503 Applied Macroeconomics Chapter 2. The Solow Growth Model Prof. M. El-Sakka Dept of Economics - Kuwait University.

503 Applied MacroeconomicsChapter 2. The Solow Growth Model

Prof. M. El-SakkaProf. M. El-SakkaDept of Economics - Kuwait UniversityDept of Economics - Kuwait University

• SOME BASIC FACTS ABOUT GROWTHSOME BASIC FACTS ABOUT GROWTH

• 1. High levels of living standards: in industrialized countries, in 1. High levels of living standards: in industrialized countries, in USUS 10 times, in 10 times, in EuropeEurope 30 times higher than 1 century ago. 50 times and 300 times 30 times higher than 1 century ago. 50 times and 300 times respectivelyrespectively than 2 centuries ago. than 2 centuries ago.

• 2. Enormous differences in standards of living, e.g., US, Germany and Japan are 2. Enormous differences in standards of living, e.g., US, Germany and Japan are 20 times Bangladesh. 20 times Bangladesh.

• 3. Growth Miracles: the cases where growth exceeds the world average for 3. Growth Miracles: the cases where growth exceeds the world average for extended periods, and moves rapidly up the world income distribution. extended periods, and moves rapidly up the world income distribution. Examples are Japan, South Korea, China.. etc. Average NIC’s growth is over 5% Examples are Japan, South Korea, China.. etc. Average NIC’s growth is over 5% since 1960s. since 1960s.

• 4. Growth Disasters: the case where growth is less than world levels, e.g., 4. Growth Disasters: the case where growth is less than world levels, e.g., ArgentinaArgentina (income was behind world leaders in 1900, now in middle income (income was behind world leaders in 1900, now in middle income countries), and countries), and Sub-Saharan countries Sub-Saharan countries (average incomes near subsistence (average incomes near subsistence levels). levels).

• Implications: Implications: vast differences are in nutrition, literacy, infant mortality, life vast differences are in nutrition, literacy, infant mortality, life expectancy and, other indicators. expectancy and, other indicators.

• The principal conclusion of the Solow model The principal conclusion of the Solow model is that the accumulation of is that the accumulation of physical capital cannot account for either the vast growth in average output physical capital cannot account for either the vast growth in average output over over time time or the vast or the vast geographic differences geographic differences in average output.in average output.

• To address the central questions of growth theory, we must move beyond the To address the central questions of growth theory, we must move beyond the Solow modelSolow model

The Solow modelThe Solow model

• The Solow model focuses on four variables: output (The Solow model focuses on four variables: output (Y Y ), capital ), capital ((KK), labor (), labor (LL), and “knowledge” or the “effectiveness of labor” (), and “knowledge” or the “effectiveness of labor” (AA).).

• Production Function: Production Function:

• YY((tt) = ) = F F ((KK((tt), ), AA((tt))LL((tt)), )), (1.1)(1.1)

• Notice also that Notice also that A A and and L L enter multiplicatively, known as enter multiplicatively, known as labor-labor-augmentingaugmenting or or Harrod-neutral, or Harrod-neutral, or effective laboreffective labor. .

• Assumptions Concerning the Production FunctionAssumptions Concerning the Production Function

• 1. constant returns to scale in its two arguments, capital and effective labor (with 1. constant returns to scale in its two arguments, capital and effective labor (with A A held fixed), i.e, increasing K, L by c held fixed), i.e, increasing K, L by c

• F F ((cKcK,,cALcAL) = ) = cF cF ((KK, , ALAL) ) for all for all c c ≥ 0. ≥ 0. (1.2) (1.2)

• 2. Other inputs are relatively unimportant.2. Other inputs are relatively unimportant.

• Setting Setting c c = 1/= 1/AL AL in equation (1.2) yields;in equation (1.2) yields;

• F F ( (K/AL, 1) = K/AL, 1) = 1/1/AL F AL F ((KK, , ALAL). ). (1.3)(1.3)

• Here Here KK//AL AL is the amount of is the amount of capital per unit of effective laborcapital per unit of effective labor, and , and F F ((KK, , ALAL)/)/AL AL is is YY//ALAL, , output per unit of effective laboroutput per unit of effective labor. Define . Define k k = = KK//ALAL, , y y = = YY//ALAL, and , and f f ((kk) ) = = F F ((kk,1). Then we can rewrite (1.3) as;,1). Then we can rewrite (1.3) as;

• y y = = f f ((kk).). (1.4)(1.4)

• Output per unit of effective labor as a function ofOutput per unit of effective labor as a function of capital per unit of effective capital per unit of effective laborlabor

• f f ((kk), is assumed to satisfy, ), is assumed to satisfy, f’ f’ (•) (•) > > 0, 0, f” f” (•) (•) < < 0. 0.

