Introduction to Endogenous Growth Models

Transcription

1 Intrductin t Endgenus Grwth Mdels 55 Intrductin t Endgenus Grwth Mdels Paul Rmer s 1986 mdel and Rbert Lucas s (1988) human capital mdel. These mdels get arund the diminishing marginal returns t capital assumptin by bradening the definitin f capital t include knwledge r human capital, bth f which may have psitive externalities. We need sme kind f external effects in rder t have a mdel in which individual firms d nt have increasing verall returns t scale, s they d nt expand infinitely and becme ecnmy-wide mnplies the ecnmy as a whle has increasing returns t scale, s that returns t capital r prduced inputs can be cnstant Endgenus grwth mdel have several features that ecnmists have fund attractive They endgenize key parameters f the mdel such as g They can explain lack f cnvergence They allw s and related plicy variables t affect the grwth rate f GDP, nt just the level f the grwth path Text bk begins with a simplified mdel f knwledge prductin via research and develpment in Chapter 3. Uses cnstant saving assumptin as in Slw mdel Incrprating Ramsey saving mdel des nt change basic dynamics Key characteristic leading t endgenus grwth: cnstant returns t scale in prduced inputs. In Slw and Ramsey mdels, capital was nly prduced input and had diminishing returns David Rmer s R&D mdel Dynamics and behaviral assumptins Ecnmy has tw sectrs: gds-prducing sectr and R&D (knwledge-prducing) sectr Each sectr uses labr and capital a L and a K are the shares f labr and capital allcated t the knwledge sectr These shuld be determined by chices f wners f labr and capital allcating them t their highest return Rmer simplifies the mdel by taking these t be exgenus Ecn 454 studies mdels in which the rewards t capital and labr in the tw sectrs are explicitly mdeled and these decisins are allwed t be endgenus

2 56 David Rmer s R&D mdel Capital (K) Knwledge (A) Labr (L) 1 a K a K 1 a L a L Gds-prducing sectr Knwledgeprducing sectr s Output (Y) 1 s New knwledge Investment Cnsumptin (C) We assume a Cbb-Duglas CRTS prductin functin fr gds: K L Y t a K t A t a L t Knwledge is prduced accrding t a Cbb-Duglas that may r may nt have CTRS: A t B a K t a L t A t K L A BaK K al L which means grwth rate f A (ur ld g) A depends n the amunts f K and L devted t research and is cnstant if thse amunts are cnstant. Nte that = 1 Replicatin argument cannt be used t justify CRTS here Same knwledge prduced by tw peple is nt twice as valuable Psitive spillvers culd yield increasing returns t scale Are ther discveries substitutes r cmplements fr the next discvery? K t sy t N depreciatin and cnstant saving rate mean Exgenus grwth f labr frce: L t nlt Analysis f R&D Mdel Rmer begins with a mdel in which there is n physical capital ( = = 0) We wn t analyze this mdel in detail, but nte the equatins f the mdel if = 1 and n = 0 (s L is cnstant)

3 David Rmer s R&D mdel 57 A t 1 L BaL L Y t A t a L A t Output is prprtinal t A and the grwth rate f A is a cnstant, s this mdel has a cnstant grwth rate f utput that is determined by B, a L, and L (and ). Higher R&D prductivity, mre labr being used in the labs, and a bigger ppulatin all lead t a higher grwth rate (nt just t a higher, parallel grwth path) Fr the full mdel (with K), we have tw state variables, A and K We dente the grwth rates f A and K by g A and g K Dynamics f K K t sy t s a k a L K t A t L t The term in brackets is a cnstant (ver time) that we shall call c K The grwth rate f K at every mment t is 1 K t AtLt gk t ck K t K t * We seek a steady state in which K grws at a cnstant rate g K, s we want t analyze the change in r grwth rate f the grwth rate g K t 1 g A n gk using ur rules fr grwth rates g t K g K t 0 if gk t ga t n * * gk t0 if gk t gatn gk 0 : gk ga n g K t0 if gk t gatn The g K 0 curve is a line with slpe f ne and intercept n the g K axis at n 0. Belw the line, g K 0 and abve the line g K 0, s the arrws pint vertically tward the line Dynamics f A A t g 1 A t BaKa L K t L t A t At The term in brackets is cnstant ver time and called c A

