lecture經(jīng)濟增長之一

1、IntermediatemacroeconomicsDai Guang2005 SpringDepartment of EconomicsNanjing University of Finance & EconomicsLECTURE SEVENEconomic Growth ILECTURE 7Economic Growth Islide 1learning objectives學(xué)習(xí)封閉經(jīng)濟的Solow 模型。Learn theclosed economy Solow model理解一個國家的生活水平如何依賴于它的儲蓄率和人口增長率。See how a countrysstandard of

2、 living depends on its savingand population growth rates黃金法則。Learn how to use the “GoldenRule” to find the optimal savings rate andcapital stockLECTURE 7Economic Growth Islide 2索羅模型The Solow ModelRobert Solow, MITwon Nobel Prize for contributions tothe study of economic growth已經(jīng)成為增長理論的一個范式paradigm:w

3、idely used in policy makingbenchmark against which mostrecent growth theories are compared尋找經(jīng)濟在長期內(nèi)增長的決定因素LECTURE 7Economic Growth Islide 3Solow model 和Lecture 3s model的不同1. K 不再固定is no longer fixed:投資使它增加investment causes it to grow,折舊使它減少depreciation causes it to shrink.2. L 不再固定 is no longer fixed

4、:人口增長使它增大population growth causes itto grow.3. 消費函數(shù)更簡單The consumption function issimpler.LECTURE 7Economic Growth Islide 4How Solow model is different fromChapter 3s model4. 沒有G 和T No G or T(only to simplify presentation;we can still do fiscal policy experiments)5. 外表不同Cosmetic differences.LECTURE 7

5、Economic Growth Islide 5生產(chǎn)函數(shù)The production function加總形式In aggregate terms: Y = F (K, L )定義Define: y = Y/L =人均產(chǎn)出k = K/L =人均資本假設(shè)規(guī)模收益不變 constant returns to scale:zY = F (zK, zL ) for any z 0提出L, 有：Y/L = F (K/L , 1)y = F (k, 1)y =f(k)其中 f(k)= F (k, 1)注意：這一生產(chǎn)函數(shù)的資本邊際產(chǎn)量是遞減的。Note: thisproduction function ex

6、hibitsdiminishing MPK.LECTURE 7Economic Growth Islide 6The production function人均產(chǎn)出, yf(k)1MPK =f(k +1) f(k) 人均資本, kLECTURE 7Economic Growth Islide 7國民收入恒等式The national incomeidentityY = C + I(記住我們假設(shè)沒有G )以 “人均per worker” 形式就是:y = c + i其中 c = C/L 且 i = I/LLECTURE 7Economic Growth Islide 8消費函數(shù)The consu

7、mption functions = 儲蓄率the saving rate,我們假設(shè)s 是一個外生變量is anexogenous parameterNote: s 是唯一的一個不等于它的大寫形除以L 的小寫字母the only lowercasevariable that is not equal toits uppercase version divided by L消費函數(shù)Consumption function:c = (1s)y (per worker)LECTURE 7Economic Growth Islide 9儲蓄和投資Saving and investment儲蓄（人均）

8、= sy國民收入恒等式 y = c + i就是: i = y c = sy (投資 = 儲蓄, likein chap. 3!)于是有：i = sy = sf(k)LECTURE 7Economic Growth Islide 10產(chǎn)出，消費和投資Output, consumption,and investment人均產(chǎn)出, yf(k)sf(k)人均資本, kk1y1c1i1LECTURE 7Economic Growth Islide 11折舊Depreciation人均折舊, k人均資本, k = 折舊率= 每期磨損掉的資本比率k1LECTURE 7Economic Growth Isli

9、de 12資本積累Capital accumulationThe basic idea基本思想:投資使 資本存量增大，折舊使 資本存量減少Investment makesthe capital stock bigger,depreciation makes it smaller.LECTURE 7Economic Growth Islide 13Capital accumulationChange in capital stock = investment depreciationk = i k由 i = sf(k) , 這就是:k = sf(k) kLECTURE 7Economic Gro

