Problem set 3: Endogenous Innovation - Solutions

Transcription

1 Problem se 3: Endogenous Innovaion - Soluions Loïc Baé Ocober 25, 22 Opimaliy in he R & D based endogenous growh model Imporan feaure of his model: he monopoly markup is exogenous, so ha here is no need o wrie he pro maximisaion problem of he rm o ge an expression for. In paricular, he mark-up does no depend on he ase for variey parameer anymore, a fac which will have serious implicaions once we analyse opimaliy quesions.. Geing he equilibrium growh rae involves he same seps as in he lecure noes, albei wih some adapaions o our case. The Euler equaion for aggregae consumpion The level of aggregae consumpion hrough ime is picked exacly in he same way as in he course, so ha C evolves according o he following equaion: C = r ρ C In he long-run, he growh rae of consumpion g C becomes a consan (for we are on a balanced growh pah by assumpion) : g C = r ρ () We now need anoher relaionship beween g C and r o ge he equilibrium growh rae, which will come from he equilibrium in he R&D secor and he preferences of agens. To do so, we will rs ge he prices, producions and pros for each diereniaed rm i as a funcion of he number of produc varieies N. Then, we will place ourselves on a balanced growh pah o compue a relaionship beween g C and he growh rae of he number of varieies. Finally, using he free-enry condiion ino he R&D secor, we will ge he second equilibrium relaionship. Equilibrium in he diereniaed goods markes The exogenous markup relaionship implies ha all goods are priced he same in equilibrium: i, p i = p. Then he aggregae In he long run, he rae of ineres is also consan so he ime index was dropped.

2 price index p reads: ( N p = ) p i = (p N ) = p N Le us ake he aggregae price index as he numéraire a all : Then he real wage obains as follows: p = p = N w = p = N (2) In equilibrium, prices being equal implies ha he demands (and he producions as well) are equal: i, C i = L i = L P = L L R = L N /N (3) N N N The insananeous pro made a by rm i wries: Using (2) and (3), we ge ha: π i = ( ) N π i = p i C i w L i = ( )w L i L N /N N = (L N )N 2 (4) N The balanced growh pah In he long run, he growh rae of he number of varieies becomes a consan g N. Denoe y he level of producion of each good in he long run, equaion (3) becomes: y = L g N g y = g N N The producion of each single good decreases as fas as he number of goods increases in he long run. Then aggregae consumpion evolves as follows: ( N C = g C = g y + ) C i = (y N ) = y N g N = g N + g N = g N (5) 2

3 The R&D secor Having a look a he R&D secor will now enable us o ge he expression for g N. Le V he value of holding he monopoly of producion (i.e. he paen) over a se of goods i of measure 2. Free-enry ino he R&D secor implies ha over any ime inerval, he cos of creaing new goods should be equal o he value of creaing hese new goods. During, he creaion of new goods is N, he value of which is V N. This requires o hire L R workers, a he price w L R. Then he free-enry condiion reads: V N = w L R V N = w N N V = w = N N N = N 2 (6) We can also calculae V in he long run in anoher way. Using an arbirage argumen, we ge he following dynamical equaion for V : 3 r V = π i + V (7) In he long run, he growh rae of he paen value is a consan g V so ha: π i + g V V = rv V = Collecing he expressions for V in equaions (6) and (8): π i = r g V (r g V ) (L g N)N 2 (8) (r g V ) (L g N)N 2 = N 2 ( )(L g N ) = r g V (9) From equaion (7) and using corollary of lecure noe 2, we ge ha g V = g π. Then equaions (4) and (5) yield: g π = 2 g N = (2 )g C Equaion (9) hen becomes: ( )(L ( )g C ) = r (2 )g C g C (( )( ) + 2) = ( )L r 2 Here, I could abusively dene V as he value of holding he paen for one good, which is maybe more palaable bu less accurae. 3 The insananeous dividend on he paen is π i, and V is he capial gain associaed wih holding he paen. 3

