c November 20, 207, Chrisopher D. Carroll BrockMirman The Brock-Mirman Sochasic Growh Model Brock and Mirman (972) provided he firs opimizing growh model wih unpredicable (sochasic) shocks. The social planner s goal is o solve he problem: max β n log C +n () n=0 s.. Y = A K (2) K + = Y C (3) where A is he level of produciviy in period, which is now allowed o be sochasic (alernaive assumpions abou he naure of produciviy shocks are explored below). Noe he key assumpion ha he depreciaion rae on capial is 00 percen. The firs sep is o rewrie he problem in Bellman equaion form and ake he firs order condiion: V (K ) = max C log C + βv + (A K C ) (4) u (C ) E A+ K+ u (C + ) ] A+ K ] + E C C + = β E A + K+ }{{} R + C C + where our definiion of R + helps clarify he relaionship of his equaion o he usual consumpion Euler equaion (and you should hink abou why his is he righ definiion of he ineres facor in his model). Now we show ha his FOC is saisfied by a rule ha says C = γy, where γ = β. To see his, noe firs ha he proposed consumpion rule implies ha K + = ( γ)y. The firs order condiion says E A ] +K+ γy K + E Y + K + γy γy + γy + ]
( γ) = β γ = β. Y ] K + Y ] Y C ] Y Y ( γ) ] ( γ) An imporan way of judging a macroeconomic model and deciding wheher i makes sense is o examine he model s implicaions for he dynamics of aggregae variables. Defining lower case variables as he log of he corresponding upper case variable, his model says ha he dynamics of he capial sock are given by K + = ( γ)y (5) = βa K (6) k + = log β + a + k (7) which ells us ha he dynamics of he (log) capial sock have wo componens: One componen (a ) mirrors whaever happens o he aggregae producion echnology; he oher is serially correlaed wih coefficien equal o capial s share in oupu. Similarly, since log oupu is simply y = a + k, he dynamics of oupu can be obained from y + = a + + k + (8) = (log K + ) + a + (9) = (log βy ) + a + (0) = (y + log β) + a + () so he dynamics of aggregae oupu, like aggregae capial, reflec a componen ha mirrors a and a serially correlaed componen wih serial correlaion coefficien. The simples assumpion o make abou he level of echnology is ha is log follows a random walk: a + = a + ɛ +. (2) Under his assumpion, consider he dynamic effecs on he level of oupu from a uni posiive shock o he log of echnology in period (ha is, ɛ + = where ɛ s = 0 s + ). Suppose ha he economy had been a is original seady-sae level of oupu ȳ in he prior period. Then he expeced dynamics of oupu would 2
Figure Dynamics of Oupu Wih a Random Walk Shock y ++ 2 + be given by y = ȳ + a (3) E y + ] = ȳ + a + a (4) E y +2 ] = ȳ + a + a + 2 a (5) and so on, as depiced in figure. Also ineresing is he case where he level of echnology follows a whie noise process, a + = ā + ɛ +. (6) The dynamics of income in his case are depiced in figure 2. The key poin of his analysis, again, is ha he dynamics of he model are governed by wo componens: The dynamics of he echnology shock, and he assumpion abou he saving/accumulaion process. For furher analysis, consider a nonsochasic version of his model, wih A =. The consumpion Euler equaion is C + C = (βr + ) /ρ Bu his is an economy wih no echnological progress, so he seady-sae ineres rae mus ake on he value such ha C + /C =. Thus we mus have βr = or R = /β. 3
Figure 2 Oupu Dynamics of Oupu Wih A Whie Noise Shock y + y + y + 2 y + +2 Time In he usual model he ne ineres rae r is equal o he marginal produc of capial minus depreciaion, r = F (K) δ, so he gross ineres rae is R = + F (K) δ. Bu in his case we have δ = so R = F (K). The uncondiional expecaion of he ineres rae, E(logF (K)]), is given by E(logF (K)]) = ( ) logβ] + E(logF (K)]) + Eɛ] E(logF (K)])( ) = ( ) logβ] (7) E(logF (K)]) = log/β] (8) E(logF (K)β]) = 0 (9) Previously we derived he proposiion ha Rβ = (20) F (K)β = (2) logf (K)β] = 0 (22) bu since his is a model in which R = F (K), his is effecively idenical o he seady-sae in he nonsochasic version of he model. Thus, in his special case, he modified golden rule ha applies in expecaion is idenical o he one ha characerizes he perfec foresigh model. The only difference ha moving o he sochasic version of he model makes is o add an expecaions operaor E o he LHS of he nonsochasic model s equaion. 4
References (click o download.bib file) Brock, William, and Leonard Mirman (972): Opimal Economic Growh and Uncerainy: The Discouned Case, Journal of Economic Theory, 4(3), 479 53. 5