# Working with the Solow Growth Model

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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 right-hand 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 per-worker 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 short-run effects and the second part as representing the long-run 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 steady-state 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 steady-state 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 steady-state situation, a higher saving rate leads to higher steady-state 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

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 right-hand side Therefore, the steady-state capital per worker, k, must adjust on the lefthand side to keep the steady-state 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 steady-state capital per worker; that is, k 2 is greater than k 1 worker, y, for given capital per worker, k Second, the steady-state 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 long-run or steady-state situation, a higher technology level leads to higher steady-state capital and real GDP per worker, k and y, not to changes in the growth rates (which remain at zero) 54

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 long-term time path of labor input We begin with a one-time 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 mid-1300s 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 in-migration to Portugal in the mid-1970s 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 steady-state 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 steady-state 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 long-run value of zero Moreover, the steady-state 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 steady-state 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 steady-state 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 steady-state 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 steady-state 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 steady-state value k, which is greater than (k) 58

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 steady-state 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

13 The ratio of workers to population is the labor-force participation rate, which we have assumed to be constant For example, if the ratio is around one-half, as in recent US experience, real GDP per person is about one-half 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 steady-state 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 steady-state 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 steady-state capital per worker, k In the long run, an economy s capital per worker, k, would be close to its steady-state 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 two-letter 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 steady-state 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 steady-state 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

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