4 Intrductin t Macrecnmics Intrductin t Macrecnmics Macr mdels Simplistic representatins f real-wrld phenmena Designed t illustrate ne r mre specific features f macrecnmy, nt all May be useful fr sme situatins and questins; never fr all Equatins and variables (smetimes can wrk with graphs, but ften t many dimensins) Appraches t macr mdeling Mankiw s scientist and engineer rles Engineer is mre imprtant when we have a prblem t slve Scientist is mre imprtant fr lng-run knwledge advancement Relative imprtance varies ver time Current crisis favrs engineering apprach Tp dwn Lks at relatinship amng macr variables Empirical r rule f thumb theries Usually the best shrt-term predictrs Basis fr mst cyclical plicy analysis Lst hpe fr fine-tuning Tenuus welfare implicatins because n utility functins Can make fundamental errrs if structure changes May imply implausible behavir by individuals Bttm up Build frm micrfundatins Classical and mdern thery emphasize this Assumptins abut behavir are explicit, but usually very simplistic Math can be difficult except with simplest setups Easier t cpe with structural changes May be less directly predictive and less useful fr specific plicy calibratin Lucas example: predicting my rute t the airprt Macr variables Output flw (GDP) can be thught f as prductin, incme, and expenditure Definitin: value, final, in ecnmy, in perid Real vs. nminal GDP, shw summatin frmulas Nte strengths and weaknesses f real GDP as welfare measure
Intrductin t Macrecnmics 5 Prices/inflatin Price index measurement Definitin f inflatin, deflatin, disinflatin Is inflatin bad? Emplyment/unemplyment Less directly analyzed: tends t crrelate with Y ver cycle Interest rates Term, risk, etc. Exchange rates Grwth vs. business cycles Usually mdeled separately We d grwth first, then cycles Evlutin f a macr thery Unexplained empirical phenmenn Theries (mdels) t explain basic facts Empirical testing f implicatins f theries Revisin/rejectin/acceptance f mdels based n empirical perfrmance
6 Slw Grwth Mdel: Expsitin Slw Grwth Mdel: Expsitin Mdel grew ut f wrk by Rbert Slw (and, independently, Trevr Swan) in 1956. Describes hw natural utput (Y n assuming full efficiency) evlves in an ecnmy with a cnstant saving rate Key questin: Can an ecnmy sustain perpetual grwth in per-capita incme thrugh nging increases in capital? (Answer: N) Aggregate prductin functin The center-piece f every grwth mdel is the aggregate prductin functin Des an aggregate prductin functin exist? Yes, if all firms have cnstant returns t scale and face the same prices fr labr and capital. In Slw mdel, we write as Yt F Kt, AtLt We use (t) ntatin because we are wrking in cntinuus time See Cursebk Chapter 2 fr details We will suppress the time dependence when it isn t needed A(t) is an index f technlgy r prductivity We mdel as Harrd neutral because it is cnvenient and leads t reasnable cnclusins Cnditins n prductin functin MPK is psitive and diminishing MPK = FK K, AL 0 Briefly discuss partial derivative and what it means FKK K, AL 0 MPL is psitive and diminishing MPL = FL K, AL 0 FLL K, AL 0 Increase in K raises MPL (and vice versa): FKL K, AL 0 Cnstant returns t scale: F ck, cal cf K, AL, c 0 Intensive frm f prductin functin Since c can be any psitive number, let c = 1/AL K 1 Y AL AL AL F,1 FK, AL AL is the amunt f effective labr r the amunt f labr measured in efficiency units
Slw Grwth Mdel: Expsitin 7 This is nt imprtant fr itself, but is a useful analytical magnitude. Fr interpretatin purpses, we will be mre cncerned with the behavir f Y and Y/L than with Y/AL. Y K Define y and k, and let f F,1 AL AL Then y f k expresses the intensive frm f prductin functin MPK = f k 0 f k 0 Graph f intensive prductin functin is increasing at a decreasing rate Inada cnditins lim f k k0 assures that intensive prductin functin is vertical as it leaves the rigin: MPK is infinitely large if we have n capital and finite labr. lim 0 assures that the intensive prductin functin eventually f k k becmes hrizntal as k increases t infinity: MPK becmes zer as capital is super-abundant. Equatins f mtin and structure f ecnmy Labr supply grws at cnstant exgenus (cntinuusly cmpunded) rate n L tnlt, L t n. Lt nt L t L0 e Technlgy/prductivity imprves at cnstant exgenus rate g A t gat, A t g. At gt A t A0 e Output is used fr cnsumptin gds and investment in new capital (n gvernment spending, clsed ecnmy) Y tc t I t Husehlds allcate their incme between cnsumptin and saving Y t C t St, I t St Capital accumulates ver time thrugh investment and depreciates at a cnstant prprtinal rate K t I tk t Key assumptin: Saving is cnstant share f incme s:
