14.06 Lecture N otesInterm ediate M acroeconom icsGeorge-Marios AngeletosMIT Depar tment of EconomicsSpring 2004Chapter 3The Neoclassical Growth M odel• In the Solow m odel, agents in the economy (or the dictator) follow a simplis tic linearrule for consum p tion and investmen t. In the Ramsey model, agents (or the dictator)c h oose consum p tion and investment optimally so as to maximize their individual utility(or s ocial welfare).3.1 The Social Planner• In this section, we start the analysis of the neoclassical grow th model by consideringthe optimal plan of a benevolen t social planner, who c hooses the static and intertem-poral a llocation of r esources in the econo my so as to m axim ize social welfare. Wewill late r show that the allocations that prevail in a decentralized competitiv e marketen vironment coincide with the allocations dictated by the social planner.• Together with consumption and saving, we also endogenize labor supply.43George-Marios Angeletos3.1.1 Preferences• Prefere nces are defined ov e r streams of consumption and leisure {xt}∞t=0, where xt=(ct,zt), and are represented by a utility function U : X∞→ R,whereX is the domainof xt, such thatU ({xt}∞t=0)=U (x0,x1, ...)• We sa y that preferences are recursive if there is a function W : X × R → R such that,for all {xt}∞t=0,U (x0,x1, ...)=W [x0, U (x1,x2, ...)]We can then represen t preferences as follows: A consumption-leisure stream {xt}∞t=0induces a utility stream {Ut}∞t=0according to the recursionUt= W (xt, Ut+1).That is, utility in period t is given as a function of consumption in period t and utilityin period t+1.W is called a utility aggregator. Finally, note that recursive preferences,as defined abo ve, are both time-consisten t and stationary.• We say that preferences are additively separable if there are functions υt: X → R suchthatU ({xt}∞t=0)=∞Xt=0υt(xt).We then interpret υt(xt) as the utility enjoyed in period 0 from consumption in periodt +1.• Throughout our analysis, we will assume that preferences are both recursive and addi-tively sepa rab le. In other w ords, we impose that the utility aggregator W is linear in44Lecture Notesut+1: There is a function U : R → R and a scalar β ∈ R such that W (x, u)=U(x)+βu.We can thus represen t preferences in recursiv e form asUt= U(xt)+βUt+1.Alternatively,Ut=∞Xτ=0βτU(xt+τ)• β is called th e discount factor. For preferences to be well defined (that is, for theinfinite sum to converge) w e need β ∈ (−1, +1). M onotonicit y of preferences imposesβ>0. Ther efore, we restrict β ∈ (0, 1). The discou nt rate is given by ρ such thatβ =1/(1 + ρ).• U is sometimes called the per-period felicity or utility function. We letz>0 denotethe max imal am ount of time per period. We a ccordingly let X = R+×[0, z]. We fina llyimpose that U is neoclassical,inthatitsatisfies the follow ing properties:1. U is contin uous and (although not always necessary) t wice differentiable.2. U is strictly increasing and strictly concave:Uc(c, z) > 0 >Ucc(c, z)Uz(c, z) > 0 >Uzz(c, z)U2cz<UccUzz3. U satisfies the Inada conditionslimc→0Uc= ∞ and limc→∞Uc=0.limz→0Uz= ∞ and limz→zUz=0.45George-Marios Angeletos3.1.2 Tec hnology and the Re source Constrain t• We abstract from population grow th and exogenous technological change.