Utilitarianism. Jeffrey Ely. June 7, This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

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1 Utltaransm June 7, 2009 Ths work s lcensed under the Creatve Commons Attrbuton-NonCommercal-ShareAlke 3.0 Lcense. Utltaransm

2 Why Utltaransm? We saw last tme that any standard of socal welfare s problematc n a precse sense. If we want to proceed, we need to compromse n some way. We wll compromse by focusng on envronments where ntensty of preference can be measured. Utltaransm

3 Money We wll assume that monetary transfers are possble and can be enforced. A monetary transfer scheme can be represented by t = (t 1,..., t n ) where t denotes the amount of money pad to ndvdual. (could be negatve) n =1 t = t 1 + t t n s the budget defct. (could be negatve) n =1 t = 0 means that the transfer scheme has a balanced budget. Utltaransm

4 Socal Choces wth Monetary Transfers Remember that socety must choose an alternatve. Now alternatves have two components. A choce from A (e.g. who gets to attend the NIN concert and who doesn t) A monetary transfer scheme t (.e. who pays, who gets pad, and how much.) And now we must descrbe the ndvduals preferences over both components. (.e. how do they trade-off monetary payments versus better/worse alternatves.) Utltaransm

5 Money Preferences Thought experment. Ple of money. Tcket to see NIN. How large can we make the ple of money before you choose that over the NIN tcket? We equate that wth your wllngness to pay. Utltaransm

6 Money Utlty Wllngness to pay s captured by utlty functons that work as follows. Defnton The value to ndvdual from alternatve x s denoted v (x). The utlty assocated wth alternatve x together wth monetary transfer t s U (x, t ) = v (x) + t Indvdual prefers a par (x, t ) to a par (y, t ) f U (x, t ) U (y, t ) and f the nequalty s strct, we say hs preference s strct. As always n economcs, a utlty functon s just a mathematcal devce that allows us to descrbe preferences n a precse way. Let s verfy that a utlty functon lke U descrbes wlngness to pay. Utltaransm

7 Money Utlty and WTP Example Suppose there s one tcket left to the NIN show. Alternatve x s you get t, alternatve y s I get t. Suppose that you derve no value from me seeng the show, so v you (y) = 0 and that your value from seng the show s v you (x) (some number.) If you are asked to choose between seeng the show (x) and payng t dollars versus not seeng the show (y) and payng nothng, you would be wllng to pay whenever U you (x, t) U you (y, 0) whch translates to or v you (x) + ( t) 0 t v you (x) Ths says that you are wllng to pay (up to but no more than) v you (x) to see the show. Utltaransm

8 More on WTP More generally, f x and y are any two alternatves, and t s a number, ndvdual prefers (x, t) to (y, 0) whenever whch translates to U (x, t) U (y, 0) t v (x) v (y) so that v (x) v (y) measures s wllngness to pay to have x rather than y. (And ths may be negatve.) Utltaransm

9 The Utltaran Socal Welfare Functon Defnton Under the utltaran socal welfare functon, socety prefers (x, t) to (y, t ) f n =1 U (x, t ) n =1 U (y, t ). In partcular, f t and t have a balanced budget then ths reduces to n n v (x) v (y) =1 =1 Ths socal welfare functon satsfes IIA and Pareto and s not a dctatorshp. Utltaransm

10 Not Perfect Wllngness to accept vs. wllngness to pay. (and ablty to pay.) Arguably not comparable across people. Tme rather than money? Utltaransm

11 Pareto Effcency For now, we restrct attenton to monetary transfer schemes that have a balanced budget. Defnton Socal choce (x, t) s Pareto domnated by another socal choce (y, t ) f every ndvdual prefers (x, t) to (y, t ) and at least one ndvdual strctly prefers t. Defnton A socal choce (x, t) s Pareto effcent f there s no (y, t ) that Pareto domnates t. In the absence of monetary transfers, there wll typcally be many Pareto effcent alternatves. Utltaransm

12 Utltaransm and Pareto Effcency When monetary transfers are possble, there wll typcally be just one Pareto effcent alternatve. Defnton If x s an alterantve for whch v (x) v (y) for every alternatve y, then x s called the Utltaran alternatve. Proposton When monetary transfers are possble, f (x, t) s Pareto effcent, then x must be utltaran as well. To demonstrate ths we wll show that f x s not utltaran, then (x, t) s not Pareto effcent. Utltaransm

13 Utltaransm and Pareto effcency Suppose that x s not utltaran,.e. there s some y such that v (y) > v (x) Utltaransm

14 Utltaransm and Pareto effcency Suppose that x s not utltaran,.e. there s some y such that v (y) > v (x) Then t s possble to devse a monetary transfer scheme t so that (y, t) Pareto domnates (x, t). To do so, frst defne ˆt = v (x) v (y) (Note that ths s postve for those who lke x better than y, negatve otherwse.) Utltaransm

15 Utltaransm and Pareto effcency Suppose that x s not utltaran,.e. there s some y such that v (y) > v (x) Then t s possble to devse a monetary transfer scheme t so that (y, t) Pareto domnates (x, t). To do so, frst defne ˆt = v (x) v (y) (Note that ths s postve for those who lke x better than y, negatve otherwse.) And now, set t = t + ˆt. We see that everyone s ndfferent. U (y, t ) = v (y) + t Utltaransm

