Working with the Solow Growth Model


 Oswald Scot George
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1 CHAPTER 4 Monkey Business/Fotolia Working with the Solow Growth Model N ow that we have constructed the Solow growth model, we can put it into action by seeing how various economic changes affect growth in the short and long run We begin by studying variations in the saving rate, the technology level, the level of labor input, and the population growth rate Then we explore convergence, or a tendency for poor countries to catch up to rich ones We found in the Solow model that the growth rate of capital per worker, k/k, was given from equation (316) We repeat this key equation here: (41) k/k s (y/k) s n where k is capital per worker, y is real gross domestic product (real GDP) per worker, y/k is the average product of capital, s is the saving rate, is the depreciation rate, and n is the population growth rate We assumed that everything on the righthand side was constant except for y/k We found that, in the transition to the steady state, the rise in k led to a fall in y/k and, hence, to a fall in k/k In the steady state, k was constant and, therefore, y/k was constant Hence, k/k was constant and equal to zero The production function in perworker form was given in equation (32) We repeat this equation here: (42) y A f(k) If we substitute for y from equation (42) into equation (41), we get a revised version of the basic Solow equation: (43) k/k sa f(k)/k s n Up to now, we assumed that the saving rate, s, the technology level, A, and the population growth rate, n, were fixed Now we allow for changes in s, A, and n We also consider changes in the level of labor input, L We analyze the effects of these changes on the two phases of the Solow model What are the effects on the transition to the steady state, and what are the effects on the steady state? We can think of the first part as representing the shortrun effects and the second part as representing the longrun effects 51
2 A Change in the Saving Rate H ow do differences in the saving rate, s, affect economic growth? As an example of differences in saving rates, we can compare nations in which the residents regularly save at a high rate such as Singapore and South Korea, or some other East Asian countries with places in which the residents typically save at a low rate such as most countries in sub Saharan Africa or Latin America Some of the differences in saving rates result from government policies and some may stem from cultural differences The important point is that saving rates differ across societies and over time Figure 41 Effect of an Increase in the Saving Rate in the Solow Model Determinants of k/k The upper curve has a higher saving rate than the lower curve k(0) k 1 k 2 k = K/L (capital per worker) s 2 + n s 1 + n s 2 (y/k) s 1 (y/k) This graph comes from Figure 39 The curves for s (y/k) are for the saving rates s 1 and s 2, where s 2 is greater than s 1 Similarly, the horizontal lines for s n are for the saving rates s 1 and s 2 The growth rate of capital per worker, k/k, is higher at any k when the saving rate is higher For example, at k(0), when the saving rate is s 1, k/k equals the vertical distance shown by the green arrows When the saving rate is s 2, k/k equals the vertical distance shown by the red arrows In the steady state, k/k is zero, regardless of the saving rate The higher saving rate yields a higher steadystate capital per worker; that is, k 2 is greater than k 1 Figure 41 extends the Solow model from Figure 39 to consider two saving rates, s 1 and s 2, where s 2 is greater than s 1 Each saving rate determines a different curve for s (y/k) the one with s 2 lies above that for s 1 Recall that the growth rate of capital per worker, k/k, equals the vertical distance between the s (y/k) curve and the horizontal line, s n There are also two positions for this horizontal line, one for s 1 and the other for s 2 However, this shift in the horizontal line turns out to be minor Therefore, we can see from the Figure 41 that k/k is higher at any capital per worker, k, when the saving rate is s 2 rather than s 1 1 Specifically, at k(0), k/k is higher when the saving rate is s 2, rather than s 1 (We have assumed that k/k is greater than zero for both saving rates) For either saving rate, the growth rate of capital per worker, k/k, declines as capital per worker, k, rises above k(0) When the saving rate is s 1, k/k reaches zero when k attains the steadystate value k 1 in Figure 41 However, at k 1, k/k would still be greater than zero if the saving rate were higher for example, if it equaled s 2 If the saving rate is s 2, capital per worker, k, rises be yond k 1 until it reaches the higher steadystate value, k 2 Since capital per worker is higher, we also know that real GDP per worker is greater when the saving rate is s 2 that is, y 2 y 1 To summarize, in the short run, an increase in the saving rate raises the growth rate of capital per worker This growth rate remains higher during the transition to the steady state In the long run, the growth rate of capital per worker is the same zero for any saving rate In this longrun or steadystate situation, a higher saving rate leads to higher steadystate 1 The rise in the s (y/k) curve is greater than the increase in the s n line as long as y/k > ; that is, as long as real GDP per worker, y, is greater than depreciation per worker, k Thus, we need only to be sure that the net domestic product is greater than zero, as is surely the case 52
