Econ. 5b Spring 996 C. Sims I. The Single-Agen-Type Model The agen maximizes subjec o and Decenralizing he Growh Model 5/4/96 version E U( C ) C + I f( K, L, A),,.,, (2) K I + δk (3) L (4) () K, all. (5) We are following he usual (in hese noes, no necessarily he world a large) convenions ha C, L, K, and I are choice variables and ha variables daed or earlier are known when variables daed are chosen. Using he mehods we have applied earlier on problems like his, we arrive a firs-order condiions C: U ( C ) λ (6) K: λ βe λ δ + D f K, L, A + + +. (7) To faciliae comparison wih versions of he model we will ake up below, we derive here he linearizaion of his model abou is deerminisic seady sae. We use (3), (4) and (6) o reduce he sysem o one in he wo variables C and K alone. We also use he fac ha, from (7), in seady sae β f + δ. (8) Here we are inroducing he convenion ha unsubscriped variables are evaluaed a heir seadysae values and ha f Df( K, L, A). We will assume hroughou also ha f is linear homogenous in K and L joinly, and ha A eners f as a muliplicaive facor. Since we have discussed linearizaion previously in a more complicaed model, here we simply display he linearizaion:
dc dk β dc C+ K δk fuβ + U da dk ( βδ ) U U dη. (9) + The d operaor denoes deviaion from seady sae. I is no hard o verify ha his sysem has wo real roos whose produc is β. We will be comparing his soluion o oher soluions as we go along. Since in some cases analyic soluion is clumsy, we will compare wo paricular special cases numerically. One of hese is he Cobb-Douglas, % depreciaion, log uiliy case for which we know ha an analyic soluion exiss, wih CK consan. The oher will se.9 (% per period depreciaion), bu reain he Cobb-Douglas, log uiliy specificaion. In boh versions of he model our numerical calculaions will assume.95,.3, i.i.d. A wih mean A. The sable soluion for he sysem has he form dc dk For our wo cases, (case is ), we have. 7526 Case : G. 3 G dc dk + HdA, H. 475. 246 Case 2: G, H. 838 II. Arrow-Debreu Equilibrium. 664. 2724 635.. () (). (2) Now we posulae he exisence of a represenaive firm as well as a represenaive consumer. The decisions of he firm and consumer are coordinaed by marke prices, which boh ypes of agen ake as unaffeced by heir decisions. The prices are quoed a ime, and are coningen boh on he dae and on he sae of he world. We hink of as a poin in he se and as deermined by an infinie sequence of random variables X. Informaion available a is X, which can be hough of as deermining a subse of consising of X given by ω s s sequences wih he same firs + elemens. The consumer s objecive funcion is sill (), bu her budge consrain is now P (, ω) C( ω) W( ω) L( ω) y( ω), (3) ω Ω where P is he goods price, W is he wage in goods unis, and y is he profi disribuion by he represenaive firm, which is owned by he consumer. To jusify he use of he summaion sign in (3), we have o hink of he number of disinc X sequences as finie, or a leas counable, which implies ha in finie ime uncerainy vanishes. Tha is, here is some T such ha for >T, here is only one ha maches ω. To eliminae his unappealing implicaion we would have 2
o allow for an uncounably infinie, rea P (,) as a funcion over i, and replace he sum over by inegrals. This would add mahemaical complicaions, however, so we sick wih he assumpion of counable for now. The firm is insruced by is owners o maximize he value of is profi disribuions, i.e. ω The firm s consrains, indexed by and ω are P (; ω) y ( ω ). (4) y( ω) f( K ( ω ), L( ω), A( ω)) K( ω) + δk ( ω ) W( ω) L( ω). (5) We suppose ha here is a probabiliy funcion defined on, so ha () can be rewrien as ω Ω πω ( ) βuc( ω). (6) To make he firs-order condiions emerge in a simple form, we inroduce he noaion P (; ω ) ω ω p (; ω ) β. (7) πω ( ) To jusify (7), we have o assume ha π ( ω ), is denominaor, never vanishes when he numeraor is nonzero. This is he only resricion we need on. I does no have o be he rue probabiliy measure, and i is no necessary ha everyone agree ha i expresses heir beliefs. The equilibrium can be described in erms of he P s wihou inroducing s a all. The marke-clearing condiion in labor is already imposed implicily hrough he use of he same symbol L in boh he firm and consumer problems. For goods, he marke clearing condiion is ha goods used for consumpion and invesmen a mus mach goods produced a. Tha is, C( ω) + I( ω) f K, L, A This is no perceived as a consrain by he ypical consumer or firm. Each consumer sees her budge consrain (3) as allowing income a any dae o be convered ino consumpion a any dae a a rae of radeoff deermined by he P(;) values. Bu he prices mus adjus so ha in equilibrium consumers choose o consume an amoun consisen wih he amouns of oupu and invesmen chosen by producers. Firs-order condiions for he consumer are. (8) C : βπω ( ) U λβπω ( ) p(; ω ), U λp(; ω ) (9) Firs-order condiions for he firm are y : p (; ω ) µ ( ω) (2) L : (2) µ ( ω ) D f W L 3
