Econ 510a (second half)Prof: Ton y SmithTA: Theodore PapageorgiouFall 2004Yale UniversityDept. of EconomicsSolutions for Final Exam1Question1a) A Competitive Equilibrium with date-0 trading for this economy is a vectorof prices {pt}2t=0and a vector of quantities {c∗it}2t=0for i = A, B such that(1) For i = A, B,{c∗it}2t=0=argmax2Pt=0βtu (cit)s.t.2Pt=0ptcit=2Pt=0ptωit(2) cAt+ cBt= ωAt+ ωBtfor t =0, 1, 2b) From the f.o.c. of the consumer’s problem, we get:βu0(ci,t+j)u0(ci,t)=pt+jpt∀t, jThis together with budget constraint and mark et clearing condition deter-mines the competitive equilibrium. Now there are 2 way s of solving this prob-lem. The first is writing down all the f.o.c. for each of the 2 agent s, the marketclearing conditions for each of the 3 time periods and the 2 budget constraintsand solve out for the prices and quantities (you won’t need to use all of theequations to solve the system). The second and easiest is to use the fact thateach agents consumption in every period is going to be a constant share of theaggregate endowment. This follows from the homotheticity of preferences. Inother words we have that:cAt= γ (wAt+ wBt) ∀tcBt=(1− γ)(wAt+ wBt) ∀t1Now we are going to use the f.o.c. and we get (normalizing p0=1):βu0(cA1)u0(cA0)= p1⇒βcA0cA1= p1=⇒βγwoγw1= p1⇒12416= p1⇔p1=18Similarly:β2u0(cA2)u0(cA0)= p2⇒β2cA0cA2= p2=⇒14γ4γ4= p2⇔p2=14We can now plug in the prices we found in consumer A’s budget constraintand solve out for his share:γ4+18γ16 +14γ4=µ4+48+1¶⇔γ (4+2+1) =112⇔7γ =112⇔γ =11142and therefore the consumption in each period will be:c0A=11144=227c1A=111416 =887c2A=11144=227c0B=3144=67c1B=31416 =247c2B=3144=67Just to be sure lets verify that the budget constraint for consumer B holds:1 ·67+18·247+14·67=18· 12 ⇔2114=128andthusitdoeshold.c) A Competitiv e Equilibrium with sequential trading for this economy is asequence {c∗it}2t=0,©a∗i,t+1ª2t=0, {R∗t}2t=0(where R∗tmeans int erest rate from tto t +1)fori = A, B such that(1) For i = A, B,©c∗it,a∗i,t+1ª2t=0=argmax2Pt=0βtu (cit)s.t.cit+ ai,t+1= R∗tai,t+ ωitai,3≥ 0 (no-Ponzi condition)ai,0=0,cit≥ 0(2) c∗At+ c∗Bt= ωAt+ ωBtfor t =0, 1, 2(3) a∗A,t+ a∗B,t=0for t =0, 1, 2Itwillbethecasethat:ptRt= pt−1⇔Rt=pt−1ptTherefore:R1=p0p1=11/8=8R2=p1p2=1/81/4=1232Question2a) A Recursive Competitive Equilibrium 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¢suc h that:(1) k0= g¡k, k¢and v¡k, k¢solves consumer’s problem:v¡k,k¢=max{c,k0}u (c, s)+βv³k0, k0´s.t.c + k0= r¡k¢k +(1− δ)k + w¡k¢k0= G¡k¢s = θf¡k, n¢(2) Price is competitively determined:r¡k¢= f1¡k, 1¢w¡k¢= f2¡k, 1¢(3) Consistency:G¡k¢= g¡k, k¢b) Solving for consumer’s problem in the usual way, we get the Euler equa-tion:βu0(ct+1)u0(ct)¡r¡kt+1¢+1− δ¢=1Imposing equilibrium conditions and substituting in for the resource con-straint we get:βu0¡(1 − δ)kt+1+ f¡kt+1¢−kt+2¢u0¡(1 − δ)kt+ f¡kt¢−kt+1¢¡f0¡kt+1¢+1− δ¢=1Settingkt= k∗for every t, we obtain:f0³k∗´+1− δ =1⇔f0³k∗´= β−1− 1+δwhich does not depend on θ. The reason is that the firms don’t