8 Slw Grwth Mdel: Expsitin St sy t, s K t sy t K t Slw s central questin Using the equatins f the mdel: Y S I K Y Can this prcess lead t sustained grwth in utput frever? Can capital deepening alne lead t eternal imprvements in living standards? T anticipate the result f ur analysis: N. Given the path f labr input, increases in capital lead t decreasing effects n utput because we have assumed diminishing marginal returns t capital If we had a plausible mdel in which marginal returns t capital were nt diminishing, then the answer culd reverse. Centuries f ecnmic analysis uses law f diminishing marginal returns and evidence seems supprtive. Is it plausible fr marginal returns t be nn-diminishing? Perhaps fr an augmented cncept f capital Mdern theries f endgenus grwth cnsider human capital and knwledge capital alng with physical capital These theries (discussed in Rmer s Chapter 3) allw fr nndiminishing returns t a bradened cncept f capital and change the answer t Slw s questin
Slw Grwth Mdel: Steady-State Grwth Path 9 Slw Grwth Mdel: Steady-State Grwth Path Cncepts f dynamic equilibrium What is an apprpriate cncept f equilibrium in a mdel where variables like Y and K grw ver time? Must cnsider a grwth path rather than a single, cnstant equilibrium value Stable equilibrium grwth path is ne where If the ecnmy is n the equilibrium path it will stay there If the ecnmy is ff the equilibrium path it will return t it Equilibrium grwth path culd be cnstant K, cnstant rate f grwth f K, r smething cmpletely different (scillatins, explsive/accelerating grwth, decay t zer, etc.) We build n the wrk f Slw and thers wh determined the nature f the equilibrium grwth path fr ur mdels. As lng as we can demnstrate existence and stability, we knw we have slved the prblem. In Slw mdel (and thers), the equilibrium grwth path is a steady state in which level variables such as K and Y grw at cnstant rates and the ratis amng key variables are stable. I usually call this a steady-state grwth path. Rmer tends t use balanced grwth path fr the same cncept. Finding the Slw steady state In the Slw mdel, we knw that L grws at rate n and A grws at rate g. The grwth f K is determined by saving. Since Y depends n K, AL, it seems highly unlikely that utput is ging t be unchanging in steady state (a statinary state ). Easiest way t characterize Slw steady state is as a situatin where y and k are cnstant ver time. K Since k, k K A L K g n, s if k is unchanging, k 0 AL k K A L K and K must be grwing at rate g + n. Using the equatin abve and substituting fr K yields k K sy K gn g n k K K sy sy sf k g n gn gn. K k k k sf k gn k
10 Slw Grwth Mdel: Steady-State Grwth Path This is the central equatin f mtin fr the Slw mdel Graph in terms f y and k: y y=f(k) (n+g+δ)k = breakeven investment sf(k) = saving/inv per AL k 1 k* k 2 k Breakeven investment line: Hw big a flw f new capital per unit f effective labr is necessary t keep existing K/AL cnstant? Must ffset shrinkage in numeratr thrugh depreciatin and increase in denminatr thrugh labr grwth and technlgical prgress: Need fr each unit f k t replace depreciating capital Need n fr each unit f k t equip new wrkings Need g fr each unit f k t equip new technlgy The mre capital each effective labr unit has the bigger the new flw f capital that is required t sustain it: breakeven investment is linear in capital per effective wrker. At k 1 the amunt f new investment per effective wrker (n curve) exceeds the amunt required fr breakeven (n the line) by the gap between the curve and the line, s k is increasing ( k 0 ). At k 2 the amunt f new investment per effective wrker falls shrt f the amunt required fr breakeven, s k is decreasing ( k 0 ). At k* the amunt f new investment per effect wrker exactly balances the need fr breakeven investment, s k is stable: k 0. At this level f k the ecnmy has settled int a steady state in which k will nt change.