3 dothemath We can determine the steadystate capital per worker, k, algebraically from the steadystate condition given in equation (319) We repeat this result here: (44) A f(k)/k n/s An increase in s lowers the righthand side Hence, the lefthand side must be lower, and this reduction can occur only through a decrease in the average product of capital, A f(k)/k We know from Figure 38 that, if A is fixed, a decrease in the average product of capital requires an increase in capital per worker, k Therefore, an increase in s raises k capital per worker, k, not to a change in the growth rate (which remains at zero) One important extension of the Solow model carried out in the mid1960s by David Cass (1965) and Tjalling Koopmans (1965) allowed households to choose the saving rate, s To study this choice, we need the microeconomic analysis of how households determine consumption at different points in time We defer this analysis to Chapter 7 A Change in the Technology Level W e have assumed, thus far, that the technology level, A, was fixed In reality, technology varies over time and across locations For examples of improvements in technology over time, we can think of the introductions of electric power, automobiles, computers, and the Internet For differences across locations, we can think of businesses in advanced economies, such as the United States and Western Europe, as having better access to leading technologies than their counterparts in poor countries To assess the influences from differences in technologies, we begin by considering the effects in the Solow model from a change in the technology level, A The formula for the growth rate of capital per worker is again (43) k/k sa f(k)/k s n where A f (k)/k is the average product of capital, y/k Note that a higher A means that y/k is higher at a given k Figure 42 compares two levels of technology, A 1 and A 2, where A 2 is greater than A 1 Each technology level corresponds to a different curve for s (y/k) sa f(k)/k The curve with the higher technology level, A 2, lies above the other one Notice that the positions of the two curves are similar to those in Figure 41, which considered two values of the saving rate, s Hence, our analysis of effects from a change in A is similar to that for a change in s At the initial capital per worker, k(0), in Figure 42, the growth rate of capital per worker, k/k, is higher with the higher technology level, A 2, than with the lower one, A 1 In both cases, k/k declines over time For the lower technology level, k/k falls to zero when capital per worker, k, attains the steadystate value k 1 For the higher technology level, k rises beyond k 1 to reach the higher steadystate value k 2 Thus, an increase in A results in a higher k/k over the transition period In the long run, k/k still falls to zero, but the steadystate capital per worker, k, is higher That is, k 2 is greater than k 1 An increase in the technology level, A, raises the steadystate real GDP per worker, y A f (k), for two reasons First, an increase in A raises real GDP per 53
4 dothemath We can derive the effect of A on k algebraically from the condition we used before: (44) A f(k)/k n/s An increase in A does not affect the righthand side Therefore, the steadystate capital per worker, k, must adjust on the lefthand side to keep the steadystate average product of capital, A f (k)/k, the same Since the increase in A raises this average product, k must change in a way that reduces the average product As we know from Figure 38, a reduction in the average product of capital requires a rise in k Therefore, an increase in A raises k Figure 42 Effect of an Increase in the Technology Level in the Solow Model Determinants of k/k The upper curve has a higher technology level than the lower curve k(0) k 1 k 2 k = K/L (capital per worker) s + n sa 2 f(k)/k sa 1 f(k)/k The upper curve has a higher technology level than the lower curve This graph comes from Figure 39 The two curves for s (y/k) sa f (k)/k are for the technology levels A 1 and A 2, where A 2 is greater than A 1 The growth rate of capital per worker, k/k, is higher at any k when the technology level is higher For example, at k(0), when the technology level is A 1, k/k equals the vertical distance shown by the blue arrows When the technology level is A 2, k/k equals the vertical distance shown by the red arrows In the steady state, k/k is zero, regardless of the technology level The higher technology level yields a higher steadystate capital per worker; that is, k 2 is greater than k 1 worker, y, for given capital per worker, k Second, the steadystate capital per worker, k, is higher when A is higher On both counts, an increase in A raises y To summarize, in the short run, an increase in the technology level, A, raises the growth rates of capital and real GDP per worker These growth rates remain higher during the transition to the steady state In the long run, the growth rates of capital and real GDP per worker are the same zero for any technology level In this longrun or steadystate situation, a higher technology level leads to higher steadystate capital and real GDP per worker, k and y, not to changes in the growth rates (which remain at zero) 54
5 EXTENDING THE MODEL Consumption in the Solow Model Recall that real income equals real net domestic product, Y K, which equals consumption, C, plus saving, s (Y K) Therefore, in terms of quantities per worker, we have y k c s (y k) where c is consumption per worker An increase in the saving rate, s, means that c must fall for given y k However, since higher saving leads in the long run to higher real GDP, consumption may increase in the long run Here, we consider what the Solow model says about the effect of saving on consumption in the long run We found that an increase in the saving rate, s, raises the steadystate capital and real GDP per worker, k and y The rise in real GDP per worker leads to an increase in the typical person s real income However, people care about their consumption, not their income, per se Thus, we want to know how a rise in the saving rate affects steadystate consumption per person Consumption per person is given by consumption per person (consumption per worker) (workers/population) The ratio of workers to population is the laborforce participation rate, which we have assumed to be constant Therefore, consumption per person always moves along with consumption per worker, c This result means that we can focus on c to see what happens to consumption per person Since consumption equals the real income not saved, and saving per worker in the steady state is s (y k), we have (45) c y k s (y k) where c is the steadystate value of c We also know from Chapter 3 that steadystate saving per worker is just enough to provide new workers with capital to work with: (317) s (y k) nk Therefore, we can substitute nk for s (y k) on the righthand side of equation (45) to get (46) c y k nk We know that a rise in the saving