1、Working with the Solow Growth ModelC h a p t e r 4Key EquationsSolow Growth Modelk/k=s(y/k)s nlk is capital per workerly is real gross domestic product(real GDP)per workerly/k is the average product of capitalls is the saving ratel is the depreciation rateln is the population growth rate.Solow Growt
2、h ModelSteady State s(y*/k*)=s+nWe 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 a
3、nd equal to zero.Solow Growth ModelChange in savings rate(s)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.Solow Growth ModelChange in sa
4、vings 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).lAf(k*)/k*=+n/sSolow Growt
5、h Modelthe effect of s on consumptionsIn 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”Solow Growth ModelChange in technology level(A)Solow Growth Mod
6、elChange 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.Solow Growth ModelChange in technology level(A)In the long run,the growth rates
7、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).Af(k*)/k*=+n/sSolow Growth Mo
8、delChange in the labor inputSolow Growth Model Change in the labor inputIn 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.Solow Growth Model Change in the labor inputI
9、n 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.Solow
10、Growth Model Change in population growth rateSolow Growth Model Change in population growth rateSolow Growth Model Change in population growth rateIn the short run,a higher n lowers k/k and y/y.These growth rates remain lower during the transition to the steady state.Solow Growth Model Change in pop
11、ulation growth rateIn 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 a
12、nd real GDP,K/K and Y/Y.Solow Growth ModelSum up k*=k*s,A,n,L(0)(+)(+)()()(0)Solow Growth Model ConvergenceOne of the most important questions about economic growth is:whether poor countries tend to converge or catch up to rich countries.Solow Growth Model ConvergenceSolow Growth Model ConvergenceEc
13、onomy 1 starts with lower capital per worker than economy 2k(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.Solow Growth Model ConvergenceThat is,the distance marked by the red arrows is
14、 greater than that marked by the blue arrows.Therefore,capital per worker in economy 1,k1,converges over time toward that in economy 2,k2.Solow Growth Model ConvergenceSolow Growth Model ConvergenceEconomy 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(
15、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,k1 converges over time toward k2.Solow Growth Model Convergencey=A f(k)and y
16、/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.Solow Growth Model ConvergenceThe Solow model says that a poor economywith low capital
17、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.Solow Growth Model ConvergenceSolow
18、Growth Model ConvergenceSolow Growth Model ConvergenceSolow Growth Model ConvergenceSolow Growth Model ConvergenceEconomy 1 starts with lower capital per worker than economy 2lk(0)1 k(0)2.Assume that economy 1 also has a lower saving rate;ls1 s2.The two economies have the same technology levels,A,an
19、d 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.Solow Growth Model ConvergenceSolow Growth Model ConvergenceEconomy 1 sta
20、rts with lower capital per worker than economy 2lk(0)1 n2.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.Solow Growth Model ConvergenceSolow G
21、rowth Model ConvergenceEconomy 1 has a lower starting capital per workerk(0)1 k(0)2and 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:k1(the red curve)towa
22、rd k*1,And k2(the green curve)toward k*2.However,since k*1 is less than k*2,k1 does not converge toward k2.Solow Growth Model ConvergenceKey Resultslk*=k*s,A,n,L(0)(+)(+)()()(0)lk/k=?k(0),k*()(+)Solow Growth Model ConvergenceConditional convergence:la lower k(0)predicts a higher k/k,conditional on k
23、.Absolute convergencelthe prediction that a lower k(0)raises k/k without any conditioning is called.Solow Growth Model the speed of ConvergenceSolow Growth Model the speed of Convergence10/)1(AkkyAky%3,3/1nkCalibration:The half-life is roughly 18 years.Solow Growth Model Endogenous population growthMalthus(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.
