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,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Macroeconomics Chapter 4,*,Working with the Solow Growth Model,C h a p t e r 4,1,Macroeconomics Chapter 4,Key EquationsSolow Growth Model,k,/,k,=,s,(,y,/,k,) ,s,n,k,is capital per worker,y,is real gross domestic product (real GDP) per worker,y,/,k,is the average product of capital,s,is the saving rate,is the depreciation rate,n,is the population growth rate.,2,Macroeconomics Chapter 4,Solow Growth ModelSteady State,s,(,y,*/,k,*)=,s+n,We assumed that everything on the right-hand side was constant except for,y,/,k,.,In the transition to the steady state, the rise in,k,led to a fall in,y,/,k,and, hence, to a fall in,k,/,k,.,In the steady state,k,was constant and, therefore,y,/,k,was constant. Hence,k,/,k,was constant and equal to zero.,3,Macroeconomics Chapter 4,Solow Growth ModelChange in savings rate (s),4,Macroeconomics Chapter 4,Solow Growth ModelChange in savings rate (s),In the short run, an increase in the saving rate raises the growth rate of capital per worker.,This growth rate remains higher during the transition to the steady state.,5,Macroeconomics Chapter 4,Solow Growth ModelChange in savings rate (s),In the long run, the growth rate of capital per worker is the samezerofor any saving rate.,In this long-run or steady-state situation, a higher saving rate leads to higher steady state capital per worker,k, not to a change in the growth rate (which remains at zero).,A,f,(,k*,)/,k*=+n/s,6,Macroeconomics Chapter 4,Solow Growth Modelthe effect of s on consumptions,In the short run, consumption decreases and k arises.,c*=y*-,k*-s(,y*-,k*),=,y*-,k*-nk*,c*=,y*-(,+n),k* = (MPK-n),k*,In the long run, whether the consumption in the steady state increases depends on MPK.,“Golden Rule”,7,Macroeconomics Chapter 4,Solow Growth ModelChange in technology level (A),8,Macroeconomics Chapter 4,Solow Growth ModelChange in technology level (A),In the short run, an increase in the technology level,A, raises the growth rates of capital and real GDP per worker.,These growth rates remain higher during the transition to the steady state.,9,Macroeconomics Chapter 4,Solow Growth ModelChange in technology level (A),In the long run, the growth rates of capital and real GDP per worker are the samezerofor any technology level.,In this long-run or steady state situation, a higher technology level leads to higher steady-state capital and real GDP per worker,k, and,y, not to changes in the growth rates (which remain at zero).,A,f,(,k*,)/,k*=+n/s,10,Macroeconomics Chapter 4,Solow Growth ModelChange in the labor input,11,Macroeconomics Chapter 4,Solow Growth Model Change in the labor input,In the short run, an increase in labor input,L,(0), raises the growth rates of capital and real GDP per worker.,These growth rates remain higher during the transition to the steady state.,12,Macroeconomics Chapter 4,Solow Growth Model Change in the labor input,In the long run, the growth rates of capital and real GDP per worker are the samezerofor any level of labor input,L,(0).,The steady-state capital and real GDP per worker,k, and,y, are the same for any,L,.,In the long run an economy with twice as much labor input has twice as much capital and real GDP.,13,Macroeconomics Chapter 4,Solow Growth Model Change in population growth rate,14,Macroeconomics Chapter 4,Solow Growth Model Change in population growth rate,15,Macroeconomics Chapter 4,Solow Growth Model Change in population growth rate,In the short run, a higher,n,lowers,k/k,and ,y/y.,These growth rates remain lower during the transition to the steady state.,16,Macroeconomics Chapter 4,Solow Growth Model Change in population growth rate,In the steady state,k/k,and ,y/y,are zero for any,n,.,A higher,n,leads to lower steady-state capital and real GDP per worker,k, and,y, not to changes in the growth rates,k/k,and,y/y,(which remain at zero).,A change in,n,does affect the steady-state,growth rates of the levels of capital and real GDP, ,K/K,and ,Y/Y.,17,Macroeconomics Chapter 4,Solow Growth ModelSum up,k,* =,k,* ,s,A,n,L,(0) ,(+) (+) () () (0),18,Macroeconomics Chapter 4,Solow Growth Model Convergence,One of the most important questions about economic growth is:,whether poor countries tend to converge or catch up to rich countries.,19,Macroeconomics Chapter 4,Solow Growth Model