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

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1 October 7, 2012 Ths work s lcensed under the Creatve Commons Attrbuton-NonCommercal-ShareAlke 3.0 Lcense.

2 Recap 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 must abandon one of the basc prncples 1 Unversal Doman 2 Pareto 3 Independence of Irrelevant Alternatves

3 Pareto 1 Pareto s the crteron most closely ted to socal welfare. 2 So we wll nsst on Pareto 3 What f we only requre Pareto?

4 Pareto Domnance Defnton Alternatve A Pareto domnates another alternatve B f every ndvdual prefers A to B,.e. A B for every ndvdual. 1 Pareto domnance s a way of rankng alternatves. 2 But t s an ncomplete rankng: often nether alternatve Pareto domnates the other. 3 Examples: 1 The last remanng basketball tcket. 2 Publc school assgnment. 3 Desgner dress dbs.

5 Pareto So Pareto domnance rarely gves us a clear rankng But when t does, the prescrpton couldn t be more compellng. Defnton An alternatve A s Pareto effcent f there s no B that Pareto domnates t. We should not choose any alternatve whch s Pareto domnated. Ths s a foundatonal prncple of Economcs. Unfortunately that stll leaves us wth a lot of alternatves and no way to compare them.

6 But Wat 1 Let s revst the example wth the basketball tcket. 2 Let s suppose we also have the possblty of enforcng monetary transfers. 3 How much money are you wllng to pay to have the tcket?

7 Wllngness to Pay Thought experment. Ple of money. Basketball tcket. How large can we make the ple of money before you take the money rather than fly? We equate that wth your wllngness to pay.

8 Wllngness to Pay Wllngness to pay adds more nformaton about your preferences. Before we just talked about your rankng of A versus B. Now we can say somethng about how much more you lke A than B. How much money would t take to get you to favor B over A? Truthfully.

9 Pareto Domnance When Money s Involved Remember that any allocaton of the tcket s Pareto effcent. Suppose we are gong to gve the tcket to j but has a hgher wllngness to pay. Consder now the followng new alternatve. 1 We gve the tcket to nstead of j. 2 We take an amount of money x from and transfer t to j. 3 x s chosen to be n between the (hgh) wllngness to pay of and the (low) wllngness to pay of j. Ths alternatve Pareto domnates gvng the tcket to j (and no exchange of money.)

10 More Generally Proposton When money s nvolved, the only Pareto effcent alternatve s to gve the tcket to the fan wth the hghest wllngness to pay. Consder gvng the tcket to a fan wth a lower wllngness to pay. We just saw how to construct a Pareto domnatng alternatve/monetary transfer. If t s Pareto domnated then t s not Pareto effcent.

11 Money, Formally Now 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 by ndvdual. (could be negatve, a subsdy) n =1 t = t 1 + t t n s the budget surplus. (could be negatve, a defct) n =1 t = 0 means that the transfer scheme has a balanced budget.

12 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 the tcket 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.)

13 Money Utlty Wllngness to pay s captured by utlty functons. 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.

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

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

16 Maxmzng Socal Value 1 Recall the allocaton of the tcket. 2 Pareto effcency mpled gvng t to the fan wth the hghest wllngness to pay. 3 In fact that s the alternatve that maxmzes the total value n socety. 4 That was a specal problem You have postve value for the one alternatve where you get the tcket. You have zero value for everythng else. 5 In typcal problems you wll have dfferent, non-zero values for many dfferent alternatves. School assgnment Ad placement etc.

17 Maxmzng Socal Value Stll, we are lead to consder the alternatve A that maxmzes total value: v (A) Ths s called the utltaran alternatve. Just as n the smple tcket example, the utltaran alternatve s the only Pareto effcent alternatve when monetary transfers are possble.

18 Utltaransm and Pareto effcency Let A be the utltaran alternatve and B be any other alternatve. Then v (A) > v (B)

19 Utltaransm and Pareto effcency Let A be the utltaran alternatve and B be any other alternatve. Then v (A) > v (B) We wll devse a monetary transfer scheme t so that (A, t) Pareto domnates B. To do so, frst defne ˆt = v (A) v (B) (Note that ths s postve for those who lke A better than B, negatve otherwse.)

20 Utltaransm and Pareto effcency Let A be the utltaran alternatve and B be any other alternatve. Then v (A) > v (B) We wll devse a monetary transfer scheme t so that (A, t) Pareto domnates B. To do so, frst defne ˆt = v (A) v (B) (Note that ths s postve for those who lke A better than B, negatve otherwse.) Everyone s ndfferent between (A, t) and B. U (A, ˆt ) = v (A) ˆt

21 Utltaransm and Pareto effcency Let A be the utltaran alternatve and B be any other alternatve. Then v (A) > v (B) We wll devse a monetary transfer scheme t so that (A, t) Pareto domnates B. To do so, frst defne ˆt = v (A) v (B) (Note that ths s postve for those who lke A better than B, negatve otherwse.) Everyone s ndfferent between (A, t) and B. U (A, ˆt ) = v (A) ˆt = v (A) (v (A) v (B))