rate, s, raises k, say, by the amount k The change in c follows from equation (46) as c y ( n) k We can compute y by noting that it must equal k multiplied by the marginal product of capital, MPK: y MPK k Therefore, if we substitute MPK k for y, we get that the change in c is given by (47) c MPK k ( n) k c (MPK n) k We see from equation (47) that c is greater than zero if MPK is greater than n The part MPK is the net marginal product of capital that is, the MPK net of depreciation This term gives the rate of return on additional capital Hence, equation (47) says that an increase in steadystate capital per worker, k, raises steadystate consumption per worker, c, as long as the rate of return on capital, MPK, is greater than the population growth rate, n Typical estimates of rates of return on capital are around 10%, whereas population growth rates are around 0 2% Therefore, in normal circumstances, c is greater than zero The positive effect of the saving rate, s, on steadystate consumption per worker, c, and, therefore, on steadystate consumption per person, does not necessarily mean that the typical person is better off by saving more In order to achieve the higher steadystate capital per worker, k, households have to save more during the transition to the steady state Hence, levels of consumption per person during part of the transition have to be reduced Thus, there is a tradeoff less consumption per person in the short run and more consumption per person in the long run Whether the typical person is better or worse off depends on, first, how much consumption is gained in the long run in comparison with how much is lost in the short run and, second, on how patient people are about deferring consumption 55
6 Changes in Labor Input and the Population Growth Rate W e can consider two types of changes in labor input, L First, L could change at a point in time because of a sudden shift in the size of the labor force Second, a change in the population growth rate could affect the longterm time path of labor input We begin with a onetime change in L A Change in Labor Input Changes in labor input, L, can result from shifts in the labor force For example, the labor force could decline precipitously due to an epidemic of disease An extreme case from the mid1300s is the bubonic plague, or Black Death, which is estimated to have killed about 20% of the European population The potential loss of life due to the ongoing AIDS epidemic in Africa may be analogous, and there is also a lot of concern about avian flu In these examples, physical capital does not change initially, and the starting Figure 43 An Increase in the Level of Labor Input Labor input (L) L(0) L(0) 0 In year 0, labor input jumps upward from L (0) to L (0) The population growth rate, n, does not change capital per worker, k(0) K(0)/L(0), rises because of the drop in L(0) Wartime casualties are another source of decrease in the labor force However, since wartime tends also to destroy physical capital, the effect on capital per worker depends on the circumstances Migration can also change the labor force One example is the Mariel boatlift of over 100,000 Cuban refugees, mostly to Miami, in 1980 Another case is the large inmigration to Portugal in the mid1970s by its citizens who had been residing in African colonies When these colonies became independent, many residents returned to Portugal, and this inflow raised the domestic Portuguese population by about 10% Finally, in Israel in the 1990s, the roughly 1,000,000 Russian Jewish immigrants constituted about 20% of Israel s 1990 population Figure 38 showed the path of labor input, L, starting at L(0) and then growing at the constant rate n In Figure 43, we assume that the initial level of labor input rises from L(0) to L(0) while n does not change Thus, we are considering a proportionate increase in the level of labor input, L, in each year Since the initial stock of capital, K(0), does not change, the increase in L(0) decreases the initial capital per worker, k(0) K(0)/L(0) Figure 44 considers the effects of an increase in the level of labor input The rise in initial labor from L(0) to L(0) reduces the initial capital per worker from k(0) to k(0) However, a key point is that the curve for s (y/k) and the horizontal line at s n do not change The reduction in k(0) raises the initial average product of capital, y/k (see Figure 39) and leads, thereby, to a higher s (y/k) along the unchanged curve Consequently, the growth rate of capital per worker, k/k, rises initially We can see this result Time in Figure 43 because the vertical distance between the s (y/k) curve and the s n line is greater at k(0) 56
7 dothemath We can again work out the steadystate results algebraically from the condition (44) A f(k)/k n/s Note that A, s, n, and are constant, and the level of labor input, L, does not enter into the equation Therefore, the steadystate capital per worker, k, does not change when L changes (the red arrows) than at k(0) (the blue arrows) The growth rate k/k remains higher during the transition to the steady state However, k/k still declines toward its longrun value of zero Moreover, the steadystate capital per worker, k, is the same whether labor input starts at L(0) or L(0) Thus, if L(0) is twice as large as L(0), the longrun level of capital, K, is also twice as large (so that capital per worker remains the same) Since k is unchanged, we also have that real GDP per worker, y, does not change In the long run, an economy with twice as much labor has twice as much real GDP, Y To summarize, in the short run, an increase in labor input, L(0), raises the growth rates of capital and real GDP per worker These growth rates remain higher during the transition to the steady state In the long run, the growth rates of capital and real GDP per worker are the same zero for any level of labor input, L(0) Moreover, the steadystate capital and real GDP per worker, k and y, are the same for any L Thus, in the long run, an economy with twice as much labor input has twice as much capital and real GDP Figure 44 Effect of an Increase in Labor Input in the Solow Model Determinants of k/k k(0) L(0) k(0) k L(0) The increase in L(0) lowers k(0) and therefore raises k/k k = K/L (capital per worker) s + n s (y/k) This graph comes from Figure 310 If the initial level of labor input rises from L (0) to L (0), the initial capital per worker declines from k (0) K (0)/L (0) to k(0) K (0)/L (0) Therefore, the growth rate of capital per worker, k/k, rises initially Note that the vertical distance shown by the red arrows is larger than that shown by the blue arrows The steadystate capital per worker, k, is the same for the two values of L (0) A Change in the Population Growth Rate Figure 45 shows an increase in the population growth rate from n to n We assume now that the initial population and, hence, level of labor input, L(0), do not change Thus, the initial capital per worker, k(0), does not change 57
