Alemayehu Geda. Investment Based on Eklund(2013), Hibs Lectures & David Romer (2012) Book

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1 Advanced and onemporary Topics in Macroeconomics Alemayehu Geda Web Page: lass Lecure Noe 7 nvesmen Based on Eklund(203, Hibs Lecures & David Romer (202 Book Addis Ababa Universiy Deparemen of Economics PhD Program, 204

2 Lecure 7 Opimal nvesmen

3 . nroducion: nvesmen Why are abou nvesmen. Why are Abou nvesmen? nvesmen drives capial formaion, and he sock of capial is a key deerminan of oupu growh and conseuenly feasible consumpion levels. ( ( (2

4 nroducion on d nvesmen is 5% or more of GDP and i is he mos volaile componen of GDP. nvesmen herefore is probably he mos imporan proximae source of business cycles [This is for insance cenral for alecki/eynes]. Governmen policy is ypically argeed heavily on invesmen; mos ax codes favor i. nvesmen dynamics yield informaion abou invesor raionaliy, eynes. "animal spiris", capial formaion insiuions, and more. Thus, he need o focus on i.

5 2. The Neoclassicl nvesmen Theory nvesmen on d 2. Saic Neoclassical nvesmen Theory Neoclassical invesmen heory daes mainly from Dale Jorgensen s papers in he 960s (AER, 963, Hall and Jorgenson, AER, 967 However, wha Jorgenson delivered was more a heory of he opimal capial sock han a heory of opimal invesmen per se. Here is he essenial seup: (see he nex slides

6 Neoclassical nvesmen on d Recall [from ur graduae/undrgad macro] Oupu is produced by a RS producion echnology Q (, A N irms inves o achieve a capial sock ha maximizes ne real profi,π, each period (, A N r wn where everyhing is priced in unis of oupu** p r p r p is he user cos of capial given he real price of capial unis p ; a fixed marke ineres rae r; and depreciaion rae δ, and w is he real wage. rk/pk gives us he real user cos of capial.

7 Neoclassical nvesmen on d. BROADER DERVATON O NEOLASSAL THEORY O NVESTMENT n his secion we derive he relaionship beween he neoclassical heory, acceleraor principle and Tobin s Q-- heory of invesmen. All hree heories assume opimizaion behavior on behalf of he decision maker (invesor. The neoclassical and Tobin s heory of invesmen explicily assumes profi/value maximizaion. The acceleraor heory of invesmen assumes his implicily, by assuming ha invesmen is deermined by an opimal capial sock.

8 Neoclassical nvesmen on d The saring poin for Jorgenson s (963, 967 and 97 neoclassical invesmen heory is he opimizaion problem of a firm. Maximizing profis in each period will yield an opimal capial sock. Assuming ha he producion funcion can be wrien as a convenional obb-- Douglas funcion: Y( f ( (, L( Where Y( is firm oupu, is capial and L denoes labor, all in period. A L (2

9 Neoclassical nvesmen on d The profi funcion for a represenaive firm can hen be expressed as follows: ( p( Y( s( ( w( L( ( denoes profi, p( is he price of oupu, s( is he price of capial and w( is he wage. Assuming profi maximizaion, he curren value of a firm, V(0, can be wrien as:** (3 V(0 max E 0 ( e r d S. E d d 0 0 p( Y( ( ( s( ( ( and ( 0 is given. w( L( e r d (4

10 Neoclassical nvesmen on d To maximize V(0 he firs sep is o se up a Lagrangian: L V (0 Which gives 0 r e d (5 L 0 r py s wl e d (6 rom his we obain he familiar curren value Hamilonian H pf (, L s wl ( Where he Lagrangian muliplier λ( is our cosae variable. should be noed ha λ( represens he shadow price of capial. (7

11 Neoclassical nvesmen on d Differeniaing he Hamilonian, we obain he following firs order condiions: H s 0 This condiion saes ha he opporuniy cos of capial shall be eual o he shadow price of capial. H ' pfl w 0 (9 L This condiion simply says ha he labor should be employed unil he marginal revenue of labor euaes wih he wage. (8