4 Replacing r by is value in equaion (): g C (( )( ) + 2) = ( )L g C ρ g C ( ) = ( )L ρ g C = ( )L ρ ( ) This las equaion pins down he long-run growh rae of consumpion in he privae equilibrium. Here, > (oherwise he formulaion of he aggregae consumpion index does no make any sense), so g C is an increasing funcion of. When he monopoly markup of each rm increases, exising rms increase heir pros so ha R&D becomes more proable: produc innovaion ends o increase. A second eec of higher markups is o reduce he real wage (as shown by (2)), so ha agens will consume less of each good i. This demanddepressing eec implies ha less labor is employed for producion purposes, inducing a shif owards employmen in he R&D secor. Because of he faser expansion of produc range N, aggregae consumpion increases more rapidly when increases (even hough he consumpion of each good i decreases more rapidly). 2. Firs, we need o see where he consrain on C comes from, o make some sense of his new opimizaion program. C wries as follows: ( N C = ) C i In his social planner problem, here are no markes anymore so in paricular here is no price equalizaion mechanism ensuring ha he consumpion of each good i is independen of i. To see why his is sill he case here, consider he fac ha he marginal impac on C of consuming any C i is decreasing wih C i (since ()/ < ): herefore, he marginal uiliy of each diereniaed good is decreasing. Then for a given level of aggregae consumpion C, i is opimal for he social planner o equalize consumpion across diereniaed goods: i, C i = c = L P /N. Then we ge ha: ( N C = ) C i = (c N ) C = (L L R )N = c N The problem o solve for he social planner is hen: + ( ) max ln (L L Rs )N s e ρ(s ) ds {L Rs } subjec o: N s = N s L Rs 4 = L P N N ()

5 Here, he conrol variable is L Rs, while he sae variable is he number of varieies N s. The Hamilonian of he problem wries: ( H(L Rs, N s, λ s ) = ln (L L Rs )N s ) e ρ(s ) + λ s N s L Rs The rs-order condiions read: H = e ρ(s ) + λ s N s = () L Rs L L Rs and: H N s = λ s ( )N s e ρ(s ) + λ s L Rs = λ s (2) In he long run, L Rs = N s N s = g N is consan. Equaion () rewries as: λ s = e ρ(s ) λ s = ρ g N (L g N )N s λ s Then equaion (2) can hen be rewrien as: ( )N s e ρ(s ) + g N λ s = (ρ + g N )λ s λ s = The wo las expressions of λ s can hen be equalised o yield: (L g N )N s e ρ(s ) = Using equaion (5), we nally ge: g C = ρ( )N s e ρ(s ) ρ( )N s e ρ(s ) L g N = ρ( ) g N = L ρ (3) Noice ha if he agens are oo impaien (i.e. for oo large a value of ρ), aggregae consumpion will acually decrease over ime. This implies ha g N, he growh rae in he number of varieies, could also be negaive: his is a problem here because g N < L Rs < in he long run. This is of course impossible: wha we are missing in he way he problem is formulaed is a physical consrain saing ha he workforce in each secor mus be posiive or equal o zero a all imes. Alernaively, we could jus impose a resricion on he values of he relevan parameers ensuring ha his growh rae is always posiive. Besides, he growh rae of consumpion is also posiively relaed o he size of oal workforce, which is a debaable fac in realiy (see he end of lecure noe 6). 5

