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1、MACROECONOMICS 2010 Worth Publishers, all rights reserved 2010 Worth Publishers, all rights reservedS E V E N T H E D I T I O NPowerPoint Slides by Ron CronovichN. Gregory MankiwC H A P T E REconomic Growth I:Capital Accumulation and Population Growth7In this chapter, you will learn:the closed econo
2、my 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 stock2CHAPTER 7 Economic Growth IThe Solow modeldue to Robert Solow,won Nobel Prize for contributions to the study of economic gr
3、owtha major paradigm:widely used in policy makingbenchmark against which most recent growth theories are comparedlooks at the determinants of economic growth and the standard of living in the long run3CHAPTER 7 Economic Growth IHow Solow model is different from Chapter 3s model1.K is no longer fixed
4、:investment causes it to grow, depreciation causes it to shrink2.L is no longer fixed:population growth causes it to grow3.the consumption function is simpler4CHAPTER 7 Economic Growth IHow Solow model is different from Chapter 3s model4.no G or T(only to simplify presentation; we can still do fisca
5、l policy experiments)5CHAPTER 7 Economic Growth IThe production functionIn aggregate 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) =
6、F(k, 1) 6CHAPTER 7 Economic Growth IThe production functionOutput per worker, y Capital per worker, k f(k)Note: this production function exhibits diminishing MPK. 1MPK = f(k +1) f(k)7CHAPTER 7 Economic Growth IThe national income identityY = C + I (remember, no G )In “per worker” terms: y = c + i wh
7、ere c = C/L and i = I /L 8CHAPTER 7 Economic Growth IThe consumption functions = the saving rate, the fraction of income that is saved (s is an 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 worke
8、r)9CHAPTER 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 (investment = saving, like in chap. 3!) Using the results above, i = sy = sf(k)10CHAPTER 7 Economic Growth IOutput, consumption, and investmen
9、tOutput per worker, y Capital per worker, k f(k)sf(k)k1 y1 i1 c1 11CHAPTER 7 Economic Growth IDepreciationDepreciation per worker, k Capital per worker, k k = the rate of depreciation = the fraction of the capital stock that wears out each period1 12CHAPTER 7 Economic Growth ICapital accumulationCha
10、nge in capital stock= investment depreciation k = i kSince i = sf(k) , this becomes: k = s f(k) k The basic idea: Investment increases the capital stock, depreciation reduces it. 13CHAPTER 7 Economic Growth IThe equation of motion for kThe Solow models central equationDetermines behavior of capital
11、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 14CHAPTER 7 Economic Growth IThe steady stateIf investment is just enough to cover depreciation sf(k
12、) = 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 15CHAPTER 7 Economic Growth IThe steady stateInvestment and depreciation Capital per worker, k sf(k) kk* 16CHAPTER 7 Economic Growth IMoving tow
13、ard the steady stateInvestment and depreciation Capital per worker, k sf(k) kk* k = sf(k) kdepreciation kk1investment17CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker, k sf(k) kk* k1 k = sf(k) k kk218CHAPTER 7 Economic Growth IMoving toward the
14、 steady stateInvestment and depreciation Capital per worker, k sf(k) kk* k = sf(k) kk2investmentdepreciation k20CHAPTER 7 Economic Growth IMoving toward the steady stateInvestment and depreciation Capital per worker, k sf(k) kk* k = sf(k) kk2 kk321CHAPTER 7 Economic Growth IMoving toward the steady
15、stateInvestment and depreciation Capital per worker, k sf(k) kk* k = sf(k) kk3Summary:As long as k k*, investment will exceed 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,
16、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 the steady state or away from it?23CHAPTER 7 Economic Growth IA numerical exampleProduction function (aggregate):To derive the per-worker production function, divide
17、 through by L:Then substitute y = Y/L and k = K/L to get24CHAPTER 7 Economic Growth IA numerical example, cont.Assume:s = 0.3 = 0.1initial value of k = 4.025CHAPTER 7 Economic Growth IApproaching the steady state: A numerical exampleYearYear kk y y c c i i kk kk 1 14.0004.0002.0002.0001.4001.4000.60