16 Utltaransm and Pareto effcency Suppose that x s not utltaran,.e. there s some y such that v (y) > v (x) Then t s possble to devse a monetary transfer scheme t so that (y, t) Pareto domnates (x, t). To do so, frst defne ˆt = v (x) v (y) (Note that ths s postve for those who lke x better than y, negatve otherwse.) And now, set t = t + ˆt. We see that everyone s ndfferent. U (y, t ) = v (y) + t = v (y) + t + ˆt Utltaransm

17 Utltaransm and Pareto effcency Suppose that x s not utltaran,.e. there s some y such that v (y) > v (x) Then t s possble to devse a monetary transfer scheme t so that (y, t) Pareto domnates (x, t). To do so, frst defne ˆt = v (x) v (y) (Note that ths s postve for those who lke x better than y, negatve otherwse.) And now, set t = t + ˆt. We see that everyone s ndfferent. U (y, t ) = v (y) + t = v (y) + t + ˆt = v (y) + t + v (x) v (y) Utltaransm

18 Utltaransm and Pareto effcency Suppose that x s not utltaran,.e. there s some y such that v (y) > v (x) Then t s possble to devse a monetary transfer scheme t so that (y, t) Pareto domnates (x, t). To do so, frst defne ˆt = v (x) v (y) (Note that ths s postve for those who lke x better than y, negatve otherwse.) And now, set t = t + ˆt. We see that everyone s ndfferent. U (y, t ) = v (y) + t = v (y) + t + ˆt = v (y) + t + v (x) v (y) = v (x) + t Utltaransm

19 Utltaransm and Pareto effcency Suppose that x s not utltaran,.e. there s some y such that v (y) > v (x) Then t s possble to devse a monetary transfer scheme t so that (y, t) Pareto domnates (x, t). To do so, frst defne ˆt = v (x) v (y) (Note that ths s postve for those who lke x better than y, negatve otherwse.) And now, set t = t + ˆt. We see that everyone s ndfferent. U (y, t ) = v (y) + t = v (y) + t + ˆt = v (y) + t + v (x) v (y) = v (x) + t = U (x, t ) Utltaransm

20 Utltaransm and Pareto effcency But notce that t has a budget surplus: t = (t + ˆt ) Utltaransm

21 Utltaransm and Pareto effcency But notce that t has a budget surplus: t = (t + ˆt ) = t + ˆt Utltaransm

22 Utltaransm and Pareto effcency But notce that t has a budget surplus: t = (t + ˆt ) = = t + ˆt ˆt Utltaransm

23 Utltaransm and Pareto effcency But notce that t has a budget surplus: t = (t + ˆt ) = = = t + ˆt ˆt [v (x) v (y)] whch s negatve by our orgnal assumpton. Thus t s possble to add a small amount to every ndvdual s transfer so that y (and the transfer) wll Pareto domnate (x, t). Utltaransm

24 Utltaransm and Pareto effcency Proposton When monetary transfers are possble, f x s utltaran and t s a budget-balanced transfer scheme, then (x, t) s Pareto effcent. Utltaransm

25 Utltaransm and Pareto effcency Proposton When monetary transfers are possble, f x s utltaran and t s a budget-balanced transfer scheme, then (x, t) s Pareto effcent. Consder any alternatve y and balanced-budget transfer scheme ˆt. Suppose (y, ˆt) were Pareto superor to (x, t). That means, v (y) + ˆt v (x) + t for all wth at least one strct nequalty. Summng over Utltaransm

26 Utltaransm and Pareto effcency Proposton When monetary transfers are possble, f x s utltaran and t s a budget-balanced transfer scheme, then (x, t) s Pareto effcent. Consder any alternatve y and balanced-budget transfer scheme ˆt. Suppose (y, ˆt) were Pareto superor to (x, t). That means, v (y) + ˆt v (x) + t for all wth at least one strct nequalty. Summng over n (v (y) + ˆt ) > (v (x) + t ) =1 Utltaransm

27 Utltaransm and Pareto effcency Proposton When monetary transfers are possble, f x s utltaran and t s a budget-balanced transfer scheme, then (x, t) s Pareto effcent. Consder any alternatve y and balanced-budget transfer scheme ˆt. Suppose (y, ˆt) were Pareto superor to (x, t). That means, v (y) + ˆt v (x) + t for all wth at least one strct nequalty. Summng over (v (y) + ˆt ) > v (y) + ˆt > n =1 n =1 (v (x) + t ) v (x) + t Utltaransm

28 Utltaransm and Pareto effcency Proposton When monetary transfers are possble, f x s utltaran and t s a budget-balanced transfer scheme, then (x, t) s Pareto effcent. Consder any alternatve y and balanced-budget transfer scheme ˆt. Suppose (y, ˆt) were Pareto superor to (x, t). That means, v (y) + ˆt v (x) + t for all wth at least one strct nequalty. Summng over (v (y) + ˆt ) > v (y) + ˆt > n =1 n =1 (v (x) + t ) n v (y) > v (x) =1 whch would mean that x s not utltaran. Utltaransm v (x) + t