• the assumptions that the assumptions that f’ f’ ((kk) is positive and ) is positive and f” f” ((kk) is negative imply that the ) is negative imply that the marginal product of capital is positivemarginal product of capital is positive, but that it declines as capital (per , but that it declines as capital (per unit of effective labor) rises.unit of effective labor) rises.

• A specific example of a production function is the A specific example of a production function is the Cobb–DouglasCobb–Douglas function,function,

• F F ((KK, , ALAL) = ) = KKα α ((ALAL))1−1−αα, , 00< α < < α < 1. 1. (1.5)(1.5)

• Multiplying both inputs by Multiplying both inputs by c c gives us;gives us;

• F F ((cKcK, , cALcAL) = () = (cK cK ))αα((cALcAL))1−1−αα

• = = ccααcc1−1−ααKKαα((ALAL))1−1−αα

• = = cF cF ((KK, , ALAL).).

The intensive form;The intensive form;•divide both inputs by divide both inputs by ALAL; this yields; this yields

•f f ((kk) ≡ ) ≡ FF ( (K/ALK/AL, 1) , 1)

= (= (K /ALK /AL))αα (1.7)(1.7)

= = kkαα..

•Equation (1.7) implies that Equation (1.7) implies that f’ f’ ((kk) = ) = αkαkαα−1−1. . this expression is this expression is positivepositive, that it , that it approaches infinity as approaches infinity as k k approaches zero, and that it approaches zero as approaches zero, and that it approaches zero as k k approaches infinity. approaches infinity.

Finally, Finally, •f” f” ((kk) = −(1 − ) = −(1 − αα))αkαkαα−2−2, ,

which is which is negativenegative

The Evolution of the Inputs into Production

• How the stocks of labor, knowledge, and capital change over time?How the stocks of labor, knowledge, and capital change over time?

• Given capital, labor, and knowledge, and labor and Given capital, labor, and knowledge, and labor and knowledge grow at constant knowledge grow at constant ratesrates::

• where where n n and and g g are exogenous parameters and where a dot over a variable are exogenous parameters and where a dot over a variable denotes a denotes a derivative with respect to timederivative with respect to time

• since ln since ln X X is a function of is a function of X X and and X X is a function of is a function of tt, we can use the chain rule to , we can use the chain rule to write;write;

• the the rates of change of the logs of rates of change of the logs of L L and and A A are constant are constant and that they and that they equal equal n n and and gg, respectively. Thus,, respectively. Thus,

• where where LL(0) and (0) and AA(0) are the values of (0) are the values of L L and and A A at time 0. Exponentiating at time 0. Exponentiating gives us gives us

• Output is divided between consumption and investmentOutput is divided between consumption and investment, output devoted , output devoted to investment, to investment, ss, and existing capital depreciates at rate , and existing capital depreciates at rate δδ. . Thus

The Dynamics of the Model

• The Dynamics of The Dynamics of kk• Since Since k k = = KK//ALAL, we can use the chain rule to find, we can use the chain rule to find

• KK//AL AL is simply is simply kk. . • Substituting into (1.16)Substituting into (1.16)•

• Break even investment is the amount of investment that must be done Break even investment is the amount of investment that must be done just to keep just to keep k k at its existing levelat its existing level

• Finally, using the fact that Finally, using the fact that YY//AL AL is given by is given by f f ((kk), we have), we have

• Equation (1.18) is the key equation of the Solow modelEquation (1.18) is the key equation of the Solow model. It . It states that the rate of change of the capital stock per unit of states that the rate of change of the capital stock per unit of effective labor is the difference between two terms. effective labor is the difference between two terms.

• - The first, - The first, sf sf ((kk), is actual investment per unit of effective ), is actual investment per unit of effective labor:labor: output per unit of effective labor is output per unit of effective labor is f f ((kk), ), and the and the fraction of that output that is invested is fraction of that output that is invested is ss. .

• - The second term, (- The second term, (n n + + gg++δδ))kk, is , is breakeven investment, breakeven investment, the the amount of investment that must be done just to keep amount of investment that must be done just to keep k k at its at its existing levelexisting level..

• Two reasons that some investment is needed to Two reasons that some investment is needed to prevent prevent k k from falling (break even)from falling (break even)..

