. I’m in­ter­ested in calcu­lat­ing the max­i­mum price at which we should be in­ter­ested in buy­ing hu­man lives.

One way of defin­ing this “buyer’s price” of a hu­man life is the max­i­mum price at which we as a so­ciety would rather save some­one’s life than just keep the money. Peo­ple are hap­pier when they have higher con­sump­tion. If I could save the life of one Amer­i­can, but re­duce Amer­i­can GDP by 10%, I’m pretty sure that would over­all not be a good trade.

So we need to have a way of com­par­ing the dam­age done by re­duc­ing the con­sump­tion of an econ­omy by some amount of money to the dam­age done by some­one in the econ­omy dy­ing.

Log­a­r­ith­mic happiness

It’s pretty com­mon to ap­prox­i­mate hap­piness as lin­ear in the log­a­r­ithm of in­come, which I’m go­ing to equiv­o­cate with con­sump­tion for the rest of this post. Con­sump­tion just means “the to­tal value of all the things you con­sume in a given year”. Aver­age hap­piness in a pop­u­la­tion where ev­ery­one has con­sump­tion cc can be writ­ten as:

av­er­age hap­piness = logc+klog⁡c+k

where cc is con­sump­tion and kk is a con­stant that tells us about the con­sump­tion level at which a life is so mea­gre and de­prived that it isn’t worth liv­ing, and also tells us how rapidly hap­piness in­creases rel­a­tive to wealth. If peo­ple have a life ex­pec­tancy of tt years, then we have the to­tal hap­piness-years of a per­son as tlogc+tt­log⁡c+tk.

Any­way, back to the math. The deriva­tive of this func­tion with re­spect to con­sump­tion is 1y1y. So if we re­duce the size of an en­tire econ­omy by some small amount ΔcΔc evenly spread across the whole pop­u­la­tion of n peo­ple, then the re­duc­tion in hap­piness per per­son is:

Δhap­piness per per­son=ΔnΔhap­piness per per­son=Δccn

but there are nn peo­ple, so the to­tal re­duc­tion in hap­piness is:

Δto­tal hap­piness=nΔcnc=ΔcΔ­to­tal hap­piness=nΔcnc=Δcc.

So if we have the op­tion to spend ΔcΔc to save a life, we should be in­differ­ent to do­ing so if the to­tal re­duc­tion of hap­piness caused by re­duc­ing the to­tal con­sump­tion by ΔcΔc would be the same as the hap­piness of a given per­son:

t(logc+k)=Δcct(log⁡c+k)=Δcc

We can solve this for ΔcΔc:

Δc=ct(logc+k)Δc=ct(log⁡c+k)

Now we can sub­sti­tute the value that we end up choos­ing for kk be­low to figure out that Amer­ica should be will­ing to spend up to about $10 mil­lion to save an Amer­i­can baby. You can see the calcu­la­tion here.

That’s all the math. Now, let’s spend two thou­sand words try­ing to figure out pre­cisely how much fun it is to be ex­tremely poor!

Think­ing the­o­ret­i­cally about kk

The biggest judge­ment call in this es­say is that kk con­stant. The con­sump­tion at which a life has 0 value ac­cord­ing to the above for­mula is 10k10−k. So if you think that life isn’t worth liv­ing if you’re con­sum­ing less than $1000 a year, then you think kk is −3.

That’s the literal mean­ing of kk. How­ever, it seems plau­si­ble to me that our log model breaks down when peo­ple are in­cred­ibly poor. So I think we should use two differ­ent strate­gies to think about kk. Firstly, we should think about it as­sum­ing the model is cor­rect, look­ing for the level of con­sump­tion at which life seems to not be worth liv­ing. Se­condly, we should try ig­nor­ing its literal mean­ing and just try­ing to di­rectly es­ti­mate it by look­ing at hap­piness vari­a­tion across rel­a­tively small con­sump­tion vari­a­tions. This should hope­fully provide a good es­ti­mate over the kind of range we’re in­ter­ested in.

Imag­ine we de­cided that liv­ing on $100,000 a year is twice as much fun as liv­ing on $10,000 a year. This would mean that log100000+klog100000+k is twice as much as log10000+klog10000+k. So kk would be −3. If we thought poverty was not as bad as that, then maybe we’d be say­ing that liv­ing on $1,000 a year is half as good as $10,000. In that case, kk would be −2. I think that the first of those is prob­a­bly truer, so this is an­other ar­gu­ment for es­ti­mat­ing kk as about −3.