Slw Grwth Mdel: Steady-State Grwth Path 11 Shw graph with k n vertical axis. k 1 k* k 2 k In this graph, k 1 and k 2 have same interpretatin as in earlier graph. Existence and stability Will there always be a single, unique intersectin f the line and curve? Yes. Diminishing returns assumptin assures that curve is cncave dwnward. Inada cnditins assure that curve is vertical at rigin and hrizntal in limit. Fr any finite slpe f the breakeven line, there will be ne intersectin with curve. Because k k* k 0 and k k * k 0, if ecnmy begins at any level f k ther than k* it will cnverge ver time tward k*. Steady-state grwth path exists, is unique, and is stable. Characteristics f steady-state grwth path We nw cnsider the behavir f macrecnmic variables when a Slw ecnmy is n its steady-state grwth path. These are the crucial utcmes f the Slw analysis. k and y are cnstant Capital per effective labr unit k is unchanging ver time in the steady state. Since utput per effective labr unit y depends n k thrugh the prductin functin, it is als unchanging. K grws at rate n + g
12 Slw Grwth Mdel: Steady-State Grwth Path A L g, n, s AL grws at n + g. A L K kal must grw at n + g (r numeratr and denminatr f k must grw at same rate fr it t stay cnstant) Y grws at rate n + g Can make the same argument fr y and Y as fr k and K Alternatively, Y F K, AL with cnstant returns t scale. Bth K and AL are grwing at n + g, s each factr expands by n + g each year and ttal utput must als expand at n + g. Y/L grws at g This is the mst imprtant f the utcmes fr standards f living. Y grws at n + g and L grws at n, s the qutient grws at the difference: g This means that in the steady state, living standards (utput per persn) grw at the rate f technlgical prgress. Capital deepening alne cannt sustain nging grwth in per-capita utput Only technlgical prgress can lead t imprvements in utput per wrker Grwth in this mdel is exgenus : utput grwth = n + g, bth f which are taken as given frm utside the mdel Determinants f steady-state path What determines the value f k* and therefre y*? Anything that shifts the curve r the line in the diagram will change k* and y*. Increase in s Intuitively: we think that an increase in saving shuld lead t mre capital accumulatin and higher steady-state k and y Graphically: An increase in s shifts sf (k) curve upward, leading t k 0 at riginal k* and mvement t the right Ecnmy cnverges t a new steady-state grwth path with a higher k* and y* Hwever, nte that the grwth rate n the new path is still the same: K and Y grw at n + g and Y/L grws at g Changes in the saving rate have a level effect nt a grwth effect in the Slw mdel Shw parallel grwth paths Increase in Intuitively: we think that mre depreciatin shuld lead t less capital accumulatin and lwer steady-state k and y Graphically: An increase in makes the break-even line n g k steeper, leading t k 0 at riginal k* and mvement t the left
Slw Grwth Mdel: Steady-State Grwth Path 13 Ecnmy cnverges t new steady-state grwth path with a lwer k* and y* Again, these are level effects, nt grwth effects: the grwth rates are the same Increase in n Intuitively: Mre rapid ppulatin grwth shuld allw ecnmy t grw faster because labr input is grwing faster, but given the saving rate it will be harder t accumulate capital per wrker because the higher birth rate means mre new wrkers must be equipped Graphically: Higher n makes the break-even line steeper, leading t k 0 at riginal k* and mvement t the left Ecnmy cnverges t a new steady-state grwth path with a lwer k* and y* Hwever, the grwth rate f Y and K n this new path is greater than the riginal grwth rate because n + g has increased. Shw new path fr Y that is lwer but steeper than ld ne. Grwth rate f Y/L has nt changed: it still grws at g. Level effect causes new grwth path fr Y/L t be parallel but lwer Increase