Lecture Notes 5: Investment

Transcription

1 Lecure Noes 5: Invesmen Zhiwei Xu Invesmen decisions made by rms are one of he mos imporan behaviors in he economy. As he invesmen deermines how he capials accumulae along he ime, i plays a cenral role in economic growh. Meanwhile, he invesmen is highly volaile (is volailiy is hree imes larger han oupu) over he business cycle, i poenially provides an imporan propagaion mechanism o amplify he shor-run ucuaions. In his noe, we will use sylized dynamic models o invesigae rms opimal invesmen decisions. The Value of he Firm In he marke economy, households are poenial owners of rms. They purchase rms share from he marke. The value of he rm is he presen value of cash ows i generaes. To see his, we sar wih households opimal porfolio choice problem. Suppose ha here are wo asses households can purchase: one is he deposi paid by ineres rae r ; he oher is he sock share wih price V. The household chooses consumpion C, saving S + ; amoun of sock share a + o maximize life-cycle uiliy. The opimizaion problem is max fc ;S + ;a + g u (C ) () =0 subjec o C + S + + V a + = (V + D ) a + W + ( + r ) S ; (2) where D is he dividend disribued by rms, W is he wage income. Le be he Lagrangian muliplier for he budge consrain (2). FOCs w.r. fc ; S + ; a + g give u 0 (C ) = ; (3) = + ( + r + ) ; (4) V = + (V + + D + ) : (5) Equaion (5) is indeed he asse pricing formula. Ierae i forwardly, we can express he sock price as he presen value of dividends V = j= j +j D +j + lim T! T T V T ; (6)

2 2 where j +j = j u0 (C + ) is he discoun facor. u 0 (C ) lim T! T T V T = 0: Finally, we have The ransversaliy condiion ensures ha V = j= j +j D +j : (7) Noe ha V is he value of share afer he dividend paymen. De ne he value of share before he dividend paymen as V ; we have V = D + V = j=0 j +j D +j : (8) Las equaion is equivalen o V = + V+ : (9) Since here is no uncerainy, (4) and (5) imply ha he rae of reurn on sock and deposi in he equilibrium mus be he same, i.e., + r + = V + + D + V : (0) In he uncerainy case, as he deposi pays risk-free rae and he sock reurn is in general risky, he rae of reurn on he sock mus be larger han he risk-free rae r + : The di erence of wo raes is he risk premium. have Consider a special case where households are risk-neural, ha is, u 0 (c ) is consan. We hen where r = V = j= : The value of share before he dividend paymen is V = V + D = ( + r) j D +j; () j=0 ( + r) j D +j: (2) 2 Firm s Problem There is a represenaive rm ha combines labor L and capial K o produce consumpion goods according o Cobb-Douglas echnology Y = A K L : Di eren from previous lecures, we assume ha he rm accumulaes capial by iself insead of rening capial from he capial

3 3 marke. As a resul, he rm needs o make invesmen decision I. The rm aims o maximize he value of share before he dividend paymen. The opimizaion problem is de ned as V 0 (K 0 ) = subjec o he ow of funds consrain: and capial accumulaion process: max fi ;K + ;L g =0 0 D ; (3) D = A K L W L I [ + (I =K )] ; (4) K + = ( ) K + I ; (5) where (I ) describes he invesmen adjusmen cos which is convex funcion wih properies 0 (:) 0; and 00 (:) 0: In he rm s problem (3), he labor decision is saic. The rm chooses opimal labor L o maximize he capial reurn A K L W L ; he FOC is W = ( ) A K L : (6) The labor demand (6) implies ha capial reurn is linear in K ; i.e., A K L W L = R K ; (7) where R A W = Y K is he rae of reurn on capial (or marginal produc of capial). as Le q be he Lagrangian muliplier for (5). The rm s opimizaion problem can be rewrien V 0 (K 0 ) = max fi ;K + g =0 The FOCs w.r..fi ; K + g are given by q = + " q = + 0 ( I K R + + ( ) q + + R K I [ + (I =K )] +q [( ) K + I K + ] ) : (8) + 0 I I ; (9) K K I+ K + 2 # 0 I+ : (20) Remember ha V = D + V ; V is he value of he rm afer he dividend paymen, which is given by K + V = + V+ (K + ) : (2) I can be shown ha V is linear in K +. More speci cally, we have V = q K + : (22)

