Polcal Economy of Insuons and Developmen: 14.773 Problem Se 2 Due Dae: Thursday, March 15, 2019. Please answer Quesons 1, 2 and 3. Queson 1 Consder an nfne-horzon dynamc game beween wo groups, an ele and a group of czens. The ele s currenly n power n a non-democrac regme, N. In he nondemocrac regme, he ele receves ncome y N, whle czens receve ncome w N. A he begnnng of each perod, he ele (acng as a sngle agen n hs game) chooses how much o spend o defend he regme, denoed by x 0. If x v, he ele are able o manan he non-democrac regme. Here v s a random erm represenng he sochasc power of he czens o cones he regme. Assume ha v s dsrbued accordng o connuous, smooh densy f, whch s n addon sngle peaked and symmerc around zero. If he czens are able o change he regme, hen a new democrac regme, D, s nsued, and n hs regme, he ele receves ncome y D < y N, whle each czen receves w D > w N (so ha he ele prefers non-democracy and czens prefer democracy). In democracy, he mng of evens s smlar: a he begnnng of each perod, he ele decde how much o nves n acves o subver democracy back o non democracy, agan denoed by x. The only dfference s ha because n democracy he czens are more powerful, he ele wll succeed only f x v + γ, where γ > 0, and v have he same dsrbuon, f. 1. Wre down he dynamc maxmzaon problem of he ele n non-democracy and separaely n democracy as a dynamc programmng problem (wh maxmzaon over x ). 1
2. Assume ha he maxmzaon problem of he ele has an neror soluon n boh non-democracy and democracy, and derve he frs-order condon for boh problems. Inerpre he condons. 3. Show, agan under he assumpon of neror soluon, ha he equlbrum probably ha nex perod here wll be non-democracy s he same sarng from democracy and nondemocracy. Inerpre hs resul. 4. Wha happens f γ ncreases (sll assumng an neror soluon)? Inerpre hs resul. 5. Wha are he mplcaons of hese resuls for welfare (Hn: compue he expeced dscouned ne presen value of boh he ele and czens sarng from he wo regmes and compare hem, and hen deermne he mplcaons of a greaer γ for welfare). 6. Explan why he resul obaned n hs model s specal and provde hree dfferen modfcaons of he model (whou dong he mah) ha wll change hs resul. Do you hnk here wll be an overlap n erms of he qualave effec of nsuons beween hese modfed models and he one n hs queson? Queson 2 Ideas dang back o de Tocquevlle sugges ha greaer socal mobly makes democracy work beer or become more sable. Consder a smple wo-perod model wh hree groups, poor, mddle-class and rch (ndvduals whn each group are compleely homogeneous). Preferences sasfy sngle crossng and sngle peakedness, and he deal polces of he hree groups are dsnc. Suppose here are also hree polcal nsuons: democracy n whch he mddle class s he medan voer, a lef dcaorshp n whch he poor make all he decsons, and an ele dcaorshp n whch he rch make all he decsons. In he second perod, he relevan decson-maker jus decdes polcy. In he frs perod, he relevan decson-maker decdes boh polcy and wha omorrow s nsuons should be. Suppose ha we sar n democracy, so ha he frs-perod decsons wll be made by he mddle class. Suppose also ha here are 200 rch ndvduals, 150 mddle-class ndvduals and 300 poor ndvduals, so ha n democracy a member of he mddle class s he medan voer. Suppose ha here s he followng ype of socal mobly: K 150 mddle-class ndvduals become poor n he second perod, whle K poor ndvduals become mddle-class (leavng he overall dsrbuon of socey across he hree groups unchanged, wh no mobly for he rch). All ndvduals know her new socal group before he vong sage n he second perod. Show ha f K < 75, democracy s sable (n he sense ha he socey remans democrac n he second perod), whle f K > 75, democracy s unsable. Explan he nuon for hs resul and dscuss de Tocquevlle s hypohess n hs lgh. 2