8 Figure 45 An Increase in the Population Growth Rate Labor input (L) L(0) 0 Population growth rate = n Population growth rate = n The population growth rate rises in year 0 from n to n The initial level of labor input, L (0), does not change Time In Figure 46, the higher population growth rate corresponds to a higher horizontal line for s n Recall that the growth rate of capital per worker, k/k, equals the vertical distance between the s (y/k) curve and the s n line Therefore, k/k is lower at any capital per worker, k, when the population growth rate is n rather than n We can also see this result from the formula for the growth rate of capital per worker: (43) k/k sa f(k)/k s n Figure 46 Effect of an Increase in the Population Growth Rate in the Solow Model Determinants of k/k k(0) (k) The upper line has a higher population growth rate than the lower line k k = K/L (capital per worker) s + n s + n s (y/k) This graph comes from Figure 310 An increase in the population growth rate from n to n raises the horizontal line from s n to s n The growth rate of capital per worker, k/k, is lower at any k when the population growth rate is higher For example, at k(0), when the population growth rate is n, k/k equals the vertical distance given by the red arrows When the population growth rate is n, k/k equals the vertical distance given by the blue arrows In the steady state, k/k is zero, regardless of the population growth rate A higher population growth rate yields a lower steadystate capital per worker; that is, (k) is less than k A higher n the yellow shaded term lowers k/k for given k The reason that k/k is lower when n is higher is that a larger portion of saving goes to providing the growing labor force, L, with capital to work with For either population growth rate in Figure 46, the growth rate of capital per worker, k/k, declines as capital per worker rises above k(0) When the population growth rate is n k/k reaches zero when k attains the steadystate value (k) However, at (k), k/k would still be greater than zero if the population growth rate were lower in particular, if it equaled n Thus, if the population growth rate is n, capital per worker rises beyond (k) k increases until it reaches the steadystate value k, which is greater than (k) 58
9 dothemath As usual, we can find the effect of n on k algebraically from the condition (44) A f(k)/k n/s An increase in n raises the righthand side of the equation Hence, the steadystate average product of capital, A f (k )/k y /k, has to rise on the lefthand side Because of diminishing average product of capital (Figure 39), this change requires a decrease in k Therefore, as we already found, an increase in n reduces k Figure 46 shows that, at the initial capital per worker, k(0), an increase in the population growth rate from n to n lowers the growth rate of capital per worker, k/k The growth rate of real GDP per worker, y/y, falls correspondingly Thus, in the short run, a higher n lowers k/k and y/y These growth rates remain lower during the transition to the steady state However, in the steady state, k/k and y/y are zero for any n That is, a higher n leads to lower steadystate capital and real GDP per worker, k and y, not to changes in the growth rates, k/k and y/y (which remain at zero) A change in n does affect the steadystate growth rates of the levels of capital and real GDP, K/K and Y/Y An increase in n by 1% per year raises the steadystate values of K/K and Y/Y by 1% per year We can see from Figure 46 that an increase in the depreciation rate,, affects the steadystate capital per worker in the same way as an increase in the population growth rate, n This result follows because equation (43) involves the term s n, which can rise either from an increase in n or an increase in The kind of analysis that we carried out for an increase in n tells us that an increase in lowers the growth rates of capital and real GDP per worker, k/k and y/y, in the short run In the steady state, an increase in leads to lower capital and real GDP per worker, k and y, not to changes in k/k and y/y, which remain at zero 2 Convergence O ne of the most important questions about economic growth is whether poor countries tend to converge or catch up to rich countries Is there a systematic tendency for lowincome countries like those in Africa to catch up to the rich OECD countries? We will start our answer to this question by seeing what the Solow model says about convergence Then we will look at how the facts on convergence match up with the Solow model Convergence in the Solow Model To study convergence, we focus on the transition for capital per worker, k, as it rises from its initial value, k(0), to its steadystate value, k In Figure 311, we see that k works like a target or magnet for k during the transition Therefore, an important part of our analysis of convergence concerns the determination of k We have studied how k depends on the saving rate, s, the technology level, A, the population growth rate, n, the depreciation rate,, and the initial level of labor input, L(0) We can summarize these results in the form of a function for k: (47) k k[s, A, n,, L(0)] ( )( )( )( )(0) 2 One difference is that an increase in n raises the steadystate growth rates of the levels of capital and real GDP, ( K /K) and ( Y /Y ), whereas an increase in ffl does not affect these steadystate growth rates 59