• FirstFirst, existing capital is depreciating. , existing capital is depreciating. δk δk term in (1.18). term in (1.18).

• SecondSecond, since the quantity of effective labor is growing at , since the quantity of effective labor is growing at rate rate n n + + gg, the capital stock must grow at rate , the capital stock must grow at rate n n + + g g to hold to hold k k steady. This is the (steady. This is the (n n + + gg))k k term in (1.18). term in (1.18).

Note that: Note that: • When When sf sf ((kk)) exceeds exceeds ((n n + + gg++δδ))kk, , , , k k is rising. is rising. • When When sf sf ((kk)) falls short of falls short of ((n n + + gg++δδ))kk, , k k is falling. is falling. • And when the two are equal, And when the two are equal, k k is constantis constant..

Figure 1.4.

• If If k k is initially less than is initially less than kk , actual investment exceeds break-even ∗, actual investment exceeds break-even ∗investment, and so investment, and so ˙̇k k is positive—that is, is positive—that is, k k is rising. is rising.

• If If k k exceeds exceeds kk , ∗, ∗ ˙̇k k is negative. is negative. • Finally, if Finally, if k k equals equals kk , then ∗, then ∗ ˙̇k k is zero. Thus, regardless of where is zero. Thus, regardless of where k k starts, starts,

it converges to it converges to kk and remains there.∗ and remains there.∗

• The Balanced Growth PathThe Balanced Growth Path• the Solow model implies that, the economy converges to a the Solow model implies that, the economy converges to a balanced balanced

growth pathgrowth path—a situation where —a situation where each variable of the model is growing at each variable of the model is growing at a constant ratea constant rate. .

• On the balanced growth path, On the balanced growth path, the growth rate of output per worker the growth rate of output per worker is is determined solely by the determined solely by the rate of technological progressrate of technological progress..

• The Impact of a Change in the Saving RateThe Impact of a Change in the Saving Rate• suppose that there is a permanent increase in suppose that there is a permanent increase in ss..

• The Impact on OutputThe Impact on Output• ↑ ↑ s s shifts the actual investment line up, and so shifts the actual investment line up, and so kk ↑∗ ↑∗ . This is . This is

shown in Figure 1.4. At this level, actual investment now shown in Figure 1.4. At this level, actual investment now exceeds break-even investment and so exceeds break-even investment and so ˙̇k k is positive. Thus is positive. Thus k k begins to rise until it reaches the new value of begins to rise until it reaches the new value of kk∗∗

Figure 1.4

• 1. s 1. s jumps up at time jumps up at time tt0 and remains constant thereafter.0 and remains constant thereafter.• 2. ˙̇k k jumps from zero to a strictly positive amount. jumps from zero to a strictly positive amount. k k rises rises

gradually from the old value of gradually from the old value of kk to the new value∗ to the new value∗ , and , and ˙̇k k falls gradually back to zero. falls gradually back to zero.

• 3. When When k k is constant, is constant, YY//L L grows at rate grows at rate gg. When . When k k is is increasing, increasing, YY//L L grows both because grows both because A A is increasing and is increasing and because because k k is increasing. Thus its growth rate is increasing. Thus its growth rate exceeds exceeds gg. . When When k k reaches the new value of reaches the new value of kk , however, again ∗, however, again ∗ only only the growth of the growth of A A contributes to the growth of contributes to the growth of YY//LL, and so the , and so the growth rate of growth rate of YY//L L returns to returns to gg. The . The growth rate growth rate of output per of output per worker, which is initially worker, which is initially gg, jumps upward at , jumps upward at tt0 and then 0 and then gradually returns to its initial level. gradually returns to its initial level.

Figure 1.5

Figure 1.5

• In sum, a change in the saving rate has a In sum, a change in the saving rate has a level effect level effect but not but not a a growth effectgrowth effect: : it does not affect the growth rate of output it does not affect the growth rate of output per worker on the balanced growth pathper worker on the balanced growth path..

• The Impact on ConsumptionThe Impact on Consumption• Consumption per unit of effective labor equals output per unit Consumption per unit of effective labor equals output per unit

of effective laborof effective labor, , f f ((kk), ), times the fraction of that output that is times the fraction of that output that is consumed, consumed, 1 − 1 − ss. Thus, since . Thus, since s s changes discontinuously at changes discontinuously at tt0 and 0 and k k does not, initially consumption per unit of effective does not, initially consumption per unit of effective labor jumps downward. labor jumps downward. Consumption then rises gradually as Consumption then rises gradually as k k rises and rises and ss remains at its higher level. remains at its higher level.

