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Date Created: 09/23/15
The Neoclassical Growth Model also known as the Solow Growth Model There are many factors that lrl uerlce the growth rate of economles Amon ch we use our mputs A typlcally referred to as technology We can reprer sent the output of the economy the SD P m a produchon functhD Y AF K HL R l whlch captures the ldea of usmg all our mputs mto productlon mcludmg labor L through the functlon F whlch ls rhodl ed by the productlvlty A or ef clency Wlth whlch we comlome those mputs to produce output Y As wntten above FK HLR ls a general functhD where the specl cs have not been e ned Productlon functhrls hke thls are commonly constructed from data and mry m thelr comp enty A slmph ed and specl c Verslorl of the productlon functhD takes the follow form Y AKlSLQS 2 where we abstract from human capltal and natural resources to focus on the for GDP ln the data approxlmately 13 of mcome goes to capltal and 23 to r ltlorl 1f we graph how output changes as the level of physlcal capltal mcreases the graph ls Flguxe 1 GDP and capnal e 1 showsthat holdmg A and L constant as the level ofphyslcal capltal ver the bowed down concave mshmg margmal returns to dJmszhmg rate Flgure 2 shows how the morgch pmduct 0 f Capital chahges w1th the level of capltal As the level of capltal rhcreases the adetlonal amount of output from an adetlonal umt of capltal decreases cohstruct graphs for 1ahor agamst output and the margmal product of 1ahor they W111 have the same shapes as m Flgures 1 and 2 he graphs would show that output mcreases w1th 1ahor but at a decreasmg rate Figure 2 mmmismng Marginal Fromm of capnax Wheh dJscussmg growth rates we are frequently more concerned w1th how rh perpersoh changesover tlme as thJsconce t 1ower case 1etters are used for per caplta erables From equatlon 2 omde hoth sldes by the slze of the 1ahor force L Y AKlSLQS f 7 T On the rlghtrhand srde the L m the numerator and dehorruhator can he corhe bmed and we can rewrlte the equatlon as 13 y A Akla or simply y A161 3 The pereworker production function simply states that income per worker depends on productivity A and k T which is the capitalelabor ratio ie the amount of physical capital available per worker Note that because k has the same exponent as K in equation 2 if we graph y against k it will look just like Figure 1 and if we graph the marginal product of the capitalelabor ratio against k it will look like Figure 2 That is income per worker rises with the capitalelabor ratio but at a decreasing rate because there are diminishing returns to the capitalelabor ratio THOUGHT EXERCISE 1 Suppose that the aggregate production funce tion takes the form Y AKlZLlZ What is the per worker production function THOUGHT EXERCISE 2 Suppose that we include human capital in the aggregate production function and it takes the form Y AK13L13H13 What is the per worker production function Let h HL represent the average level of human capital per worker How does the average level of human capital affect GDP per capita Explain this relationship using graphs SIDE NOTE Note that y here represents output per worker We frequently talk about output per person GDP per capita If we assume that the level of population is equal to the size of the labor force then y represents GDP per capita Now we do know that only a fraction of the population is in the workforce but over a long period of time that fraction is roughly constant Call that fraction p for participation rate and let P be population such that L p Then I lt or pP w y Y 7 Y7 P ipLipy E GDP per capita py GIVE and thus income per capita is a fraction of income per worker Prof vided p is roughly constant over a long period of time any change in income per worker will be re ected directly in income per capita For simplicity we assume that p 1 which makes interpretation easier but does not change any results or intuition that we derive Now to understand how income per capita changes we need to understand what changes the capitalelabor ratio k and how the capital stock K changes Investment 1 is used for two broad purposes 1 to introduce new capital equipment and 2 to replace and repair existing capital equipment Thus we can express this as IAK6K 4 where AK represents the change in the capital stock and 6K is the depreciation of the capital stock 6 is the rate of depreciation 7 that is the fraction of the value of machinery equipment etc that is lost over a period of time For example computers depreciate at a rate of roughly 25730 per year Other equipment loses value through being used worn out or becoming obsolete and generally less productive Thus some portion of gross investment 1 goes towards simply replacing or xing the existing capital we already