K : + + K + ω ω πω ( ) µ ( ; ω) πω ( ) µ ( + ; ω ) δ+ D f + πω ( ) E µ ( + ) δ+ D f K +. (22) In deriving (2)-(22) we are using he convenion ha in he Lagrangian he muliplier on he,ω consrain of he firm is βπω ( ) µ ( ; ω). Also, we are using he fac ha π( ω+) πω is he condiional probabiliy of ω + given ha ω has occurred. The FOC s (6) and (7) from he single-agen problem can be solved o eliminae, and he resuling equaion can also be derived from (9), (2), and (22). Thus an Arrow-Debreu equilibrium saisfies he consrains and FOC s of he single-agen problem. III. Auonomous Firms We now consider he opposie exreme case, in which he represenaive firm ges no guidance from asse prices in making invesmen decisions. We posulae ha he firm s objecive funcion is increasing in is profi disribuions, bu ha i has is own uiliy funcion for hose disribuions ha does no have any necessary link o he represenaive consumer s uiliy funcion. Tha is, we posulae ha firms maximize E θφ( y ), (23) wih φ >, φ. The firm s discoun facor is allowed o differ from ha of consumers. A version of his assumpion ha appears regularly in he applied lieraure on invesmen is ha (23) holds wih θ β and he ideniy funcion, so ha firms maximize expeced curren and fuure profis discouned a he fixed rae. The firm s consrains are y f( K, L, A) K + δk WL K, all. (24) The consumer sill maximizes he objecive funcion (), bu now wih he consrains C WL + y, all. (25) L Noe ha he consumer here has no ineremporal decision o make a all. Because we have assumed leisure has no uiliy, L will always be a is upper bound of. Individuals ake wages W and profi disribuions y as beyond heir conrol, so (25) deermines C wihou any reference o he consumer s objecive funcion. Obviously here he only ineresing economic decision in he economy, he choice of how much o inves and how much o consume each period, is being made by he firm, using an objecive funcion ha does no mach ha of he people in he economy. We should no expec he resuling equilibrium o be close o opimal in general. I is ineresing o ask, hough, wheher wih he firm s discoun facor maching he individual s discoun facor, here migh 4
be s for which he soluion o his problem would mach or come close o he social opimum. The firm s FOC s are y: φ ( y ) µ (26) L: W DLf( K, L, A) (27) K: µ θ E µ D f( K, L, A ) + δ (28) + K + + where µ is he Lagrange muliplier on he consrain. Noe ha in he deerminisic seady sae we will have θ DK f ( K,, ) + δ. (29) By comparing (29) o (7), we see ha he values of K, and herefore C, mach hose of he single-agen model if θ β, bu no oherwise. When, as in our case, δ, and in addiion is logarihmic and θ β, his auonomousfirm model can be shown o have an exac soluion in which y remains proporional o K, and his gives he same C ime pahs as he complee markes soluion. However ouside his special case, as in our case 2, he model does no have an analyic soluion. To sudy is behavior we linearize. Here we reduce he model o a wo-variable sysem in y and K, using (27) o eliminae W, arriving a dy dk θ + Kf y+ K δk A dy + da f θ φ φ dk ( θδ ) φ φ A dη. (3) + To give his sysem he greaes possible chance of maching he complee markes soluion, we will examine i numerically for he case where is he log funcion and, wih he res of he parameers se as in cases and 2 of secion I. Noe ha wih he maching discoun raes, he seady saes of his model do mach hose of he complee markes model. This leads o G and H marices as follows: 58 Case : G. 88 H 3., 664. (3)... 473 Case 2: G, H. 8985. 2. (32). 387 Because his sysem is in erms of y and K insead of C and K, he marices displayed here would no mach hose of secion I even if C and K followed he same pahs. However, since he second row boh here and in secion I is a difference equaion in K alone, we can see immediaely wheher he soluions mach, for eiher case. If boh he second row of G and he lower elemen of H mach, hen he soluions are firs-order equivalen, as he K pahs will be and he social resource consrain deermines he C pah from he K pah. We can see ha for case he auonomous firm soluion does mach he complee markes soluion o firs order, as we would hope given ha we know his is rue of he exac soluion. However in case 2 he soluions do no 5