Maybe we should be con­cerned by this, be­cause more than a billion hu­mans have less than $1000 an­nual con­sump­tion. I have a few thoughts on this. To start with, it seems plau­si­ble that the pur­chas­ing power par­ity ad­just­ment I’m us­ing isn’t pow­er­ful enough. It’s cheap to have a place to sleep in ru­ral Malawi, much more than it is in Amer­ica. If there were a Malaw­ian ru­ral town within com­mut­ing dis­tance of my office in SoMa, I’d be ec­static to pay $200 a month to live in a thatch-roofed hut there. (I’m not kid­ding: I lived on an air mat­tress on the floor of my office for six months last year.) There are lots of re­ally cheap goods available to Malaw­ians, like thatched roof huts and re­ally cheap shitty rice, which are un­available to me but make it much more easy to live cheaply.

To some ex­tent, I’m will­ing to buy that liv­ing in SF counts as bonus con­sump­tion, be­cause I’m closer to fun things, but mostly I just live here so that I can work here, which feels more like an em­ploy­ment-re­lated ex­pen­di­ture than con­sump­tion to me. So maybe I don’t buy that as some­one with an an­nual con­sump­tion of $30k, I’m re­ally get­ting 40 times the con­sump­tion in my life as the av­er­age res­i­dent of Malawi.

The book Poor Eco­nomics tells the story (start­ing on page 20) of this ex­tremely poor In­done­sian guy called Pak Solhin. He used to live on $2 USD PPP a day, un­til he lost his job, af­ter which he lived like this:

Pak Solhin him­self sur­vived on about 9 pounds of sub­si­dized rice he got ev­ery week from the gov­ern­ment and on fish that he caught from the edge of a lake (he could not swim). His brother fed him once in a while. In the week be­fore we last spoke with him, he had had two meals a day for four days, and just one for the other three.

That life is be­ing clas­sified as sig­nifi­cantly less than $2 PPP a day, which I don’t think in­cludes the benefits of free rice, his brother’s food, or fish­ing in the river.

I’m not try­ing to triv­ial­ize ex­treme poverty here: that life sounds pretty un­pleas­ant, and I’m glad I don’t have it. But I don’t think it’s en­tirely sen­si­ble to call that “liv­ing on $1 a day”.

On the other hand, I get the benefits of a lot of gov­ern­ment spend­ing which Malaw­ians don’t: I have sub­si­dized pub­lic trans­port, and rea­son­ably good po­lice, and good roads, and so on.

Here’s an­other fact about con­sump­tion lev­els and hap­piness. Un­der most nat­u­ral cir­cum­stances, if you con­sume ex­tremely small amounts (like $1 a year), then you aren’t ex­tremely sad, you just die from hunger or ex­po­sure. In fact, the con­sump­tion lev­els at which my hap­piness func­tion pre­dicts your life isn’t worth liv­ing is ac­tu­ally pretty close to where I’d imag­ine that you’d die from hunger if we were re­ally ad­just­ing for PPP cor­rectly. This is ei­ther in­cred­ibly in­ter­est­ing or a sur­pris­ing co­in­ci­dence. I am in­ter­ested in hear­ing spec­u­la­tion about this.

The vari­abil­ity of con­sump­tion also plays into this. If it costs you a dol­lar a day to not die of hunger, then if your monthly con­sump­tion has a stan­dard de­vi­a­tion of 50c per day, I sus­pect you’d die in a few months. So even if the low­est sur­viv­able level of poverty is bad enough that your life is barely worth liv­ing, per­haps not many peo­ple will live at that level of poverty for long.

Marie La points out that an­other way of es­ti­mat­ing this would be to look at the mor­tal risks peo­ple take when they are starv­ing. We could look at situ­a­tions where peo­ple had a choice be­tween re­main­ing in a place suffer­ing from famine, or do­ing some­thing ex­tremely dan­ger­ous to es­cape. She points out the ex­am­ple of post-war Viet­nam, where the South Viet­namese who tried to es­cape faced about a 50% chance of death and tried to es­cape any­way, par­tially be­cause they were so enor­mously hun­gry. One par­tic­u­larly good way of es­ti­mat­ing this would be to look at neigh­bor­ing re­gions where es­cap­ing is roughly as risky but the lev­els of famine were differ­ent, and com­par­ing the rates at which peo­ple tried to es­cape. There would ob­vi­ously be a mil­lion con­founders here, like how un­pleas­ant the regime was or how much peo­ple ex­pected the situ­a­tion to im­prove, but we might get some use­ful data re­gard­less.