6 3. The opimal markup equalizes he wo formula for g C in quesions and 2 (equaions () and (3)): ( )L ρ ( ) = L ρ ( )L ρ = L ρ( ) ρ( ) = L + ρ = L + ρ ρ( ) (4) Again, we should impose a resricion on he parameers of he model o ensure ha > (oherwise all rms would be making negaive pros a all ime, which does no make any sense). If >, hen he growh rae of consumpion is oo large in he long run 4. Invesmen in R&D is oo large a all imes, so ha he acual producion (hence consumpion oo) of he diereniaed goods is oo low in he privae equilibrium. On he oher hand, if <, he growh rae of consumpion is oo low. This comes from he fac ha R&D is insucien in equilibrium: he range of varieies available for consumpion does no expand fas enough. 2 A semi-endogenous growh model. The law of moion for he ne sock of asses held by he represenaive dynasy is: W = r W + Y C The solvency condiion on asse holdings wries: lim W e rsds + Inegraing he law of moion yields he following ineremporal budge condiion (see problem se for he mehod on a similar case): W + + (Y C )e rsds d 4 Since he social planner maximises he discouned ow of uiliy from aggregae consumpion beween and +, and no he consumpion a + only, here is such a hing as a oo large growh rae of consumpion compared o opimaliy. 6

7 2. The Lagrangian of he maximisaion problem is: + ( ) ( C L({C }, λ) = ln e ρ d + λ W + The rs-order condiion wih respec o C reads: L + L = e ρ λe rsds = C C ) (Y C )e rsds d λ = e ρ C e rsds Log-diereniaing he previous relaionship wih respec o ime yields: = ρ C C + r C = r ρ (5) C As for per-capia consumpion, he following Euler equaion obains: c = C c C = L = r ρ n (6) L c C L 3. Saring from he formulaion of he consumer problem, we will derive a price index for he aggregae good. 5 Consumers faced wih he se of prices {p i } i maximise: subjec o he budge consrain: C = [ N ] /() C ()/ i di N p i C i di E where E denoes oal expendiure on consumpion a and is considered given here. The Lagrangian of his problem is: 6 L(C i, λ) = N N C ()/ i di + λ(e p i C i di) 5 See lecure noe on he Dixi-Sigliz preference srucure 6 Here he Lagragian acually relaes o he ask of maximising C ()/ I is, however, sricly equivalen o maximise C ()/ maximising C ()/ have he advanage of being much simpler. 7 under he same budge consrain. or C, and he analyical expressions for he problem of

8 The FOC on C i yields: C / i = λp i C i = ( ) (7) p i λ Sauraing he budge consrain a opimaliy and using (7), he following equaion is obained: N ( ) N E = p i C i di = p i di (8) λ Now dene he price index p as follows: p = ( N ) p i di (9) Equaions (7) and (8) now help o rewrie he consumpion of each individual good and aggregae consumpion as follows: C i = E p i p = E ( ) pi (2) p p C = [ N ] /() C ()/ i di = E p [ N ( pi p ) ] /() di = E (2) p Equaion (2) shows he ineres of dening p as in (9): consumpion of he aggregae good is equal o aggregae expendiure divided by he price index, which gives a synheic formula which is he analogue of he more deailed budge consrain. Then equaion (2) shows ha he opimal consumpion paern is o spend he same amoun on each good if prices are equalized, and ha he absolue value of he elasiciy o price deviaions is (hence he deniion of as an elasiciy of subsiuion beween o inniesimal goods). 4. Firms maximise pros: Π i = p i C i w L i subjec o he demand funcion (equaion (2)) and he producion echnology: y i = C i = L i. Taking he rs-order condiion wih respec o p i yields: ( )p i = w p i p i = w = w (22) where = /( ) is he endogenous mark-up charged by he rms on wages. 5. From equaion (22), i can be seen ha prices are equalized across rms: p i = w, i. Similarly o exercise of his problem se, he price index collapses ino he following formula (where he aggregae good is he numeraire): p = = w N 8