• whether the increase raises or lowers consumption in the whether the increase raises or lowers consumption in the long run depends on whether long run depends on whether

• f f ((kk )—the marginal product of capital—is more or less than ∗)—the marginal product of capital—is more or less than ∗nn++gg++δδ. Intuitively, . Intuitively,

• when when k k rises, investment must rise by rises, investment must rise by n n + + gg++δ δ times the times the change in change in k k for the increase to be sustained. for the increase to be sustained.

• If If f f ((kk ) is less than ∗) is less than ∗ n n + + g g ++δδ, then the additional output from , then the additional output from the increased capital is not enough to maintain the capital the increased capital is not enough to maintain the capital stock at its higher level. In this case, consumption must fall stock at its higher level. In this case, consumption must fall to maintain the higher capital stock. to maintain the higher capital stock.

• If If f f ((kk ) exceeds ∗) exceeds ∗ n n + + g g ++δδ, on the other hand, there is more , on the other hand, there is more than enough additional output to maintain than enough additional output to maintain k k at its higher at its higher level, and so consumption rises.level, and so consumption rises.

Figure 1.6.

• c c is the distance between ∗ is the distance between ∗ f f ((kk) and () and (nn++gg++δδ))k k at at k k = = kk . ∗. ∗Figure 1.6 shows the determinants of Figure 1.6 shows the determinants of c c for ∗ for ∗ three different three different values of values of ss (and hence three different values of (and hence three different values of kk ). ∗). ∗

• In the top panelIn the top panel, , s s is high, and so is high, and so kk is high and ∗ is high and ∗ f f ((kk )∗)∗ is less is less than than n n + + g g + + δδ. As a result, an increase in the saving rate . As a result, an increase in the saving rate lowerslowers consumption even when the economy has reached its consumption even when the economy has reached its new balanced growth path. new balanced growth path.

• In the middle panel, In the middle panel, s s is low, is low, kk is low, ∗ is low, ∗ f f ((kk )∗)∗ is greater than is greater than n n + + g g + + δδ, and an increase in , and an increase in s s raises consumption in the long raises consumption in the long run.run.

• In the bottom panelIn the bottom panel, , s s is at the level that causes is at the level that causes f f ((kk )∗)∗ to just to just equal equal n n + + gg++δδ—that is, the —that is, the f f ((kk) and () and (n n + + gg++δδ))k k loci are loci are parallel parallel at at k k ==kk . ∗. ∗ In this case, a marginal change in In this case, a marginal change in s s has no has no effect on consumption in the long run, and effect on consumption in the long run, and consumption is at consumption is at its maximum possible its maximum possible level among balanced growth paths.level among balanced growth paths.

Figure 1.6.

Figure 1.6.

• Quantitative ImplicationsQuantitative Implications• The long-run effect of a rise in saving on output is given byThe long-run effect of a rise in saving on output is given by

• After manipulation

• kk∗∗f f ((kk )/∗)/∗ f f ((kk ) is the elasticity of output with respect to ∗) is the elasticity of output with respect to ∗capital at capital at k k = = kk . Denoting this by ∗. Denoting this by ∗ αK αK ((kk ), we have∗), we have∗

• In most countries, the share of income paid to capital is In most countries, the share of income paid to capital is about one-thirdabout one-third. If we use this as an estimate of . If we use this as an estimate of αK αK ((kk ), it ∗), it ∗follows that the elasticity of output with respect to the saving follows that the elasticity of output with respect to the saving rate in the long run is rate in the long run is about one-halfabout one-half. Thus, for example, a 10 . Thus, for example, a 10 percent increase in the saving rate (from 20 percent of percent increase in the saving rate (from 20 percent of output to 22 percent, for instance) raises output per worker in output to 22 percent, for instance) raises output per worker in the long run by about 5 percent relative to the path it would the long run by about 5 percent relative to the path it would have followed. Even a 50 percent increase in have followed. Even a 50 percent increase in s s raises raises yy ∗ ∗only by about 22 percent. only by about 22 percent. Thus significant changes in saving Thus significant changes in saving have only moderate effects on the level of output on the have only moderate effects on the level of output on the balanced growth pathbalanced growth path..

• The Solow Model and the Central Questions of Growth The Solow Model and the Central Questions of Growth TheoryTheory

• The Solow model identifies two possible sources of variationThe Solow model identifies two possible sources of variation—either —either over time or across parts of the worldover time or across parts of the world—in output per —in output per worker: worker:

• 1. 1. differences in capital per worker (differences in capital per worker (KK//LL) and differences in ) and differences in the effectiveness of labor (the effectiveness of labor (AA). ).