internalizethe (negative) externality caused by their production.4c) The important thing to realize here is that the speed of convergence equalsthe inverse of the slope of the decision rule. The way one could calculate thespeed of convergence of the aggregate capital stock to its steady state in aneighborhood of the steady state, is the follo wing. We write the Euler equationas follows:βu0((1 − δ) g (k)+f (g (k)) − g (g (k))) (1 − δ + f0(k)) = u0((1 − δ) k + f (k) − g (k))Then we differentiate with respect to k.We evaluate at the steady state (g (k∗)=k∗)What we will end up with is a quadratic equation is g0(k∗) . One of thesolutions can be proven to be in the interval (−1, 1) . Giventhatwehaveanexpression for g0(k) we can easily approximate g (k) by linear approximationaround k∗.d) The planning problem for this economy will be:max{ct,kt+1,st}∞Xt=0βtu (ct,st)s.t.ct+ kt+1=(1− δ) kt+ f (kt)st= θf (kt)After substituting in the constraints and taking the f.o.c., we are able toderive the planner’s Euler equation:β(1 − δ + f0(kt+1)) u1(ct+1,st+1)+θf0(kt+1) u2(ct+1,st+1)u1(ct,st)=1It easy to see that the steady state level of capital from the planning problemwill depend on θ and th us will differ from the competitive equilibrium. Thereforethe competitive equilibrium is Pareto inferior. The reason is that the socialplanner takes into account the negative externality and internalizes it in hisproblem.3Question3a) Note: there is more than one way to state the planning problem for thiseconomy.We know that:k0=(1− δ) k + Bki⇔ki= B−1[k0− (1 − δ) k]5Noting the above, we can write the social planning problem for this economy(after normalizing n =1)as:v (k)=maxk0{log£k − B−1(k0− (1 − δ) k)¤a+ βv (k0)}b) The f.o.c. is:−aB−1k − B−1(k0− (1 − δ) k)+ βv0(k0)=0⇔βv0(k0)=aB−1kcThe envelope condition is:v0(k)=a¡1+B−1(1 − δ)¢kcUpdating the envelope condition one period and substituting in the f.o.c. weget:βa¡1+B−1(1 − δ)¢k0c=aB−1kc⇔k0ckc= β (B +1− δ) ⇔k0c= β (B +1− δ) kcThus if g is the growth rate of kcalong the balanced growth path, then:eg= β (B +1− δ) ⇔g =ln(β (B +1− δ)) ⇔g ' β (B +1− δ) − 1where the last equation follows from the fact that ln (1 + x) ' x. Since:kt+1=(1− δ) kt+ B (kt− kct)k is growing at rate g and so is ki(since ki= k−kc). Notice that consumptionis growing at lower rate:c = Akacc0= Ak0acTherefore:c0c=µk0ckc¶a⇒egc=(eg)a⇔gc= ga6Alternative you can set up the planner’s problem as follows:v (k)=maxkc(log Akac+ βv ((1 − δ) k + B (k − kc)))Take the f.o.c and get:ak−1c− βBv0(k0)=0⇔v0(k0)=aB−1β−1kcThe envelope condition is:v0(k)=β (1 − δ + B) v0(k0)Substituting the f.o.c. condition into the envelope condition abov e gives us:v0(k)=β (1 − δ + B)aB−1β−1kc⇔v0(k)=(1− δ + B)aB−1kcWe update one period:v0(k0)=(1− δ + B)aB−1k0cAnd finally we plug into the f.o.c. to get:(1 − δ + B)aB−1k0c=aB−1β−1kc⇔k0c= β (1 − δ + B) kcwhich is exactly what we got with our original setup. From now on we justproceed in exactly the same way.4Question4a) The consumer’s problem