in g Intuitively: Faster technlgical prgress shuld allw ecnmy t grw faster, bth in aggregate and in per-capita terms. Less intuitively: Because f the way that the prductin functin incrprates technlgy, an increase in technlgical prgress means that mre investment is needed t keep up with the grwth in AL, thus making it harder t accumulate capital vis a vis AL. Graphically: Higher g makes the break-even line steeper, leading t k 0 at riginal k* and mvement t the left Ecnmy cnverges t a new steady-state grwth path with a lwer k* and y* Just as with increase in n, the grwth rate f Y and K n this new path is greater than the riginal grwth rate because n + g has increased. Shw new path fr Y that is lwer but steeper than ld ne. Nw the grwth rate f Y/L has increased as well. Its new path is als steeper but starting frm a lwer level. Glden-Rule grwth path Given that the saving rate affects k* and y*, we might cnsider asking the questin What is the ptimal saving rate? What d we mean by ptimal? We dn t have utility functins, s we cannt really cnduct welfare analysis. Is highest pssible y* ptimal? This wuld imply s = 1 is best.
14 Slw Grwth Mdel: Steady-State Grwth Path If s = 1, then cnsumptin = 0, which desn t seem like high utility Perhaps it makes sense t maximize C/L? Yes, because cnsumptin yields utility But what C/L? Setting s = 0 maximizes current cnsumptin given the current capital stck, but k will fall ver time s y will fall and C cannt be maintained Glden-Rule criterin: ignre current cnsumptin and fcus n sustainable steady-state level: maximize path f (C/L) * This is called the Glden Rule path because it gives equal pririty t future generatins Given that A is exgenus, we can maximize c* = (C/AL) * and it leads t the same result. Graphical analysis f Glden Rule As we cnsider alternative values f s, sf (k) pivts upward r dwnward, causing k* and y* t take n higher r lwer values At any steady-state k* crrespnding t a particular s, the level f c* is measured by 1 s f k *, which is the vertical gap between f k * and sf k * As we pivt the sf k curve by changing s, hw d we make the vertical gap as large as pssible? This happens when the prductin functin is parallel t the break-even line at the pint where the saving curve intersects the break-even line Prperties f Glden-Rule path Slpe f prductin functin is f k and slpe f break-even line is n g, s mathematical cnditin fr Glden Rule grwth path is: Set s s that f k * GR n g In capital-market equilibrium (which we ll study later n), prfit-maximizing firms want t hire capital up t the pint where the net marginal prduct f capital (MPK ) equals the interest rate, s r f k Thus, n Glden-Rule path: r n g, r the real interest rate equals the real grwth rate f the ecnmy. If r n g, then the ecnmy is t the right f the GR path (has higher k* than GR) and is verinvesting in capital: the rate f return n capital has been driven dwn t lw and everyne (current and future generatins) wuld be better ff with a lwer s. Such an ecnmy is dynamically inefficient. If r n g, then ecnmy is t the left f GR path (has lwer k*). In this case, steady-state k* falls belw the level that maximizes c*. The
Slw Grwth Mdel: Steady-State Grwth Path 15 ecnmy is nt saving enugh t prvide the best pssible living standard in perpetuity. It is, hwever, enjying a higher level f cnsumptin nw than it wuld n the Glden Rule path: increasing the saving rate wuld lwer the cnsumptin f current peple but raise that f future generatins in the steady state. Because sme peple are better ff and thers wrse ff under this situatin, we d nt knw if it is efficient r inefficient. It depends n hw family dynasties value current vs. future cnsumptin. We need an explicit intertempral utility functin t determine the efficient level f saving We will d this in Rmer s Chapter 2 in the Ramsey grwth mdel.