10、wth Islide 14K 的變動方程式k = sf(k) kSolow 模型的核心方程式?jīng)Q定了資本隨時間變化的行為這樣，它也就決定了所有其它 內(nèi)生變量的行為，因為它們都依賴于k. 比如.,人均收入: y = f(k)人均消費: c = (1s)f(k)LECTURE 7Economic Growth Islide 15穩(wěn)態(tài)The steady statek = sf(k) k如果投資正好可以補償折舊If investment is justenough to cover depreciation 即sf(k) = k ,那么人均資本就保持不變:k = 0.這一不變的值 k* 就是 穩(wěn)態(tài)資本

11、存量。Thisconstant value, denoted k*, is called the steadystate capital stock.LECTURE 7Economic Growth Islide 16The steady state投資和折舊人均資本, ksf(k)kk*LECTURE 7Economic Growth Islide 17向穩(wěn)態(tài)的移動Moving toward the steadystateInvestmentanddepreciationCapital perworker, ksf(k)kk*k = sf(k) kk折舊depreciationk1投資inv

12、estmentLECTURE 7Economic Growth Islide 18Moving toward the steady stateInvestmentanddepreciationCapital perworker, kkk*k1k = sf(k) ksf(k)kLECTURE 7Economic Growth Islide 19向穩(wěn)態(tài)的移動Moving toward the steadystateInvestmentanddepreciationCapital perworker, kkk*k = sf(k) ksf(k)kk1 k2LECTURE 7Economic Growt

13、h Islide 20向穩(wěn)態(tài)的移動Moving toward the steadystateInvestmentanddepreciationCapital perworker, ksf(k)kk*k = sf(k) kk2投資investment折舊depreciationkLECTURE 7Economic Growth Islide 21向穩(wěn)態(tài)的移動Moving toward the steadystateInvestmentanddepreciationCapital perworker, kkk = sf(k) ksf(k)kk2 k3 k*續(xù)向k*.增長。LECTURE 7Econ

14、omic Growth Islide 22向穩(wěn)態(tài)的移動Moving toward the steadystateInvestmentanddepreciationsf(k)Capital perworker, kkk = sf(k) kk3 k*,總結(jié):只要k k* *,投資就會超過折舊，且k 繼續(xù)向k*.增長。LECTURE 7Economic Growth Islide 23Have a try!畫出SOLOW模型，并標(biāo)出穩(wěn)態(tài)時的k*.在水平軸上，找出一個比k* 大的經(jīng)濟初始資本存量，記為k1.用剛才的方法看看 k 會隨時間怎么變化。k 會向穩(wěn)態(tài)移動還是其它方向移動呢？Y K LL LK

15、= LECTURE 7Economic Growth Islide 24例子A numerical example生產(chǎn)函數(shù)Y = F (K ,L) = K L = K 1/2L1/2集約形式:=1/21/2 1/2 L 將 y = Y/L and k = K/L 代入得到：y = f (k ) = k 1/2LECTURE 7Economic Growth Islide 25A numerical example, cont.假設(shè):s = 0.3 = 0.1初始資本存量 k = 4.0Year4.0004.2004.3952.0002.0492.0961.4001.4351.4670.6000

16、.6150.6290.4000.4200.4400.2000.1950.189LECTURE 7Economic Growth Islide 26向穩(wěn)態(tài)靠攏Approaching the Steady StateAssumptions:y = k; s = 0.3; = 0.1; initial k = 4.0Year112233kk4.0004.2004.395yy2.0002.0492.096cc1.4001.4351.467ii0.6000.6150.629 kk0.4000.4200.440 kk0.2000.1950.189Year10251004.0004.2004.3954.58

17、45.6027.3518.9629.0002.0002.0492.0962.1412.3672.7062.9943.0001.4001.4351.4671.4991.6571.8942.0962.1000.6000.6150.6290.6420.7100.8120.8980.9000.4000.4200.4400.4580.5600.7320.8960.9000.2000.1950.1890.1840.1500.0800.0020.000LECTURE 7Economic Growth Islide 27Approaching the Steady State:A Numerical Exam

