Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen. is assumed o be 0. The rm maximizes he presen value of pro s = Z 1 =0 e r [Z (K )k I C(I )]d Where K is he aggregae capial sock, k is he rm capial sock, (K ) are he pro s per uni of capial ( 0 (K ) > 0; 00 (K ) < 0); C(I) are he adjusmen coss ( we assume C(0) = 0; C 0 (0) = 0; C 00 > 0); and I is invesmen. Capial accumulaes according o : k = I : This is a parial equilibrium exercise, we ake r as given. 1. Inerpre he assumpions abou C(): Discuss where (K ) is coming from. (K ) is he pro per uni of capial once he res of facors are chosen opimally. The assumpion ha hey are proporional o k is appropriae if goods markes are compeiive, he producion funcion has CRS and he oher facors supply are perfecly elasic. Think in a Cobb Douglas producion funcion wih capial and labor. Afer solving for he demand for labor and replacing i in oal pro we ge exacly ha. I am going o assume ha here is a mass of one of projecs/invesors and ha each projec/invesor is in niesimal. Then, @K @k = 0. This makes he analysis simpler, saves some derivaives, and produces similar resuls. The cos funcion is a convex cos of adjusmen. I is cosly o inves, and more he bigger he adjusmen. 2. Wrie he curren value Hamilonian for he rm, where I is he conrol variable and k is he sae variable. Call he muliplier 1
(cosae variable) q; which is he shadow value of he sae variable (he marginal q): Derive he rs order condiion for he conrol variable. Derive he condiion ha says ha he derivaive of he Hamilonian wih respec o he sae variable equals he discoun rae imes he cosae variable minus he derivaive of he cosae wih respec o ime. Finally, de ne he ransversaliy condiion for he problem. The Hamilonian is The F OC wih respec o I is H(k ; I ) = Z (K )k I C(I ) + q I : and he derivaive of he cosae is 1 + C 0 (I ) = q ; Z (K ) = rq q : Finally he ransversaliy condiion is lim e r q k = 0:!1 Some discussion on hese condiions: You can se up a problem like his using wo di eren approaches. You should ge he same resul. Presen value Hamilonian In his case he Hamilonian is H P (k ; I ) = Z (K )k I C(I ) Then, he opimizing funcion is: Z 1 Max e r [Z (K )k I C(I )] + eq I d =0 In his case FOCs are: Curren value Hamilonian In his case he Hamilonian is e r (1 + C 0 (I)) = eq; e r Z (K ) = eq : H C (k ; I ) = Z (K )k I C(I ) + q I 2
Then, he opimizing funcion is: Z 1 Max = Max e r [Z (K )k I C(I ) + q I ] d =0 You have he wo FOCs above. Now le me discuss a lile bi he derivaion of he second: Wha we really wan o have is: @ @k = de ne = e r q hen = re r q + e r q So @ @k = e r Z (K ) = re r q e r q Or (K ) = rq q The key poin is ha he FOCs are exacly he same in boh ways of seing up his problem. Why we wan he second second formulaion o ge a measure of q? To answer his quesion you have o hink wha q is? If you know q, you should be able o compare he curren value (i.e. a ime ) of invesing one dolar wih respec o he is curren cos (obviously, one dollar if here are no invesmen subsidies). Noice ha only in he second case you ge ha he shadow price is direcly comparable o is cos. Moreover, he second rs order condiion in his case has a very nice inuiive inerpreaion: Z (K ) {z } Marginal Bene Equivalenly, Z (K ) + q {z } {z} Marginal Bene Capial "Gains" = rq {z} Capial Cos = rq {z} Capial Cos q {z} Capial "Gains" Finally, a couple of words on he ransversaliy condiion. Remeber ha he ransversaliy condiion always has o do wih a complemenary-slackness condiion. You wan: lim!1 eq k = 0: lim e r q k = 0:!1 Therefore, his implies ha eiher k has o go o 0 or k should have a value of 0 as! 1: 3
