宏观经济学曼昆曼昆

1、IN THIS CHAPTER, YOU WILL LEARN:the closed economy 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 stock1Why growth mattersData on infant mortality rates:20% in the poorest 1/5 of

2、all countries0.4% in the richest 1/5In Pakistan, 85% of people live on less than $2/day.One-fourth of the poorest countries have had famines during the past 3 decades. Poverty is associated with oppression of women and minorities.Economic growth raises living standards and reduces poverty. e and pov

3、erty in the world selected countries, 2010IndonesiaUruguayPolandSenegalKyrgyz RepublicNigeriaZambiaPanamaMexicoGeorgiaPerulinks to prepared graphs notes: circle size is proportional to population size, color of circle indicates continent, press “play” on bottom to see the cross section graph evolve

4、over time e per capita andLife expectancyInfant mortalityMalaria deaths per 100,000Cell phone users per 100 peopleWhy growth mattersAnything that effects the long-run rate of economic growth even by a tiny amount will have huge effects on living standards in the long run. 1,081.4%243.7%85.4%624.5%16

5、9.2%64.0%2.5%2.0%100 years50 years25 yearsincrease in standard of living afterannual growth rate of e per capitaWhy growth mattersIf the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher from 20002010, the average person would have earned $2,782 more during

6、 the decade.The lessons of growth theorycan make a positive difference in the lives of hundreds of millions of people.These lessons help usunderstand why poor countries are poordesign policies that can help them growlearn how our own growth rate is affected by shocks and our governments policiesThe

7、Solow modeldue to Robert Solow,won Nobel Prize for contributions to the study of economic growtha 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 runHow Solo

8、w model is different from Chapter 3s model1.K is no longer fixed: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 simplerHow Solow model is different from Chapter 3s model4.no G or T(only to simplify

9、 presentation; we can still do fiscal policy experiments)5.cosmetic differencesThe 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 =

10、F (k, 1) y = f(k)where f(k) = F(k, 1) The production functionOutput per worker, y Capital per worker, k f(k)Note: this production function exhibits diminishing MPK. 1MPK = f(k +1) f(k)The national e identityY = C + I (remember, no G )In “per worker” terms: y = c + i where c = C/L and i = I /L The co

11、nsumption functions = the saving rate, the fraction of e 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 worker)Saving and investmentsaving (per worker)= y c = y (1s)y = syNa

12、tional e 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)Output, consumption, and investmentOutput per worker, y Capital per worker, k f(k)sf(k)k1 y1 i1 c1 DepreciationDepreciation per worker, k Capital per worker, k

13、k = the rate of depreciation = the fraction of the capital stock that wears out each period1Capital accumulationChange in capital stock= investment depreciationk = i kSince i = sf(k) , this es:k = s f(k) k The basic idea: Investment increases the capital stock, depreciation reduces it. The equation

14、of motion for kThe Solow models central equationDetermines behavior of capital over timewhich, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g., e per person: y = f(k)consumption per person: c = (1 s) f(k) k = s f(k) k The steady stateIf invest

15、ment is just enough to cover depreciation sf(k) = 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 The steady stateInvestment and depreciation Capital per worker, k sf(k)kk* Moving toward the stead

16、y stateInvestment and depreciation Capital per worker, k sf(k)kk* k = sf(k) kdepreciationkk1investmentMoving toward the steady stateInvestment and depreciation Capital per worker, k sf(k)k* k1kk2k = sf(k) kkMoving toward the steady stateInvestment and depreciation Capital per worker, k sf(k)k* k2inv

17、estmentdepreciationkk = sf(k) kkMoving toward the steady stateInvestment and depreciation Capital per worker, k sf(k)k* kk2k = sf(k) kkMoving toward the steady stateInvestment and depreciation Capital per worker, k sf(k)k* k2kk3k = sf(k) kkMoving toward the steady stateInvestment and depreciation Ca

18、pital per worker, k sf(k)k* k3Summary:As long as k k*, investment will exceed depreciation, and k will continue to grow toward k*.k = sf(k) kkNOW YOU TRYApproaching k* from aboveDraw the Solow model diagram, labeling the steady state k*. On the horizontal axis, pick a value greater than k* for the e

19、conomys initial capital stock. Label it k1. Show what happens to k over time. Does k move toward the steady state or away from it?28A numerical exampleProduction function (aggregate):To derive the per-worker production function, divide through by L:Then substitute y = Y/L and k = K/L to getA numeric

20、al example, cont.Assume:s = 0.3 = 0.1initial value of k = 4.0Approaching the steady state: A numerical exampleYear k y c i k k 14.0002.0001.4000.6000.4000.200 24.2002.0491.4350.6150.4200.195 34.3952.0961.4670.6290.4400.189 44.5842.1411.4990.6420.4580.184 105.6022.3671.6570.7100.5600.150 257.3512.706

21、1.8940.8120.7320.080 1008.9622.9942.0960.8980.8960.002 9.0003.0002.1000.9000.9000.000NOW YOU TRYSolve for the steady state32Continue 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. ANSWERSSolve for the steady state33

22、An increase in the saving rateInvestment and depreciationkks1 f(k)An increase in the saving rate raises investmentcausing k to grow toward a new steady state: s2 f(k)Prediction:The Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and e

23、 per worker in the long run. Are the data consistent with this prediction?International evidence on investment rates and e per person e per person in 2010 (log scale) Investment as percentage of output (average 1960-2010) The Golden Rule: IntroductionDifferent values of s lead to different steady st

24、ates. 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* reduces consumptions share of e (1s), which lowers c*. So, how do we find the s and k* that maxi

25、mize c*?The 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* = y* i*= f (k*) i* = f (k*) k* In the steady state:i* = k* because k = 0.Then, graph f(k*) and k*, look for the point where th

26、e gap between them is biggest. The Golden Rule capital stocksteady state output and depreciationsteady-state capital per worker, k* f(k*) k*The 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

27、per worker, k* f(k*)MPK = k*The transition to the Golden Rule steady stateThe economy 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 happ

28、ens to consumption during the transition to the Golden Rule? Starting 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.timet0ciyStarting with too little capitalthen increasing c* requires an increase in s.

29、 Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption.timet0ciyPopulation 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 = n

30、L = 0.02 1,000 = 20,so L = 1,020 in year 2.Break-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, k would fall as the existing ca

31、pital stock is spread more thinly over a larger population of workers.)The equation of motion for kWith population growth, the equation of motion for k is:break-even investmentactual investmentk = s f(k) ( + n) kThe Solow model diagramInvestment, break-even investmentCapital per worker, k sf(k)( + n

32、 ) kk* k = s f(k) (+n)kThe impact of population growthInvestment, break-even investmentCapital per worker, k sf(k)( +n1) kk1* ( +n2) kk2* An increase in n causes an increase in break-even investment,leading to a lower steady-state level of k.Prediction:The Solow model predicts that countries with hi

33、gher population growth rates will have lower levels of capital and e per worker in the long run. Are the data consistent with this prediction?International evidence on population growth and e per person e per person in 2010 (log scale) Population growth (percent per year, average 1961-2010) The Gold

34、en Rule with population growthTo find the Golden Rule capital 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.Alternative perspectives on population 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 hi