1、S E V E N T H E D I T I O NModified for EC 204 by Bob Murphythe closed economy Solow model how a countrys standard of living depends on its saving and population growth rateshow to use the“Golden Rule”to find the optimal saving rate and capital stock3CHAPTER 7 Economic Growth IWhy growth mattersData
2、 on infant mortality rates:20%in the poorest 1/5 of all countries0.4%in the richest 1/5In Pakistan,85%of people live on less than$2/day.One-fourth of the poorest countries have had famines during the past 3 decades.Poverty is associated with oppression of women and minorities.Economic growth raises
3、living standards and reduces poverty.Income and poverty in the world selected countries,2000MadagascarIndiaBangladeshNepalBotswanaMexicoChileS.KoreaBrazilRussian FederationThailandPeruChinaKenyalinks to prepared graphs Gapminder.orgclick here for one-page instruction guideLife expectancyInfant morta
4、lityMalaria deaths per 100,000Adult literacyCell phone users per 100,0006CHAPTER 7 Economic Growth IWhy growth mattersAnything that effects the long-run rate of economic growth even by a tiny amount will have huge effects on living standards in the long run.1,081.4%243.7%85.4%624.5%169.2%64.0%2.5%2.
5、0%100 years50 years25 yearspercentage increase in standard of living afterannual growth rate of income per capita7CHAPTER 7 Economic Growth IWhy growth mattersIf the annual growth rate of U.S.real GDP per capita had been just one-tenth of one percent higher during the 1990s,the U.S.would have genera
6、ted an additional$496 billion of income during that decade.8CHAPTER 7 Economic Growth IThe lessons of growth theorycan make a positive difference in the lives of hundreds of millions of people.These lessons help usunderstand why poor countries are poordesign policies that can help them growlearn how
7、 our own growth rate is affected by shocks and our governments policies9CHAPTER 7 Economic Growth IThe Solow modeldue to Robert Solow,won Nobel Prize for contributions to the study of economic growtha major paradigm:widely used in policy makingbenchmark against which most recent growth theories are
8、comparedlooks at the determinants of economic growth and the standard of living in the long run10CHAPTER 7 Economic Growth IHow Solow model is different from Chapter 3s model1.K is no longer fixed:investment causes it to grow,depreciation causes it to shrink2.L is no longer fixed:population growth c
9、auses it to grow3.the consumption function is simpler11CHAPTER 7 Economic Growth IHow Solow model is different from Chapter 3s model4.no G or T(only to simplify presentation;we can still do fiscal policy experiments)5.cosmetic differences12CHAPTER 7 Economic Growth IThe production functionIn aggrega
10、te terms:Y =F(K,L)Define:y=Y/L=output per worker k=K/L=capital per worker Assume constant returns to scale:zY =F(zK,zL)for any z 0Pick z=1/L.Then Y/L =F(K/L,1)y =F(k,1)y =f(k)where f(k)=F(k,1)13CHAPTER 7 Economic Growth IThe production functionOutput per worker,y Capital per worker,k f(k)Note:this p
11、roduction function exhibits diminishing MPK.1MPK=f(k+1)f(k)14CHAPTER 7 Economic Growth IThe national income identityY=C+I (remember,no G)In“per worker”terms:y=c+i where c=C/L and i=I/L 15CHAPTER 7 Economic Growth IThe consumption functions=the saving rate,the fraction of income that is saved (s is a
12、n exogenous parameter)Note:s is the only lowercase variable that is not equal to its uppercase version divided by LConsumption function:c=(1s)y(per worker)16CHAPTER 7 Economic Growth ISaving and investmentsaving(per worker)=y c =y (1s)y =syNational income identity is y=c+iRearrange to get:i =y c =sy
13、(investment=saving,like in chap.3!)Using the results above,i =sy =sf(k)17CHAPTER 7 Economic Growth IOutput,consumption,and investmentOutput per worker,y Capital per worker,k f(k)sf(k)k1 y1 i1 c1 18CHAPTER 7 Economic Growth IDepreciationDepreciation per worker,k Capital per worker,k k=the rate of dep
14、reciation =the fraction of the capital stock that wears out each period119CHAPTER 7 Economic Growth ICapital accumulationChange in capital stock=investment depreciationk=i kSince i =sf(k),this becomes:k =s f(k)k The basic idea:Investment increases the capital stock,depreciation reduces it.20CHAPTER