4 58 David Rmer s R&D mdel As abve, the grwth rate f the grwth rate at every mment t is g A t gk tn1gat g t A n 1 g At0 if gk t gat n 1 * n 1 * gat0 if gk t gat ga 0: gk ga t n 1 g At0 if gk t gat The g A 0 curve is a line with slpe 1 and intercept n the vertical (g K ) axis at n/ 0. T the left f the line g A 0 and t the right f the line g A 0, s the arrws pint hrizntally tward the line Equilibrium dynamics The nature f the equilibrium depends crucially n tw prperties f the parameters: n > 0 vs. n = 0 This determines whether there is any exgenus surce f grwth in the mdel If n > 0 as in the Slw and Ramsey mdels, sustained grwth in ttal GDP is pssible thrugh exgenus grwth in L + = 1 vs. + < 1 (r + > 1) This determines returns t scale in prduced inputs Nte that K and A are prduced in the mdel The prductin functin fr gds always has cnstant returns in prduced inputs because K has expnent and A has expnent 1 The prductin functin fr knwledge has returns t scale in the tw prduced inputs equal t the sum f their expnents: + If + = 1, then the mdel can sustain nging endgenus grwth even if n = 0 because increases in bth K and A tgether are nt subject t diminishing returns Dynamics with n > 0 and n = 1 With n > 0, the g K 0 line intercept is psitive and the g A 0 line intercept is negative If n = 0, bth lines pass thrugh the rigin Case I: + < 1 (diminishing returns in prduced inputs)

5 David Rmer s R&D mdel 59 Slpe f g A 0 line is 1 1, s it is steeper than the g K 0 line g K g K * n 0 g A * g A Ecnmy cnverges t unique equilibrium frm all pints in space Slving algebraically, we can shw g g * A * K n 1 1 n 1 Grwth here is exgenus in the sense that if n = 0, bth K and A stp grwing. (Nte that bth lines intercept at the rigin if n = 0.) This case replicates the dynamics f the Slw mdel with g determined endgenusly as a functin f n Case II: + = 1 In this case, the slpe f the g A 0 line 1 1 parallel (r cincident) If n > 0, then they are parallel and the tw lines are

6 60 David Rmer s R&D mdel g K n 0 g A The ecnmy will mve int the channel between the lines and then grwth in bth K and A will accelerate frever. Intuitively, a bigger ecnmy means mre scientists means mre discveries means faster grwth. As lng as n > 0, the exgenus grwth in the labr frce leads t accelerating grwth. If n = 0, then the tw lines cincide g K 0 g A Ecnmy cnverged t the line and n the line, bth grwth rates are cnstant and equal (because the line has slpe f ne)

7 Micrecnmics f R&D 61 * * Because gk ga, K/A is cnstant in the steady state There is a unique K/A* that will sustain equal grwth in K and A and a unique cmmn grwth rate g* that is cnsistent with that K/A* Yu will wrk ut the algebra in Prblem 3.5. Example f this case: Suppse that B s that c A increases. This raises g A and mves the ecnmy t a pint t the right f the riginal equilibrium. Ecnmy cnverges back up and t the left t a new equilibrium that is higher than riginal. Endgenus grwth ccurs in this case: ecnmy sustains psitive grwth even when there is n exgenus surce (n = 0) Grwth rate depends (psitively) n s, B, a K, a L, and L Case III: + > 1 In this case, the g A 0 line is flatter than the g K 0 because 1 1 This case lks like Case II, but the lines are nt parallel. In this case, we get explsive grwth even when n = 0. Micrecnmics f R&D The key questin that we have ddged in Rmer s R&D mdel: What determines a K? Capital wners must decide whether t build factries r labs Ecnmists wuld assume that the chse the use f their capital that prvides the higher rate f return S in equilibrium the amunt f capital in the tw sectrs wuld have t balance the marginal rates f return Rate f return n factries is straightfrward: They prduce utput that is sld t earn revenue Hw d labs earn mney? In the real wrld, there are lts f funding surces fr R&D Crprate funding Gvernment grants Tuitin frm university students Since we dn t mdel gvernment r university research, we are interested mstly in crprate-funded research and develpment In ur mdel, knwledge is purely nn-rival and nn-excludable Any discvery is immediately useful t all prducers

