1、1Romer(2001), Advanced Macroeconomics, Chapter 1Blanchard, Olivier J., 2nd edition, Macroeconomics, Prentice Hall曼昆， 宏观经济学 ，人民大学，2000 年巴罗，萨拉伊马丁， 经济增长 ，中国社会科学出版社，2000 年Some Basic Facts about Economic GrowthAssumptions of One-Sector Growth ModelsThe Dynamics of the ModelThe Impact of a Change in the S
2、aving RateThe Speed of ConvergenceThe Solow Model and the Central Questions of Growth TheoryEmpirical ApplicationsLecture 1 The Solow Growth Model2一、 Some Basic Facts about Economic Growth: the stylized facts of growth1. Economic growth through deep time(F2)2. The stylized facts （F3）0,yLY (F4)okK, (
3、F5)tcnsYa, tconsYPWtconsYPtKa1 ,a,3. international differences in the standard of living(F6)4. The Solow growth model The ultimate objective of research on economic growth is to determine whether 3there are possibilities for raising overall growth or bringing standards of living in poor countries cl
4、oser to those in the world leaders. The Solow model is the starting point for almost all analyses of growth. The principal conclusion of the Solow model is that the accumulation of physical capital cannot account for either the vast growth over time in output per person or the cast geographic differ
5、ences in output per person. The model treats other potential sources of differences in real incomes as either exogenous and thus not explained by the model.( in the case of technological progress) or absent altogether(in the case of positive externalities from capital).二、 Basic assumptions of one-se
6、ctor growth models1 Input and Output4),(tttt LAKFY1)Labor-augmenting or Harrod-neutralcapital ),(tttt LKAFYaugmentingHicks-ueutral),(tttt2)The ratio of capital to output, K/Y, eventually settles down3)Constant returns to scale (first-degree homogeneity)The economy is big enough that the gains from s
7、pecialization have been exhausted.Inputs other than capital, labor, and knowledge are relatively unimportant.4)intensive form : )()1,(tt kfALKFYyThe amount of output per unit of effective labor depends only 5on the quantity of capital per unit of effective labor, and not on the overall size of the e
8、conomy.output per worker )(kAfLY5)Assumptions0)(f,f( Inada conditions)0)(lim0kfkThe path of the economy does not divergeCobb-Douglas functionkALKkf)()(2 the evolution of the input into production1)Labor and knowledge grow at constant rates:6ntLLneLtttntt 00l)l( gtLAgeAtttgtt 00ln)ln(2)Investment and
9、 capital formationttttt sYSIdtKdtISGTttt Ks三、 The simple dynamics of growth model1 The Harrod-Domar model71)The fixed-coefficients production functiontt LKY,min2)Saving, investment, and the warranted rate of growthsIY3)Labor force growth: gnY4)The Harrod-Domar condition:sgn2 The Solow model1)Cobb-Do
10、uglas function(F7)kyALKYkkMPkALYyK)(1 0)1()( 0)( 211 82)The dynamics of kgyLYkfLeKfeEKFeLgtgtttt tgnt)( )()(),()(0One way: actual and break-even investment kgnksfdtkt )(), k is rising0() gsft , k is falling)kkt, k is constant()nsft *)*gThe second way: speedkgnksfdtkt )() )()ln( tt)(*)(gksf )()ln(1ns
11、dttt )()l()()l(1 gnskdtkdtklity t93)the balanced growth path*)(kALKyYkKLAYYkfytttttttttttt*)( ksgnkfnLgICKY 0)()()()( AIAL g)()()()( )()()()(,ttttttttt ttt kfkfAkfLKkfAdFw)(),( tttt fdFr ttttt YKrLw10ttttLK LkfkfAKwrS*)(*)(constant)()(ffThe Solow model implies that, regardless of its starting point,
12、 the economy converges to a balanced growth path - a situation where each variable of the model is growing at a constant rate. On the balanced growth path, the growth rate of output per worker is determined solely by the rate of technological progress.四、 The impact of a change in the saving rateThe
13、division of the governments purchanses between consumption and investment goods, the division of its revenues between taxed and borrowing, 11and its tax treatments of saving and investment are all likely to affect the fraction of output that is invested.1. the impact on output(F24,F25)A change in th