3. Using your second equaion, solve for he di erenial equaion in q: This will give you he expression for he marginal q: You will have o make use of he ransversaliy condiion o cancel some of he erms. Inerpre his expression. Muliply boh sides of he second equaion by e r o ge e r (K ) = e r (rq q ) = (e r q ) Inegrae beween and T o ge e r q + B Z T T = e r Z (K )d + C, Z e r T T q = e r Z (K )d + D q = Z T e r( ) Z (K )d + e r(t ) q T + D where B D and C are consan erms. Using his condiion a ime T; we can conclude D = 0: q = Z T Taking he limi of his expression e r( ) Z (K )d + e r(t ) q T : q = Z 1 e r Z (K )d; where one can proof ha he las erm disappears because of he ransversaliy condiion. And his is wha we saw in class. The shadow value of an exra uni of capial is he NPV of all he fuure marginal producs (in ha case he rae of pro per uni of capial). 4. Going back o 2, he rs 2 equaions de ne a sysem of di erenial equaions. Draw he phase diagram for i in K q space. Is here an sable pah? Is here a long run equilibrium for K and q? Inerpre. When is invesmen posiive? Le s look rs a he equaion 1 + C 0 (I ) = q, I = f(q ); f 0 > 0; f(1) = 0 Using ha k : = I ; and rms are all exacly idenical (and we have a mass : of rms of size 1), K = I which implies : K = f(q ); 4
so when q = 1 capial is consan, and when q > 1 (<1) capial increases (decreases). Looking a he second equaion, rq Z (K ) = q ; q is consan when rq = Z (K ) which de nes a downward sloping se of poins (locus) given he assumpion abou (): Fora given q, if we ake a level of capial higher (lower) han ha makes q consan we have ha is smaller (bigger) and hus q > 0 (<0). We can summarize all his informaion in a graph. q dk/d=0 Saddle pah dq/d=0 Capial here is he sock variable while marginal q is he jump variable. For a given value of capial sock q jumps o he saddle pah and converges o he seady sae (you have o use he ransversaliy condiion o rule ou oher pahs). Where K is he aggregae capial sock, k is he rm capial sock, Z (K ) are he pro s per uni of capial ( 0 (K ) > 0; 00 (K ) < 0); C(I) are he adjusmen coss ( we assume C(0) = 0; C 0 (0) = 0 C 00 > 0); and I is invesmen. Capial accumulaes according o : k = I : This is a parial equilibrium exercise, we ake r as given. 1. (K ) is he pro per uni of capial once he res of facors are chosen opimally. The assumpion ha hey are proporional o k is appropriae k 5
if goods markes are compeiive, he producion funcion has CRS and he oher facors supply are perfecly elasic. Think in a C-D producion funcion wih capial and labor. Afer solving for he demand for labor and replacing i in oal pro we ge exacly ha. The cos funcion is a convex cos of adjusmen. I is cosly o inves, and more he bigger he adjusmen. 2. The Hamilonian is The F OC wih respec o I is H(k ; I ) = (K )k I C(I ) + q I : and he derivaive of he cosae is 1 + C 0 (I ) = q ; (K ) = rq Finally he ransversaliy condiion is q : lim e r q k = 0:!1 3. Muliply boh sides of he second equaion by e r o ge e r (K ) = e r (rq q ) = (e r q ) Inegrae beween and T o ge e r q + B T = q = Z T e r (K )d + C, Z e r T T q = e r (K )d + D Z T e r( ) (K )d + e r(t ) q T + D where B D and C are consan erms. Using his condiion a ime T; we can conclude D = 0: q = Z T Taking he limi of his expression e r( ) (K )d + e r(t ) q T : q = Z 1 e r (K )d; where one can proof ha he las erm disappears because of he ransversaliy condiion. And his is wha we saw in class. The shadow value of an exra uni of capial is he NPV of all he fuure marginal producs (in ha case he rae of pro per uni of capial). 6
4. Le s look rs a he equaion 1 + C 0 (I ) = q, I = f(q ); f 0 > 0; f(1) = 0 Using ha k : = I ; and rms are all exacly idenical, implies : K = f(q ); : K = I which so when q = 1 capial is consan, and when q > 1 (<1) capial increases (decreases). Looking a he second equaion, rq Z (K ) = q ; q is consan when rq = (K ) which de nes a downward sloping se of poins (locus) given he assumpion abou (): Fora given q, if we ake a level of capial higher (lower) han he one ha makes q consan we have ha is smaller (bigger) and hus q > 0 (<0). We can summarize all his informaion in a graph. Capial here is he sock variable while marginal q is he jump variable. For a given value of capial sock q jumps o he saddle pah and converges o he seady sae (you have o use he ransversaliy condiion o rule ou oher pahs). 5. Explain how each of he following will a ec he long run equilibrium and discuss he dynamics: (a) A permanen upward shif on : Pro s per uni of capial increase for ever and hus for a given capial sock q increases. Today capial can change so q jumps o he new saddle pah. As q > 1 capial is accumulaed. As capial accumulaes q decreases because is decreasing in K: This process coninues unil we converge o he new 7