8 62 Mdel f Learning by Ding There is n apprpriability f knwledge fr private benefit New knwledge cannt be sld r used prfitably Why wuld capital wners put mney int labs that earn nthing? They wuldn t, s we wuld need t build a mdel f hw lab wners can earn mney frm R&D in rder t pay fr the capital and labr that is used. Mdels f a K Crprate R&D is prfitable if there is an effective way fr the cmpany t apprpriate the knwledge This usually ccurs by preventing ther firms frm using the knwledge created thrugh sme kind f apprpriability mechanism May als invlve licensing Nte that either is inefficient, because nce created the knwledge is nnrival and shuld be universally used fr free Tw cmmn apprpriability mechanisms are intellectual prperty rights (patents) and secrecy Bth are flawed Sme kinds f intellectual prperty are better prtected by patents, sme by secrecy, and thers are virtually unprtectable Effective patent prtectin r secrecy gives an effective (but usually temprary) mnply n the use f the knwledge t the firm ding the R&D Tw cmmn mdels fr a K are based n this: A mdel f prduct innvatin in which R&D can prduce new varieties f (intermediate) gds n which the innvating firm hlds a mnply A mdel f prcess innvatin in which R&D can advance prductive efficiency f ne (intermediate) gd (f many) and have a cst advantage in prductin until anther firm leap-frgs it Bth mdels add cmplexity t Rmer s R&D mdel because bth require multiple gds in rder t have mre than ne firm We study bth mdels in Ecn 454 Mdel f Learning by Ding Rmer s shrt sectin n learning by ding develps the essence f Paul Rmer s first (1986) endgenus grwth mdel. Kenneth Arrw develped a mdel in the 1960s based n the idea that a firm s A wuld be increased as it prduced utput, s A Y Paul Rmer s versin f this was slightly different Firms learning is related t capital accumulatin rather than utput Knwledge is nn-apprpriable

9 Mdel f Learning by Ding 63 New knwledge ccurs as a by-prduct f capital investment Firms have (sme) incentive t invest, s knwledge creatin happens despite pure nnrivalry Learning by ding with a cnstant saving rate BK t Y t K t A t L t A t Y t K t B K t L t Slving ut A yields K t sy t sb Lt Kt This mdel cnverges, has endgenus grwth, r expldes as < 1, = 1, > 1 Case f = 1 is the endgenus-grwth case Let n = 0 s there is n exgenus grwth 1 YtBL Kt bkt K t sy t sbk t K t sb K t Thus, grwth in the capital stck and utput is cnstant at rate sb Any increase in saving, in the prductivity f learning, r in the labr frce wuld increase grwth Ramsey cnsumers in the learning-by-ding mdel (nt dne this way in 4 th editin) Assume = 1 and n = 0, s we have the endgenus-grwth case Aggregate knwledge is prprtinal t aggregate capital stck (but this is nt the case at the firm level) Firms take aggregate knwledge as given and d nt cnsider hw their wn investment will add t it because they are small 1 BK t 1 Yi t Ki t A t Li t A t 1 1 i i i Y t B K t K t L t The private marginal prduct f capital is 1 Yi t 1 1 Ki t Ki t Li t B K t r t Ki Each firm sets its Li ecnmy-wide interest rate r (there is n depreciatin) s that the private marginal prduct equals the

10 64 Mdel f Learning by Ding This means that all firms have the same K/L Setting K i /L i = K/L, 1 r t B K t 1 1 Kt Lt B K t K t L t BL t BL r Ki L i that is equal t the aggregate This rate f return r is cnstant ver time (with n = 0) and depends n the rate f knwledge accumulatin thrugh investment B,, and the size f the labr frce Nte that the marginal scial prduct f capital (varying K as well as K i ) is larger than the marginal private prduct Y MSPK BL MPP K i 1 i Ki K This means that individual firms will underinvest in capital They d nt take int accunt the psitive scial externality that their investment cnveys n all firms thrugh increased knwledge This means that the privately generated grwth rate will be lwer than the scially ptimal grwth rate Ramsey cnsumers, as usual, chse a cnsumptin path that satisfies the Euler C t rt equatin. C t In this case, rt C t r and C t K BL 1 r g T satisfy the ecnmy s budget cnstraint, Y must grw at the same rate as C, s the ecnmy grws at g at every instant. There are n cnvergence dynamics: wherever an ecnmy is, it just grws at g frm there. (Pr cuntries with same parameters d nt catch up.) Grwth rate g depends n parameters f ecnmy:, B, L,,. Once again, we have scale effects because a larger L means a faster grwth rate. If we allw n > 0, then we have bth endgenus and exgenus grwth and the grwth rate accelerates ver time.