12 Neoclassical nvesmen on d Recalling he maximum principle (nriligaor, 97 we ge:** H (0 (his is correced now hanks o Mari s commen in class Which says ha in euilibrium ne invesmen should be zero and gross invesmen eual o he depreciaion of. inally, he marginal condiion for capial is: H ' pf 0 ( The canononical euaion (nriligaor, 97, hinag 2000 H reuires ha, where y is he conrol variable such r ha e a ime. [**Recall he 2 nd O in Lecure 2]

13 Neoclassical nvesmen on d Thus: H d r e ( r d This means ha euaion ( can be wrien as: ' pf (2 r (3 rom euaion (8 we know ha s=λ, which implies ha This also means ha H can be saed in he following way: s ' pf s s rs (4

14 Rearranging he above we obain: Since is he marginal rae of reurn on capial, mrr k, euaion (5 can be rewrien as he marginal produc of capial: (5 / s ' s s r pf ' pf (6 /p / s ' s s r f Neoclassical nvesmen on d

15 Neoclassical nvesmen on d ' Noe ha f Y.Thus Jorgenson s (963 user cos of capial, c, is defined as: s[ r ( s / s], which means ha: pf ' c or f ' c / real erms (7 This can now be used o derive he opimal capial sock, *, and he invesmen funcion. Using obb-douglas echnology he marginal produc of capial becomes: Y f ' p L in (8 Which in urn can be expressed as: Y Y (9

16 Neoclassical nvesmen on d Muliplying by p and recalling euaion (7 we ge: H Y p c (20 Solving for we obain an expression for he opimal capial sock: py * c (2 We see ha * depends on oupu, price of oupu and he user cos of capial, c. Thus, invesmen become he change in capial beween wo periods: py c * ( (22

17 Neoclassical nvesmen on d Noe ha his assumes ha ( aduss insananeously and fully o *(. Assuming ha he adusmen o he opimal capial sock is only parial each period his can be incorporaed ino euaion (22 by inroducing an adusmen parameer ha depends on he difference beween acual and desired capial, see e.g. Mueller (2003. Since he neoclassical heory assumes ha he capial aduss immediaely and compleely o he desired capial sock he invesmen funcion is essenially eliminaed. has herefore been suggesed ha Jorgenson s heory is in fac a capial heory and no an invesmen heory.

18 Neoclassical nvesmen on d 2.2 AELERATOR THEORY The acceleraor approach is ofen associaed wih a eynesian approach which is primarily due o is assumpion of fixed prices. The acceleraion principle was however firs suggesed by lark (97 and is well known for is applicaions by Samuelsson (939 o business cycles. The acceleraor is, in fac, merely a special case of he neoclassical heory of invesmen where he price variables have been reduced o consans. Recall he Euaion [2] which is he opimal level of capial sock: * py c (2

19 Neoclassical nvesmen on d f he price of oupu is assumed o be consan and he price variables s and r in Jorgenson s (963 user cos of capial, ( c s[ r ( s / s] following: * Y, are fixed, euaion (2 reduces o (23 This is simply he well known acceleraor principle where he desired capial sock is assumed o be proporional o oupu. nvesmen in any period will herefore depend on he growh in oupu: Y or in discreese up Y (24 Given flexible prices and parial adusmen oward he desired capial sock each period, invesmen depend on prices of oupu and inpu and ineres raes (cos of capial.

20 Neoclassical nvesmen on d Smih (96 demonsraes wha he calls he: logical inseparabiliy of marginal efficiency and he acceleraor deerminan of invesmen expendiures. He used calculus of variaion o derive his resuls. Again, his version of he acceleraor assumes a complee and insananeous adusmen of he capial sock An alernaive is he so-- called flexible acceleraor ha includes lags in he capial sock. Eisner and Sroz (963 sugges ha hese lags are because he uni price of capial, s(,increases wih he adusmen speed,(see also Lucas,967. Allowing for lags in he adusmen of he capial sock, however, make he neoclassical heory virually indisinguishable from he acceleraor heory (see Eisner and Sroz, 963, and Eisner and Nadiri, 968. urhermore, i is worhwhile o noe ha even hough he acceleraor principle is ofen coupled wih a eynesian approach, eynes himself, was very skepical owards approaches like his. irs, eynes was very criical owards formal models of economic behavior. Second, and more fundamenally, eynes did no believe ha invesmen is deermined as adusmen owards euilibrium.