14、e saving rate has a level effect but not a growth effect: it changes the economys balanced growth path, and thus the level of output per worker at any point in time, but it does not affect the growth rate of output per worker in the balanced growth path. In the Solow model only changes in the rate o
15、f technological progress have growth effects; all other changes have only level effects.2. saving and consumption (welfare)()(1tt tt ksfkfc12*)(*)(*)()1(*;kgnkfksfcsk sgnkgnsfsc ),(*)(),(if , )(),(*gnskf 0scif , )(),( gnf *if , )(),(*gnskf 0scthe golden-rule level of the capital stock:Consumption is
16、 at its maximum possible level among balanced growth paths. gngnskfMPK),(*Quantitative Implications1 The effect of s on output in the long runsgnkfsy ),(*)(* ),(*)(),( gnskgnkf13?0*)()(*kfsgnfsk *)()( )()(* kfsgnffsy *)(/*)(*1* kfkfksyys is the elasticity of )(/*)(fkfoutput with respect to capital a
17、t . *kDenoting this by *)(kK)(1*sysKIf markets are competitive and there are no externalities, is *)(/)(*kffkthe share of total income that goes to capital on the balanced growth path.31*)(kK 21*syA 10% increase in the saving rate (from 20% of output to 22%) raises output per worker 14in the long ru
18、n by about 5% relative to the path it would have followed.Significant changes in saving have only moderate effects on the level of output on the balanced growth path.2 The speed of convergence1)One way )()( gnksfk资本的平均产品。资本报酬递减kf)(+Inada 条件使 0)(dksf：所有者的人均收入占人kykf)(均总收入的份额 ckyk gns )()1(2)The second
19、 way kgnksfk)()()( 15When k equals k*, . A first-order 0Taylor-series approximation of around kk=k* yields: *)(*)()(* ktkkk In the vicinity of the balanced growth path, k moves toward k* at a speed approximately proportional to its distance from k*. In addition, one can show that y approaches y* at
20、the same rate that k approaches k*. *0kektt )*)(1)(* gnkkKk *0yyeyy tt The half-life16The half-life is roughly equal to 70 divided by its growth rate in percent.More exactly, 5.0*te/).ln(t %443%;21;21 ygntherefore k and y move 4% of the remaining distance toward k* and y* each year, and take approxi
21、mately 18 years to get halfway to their balanced-growth-path values.Cobb-Douglas 函数 k, y 的收敛速度：)()1( gnNot only is the overall impact of a substantial change in the saving rate modest, but it does not occur very quickly3 D41)172)绝对收敛与条件收敛（F42）2. 内生增长理论1)AK model (F43)：s，A 影响人均cykgnsAdKY)(增长率；没有趋同。2)
22、规模报酬递增：(F44)(F45)(/)0lim1nksfAMPKLBYk3)CES function linearDouglsCbLKYabaY 10)1(,mi01)()(F46)(*/)1nsAbakfkCase 1 nsAba1,0Case 2 Case 3 s1,3. 贫困陷阱（F47）五、 The Solow Model and the Central 18Questions of Growth TheoryThe Solow model identifies two possible sources of variation-either over time or across
23、parts of the world-in output per worker: differences in capital per worker and differences in the effectiveness of labor. We have seen, however, that only growth in the effectiveness of labor can lead to permanent growth in output per worker, and that for reasonable cases the impact of changes in ca
24、pital per worker on output per worker is modest. As a result, only differences in the effectiveness of labor have any reasonable hope of accounting for the vast differences in wealth across time and space.If output per worker differs by a factor of X, and the elasticity of output per worker with res
25、pect to capital per worker is , log Kcapital per worker must differ by . XlnThat is, capital per worker differs by a factor of or .KXe)(lnK119103,10KXIn sum, differences in capital per worker are far smaller than those needed to account for the differences in capital per worker that we are trying to
26、 understand. The other potential source of variation on output per worker in the Solow model is the effectiveness of labor.Empirical Applications1. Growth accounting(F38)ALKY)1( Solow residual： （F48,39）LKY)1(It reflects all sources of growth other than the contribution of capital accumulation vir it