mach. The coefficien on lagged K is slighly larger in he auonomous-firm model, bu more imporanly he responsiveness of K o shocks in A is several imes bigger in he compleemarkes model. The firm ends o smooh K s ime pah more han is opimal, leing shocks have larger curren impac on C. IV. Incomplee Asse Markes As you should know from your micro heory course, he real allocaions in an Arrow-Debreu equilibrium can generally be duplicaed wihou a complee marke for claims arbirarily far in he fuure, if insead here is a one-period-ahead complee coningen claims marke a each dae. In a model like he sochasic growh model, where he disurbance A is generally hough of as coninuously disribued, his would require an uncounable infiniy of asses o be raded a each dae. Since in realiy a finie number of asses are raded, i is ineresing o explore how a compeiive equilibrium wih a few asses compares o one where here are complee markes. We consider he special case where here is a single raded asse, denominaed in shares, wih he reurn a per share purchased a ime - denoed z. We do no ye ake a posiion on wha he sochasic process of z will be, hough we will assume ha i is regarded by agens as unaffeced by individual agens acions. We denoe by Q he price of he asse a ime. Because consumers can no longer insruc firms o maximize he value of he sream of profis (because here are no quoed prices o use o value his uncerain sream), here mus be some oher objecive funcion given he firm. The consumers objecive funcion is sill (), bu her budge consrain becomes C + QS WL + Q + z S + y. (33) We need also o impose a consrain, which we hope does no bind in equilibrium, ha indefinie borrowing is no possible, e.g. ha. for some consans B> and ν (, β ) QS Bν (34) The firm sill has he objecive funcion (23), bu now wih he consrain y QS f( K, L, A ) K + K WL Q + z S δ. (35) Noe ha we use he same symbol S for boh he purchases of he securiy by he consumers and he sales of he securiy by he firm, implicily imposing marke clearing wih zero ne supply of he securiy. The firm also mus have a limi on is borrowing, which becomes (since S is securiies issued by he firm) Now he consumer s FOC s are (6) and QS Bν. (36) Tha is, wih a densiy funcion on he real line, like he log-normal or exponenial for example. 6
λ Q βe λ Q + z + + +. (37) The firm s FOC s are y : φ ( y ) µ (38) L : µ W D L f K : µ θ E µ δ + (39) + D f + (4) + + +. (4) K S : µ Q θ E µ Q + z This is more promising han he auonomous firm model. I is sill rue ha (4) uses insead of as he sochasic discoun facor, bu now we have (4) and (37), which seem o require a leas some similariy in behavior beween he discoun facors θµ + µ and βλ + λ. From (6) we know ha in seady sae is consan. Then from (37) we conclude ha Q is also consan, a he value z ( β ). Then (4) implies µ + µ β θ. (42) Using his in (4) les us conclude ha in his model we have he same equaion (7) deermining seady sae capial sock as in he single agen model, even when β θ. Suppose we linearize he sysem abou seady sae. We now have o keep rack of more variables, because y, C, K, Q and S all inerac. We also sick o he case β θ o avoid having o deal wih rends in y and S in seady sae. We wrie he sysem as Γdx Γdx + Ψdε + Πη, (43) where he vecor are endogenous predicion errors and he vecor is exogenous disurbances. We define x y Q S K C A, ε (44) z and arrange he equaions in he order (35), (35)+(33) (he social resource consrain), (37), (4), and (4). The resul is Γ φ Q ( Q+ ) β + Kf β β Qγ U, Γ γ UQ, (45) γ φ γ φ f β γ Q β γ Q φ 7
where γ φ φ φ and γ U U U. We also have Ψ S β β αf f Π,. (46) I βf 5 3 To keep our numerical soluions comparable, we will ake he seady sae o have S. The soluions for his model in our wo cases are Case G, H.3.7526 -.58 33.532 Case 2 G -.32, H.838.246 -.473 3.86. 88 8. 64. 664. 475. 2 3839.. 359 635.. 2724 Comparing (47) wih () and (48) wih (2), we see ha he las wo rows of he larger sysem mach he smaller sysem exacly. This sysem, like ha wih auonomous firms, exacly reproduces he complee-markes equilibrium in case. Boh y and C in ha case move exacly in proporion o K, so ha φ and U move exacly in proporion. This is signaled in he linearized soluion by he fac ha he hird elemen of H, he effec of a disurbance in A on S, is zero. Tha is, he linearizaion implies ha if we sar wih S, random shocks do no generae non-zero S. Wihou his condiion, y and C could no remain proporional, as he hird row of G implies ha S, once perurbed away from zero, will end o drif. (The uni coefficien on lagged S implies ha S has no endency o reurn o seady sae.) The firs row of G implies ha y will end o follow (wih opposie sign) any drif in S. In case 2, we see from he hird elemen of H ha random disurbances do affec S, so for his case S and y will drif away from heir seady sae values and y and C do no remain sricly proporionae. I is herefore no possible in case 2 ha he random discoun facor ha firms use o evaluae invesmens, βφ + φ βy y+, is exacly he same as wha consumers would use, i.e. βu + U βc C+. There will be differences in he ime pahs of K and C beween he complee and incomplee markes economies. The mach beween he linearized soluions implies, hough, ha he differences will be small as a proporion of variaion in he economy when he sochasic disurbances o he economy are small. (47) (48) 8