21 Neoclassical nvesmen on d Remarks: The saic neoclassical seup implies ha invesmen needed o ge o D occurs all a once -in one discree ump. n general his would seem highly unrealisic. Jorgenson has acual oupu on he righ-side of he invesmen euaion. This is wrong. Wha should appear is desired oupu, QD. 960s neoclassical heory is pre-raional expecaions and, herefore, expecaions of fuure profiabiliy of incremens o he capial sock oday play no role in explaining oday s invesmen behavior.

22 3. Opimal nvesmen wih onves Adusmen oss 3. Opimal nvesmen wih onvex Adusmen oss The Sae of he World: The firm is a price aker in compeiive markes. Labor is flexible (can be adused wihou cos. The only barrier o full and fas deploymen of he profi maximizing sock of capial are he convex coss of adusing he capial sock. irms can always obain he capial desired (he marke for invesmen capial is perfec. The are no axes. The producion echnology has RS. As we shall see, inroducing adusmen coss yields rich dynamics o invesmen behavior.

23 Opimal nvesmen on d 3. The Producer s Obecive The represenaive firm aims o maximize is financial value, as he expeced presen discouned value of is ne real revenue flow - revenue from sales less capial invesmen coss and labor (and oher coss {,, } 0 π Max [ Subec o E 0,( A N ] (, w N (8 (9 (20

24 Opimal nvesmen on d implying ha invesmen saisfies ( (2 Where: is he real cos of capial goods (, represens real adusmen coss in he form of diminished revenue from oupu sales, ( r, r is he discoun facor.

25 Opimal nvesmen on d n order o ge ransparenly o he famous "Tobin s ", we se up he producer s problem as a curren value Lagrangian funcion where E denoes condiional expecaions informed by all oucomes a period and earlier, and ( + gives he value (shadow price of a marginal change in a ime + ( + + in ime ( + revenue (22 ] ( [, ( ],( [ 0 }, { 0 N w AN E Max L i

26 Opimal nvesmen on d 3.2 The Os Because labor can be adused wihou cos, we need wo Os and +. ( The nvesmen O L (23 Hence we obain he shadow price of a new uni of capial as (he normalized price of a uni of capial plus he marginal cos of is insallaion: Marginal Benefi( 0 (24 Marginal os(

27 Opimal nvesmen on d To be more explici we need o specify he cos funcion, (,. A common specificaion is he sricly convex uadraic funcion (, b 2 where is he cosless level of insallaion (which could of course be zero and b is us a scale parameer. [[ See nex slide for Diagram]] 2 (25

28 Opimal nvesmen on d

29 Opimal nvesmen on d n his fairly flexible case he shadow price is** As iniially proposed by Tobin (JMB, 969, is herefore a sufficien saisic for invesmen. The O for in (26 implies he invesmen euaion (invesmen relaive o he exising sock of capial b (26 (27 b

30 Opimal nvesmen on d Euaion 27 is a famous euaion, which is someimes wrien wih b= and =0. means ha if he firm is maximizing i should inves up o he poin a which he marginal cos of a new uni of capial euals is value (shadow price o he PDV of is ne real revenues. When he value of a new uni of capial exceeds is price (, invesmen should rise above he cosless level, and (of course conversely.