18、pleAssumptions:y = k; s = 0.3; = 0.1; initial k = 4.0Year11223344 10 25 100 kk4.0004.2004.3954.5845.6027.3518.9629.000yy2.0002.0492.0962.1412.3672.7062.9943.000cc1.4001.4351.4671.4991.6571.8942.0962.100ii0.6000.6150.6290.6420.7100.8120.8980.900 kk0.4000.4200.4400.4580.5600.7320.8960.900 kk0.2000.195

19、0.1890.1840.1500.0800.0020.000LECTURE 7Economic Growth Islide 28練習(xí)：解出穩(wěn)態(tài)繼續(xù)假設(shè)：s = 0.3, = 0.1, and y = k 1/2使用動態(tài)方程k = sf(k) k解出穩(wěn)態(tài)的k, y, 和 c.LECTURE 7Economic Growth Islide 29答案k = 0s f (k *) = k *穩(wěn)態(tài)的定義當(dāng) k = 0時的方程由假設(shè)而來0.3 k * = 0.1k *k *k *and y* = k * = 3解出有: k * = 9最后, c * = (1 s)y * = 0.7 3 = 2.1LECT

20、URE 7Economic Growth Islide 30一次儲蓄率的增加An increase in thesaving rate一次儲蓄率的增加提高了投資這導(dǎo)致資本存量向新的穩(wěn)態(tài)調(diào)整:Investmentanddepreciationkk*1ks2 f(k)s1 f(k)k*2LECTURE 7Economic Growth Islide 31預(yù)測Prediction:Higher s higher k*.And since y = f(k) ,higher k* higher y* .這樣，Solow模型預(yù)測具有更高儲蓄率和投資率的國家在長期內(nèi)具有更高的資本水平和更高的人均收入。Thu

21、s, the Solow model predicts that countrieswith higher rates of saving and investment willhave higher levels of capital and income perworker in the long run.LECTURE 7Economic Growth Islide 32EgyptChadPakistanIndonesiaZimbabweKenyaIndiaCameroonUgandaMexicoIvoryCoastBrazilPeruU.S.GermanyCanadaDenmarkU.

22、K.IsraelFranceItalyJapanFinlandSingapore10,0001,000100051015Investment as percentage of output(average 1960 1992)2025303540投資率和人均收入的國際證據(jù)Income perperson in 1992(logarithmic scale)100,000LECTURE 7Economic Growth Islide 33黃金法則：介紹The Golden Rule:introduction我們已經(jīng)知道不同的 s 會導(dǎo)致不同的穩(wěn)態(tài)， 但是我們怎樣才能知道哪個才是“最好”的穩(wěn)態(tài)呢？

23、經(jīng)濟福利依賴于消費，因此“最好”的穩(wěn)態(tài)應(yīng)該具有最高的人均消費水平: c* = (1s) f(k*)而一次s 的提高： 導(dǎo)致了更高的k* and y*, 這將導(dǎo)致一個更高的 c* 另一方面，它又減少了收入中消費的份額 (1s),這將導(dǎo)致一個更低的 c*那么, 我們怎樣才能找到一個 s 和 k* 最大化 c*LECTURE 7Economic Growth Islide 34黃金水平的資本存量*穩(wěn)態(tài)時 k 的值，它最大化消費.為了找出這一黃金水平, 首先將 c* 表達成k* 的函數(shù):c*= y* i*= f(k*) i*= f(k*) k*一般地:i = k + k穩(wěn)態(tài)中有:i* = k*因為 k

