Suggested Solutions to Homework #3Econ 511b (Part I), Spring 20041. Consider an exchange economy with two (t ypes of) consumers. T ype-Aconsumers comprise fraction λ of the economy’s population and type-Bconsumers comprise fraction 1 − λ of the econom y’s population. Eac h con-sumer has (constant) endowment ω in eac h period. A consumer of type i haspreferences o ver consump tion streams of the formP∞t=0βtilog (ct). Assumethat 1 >βA>βB> 0:type-A consumers are more patient than type-Bconsumers. Consumers trade a risk-free bond in each period. There isno restriction on borro w ing except for a no-Ponzi-game condition. Eachconsumer has zero assets in period 0.(a) Carefully define a sequen tial competitive equilibrium for this economy.A sequential competitiv e equilibr ium for the econ omy {uA,uB,ω} , is a sequence{c∗it}∞t=0,©a∗i,t+1ª∞t=0, {q∗t}∞t=0(where q∗tmeans price of Arrow security) for i = A, Bsuc h that(1) For i = A, B,©c∗it,a∗i,t+1ª∞t=0=argmax∞Pt=0βtilog (cit)s.t.cit+ q∗tai,t+1= ai,t+ ωlimt→∞ai,t+1µ∞Qt=0qt¶≥ 0ai,0=0,cit≥ 0(2) λc∗At+(1− λ) c∗Bt= ωfort=0, 1, 2...(3) λa∗A,t+1+(1− λ) a∗B,t+1=0for t =0, 1, 2...(b) C ar efu lly define a recur s ive com petitive equilibrium for this eco n o my.A Recu rsive Competitive Eq uilib r ium for the econo my {uA,uB,ω} is a set offunction s:price f unction : q (A)policy function : a0i= gi(ai,A)value functions : vi(ai,A)transition function : A0= G (A)such that:(1) For i = A, B, a0i= gi(ai,A) and vi(ai,A) solvesvi(ai,A)= max{ci,a0i}log (ci)+βivi³a0i,A0´s.t.ci+ q (A) a0i= ai+ ωA0= G (A)1(2) Consisten cy:G (A)=gA(A, A)−λ1 − λG (A)=gBµ−λ1 − λA, A¶(c) Show that this economy has no steady state: in particular, sho w thatthe type-B agen ts become poorer and poorer o ver time and consumezero in the limit.To solv e this problem, we first get Euler equation. In recursive formulation,consumers solv evi(ai,A)= max{ci,a0i}log³ai+ ω − q (A) a0i´+ βivi³a0i,A0´s.t.A0= G (A)Solve for F.O.C. and use envelope condition, we get the Euler equatio n:βiu0(ci,t+1)u0(ci,t)= q (A)or equ ivalen tly,βAu0(cA,t+1)u0(cA,t)= βBu0(cB,t+1)u0(cB,t)Now w e can see that ci,t+16= ci,t(∀i, ∀t). Suppose not, without loss of gen eralitylet cA,t+1= cA,t. By feasibility condition, we know that cB,t+1= cB,t.Plugintothe equa tion w e get βA= βA, a contradiction. As a resu lt, there cannot be anysteady state in this econom y.We start to prove the convergen ce property of consumption path. First, w e wa ntto show that {cAt}∞t=0({cBt}∞t=0) is an increasing (decreasing) sequence. We al-ready know that cA,t+16= cA,t(∀t). No w suppose that cA,t+1<cA,tfor some t.Bythe feasibility condition, we know that cB,t+1>cB,t. From the strict concavity offelicity functio n, we haveu0(cA,t+1)u0(cA,t)> 1 >u0(cB,t+1)u0(cB,t)⇒ βAu0(cA,t+1)u0(cA,t)>βBu0(cB,t+1)u0(cB,t)which contradicts Euler equ atio n.Since bounded monotone sequence has a limit, we hav e cAt→ c for t →∞. Butwe ha ve show n that the economy has no steady state, so cAtcan converge tono where but the boundary, i.e. cAt→ ω and cBt→ 0.22. Consider an infinite-horizon one-sector grow th model with