31 Opimal nvesmen on d apial Sock Dynamics Since (2 The apial O The O is found for + because ha is wha is driven by. ( is sunk.** (28 ( b (30 ] ( [ (29 0 ( ] [ E β E L

32 Opimal nvesmen on d The above euaion is an invesmen Euler euaion. Noe well i s inerpreaion. says ha he marginal value (shadow price of a new uni of capial sock oday ( mus eual he expeced discouned value ( β= /(+r of he he sum of hree effecs: (i he new uni s marginal conribuion o real revenue, ( +, (ii he conribuion of he new uni o he marginal decline of fuure capial insallaion coss,- + (iii he depreciaed shadow value of capial in he nex period, + ( -δ f hese condiions do no hold, he firm is missing a profiable ineremporal arbirage opporuniy.

33 Opimal nvesmen on d Now le s re-arrange erms in (30 o ge + on he lef-side:** Now solve for : (32 ] [ ] ( [ (3 ] [ ( E L E E E (33 ] ( [ ] [ L E

34 Opimal nvesmen on d Enumeraing he forward polynomial,** [ ( L we obain: Again we have a beauiful euaion, and you should undersand well is inerpreaion. The euaion says ha he marginal value (shadow price of an exra uni of capial ( mus eual he exra uni s conribuion o he expeced PDV of real ne revenues, where he operaive discoun facor incorporaes he rae a which capial depreciaes each period: ] [ ( E ( r 0 ( r ( ( r ( L ( r [ ( r ] ] (34

35 4. Tobin s and Sock Prices nvesmen on d 4 Tobin s and Sock Prices Hayashi (982, Economerica proved ha if he producion echnology and he adusmen cos funcion are homogeneous of degree (RS, as we have assumed here, hen Tobin s marginal euals average, and average gives he marke value of he firm relaive o he replacemen cos of is capial sock. To see his, engage in he following exercise. Le us go hrough all he algebric seps (mos of hem redundan for you o obain he resul.

36 Tobin on d irs, muliply he soluion in e.(30 for by +. This gives By RS we know ha facor paymens exhaus oupu (35 ( E,, ( N N AN Q

37 Tobin on d so he firs righ-side erm in brackes in e.(35 is Hence e.(35 may be expressed, ( (36, ( N w AN N N AN (37 (, ( N w AN E

38 Tobin on d rom he capial accumulaion euaion evaluaed a period +2 we know ha And so Therefore by definiion 2 2 ( ( ( 2

39 Tobin on d Now subsiuing for n e.(37: ( 2 ( 2 2, ( (38 (, ( N w AN E N w AN E

40 Tobin on d rom he invesmen O [En 24] we know ha Make his subsiuion for he las erm in e.(38: ' ( 2 2, ( (39 ] [, ( N w AN E N w AN E

41 Tobin on d Nex remember our assumpion (necessary for Hayashi s proof ha he adusmen cos funcion, like, he producion funcion, is homogenous of degree, ( ( + + = ( + ; +. Make his subsiuion in e.(39, giving (40,, ( 2 N w AN E

42 Tobin on d We have obained a difference euaion for + wih driving (forcing funcion Noe ha he driving funcion is he ne real revenue of he firm [e.(9]: oupu less wage coss less he coss of insallaion of capial less direc capial invesmen coss., (, ( N w AN E

43 Tobin on d Use he usual mehods o solve (40 by firs moving o he lef hand side and hen divide hrough by and enumerae he forward polynomial operaor ** [ 2] E (4, (, ( L (- - N w AN E E : ( L (42, (, ( 0 N w AN E

44 Tobin on d Nex divide hrough by + inally we have arrived a ye anoher imporan euaion of he Tobin- model! says ha is he raio of he expeced presen discouned value of he firm s revenue flows relaive o he marke value of is capial sock. (Recall he price of capial was normalized o.0. (43, (, ( 0 N w AN E

45 Tobin on d The numeraor gives he fundamenal marke value of a firm: he expeced presen value of is revenue flow (eual o is price per share imes he number shares ousanding, if asse markes are efficien. The denominaor is he replacemen cos of is capial sock. f producer s are invesing opimally, he shadow price of an exra uni of invesmen (an exra uni of capial should be euaed o he firm s marke value per uni of capial value [ie. is =f(] he Tobin- invesmen model Read on nvesmen and he growh of capial sock using phase diagram in your Texbook.

46 Thank You The End Thank You