4 4 To see his, we use guess-and-verify sraegy: We guess V akes he form of (22). To verify i, we muliply boh sides of (20) by K + and have " q K + = + = + = + " 8 R + K + + q + ( ) K + + R + K + + q + K +2 q + I + + I+ K + I+ 2 # 0 I+ K + K + >< I+ >= R + K + I q + K +2 K + >: {z } >; D + K + 2 # 0 I+ K + 9 K + = + (D + + q + K +2 ) : (23) The hird line is obained by replacing he q + before he I + by (9). Las equaion implies ha q K + = j= +j D +j = V : (24) In he lieraure, q is so-called marginal Q. The opimal condiion (9) indicaes ha he marginal value of one uni exra invesmen should be equal o is marginal cos. More imporanly, (9) implies ha he marginal Q is a su cien saisics for he invesmen decision, because i fully deermines he invesmen rae I K : Furhermore, (20) implies ha he marginal Q (q ) conains su cien informaion abou he discouned value of fuure rae of reurn on capial. Since in our model here is only one good, he price of invesmen goods (or he replacemen cos of capial) is jus. Las equaion implies ha he average Q is he raio beween marke value of one uni insalled capial (V =K + ) and he replacemen cos of capial (which is ). Noe ha he marginal Q equals he average Q only under he assumpion ha pro funcion is linear in K ; oherwise (e.g., he rm has some monopoly power, or decreasing reurn-o-scale echnology), he equaion (22) may no hold anymore. If here is no adjusmen cos, or (:) = 0; hen (9) implies ha q = ; and he rm s problem degeneraes o he sandard neoclassical growh model (equivalen o a social planner s problem wihou any fricions). 2. Q-Theory: A Concree Example To beer undersand he Q heory, we now ake a concree example. To discuss he dynamics analyically, we look a he coninuous-ime version. For simpliciy, we assume he household is

5 5 risk neural, implying ha he discoun facor 0 = e r ; where r = : We assume ha he oal labor is xed, i.e., L = ; and he echnology A = A: The rm s opimizaion problem (3) is now given by Z V 0 (K 0 ) = max fi ;K g e r n R K I [ + (I =K )] + q I K _K o d: (25) where R = Y =K = AK, and he rm akes R as given. In addiion, we specify (x) = bx: The FOC w.r.. I gives us i = q ; (26) 2b where i is he invesmen rae I =K : Obviously, he invesmen rae is posiive if and only if q > : When he marginal value of insalled capial q increases, he rm is more willing o make invesmen. In fac, q > is quie inuiive o undersand. Due o he invesmen adjusmen cos, one uni of insalled capial is no only used in producion, bu also reduces he invesmen adjusmen cos in he fuure, see Euler equaion (20). The FOC regarding he capial K is given by Finally, we have capial accumulaion process _q = q (r + ) AK + bi 2 : (27) _K = (i ) K : (28) Equaions (26) o (28) de ne he full dynamic sysem. Replace i by (26), we can simplify he sysem o _q = q (r + ) The seady sae fq ; K g are given by " AK q _K = 2b # q 2 + b ; (29) 2b K : (30) q = + 2b; (3) K = A : r + + b (2r + ) (32) Remember ha he rs order condiion in coninuous ime model is given by f x() = df _x() : In his model, he d FOC for K is given by 2 # e "R r I I + q = d e r q : K K d

6 6 The capial sock is increasing in he produciviy A; and decreasing in he adjusmen cos parameer b and depreciaion rae : The marginal Q depends on he adjusmen cos parameer b and he depreciaion rae : Moreover, (29) implies The erm A (K ) A (K ) = q (r + ) b 2 = r + + b (2r + ) > r + : (33) is he marginal produc of capial (MPK). If here is no adjusmen cos (b = 0), we have q = ; and MP K = r + which is he rs-bes case. The invesmen adjusmen cos deviaes he economy from he rs bes because q > and MP K > r +: The wedge beween MP K and r + is which measures he exen of disorion or fricion. A (K ) (r + ) = b (2r + ) > 0; (34) Linearizing he sysem (29) and (30) around he seady sae fq ; K g gives us The locus for _q = 0 is given by _q = r (q q ) ( ) A (K ) 2 (K K ) ; (35) _K = K 2b (q q ) : (36) q q = ( ) A (K ) 2 (K K ) : (37) r Any (q; K) pair ha above he curve implies _q > 0; and hose pairs below he curve implies _q < 0: Similarly, he locus for _ K = 0 is given by q = q : (38) Thus any (q; K) above he horizonal line q = q implies _ K > 0; and hose pairs below he line