10 Table 41 Effects on SteadyState Capital per Worker, k Increase in this Variable Effect on k saving rate, s technology level, A depreciation rate, population growth rate, n level of labor force, L (0) increase increase decrease decrease no effect Note: The righthand column shows the effect of an increase in the variable in the lefthand column on the steadystate ratio of capital to labor, k These results come from equation (47) The sign below each variable indicates the effect on k Thus, equation (47) shows that k rises with s and A, falls with n and, and does not depend on the level of labor input, represented by L(0) Table 41 summarizes these results To apply the Solow model to convergence, we have to allow for more than one economy In making this extension, we assume that the economies are independent of each other Specifically, they do not engage in international trade in goods and services or in financial assets In other words, we still think of each economy as closed Think now of two economies, 1 and 2, and suppose that they start with capital per worker of k(0) 1 and k(0) 2, respectively, where k(0) 1 is less than k(0) 2 Each economy is assumed to have the same production function, y A f (k) Thus, economy 2 is initially more advanced in the sense of having higher capital and real GDP per worker, k(0) and y(0) Imagine that each economy has the same values for the determinants of k listed in Table 41, so that they have the same steadystate capital per worker, k We show this situation in Figure 47, which has been adapted from Figure 310 The only difference between the two economies is that one starts at k(0) 1 and the other at k(0) 2 Therefore, the differences in the transition paths of k depend only on the differences in these starting values The graph in Figure 47 shows that the vertical distance between the s (y/k) curve and the s n line is greater at k(0) 1 than at k(0) 2 That is, the distance marked with red arrows is greater than that marked with blue arrows Therefore, the growth rate of capital per worker, k/k, is higher initially for economy 1 than economy 2 Because k grows at a faster rate in economy 1, its level of k converges over time toward economy 2 s level Figure 48 shows the transition paths of capital per worker, k, for economies 1 and 2 Note that k(0) 1 is less than k(0) 2, but k 1 gradually approaches k 2 (At the same time, k 1 and k 2 both gradually approach k) Thus, economy 1 converges toward economy 2 in terms of the levels of k We can express the results in terms of real GDP per worker, y The capital per worker, k, determines y from the production function: (42) y A f(k) Since economy 1 starts with lower capital per worker, k(0), it must also start with lower real GDP per worker y(0) 1 is less than y(0) 2 The growth rate of real The Print Collector/Alamy 60
11 GDP per worker relates to the growth rate of capital per worker from equation (38), which we repeat: Figure 47 Convergence in the Solow Model (48) y/y ( k/k) where is the capitalshare coefficient (We assume that is the same in the two economies) We showed in Figure 47 that k/k was higher initially in economy 1 than in economy 2 Therefore, y/y is also higher initially in economy 1 Hence, economy 1 s real GDP per worker, y, converges over time toward economy 2 s real GDP per worker The transition paths for y in the two economies look like those shown for k in Figure 48 To summarize, the Solow model says that a poor economy with low capital and real GDP per worker grows faster than a rich one The reason is the diminishing average product of capital, y/k A poor economy, such as economy 1 in Figure 47, has the advantage of having a high average product of capital, y/k This high average product explains why the growth rates of capital and real GDP per worker are higher than in the initially more advanced economy, economy 2 Thus, the Solow model predicts that poorer economies tend to Determinants of k/k Figure 48 Convergence and Transition Paths for Two Economies k (capital per worker) k k(0) 2 k(0) 1 0 k 2 k 1 Economy 1 grows faster than economy 2 k(0) 1 k(0) 2 k k = K/L (capital per worker) k 1 converges over time toward k 2, while k 1 and k 2 both converge toward k Economy 1 starts at capital per worker k(0) 1 and economy 2 starts at k(0) 2, where k(0) 1 is less than k(0) 2 The two economies have the same steadystate capital per worker, k, shown by the dashed blue line In each economy, k rises over time toward k However, k grows faster in economy 1 because k(0) 1 is less than k(0) 2 (See Figure 47) Therefore, k 1 converges over time toward k 2 s + n s (y/k) This graph comes from Figure 310 Economy 1 starts with lower capital per worker than economy 2 k(0) 1 is less than k(0) 2 Economy 1 grows faster initially because the vertical distance between the s (y/k) curve and the s n line is greater at k(0) 1 than at k(0) 2 That is, the distance marked by the red arrows is greater than that marked by the blue arrows Therefore, capital per worker in economy 1, k 1, converges over time toward that in economy 2, k 2 Time 61
12 EXTENDING THE MODEL Endogenous Population Growth Our analysis treated the population growth rate, n, as exogenous determined outside of the model However, since the writings of Thomas Malthus (1798), economists have argued that population growth responds to economic variables Malthus was a British economist and minister who wrote his Essay on Population in 1798 He argued that an increase in real income per person raised population growth by improving life expectancy, mainly through better nutrition but also through improved sanitation and medical care Another influence, Malthus believed, was that higher income encouraged greater fertility He thought that birth rates would rise as long as real income per person exceeded a subsistence level, which is the amount needed to pay for the basic necessities of life We can incorporate Malthus s ideas about population growth into the Solow model In Figure 310, for a given population growth rate, n, the economy approaches a steadystate capital per worker, k, and a corresponding real NDP per worker, y A f (k ) k, which equals real national income per worker The real income per person is then real income per person (real NDP per worker) (workers/population) The last term on the righthand side is the laborforce participation rate, which we assume to be fixed When