Econ 510a (second half)Yale UniversityFall 2006Prof. Tony SmithHOMEWORK #4This homework assignment is due at 5PM on Friday, December 1 in Marnix Amand’s mailbox.1. Consider a two-period exchange economy with two types of consumers of equal measure.Each consumer maximizes u(ci0)+βE[u(ci1)], where cit, i = 1, 2, is the consumption of atype-i consumer in p e riod t. In period 0, type-i consumers are endowed with ωi0unitsof the (nonstorable) consumption good. Endowments in period 1 are random: withprobability πj, j = 1, 2, a type-i consumer receives ωi1junits of the consumption goodin period 1. In period 0, consumers trade Arrow securities whose payoffs depend onthe state of the world in period 1.(a) Carefully define a competitive equilibrium for this economy. Are markets com-plete? Explain why or why not.(b) Suppose that ¯ω0= ¯ω11= ¯ω12, where ¯ω0≡P2i=1ωi0and ¯ω1j≡P2i=1ωi1j. Provean aggregation theorem for this economy: that is, show that redistributions ofthe period-0 endowments do not affect the equilibrium prices of the Arrow securi-ties. (Note that a redistribution leaves the aggregate endowment unchanged.) Inaddition, characterize the equilibrium consumption allocation as much as possible.(c) Now suppose that ¯ω06= ¯ω116= ¯ω12. Prove an aggregation theorem for this economyunder the assumption that u has a constant elasticity of intertemporal substitutionequal to σ−1. In addition, characterize the equilibrium consumption allocation asmuch as possible.(d) Now suppose that in period 0 consumers cannot trade Arrow securities but insteadcan trade only a riskfree bond (i.e., a sure claim to one unit of the consumptiongood in period 1). Carefully define a competitive equilibrium for this economy.(e) For the economy in part (d), set ω10= ω20, ω111= ω212, ω112= ω211, and π1= π2= 1/2,but do not assume a f unctional form for u. Find the equilibrium consumptionallocation and the equilibrium bond price. How does the bond price compare tothe one in part (c)? Explain.Solution (a) As full set of Arrow securities are traded, markets are complete. A competitiveequilibrium for this economy is defined by {ci0, ci1j, qj, aij}i,j=1,2where(1)For each i, {ci0, ci1j, aij}j=1,2solvemax u(c0) + βE[u(ci1)]s.t. ci0+2Xj=1qjaij= wi0ci1j= aij+ wi1jj = 1, 2(2) market clearing:good market: c10+ c20= w10+ w20and c11j= w11j+ w21jfor j = 1, 2.asset market: a1j+ a2j= 0 for j = 1, 2.(b) We assume here that aggregate endowment is constant across time and states :W = ¯w0= ¯w11= ¯w12. The FOCs for the consumer i’s problem are:ai1: − u0(ci0)q1+ βπ1u0(ci11) = 0ai2: − u0(ci0)q2+ βπ2u0(ci12) = 0and then(1) qj=βπju0(ci1j)u0(ci0)for i, j = 1, 2But using the concavity of u, we get thatu0(ci1j)u0(ci0)= 1 for i, j = 1, 2. In fact, supposethere is some j for whichu0(c11j)u0(c10)=u0(c21j)u0(c20)> 1Then, given the concavity of u, it would imply that c11j< c10and c21j< c20, andthus at equilibrium we would have,¯w1j= c11j+ c21j< c10+ c20= ¯w0which is a contradiction (given that aggregate endowments are equal). Thereforethe Arrow securities prices are defined byqj= βπjfor j = 1, 2and so they are in fact independent of individual endowments.(c) Now aggregate endowments are not equal but the felicity function is a CES.Replacing this functional form in equation (1) obtained in part (a) we get thatqj=βπj(c11j)−σ(c10)−σ=βπj(c21j)−σ(c20)−σthen, the second equality implies thatc10c11j=c20c21jfor j = 1, 2. Imposing theequilibrium conditions in this equality, we get that for each j(2)¯w0− c20¯w1j− c21j=c20c21j⇒ c20=¯w0¯w1jc21jReplacing this in the equation for the Arrow securities prices we get thatqj=βπj(c21j)−σ(¯w0¯w1jc21j)−σ= βπj¯w0¯w1jσand thus the Arrow securities prices only depend on the aggregate endowment.Also, using (2) and the FOCs we getci0=¯w0¯w1jci1jfor i, j = 1, 2.(d) Now there are no Arrow securities available, consumers can only trade a risk freebond. A competitive equilibrium for this economy is composed of {ci0, ci1j, q, ai}i,j=1,2where(1)For each i, {ci0, ci1j, ai}j=1,2solvemax u(c0) + βE[u(ci1)]s.t. ci0+ qai= wi0ci1j= ai+ wi1jj = 1, 2(2) market clearing:good market: c10+ c20= w10+ w20and c11j= w11j+ w21jfor j = 1, 2.asset market: a1+ a2= 0 for j = 1, 2.(e) We now have an economy as the one in part (d) with no functional form for uand where w10= w20≡ w0, w111= w212≡ w1, w112= w211≡ w2and π1= π2= 0.5.The consumer i’s FOC is given by:ai: −u0(ci0)q +12βu0(ci11) +12βu0(ci12) = 0which implies(3) q = βu0(ci11) + u0(ci12)2u0(ci0)Given the symmetry in endowments and probabilities, we make the followingguess:c10= c20= w0, c111= c212and c112= c211First, plugging this guess in the budget constraints, we get that a1= a2= 0.Then, we need that c111= c212= w1and c112= c211= w2(ie, everybody consumeshis own endowment). Finally we also need both FOCs to be satisfied, and thus,we need the RHS of (3) to be independent of i. Replacing our guess in (3) we getfor i = 1, (3) ⇒ q = βu0(w1) + u0(w2)2u0(w0)andfor i = 2, (3) ⇒ q = βu0(w2) + u0(w1)2u0(w0)and thus the FOCs are satisfied. Summarizing, we have that the equilibriumallocations arec10= c20= w0, c111= c212= w1and c112= c211= w2and the bond price is given byq = βu0(w2) + u0(w1)2u0(w0)and thus it is the sum of the Arrow securities prices found in part (c), using thefact that π1= π2= 0.5 (no arbitrage condition!).2. Consider an exchange economy with two (types of) consumers. Type-A consumerscomprise fraction λ of the economy’s population and typ e-B consumers comprise frac-tion 1 − λ of the economy’s population. Each consumer has (constant) endowment ωin each period. A consumer of type i has preferences over consumption streams of theformP∞t=0βtiu(ct). Assume that 1 > βA> βB> 0: type-A consumers are more patientthan type-B consumers. Consumers trade a one-period riskfree bond in each periodThere is no restriction on borrowing except for a no-Ponzi-game condition. Assumethat each consumer has zero bonds in period 0.(a) Carefully define a sequential competitive equilibrium for this economy.(b) Show that this economy has no steady state: in particular, show that the type-Bconsumers become poorer and poorer over time and consume zero in the limit.Solution (a) A sequential competitive equilibrium for the economy {uA, uB, ω} , is a sequence{c∗it}∞t=0,b∗i,t+1∞t=0, {q∗t}∞t=0(where q∗tmeans price of Arrow security) for i = A, B