7 7 q = q implies _ K < 0: Figure below draws he phase diagram. Figure. Phase Diagram for he Q-heory Model Dynamic E ecs of Technological Process Suppose he echnology has a permanen increase from A o A new : Equaions (3) and (32) show ha he seady-sae q does no change, hus he locus _q = 0 is he same as before. The seady-sae capial will increase o K new : As A new (K new ) K new = A new r + + b (2r + ) : (39) 2 is decreasing in A new ; he curve (37) is less seeper han he original one. The following gure illusraes how he economy ransis from he old seady sae o he new seady sae. From he gure, i can be seen ha he improvemen in he produciviy will cause he rm o increase he invesmen, and evenually he capial sock increases. Paricularly, he invesmen and he marginal Q will jump o a high level (Poin C) and hen decreases gradually o he seady sae (he new seady sae is he same as Poin A because q does no change), while he capial sock accumulaes and evenually achieves new seady sae (Poin B).

8 8 Figure 2. Transiional Dynamics under Technological Process 3 Invesmen wih Borrowing Consrain So far, we have derived he opimal invesmen decision wih invesmen adjusmen cos (IAC). Due o his fricion, he capial sock is lower han he rs-bes level. However, he speci caion of IAC is somewha a reduced form. In his secion, we will remove he IAC speci caion (b = 0), insead we inroduce borrowing consrain. I can be shown ha he invesmen decision under he borrowing consrain is equivalen o he IAC case. Therefore, he nancial fricion provides one poenial micro-foundaion for he IAC, see Wang and Wen (202, Review of Economic Dynamics). The rm s opimizaion problem is V 0 (K 0 ) = subjec o capial accumulaion equaion max fi ;K + g =0 0 (R K I ) (40) K + = ( ) K + I ; (4) and borrowing consrain I V (K ) : (42)

9 9 where V (K) = j= j +j D +j : The borrowing consrain indicaes ha he rm can only pledge fracion of is own capial o nance he invesmen expendiure. give us Denoe fq ; g as he Lagrangian mulipliers for (4) and (42), respecively. FOCs for fi ; K + g q = + + = q ; + (K + ) R + + q + ( ) + + : (K + ) For simpliciy, we assume ha he borrowing consrain (42) is always binding, hus we have I = V (K ) : (45) We now decision (7), we have = q : Recall ha from he household s opimal porfolio choice V (K + ) = + [V + (K + ) + D + (K + )] : (46) Muliplying boh sides of (44) by K + yields q K + = + (K + ) R + K + + q + K + ( ) + + K (K + ) = + (K + ) R + K + I + + q + K +2 (q + ) I + + K (K + ) = + (K + ) D + + q + K +2 + V + (K + ) + + K + : (K + ) Las line is derived from (44), (45) and he de niion of dividend, D : = q ; we use guess-and-verify sraegy. We guess V (K) is linear in K; paricularly, V (K) = v K: Then (47) implies q K + = + D + + q + K +2 + v + K v + K + = + (D + + q + K +2 ) : (48) Comparing wih (46), las equaion implies ha V (K + ) = q K + and v = q : Once we have V (K) = q K; borrowing consrain implies I K = q : (49) Noe ha he above invesmen decision looks very similar o he one in he invesmen adjusmen cos case.

10 0 Euler equaion (44) can be rewrien as q = + [R + + q + ( ) + q + (q + )] : (50) The above wo equaions ogeher wih capial accumulaion (4) de ne he full sysem. In he seady sae, we have I K = q =, q = : (5) Therefore, in order o ensure a binding borrowing consrain in he seady sae, we mus assume a small such ha < ; i.e., he rm su ers a severe credi consrain. Moreover, (50) implies ha R = (r + ) q q (q ) ; (52) where r = = : The wedge beween R (he MPK) and r + is R (r + ) = r > 0: (53) Noe ha he wedge is decreasing in, which measures he exen ha his economy deviaes from he rs bes. One imporan implicaion of las equaion is ha if he nancial marke is less developed, he MPK ends o be high and he capial sock ends o be low (resource misallocaion).