seady sae wih a higher capial sock. dk/d=0 Saddle pah dq/d=0 k (b) A emporary upward shif on : If increases from o T (and i is known) q can jump o he new saddle pah, oherwise k and q would explode when a T he sysem urns o be he same han a he beginning. For he same reason i can jump o a higher level han ha of he new saddle pah. I will jump o a level of q below, bu i will jump o he only one such ha a ime T he rajecory his he old saddle pah. Wha does all ha mean? As here is an increase in pro abiliy, q increases and he capial sock rises. As capial increases, q decreases. When he rajecory crosses he K : = 0 pah capial sars decreasing again. Why? As T is close, rms are abou o wan less capial sock. As adjusmen is cosly, hey sar doing 8
i prior o ime T: A he end here is no chance in he capial sock. q dk/d=0 Saddle pah dq/d=0 k (c) A permanen upward shif on in he fuure: The reasoning follows ha of par a. Bu now q can jump o he new saddle pah because i sill doesn exis. I has o jump o a lower level, he exac level such ha a ime T (when he change happens) he rajecory his he new saddle pah. The reasoning for ha is he following. Pro s are higher, bu hey will be in he fuure and hus discouned. So q jumps by less. As we ge closer o T he higher pro s ge closer and hus q keeps increasing a he same ime han capial is accumulaed. This keeps going unil he rajecory his he new saddle pah. From 9
hen, i follows he same ha in poin a. q dk/d=0 Saddle pah dq/d=0 k (d) A emporary upward shif on in he fuure: You can easily see ha q jumps even less han in b, q and K increase for a while and when he change happens, we have he same behavior han in b from hen on. (e) The governmen axes reurns from owning rms a rae : Pro s are now (1 )(K) and his changes he q locus. Two hings happen, a a given capial q is now lower in he locus. The slope is reduced oo. So oday wha happens? Because of he ax, exising capial is less valuable which makes q o jump down o he saddle pah. This leads o disinvesmen. As capial reduces, q increases, and his process keeps going unil he new seady sae is reached, 10
wih a lower capial sock. q dk/d=0 Saddle pah dq/d=0 k (f) In he fuure, migh increase wih a probabiliy of.5: I will jus give he inuiion in his par. In he fuure we know he saddle pah migh be higher. There can be an expeced capial gain or loss a he ime he uncerainy is realized. This means ha q has o be midway (as probabiliy is.5) beween he poins hey would be in each of he saddle pahs a he ime he uncerainy is realized (he 2 seady saes). 2 Marginal and Average Q 1. See Olivier s noes for his par. 2. CRS are used when replacing he wage. If here are no CRS, he ideniy is no longer rue. Same happens if markes are no compeiive and/or facors are no paid heir marginal produc. Noe ha non-compeiive markes will a ec he maximizaion problem of he rm so none of our expressions have o be rue anymore. If rms has inangible capial or oumoded capial, ha won be re eced on he marginal q bu i will appear in he average q as a par of he value marke. Finally, if he marke doesn have a correc valuaion of he rm i is clear ha he 11
NPV of marginal producs of capial doesn have o be he same han he average q: 3. For he CRS hink abou exercise 1. There we have DRS in adjusmens coss, an increase of he same percenage in capial is more expensive he higher he capial level. This implies ha pro s, as capial is accumulaed, rise less han proporionally which implies ha marginal q is smaller han he average. For he marke srucure, assume rms face a downward slopping demand. In ha case doubling capial will less han double pro s and again marginal q is smaller han he average. If rms have oumoded capial, ha will be re eced on he marke value, bu no on he marginal condiions. In ha case marginal q is greaer han he average. Finally, assume here is a sock marke bubble. Tha would imply ha marginal q is smaller han he average. 4. Tha is di cul o say wihou an explici money behind. In he case of he sock marke bubble you can hink ha he bubble appears during and expansion and burs in he recession. In ha case he di erence beween boh q is procyclical. You can come up wih a reasonable sory for any of he examples, bu again i is di cul o say anyhing wihou an explici model. 12