an externalityin production. Leisure is not valued and the (representativ e ) consume rhas time-separable preferences with discoun t factor β ∈ (0, 1).Consumerso wn the factors of production. Capital depreciates at rate δ.Thereisalarge n umber of iden tical firmseachofwhichhasthefollowingproductiontec hnology:F¡k, l,k¢= Akαl1−αkγwhere k is the capital rented by the firm, k is the aggregate capital stock,and the param eters α and γ satisfy 0 <γ<1− α and α ∈ (0, 1).Thusthereisa productive externality from the rest of the econom y: a higher aggregatecapital stock increases th e firm’s productivity. A t y pical (small) firm takesthe aggregate capital stock as given when c h oosing its inputs.(a) Define a recu r sive competitiv e equilibrium for this economy. Be clearabout whic h variables consumers and firmstakeasgivenwhentheysolve their optimiz a tio n problems. Find th e competitive e q uilib r iu msteady-state aggregate capital stoc k as a function of primitives.A Recu rsive Com petitiv e Eq uilibrium for the economy is a set of functions:price function : r¡k¢,w¡k¢policy function : k0= g¡k, k¢value function : v¡k,k¢transition function :k0= G¡k¢such that:(1) k0= g¡k, k¢and v¡k, k¢solv es consu m er’s problem :v¡k,k¢=max{c,k0}u (c)+βv³k0, k0´s.t.c + k0= r¡k¢k +(1− δ)k + w¡k¢k0= G¡k¢(2) Price is competitiv ely determ ined:r¡k¢= F1¡k, 1, k¢= αAkα+γ−1w¡k¢= F2¡k, 1, k¢=(1− α) Akα+γ(3) Consisten cy:G¡k¢= g¡k, k¢Solve for consumer’s problem in the normal w a y, we solve for the Euler equationasβu0(ct+1)u0(ct)¡r¡kt+1¢+1− δ¢=13In steady state, we haver³kc´+1− δ =1β⇒ αA³kc´α+γ−1=1β− 1+δ⇒kc=Ã1β− 1+δαA!1α+γ−1where kcrepresents the competitive equilibrium steady-state aggregate capitalstoc k.(b) Write the planning prob lem for this economy in recursive form. Theplanner internalizes the externality in production: his production tec h-nology isF¡k, l, k¢= Akα+γl1−αFind the steady-state aggregate capital stoc k implied b y the planningpro ble m. Show that it is highe r than the com petitive equilib r iu msteady-state aggregate capital stoc k.The recursive form ulation of the planning problem isv¡k¢=max{c,k0}u (c)+βv³k0´s.t.c + k0= Akα+γ+(1− δ)kThe Euler equation isβu0(ct+1)u0(ct)³A (α + γ)kα+γ−1t+1+1− δ´=1In steady state, we haveA (α + γ)³ko´α+γ−1+1− δ =1β⇒ko=Ã1β− 1+δ(α + γ) A!1α+γ−1> kc=Ã1β− 1+δαA!1α+γ−1(since α<α+ γ<1)wherekorepresents the optimal steady-state aggregate capital stoc k.(c) N ow introduce a gov ern men t into the competitive eq u ilibr ium th at youdefine d in p ar t (a). Th e gov ern men t subsid ize s in vestm e nt e x pendi-tures at a proportional rate τ and finances these subsidies by means4of a lump-sum tax on consumers. The in vestment subsidy is constantacross time but the lump-sum tax varies o ver time so as to balance thego vernmen t’s budget in every period. Define a recursiv e competitiveeq u i librium for t his eco nomy.A Recursive Com petitiv e Equilibrium for the economy with taxation is a set offunction s:price function : r¡k¢,w¡k¢policy function : k0= g¡k, k¢value function : v¡k,k¢taxation function : T¡k¢transition