22 Utltaransm and Pareto effcency Let A be the utltaran alternatve and B be any other alternatve. Then v (A) > v (B) We wll devse a monetary transfer scheme t so that (A, t) Pareto domnates B. To do so, frst defne ˆt = v (A) v (B) (Note that ths s postve for those who lke A better than B, negatve otherwse.) Everyone s ndfferent between (A, t) and B. U (A, ˆt ) = v (A) ˆt = v (A) (v (A) v (B))

23 Utltaransm and Pareto effcency Let A be the utltaran alternatve and B be any other alternatve. Then v (A) > v (B) We wll devse a monetary transfer scheme t so that (A, t) Pareto domnates B. To do so, frst defne ˆt = v (A) v (B) (Note that ths s postve for those who lke A better than B, negatve otherwse.) Everyone s ndfferent between (A, t) and B. U (A, ˆt ) = v (A) ˆt = v (A) (v (A) v (B)) = v (B)

24 Utltaransm and Pareto effcency Let A be the utltaran alternatve and B be any other alternatve. Then v (A) > v (B) We wll devse a monetary transfer scheme t so that (A, t) Pareto domnates B. To do so, frst defne ˆt = v (A) v (B) (Note that ths s postve for those who lke A better than B, negatve otherwse.) Everyone s ndfferent between (A, t) and B. U (A, ˆt ) = v (A) ˆt = v (A) (v (A) v (B)) = v (B) = U (B, 0)

25 Utltaransm and Pareto effcency But notce that ˆt has a budget surplus: ˆt = [v (A) v (B)]

26 Utltaransm and Pareto effcency But notce that ˆt has a budget surplus: ˆt = [v (A) v (B)] = v (A) v (B)

27 Utltaransm and Pareto effcency But notce that ˆt has a budget surplus: ˆt = [v (A) v (B)] = v (A) v (B) And because A s utltaran, ths s postve. We can now construct a new transfer scheme t by reducng each ˆt by a small amount, balancng the budget and makng everybody strctly better off.

28 The Utltaran Socal Welfare Functon Wth wllngness to pay as a measure of preference, we can now defne a socal welfare functon whch utlzes that nformaton. Defnton Under the utltaran socal welfare functon, socety prefers (A, t) to (B, t ) f n =1 U (A, t ) n =1 U (B, t ). In partcular, f t and t have balanced budgets then ths reduces to n n v (A) v (B) =1 =1 Ths socal welfare functon satsfes IIA and Pareto and s not a dctatorshp.

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

30 Pareto Agan For the remander of ths lecture, we restrct attenton to monetary transfer schemes that have a balanced budget. Defnton Socal choce (A, t) Pareto domnates another choce (B, t ) f every ndvdual prefers (A, t) to (B, t ) and at least one ndvdual strctly prefers t. Defnton A socal choce (A, t) s Pareto effcent f there s no (B, t ) that Pareto domnates t.

31 Utltaransm and Pareto As we have shown, Pareto effcency mples utltaransm. Proposton When monetary transfers are possble, f (A, t) s Pareto effcent, then A must be utltaran as well.

32 Utltaransm and Pareto effcency The converse s true too. Proposton When monetary transfers are possble, f A s utltaran and t s a budget-balanced transfer scheme, then (A, t) s Pareto effcent.

33 Utltaransm and Pareto effcency The converse s true too. Proposton When monetary transfers are possble, f A s utltaran and t s a budget-balanced transfer scheme, then (A, t) s Pareto effcent. Suppose A s utltaran. Suppose there was a (B, ˆt) that would Pareto domnate (A, t). That would mean v (B) ˆt v (A) t for all wth at least one strct nequalty. Summng over

34 Utltaransm and Pareto effcency The converse s true too. Proposton When monetary transfers are possble, f A s utltaran and t s a budget-balanced transfer scheme, then (A, t) s Pareto effcent. Suppose A s utltaran. Suppose there was a (B, ˆt) that would Pareto domnate (A, t). That would mean v (B) ˆt v (A) t for all wth at least one strct nequalty. Summng over n (v (B) ˆt ) > (v (A) t ) =1

35 Utltaransm and Pareto effcency The converse s true too. Proposton When monetary transfers are possble, f A s utltaran and t s a budget-balanced transfer scheme, then (A, t) s Pareto effcent. Suppose A s utltaran. Suppose there was a (B, ˆt) that would Pareto domnate (A, t). That would mean v (B) ˆt v (A) t for all wth at least one strct nequalty. Summng over (v (B) ˆt ) > v (B) ˆt > n =1 n =1 (v (A) t ) v (A) t

36 Utltaransm and Pareto effcency The converse s true too. Proposton When monetary transfers are possble, f A s utltaran and t s a budget-balanced transfer scheme, then (A, t) s Pareto effcent. Suppose A s utltaran. Suppose there was a (B, ˆt) that would Pareto domnate (A, t). That would mean v (B) ˆt v (A) t for all wth at least one strct nequalty. Summng over (v (B) ˆt ) > v (B) ˆt > n =1 n =1 (v (A) t ) n v (B) > v (A) =1 v (A) t