9 w = N (23) Noice ha as he aggregae good is aken as numeraire, he wage rae w is he real wage in his economy. Addiionally, > implies ha he real wage increases wih he number of varieies. 6. If all varieies are priced he same in equilibrium, hen each rm serves an equal share of he marke and hires he same level of labor: C i = L i = L P N = ( β )L N (24) 7. Le us rewrie pros a equilibrium: Π i = p i C i w L i = w L i w L i = ( )w L i Π i = ( )N ( β ) L N 8. By arbirage (same hing as in exercise, quesion ): Π i = N 2 ( β )L (25) r V = V + Π (26) 9. Free enry ino he R&D secor implies ha a any poin in ime, he value of creaing new goods equaes he cos of creaing hem: V N = w L R = w β L γv N δ β L = w β L V = w γn δ = N γn δ (27). Log-diereniaing equaion (23) on a balanced growh pah yields: Besides, equaion (24) implies: c = ( N ) c i g W = g N c i = C i = [ L = β N N ( β N 9 ) ] = β N N

10 c = ( β )N (28) On a balanced growh pah, β is consan. To see his, consider rs he fac ha g β needs o be consan on a BGP. Addiionally, labor marke equilibrium saes ha: L = L R + L P = β L + L P where L R is he workforce employed in he R & D secor, while L P = N l i di denoes labor in he producion secor. Then L and L R need o grow a he same rae, which implies ha g β = on a balanced growh pah. Then log-diereniaing (28) on a balanced growh pah yields:. The producion funcion of new goods reads: On a balanced growh pah, g = g N = g W (29) N = γn δ β L N = g N N so he relaionship above is equivalen o: g N N = γn δ β L β is consan so by log-diereniaing we obain: g N = δg N + g L The growh rae of L is g L = n so: g N = n δ n g = ( δ)( ) (3) 2. Here, he higher he populaion growh rae, he more people can go ino he R&D secor: hen he populaion growh rae has a posiive impac on g N (hence on g as well). Equaion (3) ells us ha if n = hen g = in he long run: growh dies ou in he absence of an exogenous growh of populaion. The model is dubbed semi-endogenous because of his propery: he producive exernaliy of he number of goods on he process of creaing new goods (i.e. he N δ erm) fuels growh in he shor erm, bu is no srong enough o generae long-erm growh. 3. On a balanced growh pah, he Euler equaion (equaion (6)) wries: r = g + ρ + n = g = r ρ n n ( δ)( ) + ρ + n r = ρ + n + δ δ ( δ)( ) (3)

11 4. In quesion, i has been shown ha on a balanced growh pah g N = n/( δ). This shows ha he growh in he number of varieies is independen from he markup: hence R & D incenives play no role in he deerminaion of he equilibrium rajecory. Even hough equaion (3) shows here is a role for in he equilibrium rajecory (and even hough he markup is deermined endogenously as a funcion of ), he dependency of g on mus be undersood as a ase of variey eec. Commens: Naional accouning in his economy is a bi dieren from usual. From he income side, oupu is divided beween wages and he rms pro: Y = w L + Π. From he expendiure side, i can be seen ha oupu is spli beween consumpion and he creaion of new goods: Y = C + V N. Here he creaion of new goods erm is akin o invesmen in a more radiional growh model wih physical capial. To end his exercise, le us ake a look a he equilibrium on he marke for loanable funds (also known as capial marke when here is acually capial in he model). Funds are supplied by he consumers: he par of heir income which is no consumed is Y C. The demand for funds come from he enrepreneurs who creae new goods (remember hey incur an iniial cos of invening he goods, which is recouped aferwards by collecing all he fuure pros from producion): he amoun is w L R = w β L. A each poin in ime, he ineres rae r adjuss o equalise supply and demand of loanable funds: 7 w β L = Y C w β L = V N I is noeworhy ha he las equaion, which was obained by combining he loanable funds marke equilibrium equaion and an accouning equaion, is equivalen o he free-enry condiion in he R & D secor. This is an illusraion of Walras' law, which saes ha in general equilibrium heory, if all markes bu one are in equilibrium, hen he las marke necessarily is oo. 7 An increase in r induces consumers o save, as Euler equaion (5) shows. I also reduces he demand for funds by enrepreneurs, who need o borrow a a higher cos for he same fuure pros.