Queson 3 Consder an economy populaed by λ rch agens who nally hold power, and 1 λ poor agens who are excluded from power, wh λ < 1/2. All agens are nfnely lved and dscoun he fuure a he rae β (0, 1). Each rch agen has ncome θ/λ whle each poor agen has ncome (1 θ) / (1 λ) where θ > λ. The polcal sysem deermnes a lnear ax rae, τ, he proceeds of whch are redsrbued lump-sum. Each agen can hde her money n an alernave non-axable producon echnology, and n he process hey lose a fracon φ of her ncome. There are no oher coss of axaon. The poor can underake a revoluon, and f hey do so, n all fuure perods, hey oban a fracon µ () of he oal ncome of he socey (.e., an ncome of µ () /(1 λ) per poor agen). The poor canno revol agans democracy. The rch lose everyhng and receve zero payoff afer a revoluon. A he begnnng of every perod, he rch can also decde o exend he franchse o he poor, and hs s rreversble. If he franchse s exended, he poor decde he ax rae n all fuure perods. 1. Defne MPE n hs game. 2. Frs suppose ha µ () = µ l a all mes. Also assume ha 0 < µ l < 1 θ. Show ha n he MPE, here wll be no axaon when he rch are n power, and he ax rae wll be τ = φ when he poor are n power. Show ha n he MPE, here s no exenson of he franchse and no axaon. 3. Suppose ha µ l (1 θ, (1 φ) (1 θ) + φ (1 λ)). Characerze he MPE n hs case. Why s he resrcon µ l < (1 φ) (1 θ) + φ (1 λ) necessary? 4. Now consder he SPE of hs game when µ l > 1 θ. Consruc an equlbrum where here s exenson of he franchse along he equlbrum pah. [Hn: frs, o smplfy, ake β 1, and hen consder a sraegy profle where he rch are always expeced o se τ = 0 n he fuure; show ha n hs case he poor would underake a revoluon; also explan why he connuaon sraegy of τ = 0 by he rch n all fuure perods could be par of a SPE]. Why s here exenson of he franchse now? Can you consruc a smlar non-markovan equlbrum when µ l < 1 θ? 5. Explan why he MPE led o dfferen predcons han he non-markovan equlbra. Whch one s more sasfacory? 6. Now suppose ha µ () = µ l wh probably 1 q, and µ () = µ h wh probably q, where µ h > 1 θ > µ l. Consruc a MPE where he rch exend he franchse, and from here on, a poor agen ses ha ax rae. Deermne he parameer values ha are necessary for such an equlbrum o exs. Explan why exenson of he franchse s useful for rch agens? 3
7. Now consder non-markovan equlbra agan. Suppose ha he unque MPE has franchse exenson. Can you consruc a SPE equlbrum, as β 1, where here s no franchse exenson? 8. Conras he role of resrcng sraeges o be Markovan n he wo cases above [Hn: why s hs resrcon rulng ou franchse exenson n he frs case, whle ensurng ha franchse exenson s he unque equlbrum n he second?]. Queson 4 Consder he followng nfne horzon economy. Tme s dscree and ndexed by. There s a se of czens, wh mass normalzed o 1 and a ruler. Czens dscoun he fuure wh he dscoun facor β, and have he uly funcon [ ] c 1 θ u = β j +j 1 θ e +j, j= where θ > 0, c +j s consumpon and e +j s effor (he ruler exers no effor). Each czen has access o he followng Cobb-Douglas producon echnology: y = A α ( e ) 1 α, where A denoes he sae of echnology and nfrasrucure a me, whch wll be deermned by he ruler. In addon, he ruler ses a ax rae τ on ncome. Also, each czen can decde o hde a fracon z of hs oupu, whch s no axable, bu hdng oupu s cosly, so a fracon δ of s los n he process. So consumpon of agen s: c [ (1 τ ) ( 1 z ) + (1 δ) z ] y, and ax revenues are T = τ (1 z ) y d. The ruler a me decdes how much o spend on A +1 wh echnology: A +1 = G where G denoes governmen spendng on nfrasrucure, and φ < 1. Ths mples ha he consumpon of he ruler s c R = T G. The ruler s assumed o maxmze he ne presen dscouned value of hs or her consumpon. The mng of evens whn every perod s as follows: 4
The economy nhers A from governmen spendng a me 1. Czens choose nvesmen, { e }. The ruler ses ax rae τ. Czens decde how much of her oupu o hde, { z }. The ruler decdes how much spen on nex perod s nfrasrucure G. 1. Fnd he frs-bes allocaon n hs economy. 2. Defne a Markov Perfec Equlbrum (MPE) where sraeges a me depend on he payoff-relevan sae of he game a me. Be specfc abou wha he sraeges are and wha he sae s. 3. Show ha n a MPE τ = δ for all. Inerpre wha δ corresponds o. 4. Gven hs, fnd he equlbrum level of effor by czens and ax revenues as funcons of A, T (A ). 5. Explan why we could wre he value funcon of he ruler as V (A ) = max A +1 {T (A ) A +1 + βv (A +1 )}, and usng hs, fnd he equlbrum spendng on nfrasrucure, G. 6. Wha s he effec of δ on spendng on nfrasrucure and equlbrum nvesmens. Does a declne n δ always mply greaer effor by czens? If no, why no? Calculae he equlbrum level oupu and fnd he oupu maxmzng level of δ. Dscuss wha hs means, and how you could map hs o realy. 7. How would you generalze/modfy hs model so ha some counres ax a lo and use mos of he proceeds for nfrasrucure and publc good nvesmens? 5