Convergence,20,Macroeconomics Chapter 4,Solow Growth Model Convergence,Economy 1 starts with lower capital per worker than economy 2,k,(0),1,is less than,k,(0),2,.,Economy 1 grows faster initially because the vertical distance between the,s,(,y,/,k,) curve and the,s,+,n,line is greater at,k,(0),1,than at,k,(0),2,.,21,Macroeconomics Chapter 4,Solow Growth Model Convergence,That is, the distance marked by the red arrows is greater than that marked by the blue arrows.,Therefore, capital per worker in economy 1,k,1, converges over time toward that in economy 2,k,2,.,22,Macroeconomics Chapter 4,Solow Growth Model Convergence,23,Macroeconomics Chapter 4,Solow Growth Model Convergence,Economy 1 starts at capital per worker,k,(0),1,and economy 2 starts at,k,(0),2, where,k,(0),1,is less than,k,(0),2,.,The two economies have the same steady-state capital per worker,k,*, shown by the dashed blue line.,In each economy,k,rises over time toward,k,*. However,k,grows faster in economy 1 because,k,(0),1,is less than,k,(0),2,.,Therefore,k,1 converges over time toward,k,2.,24,Macroeconomics Chapter 4,Solow Growth Model Convergence,y,=,A,f,(,k,) and,y,/,y,=,(,k,/,k,),k/k,was higher initially in economy 1 than in economy 2.,Therefore, ,y/y,is also higher initially in economy 1. Hence, economy 1s real GDP per worker,y, converges over time toward economy 2s real GDP per worker.,25,Macroeconomics Chapter 4,Solow Growth Model Convergence,The Solow model says that a poor economywith low capital and real GDP per workergrows faster than a rich one. The reason is the diminishing average product of capital,y/k.,The Solow model predicts that poorer economies tend to converge over time toward richer ones in terms of the levels of capital and real GDP per worker.,26,Macroeconomics Chapter 4,Solow Growth Model Convergence,27,Macroeconomics Chapter 4,Solow Growth Model Convergence,28,Macroeconomics Chapter 4,Solow Growth Model Convergence,29,Macroeconomics Chapter 4,Solow Growth Model Convergence,30,Macroeconomics Chapter 4,Solow Growth Model Convergence,Economy 1 starts with lower capital per worker than economy 2,k,(0),1,k,(0),2,.,Assume that economy 1 also has a lower saving rate;,s,1,s,2,.,The two economies have the same technology levels,A, and population growth rates,n,.,Therefore,k,*,1,is less than,k,*,2,.,It is uncertain which economy grows faster initially. The vertical distance marked with the blue arrows may be larger or smaller than the one marked with the red arrows.,31,Macroeconomics Chapter 4,Solow Growth Model Convergence,32,Macroeconomics Chapter 4,Solow Growth Model Convergence,Economy 1 starts with lower capital per worker than economy 2,k,(0),1,n,2,.,Therefore,k,*,1,is less than,k,*,2,.,It is again uncertain which economy grows faster initially. The vertical distance marked with the blue arrows may be larger or smaller than the one marked with the red arrows.,33,Macroeconomics Chapter 4,Solow Growth Model Convergence,34,Macroeconomics Chapter 4,Solow Growth Model Convergence,Economy 1 has a lower starting capital per worker,k,(0),1,k,(0),2,and also has a lower steady-state capital per worker,k,*,1,(the dashed brown line) is less than,k,*,2,(the dashed blue line).,Each capital per worker converges over time toward its own steady-state value:,k,1,(the red curve) toward,k,*,1,And,k,2,(the green curve) toward,k,*,2,. However, since,k,*,1,is less than,k,*,2,k,1,does not converge toward,k,2,.,35,Macroeconomics Chapter 4,Solow Growth Model Convergence,Key Results,k,* =,k,*,s,A,n,L,(0) ,(+) (+) () () (0),k,/,k,=,k,(0) ,k,*,() (+),36,Macroeconomics Chapter 4,Solow Growth Model Convergence,Conditional convergence,:,a lower,k,(0) predicts a higher,k/k, conditional on,k,.,Absolute convergence,the prediction that a lower,k,(0) raises ,k/k,without any conditioning is called.,37,Macroeconomics Chapter 4,Solow Growth Model the speed of Convergence,38,Macroeconomics Chapter 4,Solow Growth Model the speed of Convergence,Calibration:,The,half-life,is roughly 18 years.,39,Macroeconomics Chapter 4,Solow Growth Model Endogenous population growth,Malthus (1798),the increase of y (or k) leads to a higher growth rate of population, which reduces the level of income per capita.,Modern growth theory:,the higher income per capita reduces the population growth rate.,40,Macroeconomics Chapter 4,
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