• The time constraint is giv en byzt+ lt≤ z.We usually norm aliz e z =1and thus interpret ztand ltas the fraction of time tha t isdev oted to leisure and production, respectiv ely.• Theresourceconstraintisgivenbyct+ it≤ yt• Let F (K, L) be a neoclassical technolog y and let f(κ)=F (κ, 1) be the intensive formof F. Ou tpu t in the economy is give n byyt= F (kt,lt)=ltf(κt),whereκt=ktltis the capital-labor ratio.• Capital accumulates according tokt+1=(1− δ)kt+ it.(Alternatively, interpret l as effective labor and δ as the effectiv e depreciation rate.)• Finally, we impose the following natural non-nega tivitly constra ints:ct≥ 0,zt≥ 0,lt≥ 0,kt≥ 0.46Lecture Notes• Combining the above, we can rewrite the resource constraint asct+ kt+1≤ F (kt,lt)+(1− δ)kt,and the time constraint aszt=1− lt,withct≥ 0,lt∈ [0, 1],kt≥ 0.3.1.3 The Ramsey Problem• The social plann e r chooses a plan {ct,lt,kt+1}∞t=0so as to maximize utility subject tothe resource constraint of the economy, taking initial k0as given:max U0=∞Xt=0βtU(ct, 1 − lt)ct+ kt+1≤ (1 − δ)kt+ F (kt,lt), ∀t ≥ 0,ct≥ 0,lt∈ [0, 1],kt+1≥ 0., ∀t ≥ 0,k0> 0 given.3.1.4 Optimal Control• Let µtdenote the La gran ge m ultip lier for the resource constraint. Th e Lagrangian ofthe social planner’s problem isL0=∞Xt=0βtU(ct, 1 − lt)+∞Xt=0µt[(1 − δ)kt+ F (kt,lt) − kt+1− ct]47George-Marios Angeletos• Define λt≡ βtµtandHt≡ H(kt,kt+1,ct,lt,λt) ≡≡ U(ct, 1 − lt)+λt[(1 − δ)kt+ F (kt,lt) − kt+1− ct]H is called the Hamiltonian of the problem.• We can rewrite the Lagrangian asL0=∞Xt=0βt{U(ct, 1 − lt)+λt[(1 − δ)kt+ F (kt,lt) − kt+1− ct]} ==∞Xt=0βtHtor, in recursive formLt= Ht+ βLt+1.• Given kt,ctand ltenter only the period t utilit y and resource constraint; (ct,lt) th usappears only in Ht. Similarly, kt,en ter only the period t and t +1utility and resourceconstrain ts; they thus appear only in Htand Ht+1. Therefore,Lemma 11 If {ct,lt,kt+1}∞t=0is the optimum and {λt}∞t=0the associated multipliers, then(ct,lt)=argmaxc,lHtz }| {H(kt,kt+1,c,l,λt)taking (kt,kt+1) as given , andkt+1=argmaxk0Ht+ βHt+1z }| {H(kt,k0,ct,lt,λt)+βH(k0,kt+2,ct+1,lt+1,λt+1)taking (kt,kt+2) as given .48Lecture NotesEqu iva lently,(ct,lt,kt+1,ct+1,lt+1, )=argmaxc,l,k0,c0,l0[U(c, l)+βU(c0,l0)]s.t. c + k0≤ (1 − δ)kt+ F (kt,l)c0+ kt+2≤ (1 − δ)k0+ F (k0,l0)taking (kt,kt+2) as given .• We henceforth assume an interior solution. As long as kt> 0, interior solution is indeedensured by the Inada conditions on F and U.• The FOC with respect to ctgives∂L0∂ct= βt∂Ht∂ct=0⇔∂Ht∂ct=0⇔Uc(ct,zt)=λtThe FOC with respect to ltgives∂L0∂lt= βt∂Ht∂lt=0⇔∂Ht∂lt=0⇔Uz(ct,zt)=λtFL(kt,lt)Finally, the FO C with respect to kt+1gives∂L0∂kt+1= βt·∂Ht∂kt+1+ β∂Ht+1∂kt+1¸=0⇔−λt+ β∂Ht+1∂kt+1=0⇔λt= β [1 − δ + FK(kt+1,lt+1)] λt+149George-Marios Angeletos• Combining the above, we getUz(ct,zt)Uc(ct,zt)= FL(kt,lt)andUc(ct,zt)βUc(ct+1,zt+1)=1− δ + FK(kt+1,lt+1).• Both conditions impose equalit y bet ween