household real income per person rose above the subsistence level, Malthus believed that the population growth rate would rise Figure 46 showed the effect from a rise in the population growth rate; this change lowered the steadystate capital and real GDP per worker According to Malthus, this process would continue until the steadystate real income per person fell to the subsistence level Malthus s view on the relation between real income per person and life expectancy is reasonable Data across countries show that higher real GDP per person matches up closely with higher life expectancy at birth 3 However, Malthus s idea about fertility seems unreasonable At least in the crosscountry data since 1960, higher real GDP per person matches up with lower fertility 4 In fact, this relation is so strong that higher real GDP per person matches up with a lower population growth rate, even though countries with higher real GDP per person have higher life expectancy We can modify the Solow model to include Malthus s idea that population growth is endogenous However, contrary to Malthus, we should assume a negative effect of real GDP per person and, hence, of capital per worker, k on the population growth rate, n The condition for the growth rate of capital per worker is again (43) k/k sa f(k/k) s n During the transition to the steady state, a rise in k reduced the average product of capital, y/k, and thereby decreased the growth rate of capital per worker, k/k Now we have that a rise in k also lowers n This change raises k/k and, thus, offsets the effect from a reduced average product of capital Hence, a declining population growth rate is one reason why rich societies can sustain growing capital and real GDP per worker for a long time converge over time toward richer ones in terms of the levels of capital and real GDP per worker Facts About Convergence The main problem with these predictions about convergence is that they conflict with the evidence for a broad group of countries We already looked, in Figure 33, at growth rates of real GDP per person from 1960 to 2000 To apply the Solow model to these data, we have to translate from amounts per worker to amounts per person The formula for real GDP per person is again real GDP per person (real GDP per worker) (workers/population) 3 Although this relation is suggestive, it does not prove that the causation is from higher real income per person to greater life expectancy, rather than the reverse In fact, both directions of causation seem to be important 4 This relation does not prove that the causation is from higher real income per person to lower fertility, rather than the reverse In fact, the reverse effect is predicted by the Solow model If a society chooses, perhaps for cultural reasons, to have higher fertility and population growth, the model predicts lower steadystate real GDP per worker In practice, both directions of causation seem to be important 62
13 The ratio of workers to population is the laborforce participation rate, which we have assumed to be constant For example, if the ratio is around onehalf, as in recent US experience, real GDP per person is about onehalf of real GDP per worker With this translation, we find that the Solow model predicts convergence for real GDP per person Specifically, the model predicts that a lower level of real GDP per person would match up with a higher subsequent growth rate of real GDP per person Figure 49 uses the data for countries from Figure 33 to plot growth rates of real GDP per person from 1960 to 2000 against levels of real GDP per person in 1960 If the convergence predictions from the Solow model were correct, we should find low levels of real GDP per person matched with high growth rates, and high levels of real GDP per person matched with low growth rates Instead, it is difficult to discern any pattern in the data if anything, there is a slight tendency for the growth rate to rise with the level of real GDP per person The sample of countries included in Figure 49 is very broad; it includes the richest and poorest economies in the world The convergence prediction of the Solow model accords better with the data if we limit the observations to economies that have more similar economic and social characteristics Figure 410 on page 65 is the same as Figure 49, except that the sample is limited to 18 advanced OECD countries For this limited sample, lower levels of real GDP per person in 1960 do match up, on average, with Figure 49 Growth Rate Versus Level of Real GDP per Person for a Broad Group of Countries Growth rate, Real GDP per person in 1960 The horizontal axis shows real GDP per person in 1960 in 2000 US dollars on a proportionate scale for 112 countries The vertical axis shows the growth rate of real GDP per person for each country from 1960 to 2000 The red line is the straight line that provides a best fit to the relation between the growth rate of real GDP per person (the variable on the vertical axis) and the level of real GDP per person (on the horizontal axis) Although this line slopes upward, the slope is in a statistical sense negligibly different from zero Hence, the growth rate is virtually unrelated to the level of real GDP per person Thus, this broad group of countries does not display convergence higher growth rates from 1960 to 2000 This pattern reflects especially the catching up of some of the initially poorer OECD countries Greece, Ireland, Portugal, and Spain to the richer ones Figure 411 on page 65 shows an even clearer pattern of convergence for a still more homogenous group of economies the states of the United States The figure plots the average growth rate of personal income per person from 1880 to 2000 against the level of personal income per person in This graph shows a dramatic tendency for the initially poorer states to grow faster than the initially richer ones over the 120 years after 1880 This convergence tendency does not reflect only the recovery of the southern states, which were defeated during the US Civil War ( ) The convergence pattern applies if we examine economic performance within any of the four main regions Northeast, South, Midwest, and West Researchers have found results similar to those in 5 We use personal income because data on gross state product are unavailable for the US states for the period since