have and the remainder AK goes towards increasing the capital stock Thus AK is referred to as net investment our investment level after netting out the fraction spent on simply replacing worn out equipment Putting this investment equation into per capita terms and to relate it to our capitalelabor ratio k let 239 represent gross investment per capita Then we have 239Akn6k 5 which represents much the same ideas as in equation lnvestment per capita is used for replacing the worneout depreciated portion of the capitalelabor ratio 6k and increasing the level of the capitalelabor ratio Ak The only addition here is n which represents the population or worker growth rate If we eep the level of capital K constant but the size of the population L increases the capitalelabor ratio falls since k f Thus in order to maintain the same level of the capitalelabor ratio we need to invest just to keep up with increases in the population growth rate Referring back to equation yAk13 to understand how 3 GDP per capita changes we want to understand how the capitalelabor ratio changes Rearrange equation 5 Aki7n6k 6 which shows that net investment in k is equal to the difference between the gross investment per capita and the total depreciation of the capitalelabor ratio which comes from the growth rate of the population and the wearing out of machinery For reasons we will examine later in the course when we cover savings and investment note that investment by rms in physical capital re ects a demand The supply of funds for that investment comes from the savings of the economy Thus savings equals investment in the market where supply meets demand Let I be the economyewide level of investment and let S be the economyewide level of savings thus S I The level of savings in the economy is that portion of income Y not used for consumption That fraction we refer to as the savings rate and we will label it 5 lower case Note that as a fractlon of lncome 0 lt s lt 1 Thus I l ved sY Investment ls the lsetlon of ncome 58 On a pen esplts basls we would wllte l 5y Usmg pe the caplta ploduetlon funetlon then we have t at l We 3 7 whleh states that gross lnvestment pel caplta l ls the saVlngs late s tlmes the mcome pel esplts y Whlch ls detelmlned by the technology level A and the Capitalrlabor lstlo k Replace l ln equstlon 5 wlth the expresslon above 7 a we have Ale We 3 7 n a 1e 8 e above equstlon 8 has the same lntelpletstlon as the equstlon ln 6 To Vlsusllze thls equstlon and the lelstlonshlp between k and y eonsldel Flgule 3 below Figure 3 Solow Growth Model PAH steadyestale level of Cunsumpnun an Nel Invesmlem AK The uppel eulve ls the pen esplts ploduetlon functlon glven ln s the lowel ll E e a e ee 5 e e g E r ll e e 2 Va if pel stlslght llne begmnmg at the pllgln leplesents effecthe depleelstlon n 5 1e we 1 n e the levels of l and y H r a 12 l7 7 n 2 2 5 Pr gt3 ll E ll ls consumptlon pel esplts e slnee the dlfference ls lncome sVlngs or the lsetlon of lncome used ln expendltules The veltlesl dlfference between the sAIelS and n 39 39 t t t Ak Now conslder the level of the capltal stock shown 1n the gure as 1a a a lt o 3 E o EL gt R a a o a o m n o m n U o n Rquot o E o n o n m n d a o m 9 e L 1ahor ratlo ls unchangmg We refer to that level of the capltalrlabor ratro as the steadyrstate level An the vanah1es w1th an astensk kg 2 a and 1ndJcate the levels of those erables when the econorn 15 at the steadyrstate when the capltalrlabor ratro 5 un anglng GDP per caplta ls unchangmg 1 e standards of hvmg do not change Restated another way at M there 15 no a mg Conslder the econorny where the capltalrlabor ratlo 15 not at Isa but at a dl erent 1eve1 kg m the gure At that level of the capltalrlabor ratlo gross dl erence between the per worker productlon functlon and the savmgs curve 15 y a z thch1s the level of per caplta consumptlon o THOUGHT EXERCISE 3 Suppose there 15 no populatlon growth n 0 How does equatlon 8 change7 Construct a dlagram for thJs case slrrular to Flgure 3 How does 1t dl er from Flgure 37 When the economyls not at k then we have econormc growth To see thJs look at Flgure 4 he1ow Figure 4 Behavior of Economy Away from SteadyState In this gure I have removed the graph of the pereworker production funce tion 31 Ak13 just to make it more clear First consider a situation where an economy has a low level of the capitalelabor ratio represented by kLOW From the graph we can see that sAk13 lies above n6k thus Ak gt 0 In words net investment is positive and the capitalelabor ratio is increasing growing If k is growing it means that the economy is moving from kLOW towards the right as k increases over time Since k is growing then from the pereworker production function y is also increasing meaning that GDP per