• Only differences in the Only differences in the effectiveness of labor effectiveness of labor have any have any reasonable hope of accounting for the vast differences in reasonable hope of accounting for the vast differences in wealth across time and space. variations in the accumulation wealth across time and space. variations in the accumulation of physical capital of physical capital do not account do not account for a significant part of for a significant part of either worldwide economic growth or cross-country income either worldwide economic growth or cross-country income differences.differences.

• Thus differences in K/L cannot account for the differences in output per Thus differences in K/L cannot account for the differences in output per worker that we observe.worker that we observe.

• 2. The other potential source of variation in output per worker is 2. The other potential source of variation in output per worker is the the effectiveness of laboreffectiveness of labor. .

• Unfortunately, however, the Solow model has little to say about the Unfortunately, however, the Solow model has little to say about the effectiveness of labor. Most obviously, theeffectiveness of labor. Most obviously, the growth of the effectiveness of growth of the effectiveness of labor is exogenouslabor is exogenous: the model takes as given the behavior of the variable : the model takes as given the behavior of the variable that it identifies as the driving force of growth. that it identifies as the driving force of growth.

• What determines the stock of knowledge over time? What determines the stock of knowledge over time? • There are other possible interpretations of There are other possible interpretations of AA: the education and skills of : the education and skills of

the labor force, the strength of property rights, the quality of the labor force, the strength of property rights, the quality of infrastructure, cultural attitudes toward entrepreneurship and work, and infrastructure, cultural attitudes toward entrepreneurship and work, and so onso on

Empirical Applications

• To see how growth accounting works, consider again the To see how growth accounting works, consider again the production functionproduction function

• YY((t t ) = ) = F F ((KK((t t ), ), AA((t t ))LL((t t )). )). This impliesThis implies

• where where ∂Y∂Y//∂L ∂L and and ∂Y∂Y//∂A ∂A denote [denote [∂Y∂Y//∂∂((ALAL)])]A A and [and [∂Y∂Y//∂∂((ALAL)])]LL, , respectively.respectively.

• Dividing both sides by Dividing both sides by YY((t t ) and rewriting the terms on the ) and rewriting the terms on the right-hand side yieldsright-hand side yields

• Here Here αLαL((t t ) is the ) is the elasticity of output with respect to labor elasticity of output with respect to labor at at time time tt, , αK αK ((t t ) is again the ) is again the elasticity of output with respect to elasticity of output with respect to capitalcapital, and , and RR((t t ) ≡ [) ≡ [AA((t t )/)/YY((t t )][)][∂Y∂Y((t t )/)/∂A∂A((t t )][ )][ ˙ ˙ AA((t t )/)/AA((t t )]. )]. Subtracting Subtracting ˙̇L L ((t t )/)/LL((t t ) ) from both sides and using the fact that from both sides and using the fact that αLαL((t t ) + ) + αK αK ((t t ) = 1 gives an ) = 1 gives an expression for the growth rate of expression for the growth rate of output per workeroutput per worker::

• Thus (1.35) provides a way of Thus (1.35) provides a way of decomposing the growth of decomposing the growth of output per worker into the contribution of growth of capital output per worker into the contribution of growth of capital per workerper worker and a remaining term, and a remaining term, the the Solow residualSolow residual..

• The Solow residual is sometimes interpreted as a measure The Solow residual is sometimes interpreted as a measure of the of the contribution of technological progress.contribution of technological progress.

• Growth accounting only examines the immediate Growth accounting only examines the immediate determinants of growth: it asks how much factor determinants of growth: it asks how much factor accumulation, improvements in the quality of inputs, and so accumulation, improvements in the quality of inputs, and so on contribute to growth while ignoring the deeper issue of on contribute to growth while ignoring the deeper issue of what causes the changes in those determinants.what causes the changes in those determinants.

• Growth accounting has been applied to many issues. For Growth accounting has been applied to many issues. For example, it has played a major role in a recent debate concerning example, it has played a major role in a recent debate concerning the exceptionally rapid growth of the the exceptionally rapid growth of the newly industrializing newly industrializing countries of East Asiacountries of East Asia. . Young (1995) Young (1995) uses detailed growth uses detailed growth accounting to argue that the higher growth in these countries than accounting to argue that the higher growth in these countries than in the rest of the world is almost entirely due in the rest of the world is almost entirely due to rising investmentto rising investment, , increasing labor force participationincreasing labor force participation, and improving , and improving labor quality labor quality (in (in terms of education), and not to rapid technological progress and terms of education), and not to rapid technological progress and other forces affecting the Solow residual.other forces affecting the Solow residual.