16 Cnvergence in the Slw Mdel Cnvergence in the Slw Mdel We nw knw hw the Slw ecnmy behaves when it is n its steady-state grwth path and have shwn that it will in sme manner cnverge t this path. What frm will this cnvergence take and hw lng will it take? We use the secnd frm f equilibrium graph abve. Apprximating the cnvergence functin We dn t knw the functinal frm f f s we can t evaluate k crrespnding t k 0 directly. using a first-rder Taylr series apprximatin arund the We can apprximate k k 0 steady state: What line (first-rder plynmial) wuld give the best apprximatin f the unknwn functin in a small neighbrhd f k*? Shuld pass thrugh the true value f k at k = k*, which is zer. Shuld have same slpe as the true functin k k* at k = k*, which is sf k * n g This is the first-rder Taylr apprximatin in a neighbrhd f k* Yu can see frm graph that apprximatin is pretty gd very clse t k* but nt s gd further away (if functin has strng curvature) Linear apprximatin t arund k*: k 0 k* k
Cnvergence in the Slw Mdel 17 Slpe f tangent line is sf k * n g, which we can trture algebraically t get smething smewhat intuitive: We knw that at the steady-state k, sfk * n g k*, s s f k* The slpe f the tangent is k sfk* n g k kk* n gk* f k* n g f k* f k* k* 1 n g f k* K 1n g1k n g. K is capital s share if wners f capital are paid its marginal prduct: f is payment t each unit f capital k* is number f units f capital (per AL) Numeratr is ttal payments t capital f is ttal utput = ttal payments t all factrs in ecnmy Rati is share f ttal factr payments that g t capital Empirically, K is abut 1/3. n g k*. Our apprximatin is the linear, first-rder differential equatin kt kt k* Slutin t this equatin (which yu dn t need t knw) invlves a time path fr k frm a given initial value k(0). k t k* e k 0 k*. t In this case, the slutin is The bracketed term k 0 k* is the initial gap between k and k* The slutin equatin says that after t perids, the remaining gap between actual t and steady-state k will be e share f the riginal gap at time 0. Same cnvergence prcess applies t y as t k, s y als cnverges at expnential rate. Cnvergence implicatins f mdel What is the pattern f cnvergence? Asympttic expnential cnvergence with a given half-life fr the gap between actual and steady state Never actually reach steady state, but after sufficient time we are arbitrarily clse
18 Cnvergence in the Slw Mdel Shw graph f cnvergence t linear path fr lg y Hw lng des cnvergence take? t 1 K n g t T determine the half-life, set e e and slve fr t. Calibratin: Rmer argues that n g 6%, K 1/3, s 0.04. 0.04t e 1 2 t 18, s the Slw mdel (with this calibratin) predicts: The ecnmy mves abut 4% f the way tward the steady-state path each year It takes abut 18 years t eliminate half f the gap between actual and steady-state per-capita incme Empirical evidence There have been dzens f studies f cnvergence acrss cuntries Mst f the evidence suggests that there is cnvergence, if ne cntrls fr variatins in such variables as the saving rate that might affect the level f a cuntry s steady-state path. Estimated cnvergence rates are lwer than 4%, typically clser t 2% per year. (As in Barr/Sala-i-Martin paper f the week) The 2% cnvergence rate wuld fit the calibratin f the mdel if K were 2/3 instead f 1/3. Mankiw, Rmer, and Weil (in ne f the papers f the week) suggest that this is plausible if ne cnsiders human capital as part f K rather than L. 1 2
Natural Resurces in the Slw Mdel 19 Natural Resurces in the Slw Mdel What are the implicatins f sustainability f natural-resurce use fr the Slw mdel? Rmer shws us a stylized mdel with finite resurces, but the cnclusins depend crucially n sme fairly arbitrary assumptins. Prductin with land and natural resurces Let R(t) be the amunt depletable natural resurces used up in prductin at time t Let T(t) be the amunt f land used at time t This mdel is difficult t slve with general functinal frm, s we assume that the prductin functin is Cbb-Duglas: 1 Y t K t R t T t A() t L t Cnstant returns t scale impsed All expnents assumed psitive Nte: Cbb-Duglas assumptin is nt inncuus N prductin is