15、7 Economic Growth IThe equation of motion for kThe Solow models central equationDetermines behavior of capital over timewhich,in turn,determines behavior of all of the other endogenous variables because they all depend on k.E.g.,income per person:y =f(k)consumption per person:c =(1s)f(k)k =s f(k)k 2
16、1CHAPTER 7 Economic Growth IThe steady stateIf investment is just enough to cover depreciation sf(k)=k,then capital per worker will remain constant:k =0.This occurs at one value of k,denoted k*,called the steady state capital stock.k =s f(k)k 22CHAPTER 7 Economic Growth IThe steady stateInvestment a
17、nd depreciation Capital per worker,k sf(k)kk*23CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker,k sf(k)kk*k=sf(k)kdepreciationkk1investment24CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker,
18、k sf(k)kk*k1k=sf(k)kkk225CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker,k sf(k)kk*k=sf(k)kk2investmentdepreciationk26CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker,k sf(k)kk*k=sf(k)kkk22
19、7CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker,k sf(k)kk*k=sf(k)kk2kk328CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker,k sf(k)kk*k=sf(k)kk3Summary:As long as k k*,investment will exceed
20、 depreciation,and k will continue to grow toward k*.NOW YOU TRY:Approaching k*from aboveDraw the Solow model diagram,labeling the steady state k*.On the horizontal axis,pick a value greater than k*for the economys initial capital stock.Label it k1.Show what happens to k over time.Does k move toward
21、the steady state or away from it?30CHAPTER 7 Economic Growth IA numerical exampleProduction function(aggregate):To derive the per-worker production function,divide through by L:Then substitute y=Y/L and k=K/L to get31CHAPTER 7 Economic Growth IA numerical example,cont.Assume:s=0.3=0.1initial value o
22、f k=4.032CHAPTER 7 Economic Growth IApproaching the steady state:A numerical exampleYear k y c i k Dk 14.0002.0001.4000.6000.4000.200 24.2002.0491.4350.6150.4200.195 34.3952.0961.4670.6290.4400.189 44.5842.1411.4990.6420.4580.184 105.6022.3671.6570.7100.5600.150 257.3512.7061.8940.8120.7320.080 1008
23、.9622.9942.0960.8980.8960.002 9.0003.0002.1000.9000.9000.000NOW YOU TRY:Solve for the Steady State Continue to assume s=0.3,=0.1,and y=k 1/2Use the equation of motion k=s f(k)k to solve for the steady-state values of k,y,and c.ANSWERS:Solve for the Steady State 35CHAPTER 7 Economic Growth IAn increa
24、se in the saving rateInvestment and depreciationkdks1 f(k)An increase in the saving rate raises investmentcausing k to grow toward a new steady state:s2 f(k)36CHAPTER 7 Economic Growth IPrediction:Higher s higher k*.And since y=f(k),higher k*higher y*.Thus,the Solow model predicts that countries wit
25、h higher rates of saving and investment will have higher levels of capital and income per worker in the long run.International evidence on investment rates and income per personIncome per person in 2003(log scale)Investment as percentage of output(average 1960-2003)38CHAPTER 7 Economic Growth IThe G
26、olden Rule:IntroductionDifferent values of s lead to different steady states.How do we know which is the“best”steady state?The“best”steady state has the highest possible consumption per person:c*=(1s)f(k*).An increase in s leads to higher k*and y*,which raises c*reduces consumptions share of income(
27、1s),which lowers c*.So,how do we find the s and k*that maximize c*?39CHAPTER 7 Economic Growth IThe Golden Rule capital stockthe Golden Rule level of capital,the steady state value of k that maximizes consumption.To find it,first express c*in terms of k*:c*=y*i*=f(k*)i*=f(k*)k*In the steady state:i*
28、=k*because k=0.40CHAPTER 7 Economic Growth IThen,graph f(k*)and k*,look for the point where the gap between them is biggest.The Golden Rule capital stocksteady state output and depreciationsteady-state capital per worker,k*f(k*)k*41CHAPTER 7 Economic Growth IThe Golden Rule capital stockc*=f(k*)k*is