• This suggests that for other countries to replicate the NICs’ This suggests that for other countries to replicate the NICs’ successes, it is successes, it is enough for them to promote accumulation of enough for them to promote accumulation of physical and human capital and greater use of resourcesphysical and human capital and greater use of resources, and that , and that they need not tackle the even more difficult task of finding ways of they need not tackle the even more difficult task of finding ways of obtaining greater output for a given set of inputs.obtaining greater output for a given set of inputs.

• Hsieh (2002), however, observes that one can do growth Hsieh (2002), however, observes that one can do growth accounting by examining accounting by examining the behavior of factor returns the behavior of factor returns rather than rather than quantities. If rapid growth comes solely from capital accumulation, quantities. If rapid growth comes solely from capital accumulation, for example, we will see either a large for example, we will see either a large fall in the return to capital fall in the return to capital or or a a large rise in capital’s share large rise in capital’s share (or a combination). Doing the growth (or a combination). Doing the growth accounting this way, Hsieh finds a much accounting this way, Hsieh finds a much larger role for the larger role for the residualresidual. Young (1998) and Fernald and Neiman (2008) extend the . Young (1998) and Fernald and Neiman (2008) extend the analysis further, and identify reasons that Hsieh’s analysis analysis further, and identify reasons that Hsieh’s analysis may may have underestimated the role of factor accumulationhave underestimated the role of factor accumulation ..

• Growth-accounting studies of the rebound suggest that Growth-accounting studies of the rebound suggest that computers computers and other types of information technologand other types of information technology are the main source of y are the main source of the rebound (see, for example, Oliner and Sichel, 2002, and the rebound (see, for example, Oliner and Sichel, 2002, and Oliner, Sichel, and Stiroh, 2007). Until the mid-1990s, the rapid Oliner, Sichel, and Stiroh, 2007). Until the mid-1990s, the rapid technological progress in computers and their introduction in many technological progress in computers and their introduction in many sectors of the economy sectors of the economy appear to have had little impact on appear to have had little impact on aggregate productivityaggregate productivity In partIn part, this was simply because , this was simply because computers, although spreading rapidly, were still only a small computers, although spreading rapidly, were still only a small fraction of the overall capital stock.fraction of the overall capital stock.

• ConvergenceConvergence• An issue that has attracted considerable attention in empirical work on An issue that has attracted considerable attention in empirical work on

growth is whether poor countries tend to grow faster than rich countries.growth is whether poor countries tend to grow faster than rich countries.

• FirstFirst, the Solow model predicts that countries converge to their balanced , the Solow model predicts that countries converge to their balanced growth paths. Thus to the extent that differences in output per worker growth paths. Thus to the extent that differences in output per worker arise from countries being at different points relative to their balanced arise from countries being at different points relative to their balanced growth paths, growth paths, one would expect poor countries to catch up to rich ones.one would expect poor countries to catch up to rich ones.

• SecondSecond, the Solow model implies that the rate of return on capital is , the Solow model implies that the rate of return on capital is lower in countries with more capital per worker. lower in countries with more capital per worker. Thus there are incentives Thus there are incentives for capital to flow from rich to poor countriesfor capital to flow from rich to poor countries; this will also tend to cause ; this will also tend to cause convergence. convergence.

• ThirdThird, if there are lags in the diffusion of knowledge, income differences , if there are lags in the diffusion of knowledge, income differences can arise because some countries are not yet employing the best can arise because some countries are not yet employing the best available technologies. These differences might tend to shrink as poorer available technologies. These differences might tend to shrink as poorer countries gain access to state-of-the-art methods.countries gain access to state-of-the-art methods.

• Baumol (1986) examines convergence from 1870 to 1979 among Baumol (1986) examines convergence from 1870 to 1979 among the 16 industrialized countries. Baumol regresses output growth the 16 industrialized countries. Baumol regresses output growth over this period on a constant and initial income.over this period on a constant and initial income.

