The Brock-Mirman Stochastic Growth Model

Transcription

1 c December 3, 208, 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 E β n log +n () n=0 s.. K + = Y (2) Y + = A + K + (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. In his model he capial sock is no useful as a sae variable: Because capial has a 00 percen depreciaion rae, all ha maers o he consumer when choosing how much o consume is how much income hey have now, and no how ha income breaks down ino a par due o K and a par do o A. The firs sep is o rewrie he problem in Bellman equaion form and ake he firs order condiion: V (Y ) = max log + β E V + (Y + ) (4) u ( ) E A+ K+ u (+ ) A+ K + E C + = β E A + K+ }{{} R + + 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 he consumpion funcion = κy, where κ = β. To see his, noe firs ha he proposed consumpion rule implies ha K + = ( κ)y.

2 The firs order condiion says E A +K + K + E Y + ( κ) κ = β. κy K + κy + Y K + Y Y C Y Y ( κ) ( κ) κy κy + 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 + = ( κ)y (5) A 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 2

3 Figure Dynamics of Oupu Wih a Random Walk Shock y 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 ˇy in he prior period. Then he expeced dynamics of oupu would be given by y = ˇy + a (3) E y + = ˇy + a + a (4) E y +2 = ˇy + 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 + = ǎ + 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 =. 3

4 Figure 2 Oupu Dynamics of Oupu Wih A Whie Noise Shock y + y + y + 2 y + +2 Time The consumpion Euler equaion is + = (βr + ) /ρ Bu his is an economy wih no echnological progress, so he seady-sae ineres rae mus ake on he value such ha + / =. Thus we mus have βr = or R = /β. We can furher derive he seady sae level of capial of a nonsochasic version of he model in which a = a from (7): k = log β + a + k (7) ( )k = log β + a (8) ( ) log β k = + a (9) The nonsochasic version of he model is of course no very ineresing, excep as a poin of comparison o he sochasic version of he model. Bu wha could be mean by he seady sae of a sochasic mdoel ha never seles down? We can define a sochasic seady sae for such models in a number of (poenially) differen ways: The locaion (if one exiss) o which he model will converge afer an arbirarily long period in which no shocks occurred a + = a (even if in every period agens expec ha shocks will occur) 4

5 The mean value of some variable in he model (say, K) The value of some sae variable, say Ǩ, such ha E K + = K if K = Ǩ. We consider here he las of hese, which we will show reduces (in his special case) o he same equaion as for he nonsochasic version of he model, K + = K. To see his, rewrie he Euler equaion as: E A + K+ }{{} + R + E A + K+ Y Y + E A + K+ A K A + K+ E K+ A K K+ K+ A K K+ where he expecaions operaor disappears because no variables are sochasic (he A +s in he numeraor and denominaor cancel, and K + is direcly chosen in so is known. For any given A, he seady sae where K + = K = Ǩ is hen where A Ǩ ( ) log β ǩ = + a which is he generalizaion of he nonsochasic soluion derived in (9). The resul ha he nonsochasic and sochasic seady saes are he same is special o he Brock-Mirman model; i is NOT rue of many oher models of growh; i occurs here because he lineariy of he consumpion funcion, among oher special assumpions. Furhermore, he hird of our possible definiions of a seady sae will generally differ a leas a lile bi from eiher of he firs wo. 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),