18、00.6000.4000.4000.2000.200 2 24.2004.2002.0492.0491.4351.4350.6150.6150.4200.4200.1950.195 3 34.3954.3952.0962.0961.4671.4670.6290.6290.4400.4400.1890.189 44.5842.1411.4990.6420.4580.184 105.6022.3671.6570.7100.5600.150 257.3512.7061.8940.8120.7320.080 1008.9622.9942.0960.8980.8960.002 9.0003.0002.1
19、000.9000.9000.000NOW 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 28CHAPTER 7 Economic Growth IAn increase in the saving rateInvestm
20、ent and depreciationkks1 f(k)An increase in the saving rate raises investmentcausing k to grow toward a new steady state: s2 f(k)29CHAPTER 7 Economic Growth IPrediction:Higher s higher k*. And since y = f(k) , higher k* higher y* . Thus, the Solow model predicts that countries with higher rates of s
21、aving and investment will have higher levels of capital and income per worker in the long run. International evidence on investment rates and income per person051015202530351001,00010,000100,000Income per person in 2003 (log scale) Investment as percentage of output (average 1960-2003) 31CHAPTER 7 E
22、conomic Growth IThe Golden 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* re
23、duces consumptions share of income (1s), which lowers c*. So, how do we find the s and k* that maximize c*?32CHAPTER 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
24、* = y* i*= f (k*) i* = f (k*) k* In the steady state:i* = k* because k = 0.33CHAPTER 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*34C
25、HAPTER 7 Economic Growth IThe Golden Rule capital stockc* = f(k*) k*is 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 = 35CHAPTER 7 Economic Growth IThe transition to the Golden Rule steady stateThe econom
26、y does NOT have a tendency to move toward the Golden Rule steady state. Achieving 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? 36CHAPTER 7 Economic Gr
27、owth IStarting with too much capitalthen increasing c* requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time.timet0ciy37CHAPTER 7 Economic Growth IStarting with too little capitalthen increasing c* requires an increase in s. Future generations enjoy
28、higher consumption, but the current one experiences an initial drop in consumption.timet0ciy38CHAPTER 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 =
29、 n L = 0.02 1,000 = 20,so L = 1,020 in year 2.39CHAPTER 7 Economic Growth IBreak-even investment( + n)k = 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,
30、 k would fall as the existing capital stock is spread more thinly over a larger population of workers.)40CHAPTER 7 Economic Growth IThe equation of motion for kWith population growth, the equation of motion for k is:break-even investmentactual investment k = s f(k) ( + n) k41CHAPTER 7 Economic Growt
31、h IThe Solow model diagramInvestment, break-even investmentCapital per worker, k sf(k)( + n ) kk* k = s f(k) ( +n)k42CHAPTER 7 Economic 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 br
32、eak-even investment,leading to a lower steady-state level of k.43CHAPTER 7 Economic Growth IPrediction:Higher n lower k*. And since y = f(k) , 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
33、in the long run. International evidence on population growth and income per personIncome per person in 2003 (log scale) Population growth (percent per year, average 1960-2003) 0123451001,00010,000100,00045CHAPTER 7 Economic Growth IThe Golden Rule with population growthTo find the Golden Rule capita
34、l 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 depreciation equals the population growth rate.46CHAPTER 7 Economic Growth IAlternative perspectives on popu
35、lation 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 increased sixfold, yet living standards are higher than ever.Malthus neglected the effects of technologic
36、al progress. 47CHAPTER 7 Economic Growth IAlternative perspectives on population growthThe Kremerian Model (1993)Posits that population growth contributes to economic growth. More people = more geniuses, scientists & engineers, so faster technological progress.Evidence, from very long historical per
37、iods: As world pop. growth rate increased, so did rate of growth in living standardsHistorically, regions with larger populations have enjoyed faster growth.Chapter Summary1.The Solow growth model shows that, in the long run, a countrys standard of living depends:positively on its saving ratenegativ
38、ely 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 growthChapter Summary3.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.