29、 biggest where the slope of the production function equals the slope of the depreciation line:steady-state capital per worker,k*f(k*)k*MPK=42CHAPTER 7 Economic Growth IThe transition to the Golden Rule steady stateThe economy does NOT have a tendency to move toward the Golden Rule steady state.Achie
30、ving the Golden Rule requires that policymakers adjust s.This adjustment leads to a new steady state with higher consumption.But what happens to consumption during the transition to the Golden Rule?43CHAPTER 7 Economic Growth IStarting with too much capitalthen increasing c*requires a fall in s.In t
31、he transition to the Golden Rule,consumption is higher at all points in time.timet0ciy44CHAPTER 7 Economic Growth IStarting with too little capitalthen increasing c*requires an increase in s.Future generations enjoy higher consumption,but the current one experiences an initial drop in consumption.ti
32、met0ciy45CHAPTER 7 Economic Growth IPopulation growthAssume the population and labor force grow at rate n(exogenous):EX:Suppose L=1,000 in year 1 and the population is growing at 2%per year(n=0.02).Then L=n L=0.02 1,000=20,so L=1,020 in year 2.46CHAPTER 7 Economic Growth IBreak-even investment(+n)k=
33、break-even investment,the amount of investment necessary to keep k constant.Break-even investment includes:k to replace capital as it wears outn k to equip new workers with capital(Otherwise,k would fall as the existing capital stock is spread more thinly over a larger population of workers.)47CHAPT
34、ER 7 Economic Growth IThe equation of motion for kWith population growth,the equation of motion for k is:break-even investmentactual investmentk=s f(k)(+n)k48CHAPTER 7 Economic Growth IThe Solow model diagramInvestment,break-even investmentCapital per worker,k sf(k)(+n)kk*k =s f(k)(+n)k49CHAPTER 7 E
35、conomic Growth IThe impact of population growthInvestment,break-even investmentCapital per worker,k sf(k)(+n1)kk1*(+n2)kk2*An increase in n causes an increase in break-even investment,leading to a lower steady-state level of k.50CHAPTER 7 Economic Growth IPrediction:Higher n lower k*.And since y=f(k
36、),lower k*lower y*.Thus,the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.International evidence on population growth and income per personIncome per person in 2003(log scale)Population growth(percent p
37、er year,average 1960-2003)52CHAPTER 7 Economic Growth IThe Golden Rule with population growthTo find the Golden Rule capital stock,express c*in terms of k*:c*=y*i*=f(k*)(+n)k*c*is maximized when MPK =+n or equivalently,MPK =nIn the Golden Rule steady state,the marginal product of capital net of depr
38、eciation equals the population growth rate.53CHAPTER 7 Economic Growth IAlternative perspectives on population growthThe Malthusian Model(1798)Predicts population growth will outstrip the Earths ability to produce food,leading to the impoverishment of humanity.Since Malthus,world population has incr
39、eased sixfold,yet living standards are higher than ever.Malthus neglected the effects of technological progress.54CHAPTER 7 Economic Growth IAlternative perspectives on population growthThe Kremerian Model(1993)Posits that population growth contributes to economic growth.More people=more geniuses,sc
40、ientists&engineers,so faster technological progress.Evidence,from very long historical periods:As world pop.growth rate increased,so did rate of growth in living standards Historically,regions with larger populations have enjoyed faster growth.1.The Solow growth model shows that,in the long run,a co
41、untrys standard of living depends:positively on its saving ratenegatively on its population growth rate2.An increase in the saving rate leads to:higher output in the long runfaster growth temporarily but not faster steady state growth3.If the economy has more capital than the Golden Rule level,then reducing saving will increase consumption at all points in time,making all generations better off.If the economy has less capital than the Golden Rule level,then increasing saving will increase consumption for future generations,but reduce consumption for the present generation.