11 (Paul) Rmer mdel (nt wrth ding the details) 65 Are scale effects realistic? Sme argue n, but Kremer s argument fr Eurasia, Australia, and Tasmania seems t prvide sme supprt. In additin, there is much evidence that grwth has accelerated ver the centuries (as ppulatin has grwn). Nn-ptimality: scial planner wuld internalize the knwledge externality and use r* BL r BL 1 1 leading t faster grwth at g* BL 1 g (Paul) Rmer mdel (nt wrth ding the details) What s different abut this mdel? We mdel the incentives fr prductin f knwledge explicitly We intrduce the Ethier prductin functin and the nw-ubiquitus mdel f a cntinuum f intermediate gds Human Capital in the Slw Mdel Distinctin between knwledge capital and human capital Latter is rival and embdied in wrker Frmer relates t nnrival ideas that all share (cstlessly) Mdel is mtivated by the dminant questin: Why are sme cuntries richer than thers? Slw mdel says differences in k Nt plausible (as Rmer shws late in Ch 1) Mankiw, Rmer, & Weil: differences in physical and human capital They argue this is plausible; thers disagree Differences in A Why wuld technlgy be different acrss cuntries? Barriers (legal and therwise) t adptin Nn-applicability f advanced technlgies in pr cuntries (climate, unreliable physical infrastructure, etc.) Differences in scial infrastructure We ll have mre t say abut this sn Hw t incrprate human capital int mdel? Many alternative ways; Rmer des ne (and thers in prblems 4.8 and 4.9) Hw des ecnmy prduce human capital? Prcess f educatin r training has tw majr csts: teachers time (fr which they are paid) and students time (fr which they are nt paid)

12 66 Human Capital in the Slw Mdel Can use a tw-sectr mdel with a prductin functin fr educatin using labr (teachers) and capital (schls) like the ne fr knwledge in the R&D mdel Can just deduct sme amunt f a cnglmerate utput as being educatin in a ne-sectr mdel (like sme utput is physical capital rather than cnsumptin). This is Rmer s 4.8. Can mdel the prcess as hlding peple ut f the labr frce during an educatin perid. This is Rmer s Sectin 4.1. This desn t mdel the cst f teachers and schls. Nte that frgne earnings may be higher than teacher/schl csts at mst schls (if maybe nt at Reed) Simple human-capital mdel setup Let H t LtGE be the amunt f human capital, which is the number f wrkers Lt times the amunt f human capital per wrker GE educatin level f current wrkers. G E 0 G E E e is a cmmnly used functinal frm, where E is the average We assume that in a steady state with educatin level E, peple live T years, ging t schl fr E years and wrking fr T E years. In general (but nt in this mdel), human capital includes nt just educatin but training, health and ther acquired characteristics that affect labr prductivity. 1 Y t K t A t H t K t sy t K t A t gat L t nlt Slving the mdel This mdel lks (and behaves) similarly t Slw mdel Define K K k AH ALG E k t sf ktn gkt s k 0 k k* n g 1 1

13 Human Capital in the Slw Mdel 67 Hw will a change in E affect the steady-state grwth path? Effects f E (r G ) n K and Y are equivalent t increase in L Ecnmy mves t higher, parallel steady-state path Level effect, but n grwth effect Y and Y/L are higher in steady-state But the imprtant variable (living standards) here is Y/N, where N is ttal ppulatin * * Y t L t y* AtGE n the steady-state path N t N t Increase in E des nt affect y* r A(t) Increase in E raises G(E) Increase in E lwers L/N because mre peple are in schl and fewer in the labr frce What will be the net effect? What is L/N? It seems like it shuld be T E/ T since that is the rati f wrking years t ttal life years fr each individual That is crrect if n = 0 If the ppulatin is grwing, then the chrt in educatin is larger than the chrt that is wrking. Rmer (and Cursebk) shws that in steady state ne nt Lt e e nt N t e 1 It is intuitively clear (and mathematically easy) that L/ N E 0 Dynamics f increase in E Initial effect lwers Y because fewer peple in labr frce but n immediate increase in the educatin f thse wh are wrking In steady state, the tw effects nted abve are in cnflict and we dn t knw which will dminate / Y / N Y / N Y / N L/ N E E L N E The first term depends mstly n G E If G E and the secnd is negative. is large, then Y/N is likely t rise with an increase in E

14 68 Human Capital in the Slw Mdel This makes intuitive sense: if educatin is highly prductive it will raise percapita incme; if it is nt, then it drains peple wh culd be wrking int useless educatin.