27、s private return. Denison(1967): different types of capital and labor Young(1995): the exceptionally rapid 20growth of the NICs Hsieh(1998a): to exam the behavior of factor returns rather than quantities Oliner and Sichel(2000), Jorgenson and Stiroh(2000), Whelan(2000): U.S. productivity growth rebo
28、und2. ConvergenceGrowth is whether poor countries tend to grow faster than rich countries.1)The Solow model predicts countries converge to their balanced growth paths2)The Solow model implies that the rate of return on capital is lower in countries with more capital per worker3)If there are lags in
29、the diffusion of knowledge, income differences might tend to shrink as poorer countries gain access to state-of-the-art methods.Baumol(1986): )ln(95.0457.8)ln()ln( 1870,)4(1870,197, iii NYNYNY21De long(1988)3. Saving and InvestmentIn the absence of barriers to capital movements, there is no reason t
30、o expect countries with high saving to also have high investment.Feldstein and Horioka(1980):ii YSYI )(87.035.0)( 4Barro, Mankiw, and Sala-I-Martin(1995): There are underlying variables that affect both saving and investment.Furthermore( Delong) 1 The Solow model )()ln(1gnskdttt )()l()()l( 1gnskdtkd
31、tlity ttttSo lets write for the capital-YKyz/output ratio22)()1()ln()()ln( 1 gnskdtydtklindtz t)()1()l( 1gsztztt )()( tt znsdz tt gnsz ttd)()1( tgntdA)()1(0 tgnt dgnsz )()1(01)(ttt zLAY1)(tttt 1)(ttzy )()()()(), ttttttttttt kffAkfLKkfAdLKFw )(),(tttt fr 23tttt YKrLwwhich makes it more clear that the
32、 real wage is growing at rate g, and the real return to capital is constant in steady-state growth. 1)()1(ttt zAw1ttzrIn other words, a share of output is going to capital- and is evenly split as the return to capital. 2 Lets look at what happens when we change the parameters of the model: 1)A chang
33、e in the savings rate s:Lets take =0.5, n=g=0.01, =0.03; s starts at 0.015 and jumps to 0.20.Start at time zero with A0=L0=1Our stesdy-state growth path jumps form z*=3, with Yt growing at 0.02 per year, to z*=4, with Yt growing at 0.02 percent per year-a 33.3% boost to steady-state 24capital per ef
34、fective worker and output per effective worker.How fast does this economy converge to its steady state? =0.025 equals 2.5 )(1Agnpercent per year. So the 1/e time-the time after the time 0 jump in the savings rate for the capital-output ratio to close the gap to its new steady-state value to 1/e of i
35、ts initial value-40 years.252)“Level effect” but not a “growth effect”Does consumption rise when the savings rate rises? At the beginning, certainly not. *)(*)1(*kgnysycdskfd26if , then the economy is called )(*)gnkfdynamically inefficient-consumption could be raised at all future dates by lowering
36、saving today.if , then the economy is said to )(*)gnkfbe at the Golden Rule.*max()()()scfksffkngk*fngsIf the production function is Cobb-Douglas, 11)(gnszytt 1)(*sc1)(gnsALCtt gngnsdsc 1)(1* 1This will be positive if and only if: 0)1(ssWhen s is sufficiently low, increases in s increase steady-state
37、 consumption.27When s hits the magic number of the capital share, you are at the Golden Rule.When s is higher, increases in s decrease steady-state consumption and the economy is dynamically inefficient.Increases in increase the level of s at which the which the economy becomes dynamically inefficie
38、nt.3)Speed of convergence: What Romer produces is much more general: that if the economy is “near” its steady-state growth path, its convergence behavior is “nearly” the same as Cobb-Douglas.4)the Solow model and the central questions of Growth theoryOnly growth in the effectiveness of labor can lea
39、d to permanent growth in output per workerThe impact of changes in capital per worker on output per worker is modest.28Only differences in the effectiveness of labor have any reasonable hope of accounting for vast differences in wealth across time and space.Variations in the accumulation of physical capital do not account for a significant part of either worldwide economic growth or cross-country income differentialsDifferences in savings rates too small to account for cross-section