That’s all been try­ing to es­ti­mate kk by look­ing at hap­piness at the low­est end: how about if we try to es­ti­mate it by look­ing at how much a 10% change in con­sump­tion changes hap­piness in a na­tion? This has fewer philo­soph­i­cal is­sues, so I’m not go­ing to dis­cuss them. This way of in­di­rectly es­ti­mat­ing kk is closer to how we’re go­ing to es­ti­mate it in the next sec­tion.

You also might be in­ter­ested in look­ing up the pa­per which had hap­piness as a func­tion of con­sump­tion within par­tic­u­lar poor coun­tries, then try­ing to solve for kk within that much smaller and eas­ier to mea­sure range. How­ever, this re­quires mak­ing a judge­ment call about how to trans­late the de­scrip­tions of hap­piness into real num­bers. Depend­ing on your feel­ings, this judge­ment call might be worse or bet­ter than what I did.

Numer­i­cally es­ti­mat­ing kk

Here’s a few data points to use to es­ti­mate kk:

  • A few weeks ago I was talk­ing to this dude who grew up in poverty in Mex­ico and ille­gally im­mi­grated to Amer­ica when he was 12. We chat­ted about his pre­vi­ous qual­ity of life for a while: not get­ting enough food, only get­ting one pair of shoes a year, and so on. My im­pres­sion is that his life in Mex­ico seemed about half as worth liv­ing as his life here is. From his de­scrip­tion of his qual­ity of life, and my un­der­stand­ing of Mex­i­can poverty, I would guess he was liv­ing on maybe $4000 a year. I wish that I’d thought to ask him for a nu­meric state­ment of how much bet­ter his life here was, but I didn’t think of this. Next time.

  • Slate Star Codex did a sur­vey try­ing to calcu­late the rel­a­tive qual­ity of life in differ­ent situ­a­tions. Re­spon­dents thought that Ethiopian life is 50% as good as Amer­i­can life, and Chi­nese life was 85% as good.

I’m a lot more averse to poverty than SSC read­ers, ap­par­ently.

From these we get an av­er­age es­ti­mate of kk as about −2.2. This num­ber is low enough that na­tions with av­er­age con­sump­tion PPP less than 102.2=158102.2=158 are clas­sified to have a nega­tive qual­ity of life. I think this is an in­ac­cu­rately high bar. I re­moved the China data point be­cause I think that the peo­ple an­swer­ing the Slate Star Codex sur­vey over­es­ti­mated how rich China is. I vaguely re­call the sur­vey im­ply­ing that you lived in a city in China or some­thing, as op­posed to liv­ing in ru­ral China as 50% of Chi­nese ac­tu­ally do.

Over­all, I think that this model of hap­piness as log­a­r­ith­mic in con­sump­tion is good but gives bad re­sults for ex­tremely poor coun­tries, be­cause we are un­der­es­ti­mat­ing the con­sump­tion of ex­tremely poor peo­ple.

Ap­pli­ca­tion to global poverty charities

One in­ter­pre­ta­tion of these num­bers is “the price at which we are in­differ­ent to de­creas­ing to­tal con­sump­tion by that much to save a sin­gle life”. Another in­ter­pre­ta­tion, though, is “the price at which sav­ing a sin­gle life is bet­ter value for money than just in­creas­ing con­sump­tion by that much”. This is re­ally im­por­tant, be­cause as philan­thropists we have the op­tion of do­ing both of these things, most ob­vi­ously through GiveDirectly and AMF. Let’s quickly re­view the lev­els of poverty of the peo­ple af­fected by these pro­grams:

AMF op­er­ates mostly in Malawi and the DRC. Bed­net dis­tri­bu­tion is slightly cheaper in Malawi. Malawi has a GDP PPP per cap­ita of about $226. Ap­par­ently, this is pretty un­evenly dis­tributed. So the peo­ple saved by AMF, who I think mostly live in ru­ral ar­eas (from look­ing up re­gions listed here), are prob­a­bly poorer than av­er­age for Malaw­ians. (AMF’s dis­trib­u­tors seem to find that most of the houses they look at need LLINs (Ex­cel file), so we don’t need to worry about sav­ing un­usu­ally poor peo­ple among ru­ral Malaw­ians.)