capita is rising On the other hand if k is larger than kquot represented by kHIGH in the gure then the depreciation of the capital exceeds the gross investment level Therefore Ak lt 0 That is the capitalelabor ratio is declining and the economy is moving to the left along the Xeaxis as the capitalelabor ratio shrinks GDP per capita is falling Whether we are at points like kLOW or kHIGH the economy is moving over time towards the steadyestate level 16 Example 1 Suppose a country suffers a major earthquake which destroys much of its capital stock eg Tokyo earthquake of 1923 The capitalelabor ratio falls signi cantly and is well below the steadyestate level Over time the country invests such that it rebuilds its capital stock and the capitalelabor ratio rises Note that initially the growth rate itself is high but declines as k increases until the growth rate of the economy reaches zero once it returns to kquot The growth rate of the capitalelabor ratio is given by Ak Growth rate of capitalelabor ratio which is just the percentage growth rate formula1 Divide equation 8 through by k to get Ak A s W7n6 9 The decline in the growth rate occurs for two reasons First as the economy grows because of diminishing returns to the capitalelabor ratio the rate at which output rises is declining This effect is captured in the rst term on the righthand side sAk23 Since the capitalelabor ratio is in the denominator as the capitalelabor ratio increases this number grows smaller Secon the effective depreciation rate is a constant fraction n 6 of the capitalistock As k increases a larger and larger amount of gross investment must go towards replacing worn out equipment Taken together the level of net investment declines as the capitalelabor ratio increases Thus the growth rate of the capitalistock Alek declines as k rises The same logic applies to GDP per capita y So the earthquake causes GDP per capita to drop immediately with the destruction of the capital stock which lowers the capitalelabor ratio Once the economy begins to invest again the GDP growth rates will be very 1 represents the same idea we used for the one year growth rates for GDP and for prices As an example for say the year 2005 would be 192005 7 192004 k72004 thh inrtieuy end dechne over tune as it returns to the eteedyetete level of the capitalrlabor retro An example of e situation where an econorny is at kHIGH 1ehor ratio tens GDP per capita dechnee es wen The rate at thch the dechne negative growth occurs is initieuy Very large but the dechne slowed down as urope approached the new eteedyetete level of the capitalrlabor ratio THOUGHT EXERCISE 4 Consider two modern day countnee in light of the model Look at Figure 5 he1ow The rs country is e developing nation with e Verylow level of the capital stock The second countryis anmdustmahzed nation but its capitalrlabor ratio is slightly 1ower then the eteedyetete level of the capitalrlabor retro M Whmh country grows teeter and why7 Figure 5 Convergence k capnaI Iahor ratio The above example in the thought exercise illustrates the concept of Une conditional Convergence De nition 1 Unconditional Convergence Poor economies mu eventually cotch up to the nch conntnes such thot lwmg stondonts mound the world mu be the same in the longrmn That follows from the idea that econornies with 1eve1s of the capitalrlabor ratio farther away from k and win grow faster and thus catch up That is rn t e sarne such as access to techno1ogy governrnent pohcies savings rates a d population growth rates arnong other things Therefore a diirerent prediction from the Neoclassical growth rnode1 is terrned Conditional Convergence De nition 2 Conditional Convergence Economies not similar thumb temstws e g somngs mtes technology mu converge to condor standards of living Conditionai convergence says that if we identify a group of countries with sirni1ar characteristics e g UK Italy Germany they win converge to sirni1ar than D the UK as it was In the early 1960s but other factors that affect growth than in the UK we would not necessariiy expect Kenya to grow faster and catch To i11ustrate how these other factors affect growth and standards of hvin consider 2 ctional countries Papadakistan PPS for short and Tsetsekostan TTS for short Suppose the countries are cornp1ete1y identica1 except that the savings rate in PPS sees is rnuch1ower than in TTS 5T Look at Figure 6 below may y39PSS Steadvsute fur Tss Because TSS has a higher savings rate the curve showing the relationship between the capitalelabor ratio k and gross investment i is shifted proportione ately upwards for TSS Thus the intersection between the effective depreciation line n6k and savings gross investment sy sAka lies much further to the right at a higher level of the