• Here ln(Here ln(YY//NN) is log income per person, ) is log income per person, ε ε is an error term, and is an error term, and i i indexes indexes countriescountries. If there is convergence, . If there is convergence, b b will be negativewill be negative: countries with : countries with higher initial incomes have lower growth. A value for higher initial incomes have lower growth. A value for b b of −1 of −1 corresponds corresponds to to perfect convergenceperfect convergence: higher initial income on average lowers : higher initial income on average lowers subsequent growth one-for-one, and so output per person in 1979 is subsequent growth one-for-one, and so output per person in 1979 is uncorrelated with its value in 1870. A value for uncorrelated with its value in 1870. A value for b b of 0of 0, on the other hand, , on the other hand, implies that growth is uncorrelated with initial income and thus that there implies that growth is uncorrelated with initial income and thus that there is is no convergenceno convergence..

• The results are:

• The numbers in parentheses are standard errors. The numbers in parentheses are standard errors. • The regression suggests almost perfect convergence. The estimate of The regression suggests almost perfect convergence. The estimate of b b

is almost exactly equal to −1is almost exactly equal to −1, and it is estimated fairly precisely; the two-, and it is estimated fairly precisely; the two-standard-error confidence interval is (0.81, 1.18). In this sample, per standard-error confidence interval is (0.81, 1.18). In this sample, per capita income today is essentially unrelated to its level 100 years ago.capita income today is essentially unrelated to its level 100 years ago.

• DeLong (1988) demonstrates, however, that Baumol’s finding is largely DeLong (1988) demonstrates, however, that Baumol’s finding is largely spuriousspurious. There are two problems. . There are two problems.

• The first is The first is sample selectionsample selection. . Since historical data are constructed Since historical data are constructed retrospectively, the countries thatretrospectively, the countries that have long data series are generally have long data series are generally those that are the most industrialized todaythose that are the most industrialized today. Thus countries that were not . Thus countries that were not rich 100 years ago are typically in the sample rich 100 years ago are typically in the sample only if they grew rapidly only if they grew rapidly over the next 100 years. Countries that were rich 100 years ago, in over the next 100 years. Countries that were rich 100 years ago, in contrast, contrast, are generally included even if their subsequent growth was only are generally included even if their subsequent growth was only moderatemoderate. Because of this, we are likely to see . Because of this, we are likely to see poorer countries growing poorer countries growing faster than richer ones in the sample of countries we consider, even if faster than richer ones in the sample of countries we consider, even if there is no tendency for this to occur on average.there is no tendency for this to occur on average.

• DeLong therefore considers DeLong therefore considers the richest countries as of 1870the richest countries as of 1870; specifically, ; specifically, his sample consists of all countries at least as rich as the second poorest his sample consists of all countries at least as rich as the second poorest country in Baumol’s sample in 1870, Finland. This causes him to add country in Baumol’s sample in 1870, Finland. This causes him to add sevenseven countries to Baumol’s list (Argentina, Chile, East Germany, countries to Baumol’s list (Argentina, Chile, East Germany, Ireland, New Zealand, Portugal, and Spain) and to Ireland, New Zealand, Portugal, and Spain) and to drop one drop one (Japan).22(Japan).22

• Figure 1.8 shows the scatterplot for the unbiased sample. The inclusion Figure 1.8 shows the scatterplot for the unbiased sample. The inclusion of the new countries weakens the case for convergence considerably. of the new countries weakens the case for convergence considerably. The regression now produces an estimate The regression now produces an estimate of of b b of −0.566of −0.566, with a standard , with a standard error of 0.144. Thus accounting for the selection bias in Baumol’s error of 0.144. Thus accounting for the selection bias in Baumol’s procedure eliminates about procedure eliminates about half of the convergence half of the convergence that he finds.that he finds.

• The second problem that DeLong identifies is The second problem that DeLong identifies is measurement errormeasurement error. . Estimates of real income per capita in 1870 are imprecise. Measurement Estimates of real income per capita in 1870 are imprecise. Measurement error again creates bias toward finding convergence. When 1870 income error again creates bias toward finding convergence. When 1870 income is overstated, growth over the period 1870–1979 is understated by an is overstated, growth over the period 1870–1979 is understated by an equal amount; and vice versa. Thus measured growth tends to be lower equal amount; and vice versa. Thus measured growth tends to be lower in countries with higher measured initial income even if there is no in countries with higher measured initial income even if there is no relation between actual growth and actual initial income.relation between actual growth and actual initial income.

• DeLong therefore considers the following model:DeLong therefore considers the following model:

• Here ln[(Here ln[(YY//NN)1870] is the true value of log income per capita in 1870 ∗)1870] is the true value of log income per capita in 1870 ∗and ln[(and ln[(YY//NN)1870] is the measured value. )1870] is the measured value. ε ε and and u u are assumed to be are assumed to be uncorrelated with each other and with ln[(uncorrelated with each other and with ln[(YY//NN)1870].)1870].

