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1、9/28/20231Chapter 3ENDOGENOUS GROWTH9/28/20232SOME BASIC INFORMATION(1)If r=f(k),and k is modest,then capital accumulation cannot account for a large part of either long-run or cross-country income differences.(2)The effectiveness of labor(A)is the only determinant of income,whose behavior is taken
2、as exogenous.(3)The effectiveness of labor(A)is explicitly interpreted as knowledge,and knowledge accumulation is endogenous.(4)Knowledge accumulation is probably central to worldwide growth.(5)Much of the variation in income across countries comes from differences in output for given amounts of phy
3、sical and human capital.9/28/202333.1 Framework and Assumptions3.1 Framework and AssumptionsAssumptions(1)The effectiveness of labor(A)represents knowledge or technology,there are two sectors,an R&D sector;a goods-producing sector.(2)Two other major simplifications:First,both the R&D and goods produ
4、ction function are C-D function;second,the model takes the s,K and L as exogenous and constant.9/28/20234Assumptions(3)Fraction ak of the capital stock is used in the R&D sector and fraction 1-ak in the goods-producing sector;fraction aL of the labor force is used in the R&D sector and fraction 1-aL
5、 in the goods-producing sector.(4)The quantity of output produced at time t is thus:)1.3(10,)()1)()()1()(1tLatAtKatYLk物质生产部门的生产函数9/28/20235知识生产（R&D）部门的生产函数()()()(),0,0,0KLA tB a K ta L tA tB知识生产部门的生产函数取消了报酬不变的假定，因而可能存在报酬递减或报酬递增两种可能性。The parameter reflects the effect of the existing stock of knowledg
6、e on the success of R&D（3.2）9/28/20236(5)As in the Solow model,the saving rate is exogenous and constant,depreciation is set to zero.()()K tsY t()()L tnL t(6)Population growth is treated as exogenous.(7)Without capital for the moment.=0,=0（3.3）（3.4）9/28/202373.2 The Model without Capital()()(1)()LY
7、tA taL t()()()LA tB a L tA tThe production for output becomes:The production for new knowledge is now:We then focus on the dynamics of A:（3.5）（3.6）9/28/202381()()()()()ALA tgtA tBa L tA t()(1)()()AAAgtngtgt2()()(1)()AAAgtngtgtTaking logs of both sides,then differentiating the two sides withrespect t
8、o time t,yields:1 or 1 or=1（3.9）（3.8）（3.7）9/28/20239AgAgAg The dynamics of the growth rate of knowledge when 1Case 1 1)10.3(10*nggggAAAAF3.19/28/202310Case 1 1,报酬递增报酬递增由由3.9式知式知,此时的技术进步速度越来越快此时的技术进步速度越来越快Ag Ag经济增长速度也越来越快这就解释了发达国家的增长路径F3.4Ever-increasing growth9/28/202311Case 1=1的情形如图的情形如图3.3AAAnggng
9、gA2)1(3.3 The General Case（一）资本存量的变化（一）资本存量的变化（K）11113.1(1)(1)(1)(1)tttkttLtkLttKs YsaKAaLsaaKAL，将代 入（3.12）（3.13）9/28/202312 1k*()()()()()()()(1)()()()()()0g=F3.5()0()0,()1kkkAkkAkAkkkkK tA t L tgtcK tK tgtgtngtgtgtngtnggtgtggt取 自 然 对 数 再 求 导当即线 上，如 图（3.14）（3.15）9/28/202313 kk()()0g,()0()0,()3()()0g
10、,()0()0,()2AkAkkkkAkAkkkkgtng tngg tg tgg tgtng tngg tg tgg t当当即线右下方。即线右上方。9/28/202314（二）知识的变化（二）知识的变化（A）()()(1)()11,0kLAkAAABaKaLAggtngtg（）令*(1)()(1)()01,0,0,F3.6kAAkAAAAgtngtgnggggg 如 图线 上9/28/202315AAAAAkAAAAAkgggggnggggggng,0,0,0)1()3(,0,0,0)1()2(（三）经济增长的路径（三）经济增长的路径Case 1：+11的情形的情形*1()AkAgnggnC
11、ase 2：+1 1的情形的情形（3.21）9/28/202316Case 2：+=1=1的情形的情形AgKg0Ag 0Kg nnAgKg0KAggF3.89/28/2023173.4 Learning by Doing 模型模型1()()()()Y tK tA t LKt有，产出效应()(),0,0A tBKK tB，有技术效应1)1(1)()()()(tLtKBtKtY)()()()()1(1tLtKsBtsYtK,1)1(,1)1(*当当KKgincreaeverYKsin会及（3.23）（3.34）（3.25）（3.22）3.5 Romer ModelP130:(3.45)影响增长的四
12、个参数：1.从事R&D的人数2.从事R&D的动力：对专利的垄断力3.R&D部门的生产率（B)4.人口规模（L)9/28/2023183.7 经验应用：人口增长与技术进步 历史上人口增长与技术进步正相关：（3.58)：人口增长推动技术进步，（3.57）：技术进步带来收入增长（3.62）：技术进步带来人口增长 (3.63)和 F3.9：人多力量大（创新是内生的），人口多则从事发明创造的人也多，技术进步越快，人口增长率越高。9/28/2023199/28/2023203.8知识积累模型与增长理论的中心问题R&D模型认为知识（A)增长是收入增长的主要原因。但知识是非排他性的，落后国家按理通过学习可以赶上先进国家。事实不然。所以国家之间收入的不同，不是因为由于技术和知识差异，而是运用这些技术和知识的能力的不同。