capitalelabor ratio Thus TSS will achieve a higher GDP per capita in longerun no matter where their initial levels of the capitale labor ratio are currently Suppose as shown in the diagram that the current levels of the capitalelabor ratios in PSS and TSS are such that kpss lt kTSS If the countries were completely identical including the savings rate then the prediction of unconditional convergence would hold and we would expect PSS to grow faster because it is farther from kquot that is PSS has more room to grow However since the savings rate in PSS is much lower even when kpss lt kTSS PSS may or may not grow faster than TSS Their growth rates will depend on how far from their own 16 levels they currently are THOUGHT EXERCISE 5 Construct a diagram for PSS and TSS where they are identical in every way including the savings rate but the technology level A is much higher in PSS than in TSS Which country will have a higher steadyestate level of GDP per capita If they both begin with the same initial capitalelabor ratio which country grows faster Explain why THOUGHT EXERCISE 6 Construct another diagram for PSS and TSS where they are identical in every way but now suppose the population growth rate is much higher in PSS than in TSS Which country will have a higher steadyestate If they both begin with the same initial capitalelabor ratio which country grows faster Explain why Important reminder 7 at the steadyestate level of the capitalilabor ratio 16 the growth rate of GDP per capita is zero because the economy is saving just enough to o set the e ectiue depreciation However for the past 150 years we have had positive levels of the growth rate of GDP per capita Standards of living have been steadily increasing and we have not observed the US economy and most other economies attening out such that GDP per capita stays the same over a long period of time Notice that while increases in the savings rate will increase the steadyestate level of GDP per capita higher 5 cannot sustain growth because the savings rate cannot be greater than 1 Can you think of any other problems associated with government policies that try to raise the savings rate to increase GDP per capita In contrast the technology parameter A has no known upper bound There is of course an argument that there does exist limits to our knowle edgetechnology and that argument has been made for nearly 200 years At this point its not clear whether that limit exists andor how far away from that limit we actually are Consider the US economy which has been at the forefront of RampD invest ment for most of the past 6070 years As production becomes more ef cient 10 through the apphoatlon of teohno1ogy ollr productwlty rlses and ls re ected ln an lnorease ln A Flgllre 71Hustrates the effect Figure 7 Effect of Technological Improvemenis e 1ower of the two Sawngs Curves ls for the US as of 1960 and the upper ls that ln 1960 our productlvlty level was A1 and l lt as A2 w ere A2 gt A1 that ls teohno gy lrnp oved The steadyrstate capltalrlabor ratlo an oaplta ln 1960 w re rnuoh1ower than l us lf the US ogy lrnproved the Savmgs Curve sthted up As the Savlngs Curve sthts up sustaln posltlve growth rates ln GDP per oaplta and standards of llvmg THOUGHT EXERCISE 7 Whloh governmth pohcy would he rnore 11ke1y to ralse GDP er oaplta growth rates and wh 7 A A oonsllrnptlon sales tax to olsoollrage oonsllrnptlon and lnorease saVlngs or B suhsloles for 9D pald for by talnng corporate pro ts Explam the effects of each on growth rates Dlstlnglllsh hetween temporary effects and perrnanent effects THOUGHT EXERCISE 8 The ease of slngapore and Hong Kong slngapore and Hong Kong are oonsldered two of the East Aslan Growth ere ao1es quot Both had Very hlgh growth rates ln the postrWorld War II era Thelr growth rates were 1arge1y the sarne slnoe 1960 If both oollntnes started at 11 roughly the same distance from the steadyestate7 what would you expect of their growth rates over time Use a diagram to illustrate your answer and relate your answer to the concepts of unconditional convergence and condie tional convergence When looking more carefully at the data7 we see that the savings rate in Singapore was considerably higher than in Hong Kong If that were the only difference between the two how would you expect their growth rates to compare Use diagrams to illustrate your answer Since their growth rates7 at least until recently were largely the same and assuming the population growth rates7 n and depreciation 6 were the same7 how could you explain them having very similar growth rates and similar initial and recent levels of GDP per capita Based on your explanation7 what would you expect for the future of these two countries
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