pssible withut R r T Any amunt f Y can be prduced with arbitrarily small amunts f R and T if nly the levels f K and L are large enugh Elasticity f substitutin amng factrs is assumed t be ne These assumptins may be valid, but it is imprtant t recgnize that we have (implicitly) made them when we chse the Cbb-Duglas frm Equatins f mtin fr resurces T t 0 R t Rt : Land is fixed b 0 If the rate f use f natural resurce is cnstant (r grwing) ver time, we will eventually run ut. The nly pssible steady state is with natural resurce use declining sufficiently rapidly that we d nt run ut. As in the standard Slw mdel: Labr grws at n Technlgical prgress at g Cnstant saving rate s leads t K t K t Y t s K t
20 Natural Resurces in the Slw Mdel Steady-state analysis Can we find a variable like k K/ AL that will be cnstant in steady state? Will there be balanced grwth in the same sense as the usual Slw mdel? All factrs cannt grw at same rate, because AL grws at n + g T grws at 0 R grws at b K grws at sme as-yet-undetermined rate Alternative strategy fr finding steady-state equilibrium: cnstant grwth rates Search fr an equilibrium in which K is grwing at a cnstant rate Fllw prcess here that will wrk fr many mdels K t Y t Frm abve, s K t K t t t K If is t be cnstant ver time n a steady-state grwth path, then K the right-hand side must be cnstant, and since s and are cnstant that Y t means that must be cnstant in the steady state. K t t t Y If is t be cnstant n the steady-state grwth path, then Y must K be grwing at the same rate as K in the steady state. Let s call that cmmn steady-state grwth rate g Y *. Using the Cbb-Duglas prductin functin (this is why we need that assumptin), we can use ur rules f grwth rates f prducts and pwers t get Y K R T A L 1, which hlds at every mment. Y K R T A L Y K * * * In the steady state, gy, s gy gy b 1 g n r Y K g * 1 Y gn b. 1 Rmer shws that under reasnable cnditins this steady-state grwth path is unique and stable. Implicatins f mdel Grwth rate f per-capita incme is * b g n gy/ L g g 1
Natural Resurces in the Slw Mdel 21 Thus there is a grwth drag intrduced by the presence f natural resurces and land in the prductin functin. Grwth drag term is zer if = = 0 (which gets us back t a Cbb-Duglas versin f the basic Slw mdel). Rmer cites Nrdhaus estimates that the grwth drag may be ~ 0.25% per year. Rmer ntes that Cbb-Duglas assumptin that elasticity f substitutin amng factrs is ne may nt be crrect. If elasticity f substitutin is bigger than ne, then it is relatively easy t substitute amng inputs and the grwth drag will be smaller: it is easier t get alng withut R and T as they becme scarce relative t K and L. If elasticity f substitutin is less than ne, then it is difficult t substitute and the grwth drag will be mre severe. Sme empirical evidence suggests elasticities f substitutin > 1, s grwth drag may nt (in neighbrhd f tday s equilibrium) be as large as Nrdhaus s estimates.
22 Summing Up the Slw Mdel Summing Up the Slw Mdel Simple mdel f evlutin f capacity utput in an ecnmy with cnstant (exgenus) saving rate. There are unique, stable steady-state grwth paths fr K, Y, Y/L t which the ecnmy cnverges asympttically at an apprximately expnential rate. Output grws in the steady state thrugh (exgenus) increases in labr frce and prductivity (g + n). Output per wrker grws at the rate f prductivity grwth g. Changes in the saving rate have level, nt grwth, effects n the steady-state grwth path. Steady-state cnsumptin per wrker is maximized n the Glden Rule grwth path where r = n + g, but this may r may nt be ptimal depending n hw ne weights current vs. future well-being. Intrducing depletable natural resurces int the mdel puts a drag n steady-state grwth in a Cbb-Duglas versin f the mdel. One crucial weakness f the Slw mdel is the ad-hc assumptin f a cnstant saving rate. Is this cnsistent with ptimal, utility-maximizing cnsumer behavir? If we had a utility functin, we culd mdel ptimal cnsumptin and saving behavir and als perfrm welfare analysis balancing present vs. future. That is the rle f the Ramsey mdel, t which we nw turn.