14 Figure 410 Growth Rate Versus Level of Real GDP per Person for OECD Countries Growth rate, POR SPA GRE IRE LUX AUS NOR ITA BEL ICE FRA CAN US NLD DEN UK SWE Real GDP per person in 1960 The horizontal axis shows real GDP per person in 1960 in 2000 US dollars on a proportionate scale for 18 of the 20 founding members of the OECD (excluding Germany and Turkey) The abbreviation identifies each country (AUS stands for Austria; Australia was not a founding member of the OECD) The vertical axis shows the growth rate of real GDP per person for each country from 1960 to 2000 The red line is the straight line that provides a best fit to the relation between the growth rate of real GDP per person (the variable on the vertical axis) and the level of real GDP per person (on the horizontal axis) The line has a clear negative slope therefore, a lower level of real GDP per person in 1960 matches up with a higher growth rate of real GDP per person from 1960 to 2000 Thus, the group of OECD countries exhibit convergence SWZ Figure 41 for regions of some other advanced countries Figures 49 to 411 tell us that similar economies tend to converge, whereas dissimilar economies display no relationship between the level of real GDP per person and the growth rate Thus, the convergence pattern is strongest for regions of an advanced country (Figure 411), next strongest among a group of rich countries (Figure 410), and weakest in fact, absent for the full worldwide sample of countries (Figure 49) istockphotocom/dustin Steller Conditional Convergence in the Solow Model The Solow model s prediction of convergence seems to explain growth patterns in similar economies, and it seems to fail when we examine a dissimilar array of economies Do these findings mean that the model is flawed? Are there changes we can make to improve its predictions? To find the flaw and try to correct it, let s reexamine the Solow model One key assumption was that the determinants of the steadystate capital per worker, k, were the same for all economies This assumption is reasonable for similar economies but is less plausible for a broad sample of countries with sharply different economic, political, and social characteristics In particular, the assumption is unreasonable for the worldwide sample of countries considered in Figure 49 To explain the lack of convergence for this group, we have to allow for differences in the steadystate positions, k 64
15 Suppose that countries differ with respect to some of the determinants of k in equation (47) and Table 41 For example, k could vary because of differences in saving rates, s, levels of technology, A, and population growth rates, n 6 Figure 412 on page 66 modifies Figure 47 to show how differences in saving rates affect convergence Economy 1 has the saving rate s 1, and economy 2 has the higher saving rate s 2 We assume, as in Figure 47, that economy 1 has lower initial capital per worker; that is, k(0) 1 is less than k(0) 2 Remember that the growth rate of capital per worker, k/k, equals the vertical distance between the s (y/k) curve and the s n line We see from Figure 412 that it is uncertain whether the distance between the s (y/k) curve and the s n line is greater initially for economy 1 or economy 2 The lower capital per worker, k(0), tends to make the distance greater for economy 1, but the lower saving rate, s, tends to make the distance smaller for economy 1 In the graph, these two forces roughly balance, so that k/k is about the same for the two economies That is, the distance marked with the blue arrows is similar to the one marked with the red arrows Therefore, the poorer economy, economy 1, does not necessarily converge toward the richer economy, economy 2 To get the result in Figure 412, we had to assume that the economy with lower k(0) economy 1 had a lower saving rate, s This assumption is reasonable because an economy with a lower s has a lower steadystate capital per worker, k In the long run, an economy s capital per worker, k, would be close to its steadystate value, k Therefore, it is likely when we examine countries at an Figure 411 Growth Rate Versus Level of Income per Person for US States, Growth rate of personal income per person, NC SC FL VA GA TN AL TX 020 AR WV MS NM KY KS UT MO WI ME IN LA NE IA MI MD MN NH VT OH SDDE IL NJCT PA WA MA ND OR NY RI ID WY CO CA The horizontal axis shows real personal income per person in US dollars on a proportionate scale for 47 US states The twoletter abbreviation identifies the state (Alaska, the District of Columbia, Hawaii, and Oklahoma are excluded) The vertical axis shows the growth rate of real personal income per person for each state from 1880 to 2000 The solid line is the straight line that provides a best fit to the relation between the growth rate of income per person (the variable on the vertical axis) and the level of income per person (on the horizontal axis) The line has a clear negative slope therefore, a lower level of income per person in 1880 matches up with a higher growth rate of income per person from 1880 to 2000 Thus, the US states exhibit convergence arbitrary date, such as date 0, that k(0) will be lower in the economy with the lower s k(0) tends to be lower in economy 1 than in economy 2 Thus, the pattern that we assumed a low saving rate matched with a low k(0) is likely to apply in practice We get a similar result if we consider other reasons for differences in the steadystate capital per worker, k Suppose that the two economies have the same saving rates but that economy 1 has a lower technology level, A, than economy 2 In this case, the two curves for s (y/k) again look as shown in Figure Therefore, it is again uncertain whether the vertical distance between the s (y/k) curve and the s n line is greater for economy 1 or economy 2 The lower capital per worker, k(0), tends to make the distance greater for economy 1, but the lower A tends to make the distance smaller for economy 1 As before, it is possible that the two forces roughly balance, so that k/k is about the same for the two economies Thus, the AZ MT NV 6 Levels of population and the labor force vary greatly across countries, but the level of labor input, represented by L (0), does not affect k in the model The depreciation rate,, probably does not vary systematically across countries 7 In this case, unlike in Figure 412, the s n lines are the same for the two countries 65
