1. Consider a pure-exchange economy with stochastic endowments. The state of the economy

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1 Answer 4 of he following 5 quesions. 1. Consider a pure-exchange economy wih sochasic endowmens. The sae of he economy in period, 0,1,..., is he hisory of evens s ( s0, s1,..., s ). The iniial sae is given. The probabiliy of reaching sae s is ( s ). Consumer i, i 1,2, has he expeced uiliy funcion ( s ) u ci ( s ), 0 s where 0 1, u () c 0and u () c 0for c 0, lim c 0 u ( c), and lim c u ( c) 0. In sae s, consumer i is endowed wih ( i s ) unis of a nonsorable consumpion good. The endowmens saisfy ( s ) ( s ) 1 2 for all s. a. Define an Arrow-Debreu equilibrium. b. Define a sequenial markes equilibrium. c. Prove ha each consumer s consumpion level is consan over ime. d. Calculae he prices of he Arrow securiies in a sequenial markes equilibrium. e. Sae and prove he Firs Welfare Theorem. 1

2 2. Consider a producion economy wih wo overlapping generaions. In period, 0,1,..., he measure of young people is N. The measure of young people evolves according o N 1 gn, where g 1. The iniial measure of old people is N 1. Each member of he iniial old generaion is endowed wih k 0 unis of capial and amoun b 0 of nominal governmen asses and 1 has he uiliy funcion uc ( 0 ). Each member of every oher generaion is endowed wih unis of labor when young and has he uiliy funcion uc ( ) uc ( 1), where 0. The resource consrain in period is N c N ( c k ) F( N k, N ), where F(,) is homogeneous of degree one. (Noice ha here is full depreciaion of capial each period.) a. Define an Arrow-Debreu equilibrium. b. Define a sequenial markes equilibrium. c. Characerize he soluions o he consumers problems in a sequenial markes equilibrium and derive a no-arbirage condiion. y o k b d. Lis all he condiions for a seady sae in he objecs { c, c, k, b, r, r, w }. e. How many seady saes are here? If here is more han one seady sae, is one beer han he oher(s) (in some sense ha you make precise)? Sae and prove your claims. 2

3 3. Consider a closed endowmen economy wih single perishable physical goods. There are wo ypes of invesors, A and B, who have idenical sochasic endowmen sreams, fy A g and fy B g, where Y A = Y B = Y for all. The endowmen growh, fy +1 =Y g, follows a rs-order Markov process. The curren period is period. Consider he wealh porfolio which delivers he endowmens as dividends. Tha is, one uni of his asse delivers Y +1 unis of physical goods a period + 1, Y +2 a period + 2, ec., as dividends. Type-A invesor solves a period " 1 # X max E log(c+) A ; 0 < < 1; fa A + g =0 for 0 < < 1, where a A + is he unis of he wealh porfolio owned by ype-a invesor from + o + + 1; and C+ A is he consumpion a period +, subjec o C A = Y a A P ; C A +1 = Y +1 + a A Y +1 + (a A a A +1)P +1 ; C A +2 = Y +2 + a A +1Y +2 + (a A +1 a A +2)P +2 ; ::: where P + is he price of one uni of he wealh porfolio a period +, for all = 0; 1; 2; ::: Assume C+ A > 0 for all = 0; 1; 2; ::: On he oher hand, Type-B invesor s decision is known in advance (a he beginning of period ) as a B = a > 0; a B +1 = a B +2 = ::: = 0, where we assume ap < Y. Tha is, his invesor will hold a unis of he wealh porfolio beween and + 1, afer which he will leave he marke. (a) Suppose ha he economy consiss of ype-a invesors only. To be speci c, a ype-a invesor is he represenaive consumer. Hence, he marke clearing condiion is a A + = 0 for all. Derive he rs-order condiion wih respec o a A, and apply he marke clearing condiion o ll he blank in P Y = E? + P +1 Y +1 If you believe ha his equaion is incorrec, hen provide he correc one. 3 :

4 (b) Now suppose ha he economy consiss of equal numbers of ype-a and ype-b invesors. To be speci c, here is one ype-a invesor and one ype-b invesor in his economy. Hence, he marke clearing is a A + + a B + = 0 for all. Follow your seps in (a) o ll he blank in P = 1 + a P " E Y Y 1? (1 + P +1 Y +1 ) wih a and P +1 Y +1 only. (Recall a B = a.) If you believe ha his equaion is incorrec, hen provide he correc one. # 4

5 4. Consider he following fundamenal equaion of asse pricing in coninuous ime: E dr j d d + E = E dr j ; where dr j is he ne reurn on asse j from period. Also, e u 0 (C ), where is a parameer represening impaience, and u(), he uiliy funcion, sais es u(c ) = C1 1 : Hence, u 0 (C ) = C. In addiion, C is consumpion, following dc C = d + dz ; where and are consan, and dz is an incremen for a Brownian moion, saisfying dz ~N(0; d). (a) Fill he blanks: Hin: For = f(; C ); (b) Show ha d =? d +? dc +? dc2 : C C 2 d + 2 f 2 d2 dc + 2 E dr 2 f dc 2 2 r f dc d =? cov dr j ; ddc : where r f is he risk-free rae. Make reasonable assumpions if required. (c) Explain he equiy premium puzzle using he resul in (b). Use he following observaions: E dr j = 8% (annual sock reurn on average), dr j = 14% (volailiy of annual sock reurns), r f d = 1% (annual reurn on governmen bonds), dc C = 1% (volailiy of annual consumpion growh). Hin: cov(x; Y ) = (X)(Y )(X; Y ) where (X; Y ) is correlaion coe cien. Also, 1 (X; Y ) 1. 5

6 5. Consider a wo period overlapping generaions model. Time is discree and he economy lass forever. N = N > 1 agens are born a ime and each is endowed wih y unis of he consumpion good when young and none when old. The good can be sored for one period and earns a reurn of 1+r. Uiliy is increasing in he gap beween he agen s own consumpion and average consumpion and is represened by a uiliy funcion of he form, u Ln( c1 1c1 ) Ln( c2 1 2c2 1) where c / 1 C1 Nis average consumpion of he young generaion a ime, c / 2 C2 N is average consumpion of he old generaion a ime, aggregae consumpion is C 1 = N c 1 for he young generaion and C 2 = N -1 c 2 for he old generaion a ime, and where 1 > 1, 2 > 0 are weighs. Of course, he individual consumer akes aggregae variables like average consumpion of his generaion and he reurn o savings as given. A. Derive he consumpion funcion for he individual consumer. Wha does curren consumpion depend on? How is i affeced by c 1? How is i affeced by c 2? B. Find he aggregae consumpion funcion in a seady sae. How does aggregae consumpion respond o an increase in? How does i respond o an increase in? C. Derive he savings funcion of he individual consumer. How does he consumer s savings depend on average consumpion of he young generaion? How does i depend on average consumpion of he old generaion? D. Find he seady sae aggregae sock of capial. Wha does i depend on? How does i respond o an increase in? How does i respond o an increase in? 6