Kenyans who re­ceive GiveDirectly grants have a me­dian nom­i­nal con­sump­tion of $0.55 per day. (I use me­dian in­stead of mean be­cause I sus­pect that con­sump­tion fits a log-nor­mal dis­tri­bu­tion; this is sug­gested by the mean be­ing higher than the me­dian.) The nom­i­nal-to-PPP con­ver­sion for Kenya seems to be about 2.18 (from com­par­ing nom­i­nal and PPP GDP es­ti­mates), so that’s yearly con­sump­tion of about $408. This is al­most twice Malawi’s mean con­sump­tion, and as I said above the Malaw­ians saved by AMF are prob­a­bly poorer than above. Ob­vi­ously, these num­bers are so bad that they’re al­most use­less. The Malaw­ian con­sump­tion es­ti­mates here try to in­clude sus­te­nance farm­ing, but it’s re­ally hard to get that right. Ac­cord­ing to the table on page 24 of this re­port, Malawi’s pro­por­tion of un­der­nour­ished peo­ple is 23.1% while Kenya’s is 30.4%. (Un­dernour­ished means that you are be­low the “min­i­mum level of dietary en­ergy con­sump­tion”.) Th­ese num­bers kinda look like they fit with my hy­poth­e­sis that GiveDirectly re­cip­i­ents have pretty similar lev­els of con­sump­tion to Malaw­ians saved through AMF.

We’ve got a plethora of ex­tra fac­tors in this par­tic­u­lar case. To start with, maybe there are flow-through effects of cash trans­fers which case them to in­crease con­sump­tion more. Also, bed­nets have other pos­i­tive effects than sav­ing lives, like pre­vent­ing de­vel­op­men­tal im­pair­ments from malaria that limit life­time earn­ing po­ten­tial, pre­vent­ing malaria death in the above-5 year old age group (which isn’t counted) and pre­ven­tion of other mosquito-borne ill­nesses. And hav­ing a marginal hu­man might in­crease the con­sump­tion of other peo­ple in their so­ciety in some situ­a­tions, but I don’t know which situ­a­tions that is. Also, GiveDirectly might save lives as well, by giv­ing peo­ple money to buy things like medicine and med­i­cal care and bet­ter food, or in­di­rectly by al­low­ing peo­ple to get e.g. metal roofs and thus hav­ing bet­ter hy­giene in their houses.

I think that my for­mula is to­tally use­less for an­swer­ing the ques­tion of how much we should be will­ing to spend per life saved by AMF if our al­ter­na­tive is giv­ing to GiveDirectly, be­cause it’s so sen­si­tive to changes in my kk. How­ever, I did get some feel­ing about it from do­ing the re­search into poverty in Malawi and Kenya I did to write this dis­cus­sion sec­tion. The peo­ple whose lives you save by giv­ing to AMF seem to be pretty in­tensely im­pov­er­ished. If 30% of Malaw­ians are un­der­nour­ished in gen­eral, and peo­ple saved by AMF are un­usu­ally poor, I sus­pect that prob­a­bly a ma­jor­ity of the lives saved there are un­der­nour­ished. That means that these peo­ple feel hun­gry all the time. I some sym­pa­thy for the per­spec­tive that those lives sound per­haps not worth liv­ing, in which case mak­ing them bet­ter is the bet­ter op­tion. I also think it’s pretty plau­si­ble that these lives are ac­tu­ally pretty okay. At the mo­ment, I am prob­a­bly in­clined to think that the lives of the ex­tremely poor are worth liv­ing.

GiveWell has ob­vi­ously thought about how to com­pare sav­ing lives to in­creas­ing con­sump­tion, but AFAICT they haven’t con­sid­ered it this ex­plic­itly. In 2012 they said that they sus­pected AMF was a much bet­ter deal than GiveDirectly. How­ever, GiveWell hasn’t tried to quan­tify the value of in­creas­ing con­sump­tion vs sav­ing lives like I have here. They prob­a­bly haven’t tried be­cause it turns out that when you do, you get rel­a­tively difficult-to-in­ter­pret re­sults, as I did above.

Carl Shul­man has also writ­ten about hap­piness as log of con­sump­tion and GiveDirectly be­fore–I hadn’t seen his post be­fore I wrote all this.

Peter Hur­ford has a great sum­mary of the lives of ex­tremely poor peo­ple in third world coun­tries here.

If I donated to global poverty causes, I re­ally don’t know which of these I’d give to.

Conclusion

The value of sav­ing a life in­creases lin­ear­ith­mi­cly with con­sump­tion. This is neat. I am pretty sure this is cor­rect, and I’m re­ally happy to have a prin­ci­pled deriva­tion for it.

I think my equa­tion for the value of sav­ing a life is quite good for richer coun­tries where it’s eas­ier to mea­sure con­sump­tion. Maybe one day, we will have ended global poverty, and the Δc=ct(logc+k)Δc=ct(log⁡c+k) equa­tion will be ac­tu­ally use­ful when we’re try­ing to de­cide whether a par­tic­u­lar hov­er­car safety mea­sure is worth it.