16 Figure 412 Failure of Convergence in the Solow Model: Differences in Saving Rates Effects on k/k As in Figure 47, economy 1 starts with lower capital per worker than economy 2 k(0) 1 is less than k(0) 2 However, we now assume that economy 1 also has a lower saving rate; that is, s 1 is less than s 2 The two economies have the same technology levels, A, and population growth rates, n Therefore, k 1 is less than k 2 In this case, it is uncertain which economy grows faster initially The vertical distance marked with the blue arrows may be larger or smaller than the one marked with the red arrows Figure 413 Failure of Convergence in the Solow Model: Differences in Population Growth Rates Effects on k/k k(0) 1 Economy 1 has a lower saving rate than economy 2 k(0) 1 k(0) 2 k1 k 2 k = K/L (capital per worker) Economy 1 has a higher population growth rate than economy 2 k(0) 2 k 1 k 2 k = K/L (capital per worker) s 2 + n s 1 + n s 2 (y/k) s 1 (y/k) s + n 1 s + n 2 s (y/k) As in Figure 412, economy 1 starts with lower capital per worker than economy 2 k(0) 1 is less than k(0) 2 The two economies now have the same saving rates, s, and technology levels, A, but economy 1 has a higher population growth rate, n; that is, n 1 is greater than n 2 Therefore, as in Figure 412, k 1 is less than k 2 It is again uncertain which economy grows faster initially The vertical distance marked with the blue arrows may be larger or smaller than the one marked with the red arrows poorer economy, economy 1, need not converge toward the richer economy, economy 2 To get this result, we had to assume that the economy with the lower starting capital per worker, k(0) economy 1 had the lower technology level, A This assumption is reasonable because an economy with a lower A has a lower steadystate capital per worker, k Therefore, it is again likely when we look at the two economies at date 0 that k(0) will be lower in economy 1 than economy 2 The same conclusions apply if we consider differences in population growth rates In Figure 413, the two economies have the same saving rates and technology levels, but economy 1 has a higher population growth rate, n Hence, the s n line is higher for economy 1 It is again uncertain whether the vertical distance between the s (y/k) curve and the s n line is greater for economy 1 or economy 2 The lower capital per worker, k(0), for economy 1 tends to make the distance greater for economy 1, but the higher n tends to make the distance smaller for economy 1 As in our other cases, it is possible that the two forces roughly balance, so that k/k would be about the same in the two economies the distance marked with the blue arrows is similar to the one marked with the red arrows Thus, economy 1 again need not converge toward economy 2 To get this result, we had to assume that the economy 66
17 with the lower starting capital per worker, k(0) economy 1 had the higher population growth rate, n This assumption makes sense because an economy with a higher n has a lower steadystate capital per worker, k Therefore, it is again likely when we look at the two economies at date 0 that k(0) will be lower in economy 1 than economy 2 Now let s generalize the conclusions from our three cases In each case, economy 1 had a characteristic lower saving rate, s, lower technology level, A, higher population growth rate, n that led to a lower steadystate capital per worker, k For a given starting capital per worker, k(0), each of the three characteristics tended to make country 1 s initial growth rate less than economy 2 s initial growth rate We see these effects in Figures 412 and 413 At a given k(0), the vertical distance between the s (y/k) curve and the s n line is smaller if s or A is lower or if n is higher Since economy 1 has lower k, it is also likely to have lower initial capital per worker, k(0) The lower k(0) tends to make economy 1 grow faster than economy 2 the convergence force shown in Figure 47 Whether economy 1 grows faster or slower overall than economy 2 depends on the offset of two forces The lower k(0) generates faster growth in economy 1, but the lower k generates slower growth in economy 1 It is possible that the two forces roughly balance, so that the two economies grow at about the same rate That is, we need not find convergence Figure 414 shows the transition paths of capital per worker, k, for the two economies We assume that economy 1 starts with a lower capital per worker k(0) 1 is less than k(0) 2 and also has a lower steadystate capital per worker k 1 is less than k 2 The graph shows that capital per worker in each economy converges Figure 414 Failure of Convergence and Transition Paths for Two Economies k (capital per worker) k 2 k 1 k(0) 2 k(0) 1 0 k 2 k 1 Economy 1 has lower k(0) and lower k therefore, k 1 does not converge toward k 2 As in Figures 412 and 413, economy 1 has a lower starting capital per worker k(0) 1 is less than k(0) 2 and also has a lower steadystate capital per worker k 1 (the dashed brown line) is less than k 2 (the dashed blue line) Each capital per worker converges over time toward its own steadystate value: k 1 (the red curve) toward k 1, and k 2 (the green curve) toward k 2 However, since k 1 is less than k 2, k 1 does not converge toward k 2 Time toward its own steadystate level k 1 toward k 1, and k 2 toward k 2 However, since k 1 is less than k 2, k 1 does not converge toward k 2 We can summarize our findings for the growth rate of capital per worker in an equation: Key equation (conditional convergence in the Solow model): (49) k/k [k(0), k] ( ) ( ) growth rate of capital per worker function of initial and steadystate capital per worker The function indicates how k/k depends on the initial capital per worker, k(0), and the steadystate capital per worker, k The minus sign under k(0) signifies that, for given k, a decrease in k(0) raises k/k The plus sign under k means that, for given k(0), a rise in k increases k/k 67
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