• Figure 1.9 is a Figure 1.9 is a convergence scatterplot convergence scatterplot analogous to Figures 1.7 and 1.8 analogous to Figures 1.7 and 1.8 for virtually the for virtually the entire non-Communist world for the period 1970–2003entire non-Communist world for the period 1970–2003. As . As the figure shows, there is the figure shows, there is little evidence of convergencelittle evidence of convergence. .

• Saving and InvestmentSaving and Investment• Now suppose that the saving rate in one country rises. If all the additional Now suppose that the saving rate in one country rises. If all the additional

saving is invested domestically, the marginal product of capital in that saving is invested domestically, the marginal product of capital in that country falls below that in other countriescountry falls below that in other countries. The country’s residents . The country’s residents therefore have incentives to invest abroad. Thus if there are no therefore have incentives to invest abroad. Thus if there are no impediments to capital flows, not all the additional saving is invested impediments to capital flows, not all the additional saving is invested domestically. Instead, the investment resulting from the increased saving domestically. Instead, the investment resulting from the increased saving is spread uniformly over the whole world.is spread uniformly over the whole world.

• Thus in the absence of barriers to capital movements, there Thus in the absence of barriers to capital movements, there is no reason is no reason to expect countries with to expect countries with high saving to also have high investmenthigh saving to also have high investment..

• Feldstein and Horioka (1980) Feldstein and Horioka (1980) examine the association between saving examine the association between saving and investment rates. They find that, contrary to this simple view, saving and investment rates. They find that, contrary to this simple view, saving and investment rates are and investment rates are strongly correlatedstrongly correlated. The results are. The results are

• The numbers in parentheses are standard errorsThe numbers in parentheses are standard errors

• Thus, rather than there being no relation between saving and investment, Thus, rather than there being no relation between saving and investment, there is an almost one-to-one relation.there is an almost one-to-one relation.

• Explanations:Explanations:

• One possibility, suggested by Feldstein and Horioka, is that there are One possibility, suggested by Feldstein and Horioka, is that there are significant barriers significant barriers to capital mobility. In this case, differences in saving to capital mobility. In this case, differences in saving and investment across countries would be associated with rate-of-return and investment across countries would be associated with rate-of-return differences. There is little evidence of such rate-of-return differences, differences. There is little evidence of such rate-of-return differences,

howeverhowever..

• Another possibility is that there are underlying variables that affect both Another possibility is that there are underlying variables that affect both saving and investment. For example, high tax rates can reduce both saving and investment. For example, high tax rates can reduce both saving and investment (Barro, Mankiw, and Sala-i-Martin, 1995). saving and investment (Barro, Mankiw, and Sala-i-Martin, 1995). Similarly, countries whose citizens have low discount rates, and thus high Similarly, countries whose citizens have low discount rates, and thus high saving rates, may provide favorable investment climates in ways other saving rates, may provide favorable investment climates in ways other than the high saving; for example, they may limit workers’ ability to form than the high saving; for example, they may limit workers’ ability to form strong unions or adopt low tax rates on capital income.strong unions or adopt low tax rates on capital income.

• Finally, the strong association between saving and investment can arise Finally, the strong association between saving and investment can arise from government policies that offset forces that would otherwise make from government policies that offset forces that would otherwise make saving and investment differ. Governments may be averse to large gaps saving and investment differ. Governments may be averse to large gaps between saving and investment—after all, a large gap must be between saving and investment—after all, a large gap must be associated with a large trade deficit (if investment exceeds saving) or a associated with a large trade deficit (if investment exceeds saving) or a large trade surplus (if saving exceeds investment). If economic forces large trade surplus (if saving exceeds investment). If economic forces would otherwise give rise to a large imbalance between saving and would otherwise give rise to a large imbalance between saving and investment, the government may choose to adjust its own saving investment, the government may choose to adjust its own saving behavior or its tax treatment of saving or investment to bring them into behavior or its tax treatment of saving or investment to bring them into rough balance.rough balance.

• Helliwell (1998) finds that the saving investment correlation is much Helliwell (1998) finds that the saving investment correlation is much weaker if we look weaker if we look across regions across regions within a country rather than across within a country rather than across countries. This is certainly consistent with the hypothesis that national countries. This is certainly consistent with the hypothesis that national governments take steps to prevent large imbalances between aggregate governments take steps to prevent large imbalances between aggregate saving and investment, but that such imbalances can develop in the saving and investment, but that such imbalances can develop in the absence of government intervention.absence of government intervention.