Lecture 8Endogenous Growth ModelsConsumption and SavingsNoah WilliamsUniversity of Wisconsin - MadisonEconomics 312Williams Economics 312An ExampleLet’s use the Solow model to analyze the effects of immigration.For simplicity suppose that there is no productivity growth,that the population grows at a constant rate n, the savings rateis the constant s, and depreciation is the constant d. Supposethat the economy is initially in the steady state.1Now suppose that there is a one-time increase in the laborforce from immigration, but n remains constant. Analyzethe short-run and long-run effects of this change for thelevels of wages and per-capita output, and the growth ratesof (total) output and per-capita output.2Now suppose that immigration is a continuing process sothat n increases to a higher value n0. Analyze the short-runand long-run effects of this change for the levels of wagesand per-capita output, and the growth rates of (total)output and per-capita output.Williams Economics 312Endogenous Growth ModelsNow briefly discuss some models which try to explainsources of growth endogenous growth models.An active research topic in late 1980s-1990s, not as muchlately. Romer (1986, 1990), Lucas (1988) most influential:models of R & D, human capital.More recently Acemoglu et al: role of institutions ingrowth.Williams Economics 312What’s in TFP? Institutions & GeographyAside from innovations (which we’ll turn to next),infrastructure, institutions, and geography are alsoimportant.Interesting comparison: experiences of former colonies.Acemoglu, Johnson and Robinson (2001).Small initial differences in income.Differences in settlers mortality influenced whether colonywas run for “extraction” or whether colonists developedinstitutions. Those colonies where institutions took holddeveloped faster.Large differences in outcomes – still today!Williams Economics 312Williams Economics 312What’s in TFP? Ideas and Human CapitalRelatively new branch of economic theory: endogenousgrowth theory seeks to explain how technical changehappens.Simple endogenous growth model (AK model): aggregateproduction function Y = AK. (Ignore labor andpopulation growth, could think of this as per capitaproduction.)Not subject to diminishing returns: MPK is constantFK=YK= A.Idea: Aggregate capital K captures not just increases inphysical capital but changes in the makeup of that capital.Williams Economics 312Human Capital as a Source of GrowthHuman capital: knowledge, skills, and training ofindividuals. As economies become richer they invest inhuman capital in the same proportion, offsetting thediminishing marginal product of physical capital alone.Explicitly: production depends on human capital H ,physical capital K :Y = zHθK1−θSay H = hK, so that human capital is constant fraction ofphysical, then letting A = zhθ:Y = z(hK)θK1−θ=hzhθiK = AKWilliams Economics 312Other InterpretationsResearch and development programs are part of capitalinvestment. They increase the stock of knowledge, whichoffsets diminishing marginal products of capitalaccumulation.Learning by doing: as economies produce more they learnbetter how to produce.Williams Economics 312Implications of the Endogenous Growth ModelAgain savings constant fraction s of output. So:˙K = sAK − δKSince Y = AK ,˙YY=˙KK= sA − δGrowth of output depends on the saving rate, even in thelong run. No steady state.Higher savings ⇒ more human capital, R&D, learning bydoing. So higher savings leads to productivityimprovements and higher growth.Important implication, some evidence that measured TFPdoes depend on savings, human capital.Williams Economics 312A More Explicit Model of Human CapitalCobb-Douglas aggregate production function:Y = KαHβ(AN )1−α−βAgain we have constant returns to scale now in (K , H , N ).Human capital and labor enter with different coefficients.Society accumulates human capital according to:˙H = shY − δHCapital accumulation equation:˙K = skY − δKTechnological progress:˙AA= g > 0.Labor force grows at constant rate:˙NN= n > 0.Williams Economics 312Analyzing the ModelDividing the production function by AN :˜y =˜kα˜hβDecreasing returns to scale in per efficiency units.The evolution of inputs is determined by:˙˜k = sk˜kα˜hβ− (n + g + δ)˜k˙˜h = sh˜kα˜hβ− (n + g + δ)˜hSystem of two differential equations determining˜k,˜h.Williams Economics 312Balanced Growth PathTo find the BGP equate both equations to zero:sk˜k∗α˜h∗β− (n + g + δ)˜k∗= 0sh˜k∗α˜h∗β− (n + g + δ)˜h∗= 0From first equation:˜h∗=(n + g + δ)sk˜k∗1−α1βPlugging it in the second equationsh˜k∗α(n + g + δ)sk˜k∗1−α− (n + g + δ)n + g + δsk˜k∗1−α1β= 0 ⇒shsk˜k∗=n + g + δsk˜k∗1−α1βWilliams Economics 312Finding the Balanced Growth Pathshsk˜k∗=n + g + δsk˜k∗1−α1β⇒˜k∗1−1−αβ=˜k∗−1−α−ββ=sksh(n + g + δsk1β⇒˜k∗=s1−βksβhn+g+δ11−α−β˜h∗=sαks1−αhn+g+δ11−α−βWilliams Economics 312The Balanced Growth PathUsing the production function:˜y =YAN=˜kα˜hβ= s1−βksβhn + g + δ!α1−α−β sαks1−αhn + g + δ!β1−α−β⇒y =YN= s1−βksβhn + g + δ!α1−α−β sαks1−αhn + g + δ!β1−α−βAGiven some initial value of technology A0we have:y = s1−βksβhn + g + δ!α1−α−β sαks1−αhn + g + δ!β1−α−βA0egtWilliams Economics 312Evaluating the ModelTaking logs:log y = log A0+ gt −α + β1 − α − βlog (n + g + δ) ++α1 − α − βlog sk+β1 − α − βlog shWhat if we have a lot of countries i = 1, ..., n?We can assume that log A0= a + εiAlso assume that g and δ are constant across countries.Williams Economics 312Williams Economics 31245 TFP growth rate, 1965-95(Labor share=0.65, no returns to education)Figure 1: Relation of TFP growth to saving rateSaving rate, 1965-950 .2 .4-.04-.020.02.04MOZUGAMDGRWAETHCAFBDIMRTZARBENTGOEGYSLVSENGHAMLINERNGABFACMRCIVAGOGTMBOLNICBGDTTOLKAPNGNPLPAKINDKENCOLHNDPRYMUSURYDOMJORMWIMARZAFIDNCHLCRIZMBPHLSYRUSATURTUNDZAARGPERBWAGBRJAMVENPANECUCOGNZLMEXPRTMYSBRASWEBELCANZWEESPITANLDCHEAUSTZAIRLHKGDNKFRAGRCAUTISRKORTHAFINJPNNORSGP TFP growth rate, 1965-95(Labor share=0.65, no returns to education)Figure 2: Relation of TFP growth to schooling rateHuman capital investment rate, 1965-950 .05 .1