24、 = 0.c goldy gold = f (kgold )igold = kgoldkgoldLECTURE 7Economic Growth Islide 35然后, 畫出 f(k*)和 k*, 并找出它們之間距離最大的那一點.The Golden Rule Capital Stock穩(wěn)態(tài)時的產(chǎn)出和折舊穩(wěn)態(tài)時的人均資本, k*k*f(k*)* * *c goldkgoldLECTURE 7Economic Growth Islide 36The Golden Rule Capital Stock穩(wěn)態(tài)時的人均資本, k*k*f(k*)*當(dāng)生產(chǎn)函數(shù)的斜率等于折舊曲線的斜率時，c* =f(k*)

25、 k*就最大化了:MPK = LECTURE 7Economic Growth Islide 37向黃金法則穩(wěn)態(tài)水平的轉(zhuǎn)移首先，經(jīng)濟并沒有向黃金法則穩(wěn)態(tài)水平靠攏的趨勢。要達到這一黃金法則要求政策制定者調(diào)整s.這一調(diào)整必導(dǎo)致一個具有更高消費水平的新的穩(wěn)態(tài)。.但是，在一調(diào)整過程中，消費發(fā)生了什么樣的變化呢？LECTURE 7Economic Growth Islide 38那么提高 c*要求降低 s.在這一向黃金穩(wěn)態(tài)的調(diào)整過程中, 消費在所有的時點上都變得更高了.以過多資本開始Starting with too muchcapital*timet0yci要提高 c* 就要求提高 s.未來的人口將享

26、受更高的消費,但現(xiàn)在的人口必須經(jīng)歷一次初始定額消費水平的下降。LECTURE 7以資本過少開始Starting with too littlecapital*t0Economic Growth Itimeslide 39yciLECTURE 7Economic Growth Islide 40小結(jié)這個簡單的solow模型不能解釋持續(xù)的經(jīng)濟增長。 它僅僅是說更高的儲蓄率會導(dǎo)致暫時的經(jīng)濟增長，但是經(jīng)濟最終會像穩(wěn)態(tài)靠近。為了解釋持續(xù)的經(jīng)濟增長，我們試著考慮其它的增長來源：人口 and 技術(shù)進步。LECTURE 7Economic Growth Islide 41人口增長Population Grow

27、th(n假設(shè)人口，即勞動力，以不變的n 增長.是外生變量)LL= n解釋: 假設(shè)在第一年 L = 1000 且增長率為2%/year (n = 0.02).那么 L = n L = 0.02 1000 = 20,因此 第二年人口 L = 1020。LECTURE 7Economic Growth Islide 42收支相抵的投資Break-even investment( +n)k =收支相抵的投資break-eveninvestment, 即使 k 保持不變的投資水平。收支相抵的投資包括:k 替代資本的磨損部分nk 補充給新工人的資本LECTURE 7Economic Growth Isli

28、de 43動態(tài)方程The equation of motion for k在有人口增長時, k 的動態(tài)方程變?yōu)閗 = sf(k) ( +n)k實際投資收支相抵的投資LECTURE 7Economic Growth Islide 44人均資本, kk*索羅模型The Solow Model diagramk =s f(k) ( +n)k收支相抵的投資水平(+n)ksf(k)LECTURE 7Economic Growth Islide 45人口增長的影響The impact ofpopulation growthInvestment,(+n2)k(+n1)ksf(k)k1* Capital pe

29、rworker, kk2*break-eveninvestment一次 n 的增加導(dǎo)致了收支相抵投資水平的下降，這導(dǎo)致了更低的穩(wěn)態(tài)資本k.LECTURE 7Economic Growth Islide 46預(yù)測Prediction:更高的 n 更低的 k*.由 y = f(k) ,更低的 k* 更低的 y*.于是，索羅模型預(yù)測具有更高人口增長率的國家在長期將會具有更低的資本水平和人均收入。BrazilLECTURE 7Economic Growth Islide 47ChadZimbabweCameroonPakistanUgandaIndiaIndonesiaIsraelPeruMexicoEgyptSingaporeU.S.CanadaFinlandJapanFranceIvoryCoastKenyaGermanyDenmarkU.K.Italy100,00010,0001,000100123 40Income perperson in 1992(logarithmic scale)Population growth (percent per year)(average 1960 1992)人