Chapter 1 The Solow Growth Model Basic Facts Assumptions The Dynamics of the Model The Impact of a Change in the Saving Rate Quantitative Implications Ths Solow Model and the Central Questions of Growth Theory Empirical Applications Basic Issues/Facts Comparisons of average real incomes (ARIs) in OECD countries TODAY to ARI one or two centries ago - 100: 10 30-200: 50 300 Productivity growth slowdown - Since easly 1990s, the average annual GDP growth per capita has been approxiately 1% point below its earlier years. Enormous differences in standards of living across different part of the world - ARI OECD vs. ARI others Growth miracles - Episodes where ARI in a country far exceeds the world ARI - eg. NICs Growth disaster - Episodes where ARI in a country falls far short of the world ARI - eg. Argentina, Sub-Saharan Africa Cross-country income differences have widened on average. - Past few decades: no strong tendency either toward continued convergence or divergence. Growth and welfare - Over time/generations - Across countries - Short-term vs. long-term fluctuations "Once one starts to think about economic growth, it is hard to think about anything else." (Lucas, R.,1988) Assumptions Benchmark Basic insight: Accumulation of physical capital cannot account for the two most standard stylized facts of economic growth. - Vast growth over time in ARI; - Vast geographic diffences in ARI. So, other models needed. Drawbacks - Other potential sources to explain those facts are regarded as exogenous in
the Solow model(e.g., technological progress) or is absent (e.g., externalities). - No optimization: Saving rate is treated as exogenous and constant. - No explicit welfare analysis. Assumptions - The production function: Yt FKt, AtLt where Y : Output K : Capital L : Labor A : "Productivity"; "Knowledge"; "Effectiveness of labor" t : Time - Output changes over time only if input to production changes. - A L is called effective labor. - Different kinds of technological progress * Y FK, AL : labor-augmenting (Harrod-neutral) * Y FAK, L : capital-augmenting * Y A FK, L : Hicks-neutral - Why assume Harrod-neutral technological progress? * Because it implies, together with the other assumptions, that the capital-output ratio eventually settles down. Properties of the production function - CRS (constant returns to scale) (in K and AL) * FcK, cal cfk, AL * Two sub-assumptions No additional gains from specialization. No other inputs more important. Production function in intensive form FcK, cal cfk, AL c 1 AL F AL K,1 1 FK, AL AL Define : k K ; y AL Y, Fk, 1 AL fk y fk; Y L Afk - The intensive form of the production function expresses output per unit of effective labor as a function of capital per unit of effective labour. - More convenient to study the behavior of the world by focussing on k, rather than K and AL. - Marginal product of capital Y fkal f K AL AL K Y f AL K AL AL f k
Assume that Moreover f0 0; f k 0; f k 0 lim k 0 f k lim k f k 0 Inada Conditions - Figure 1.1
Dynamic behavior of factors of production - Input * Capital, labor and knowledge * Initial levels taken as given * Constant growth rates of labor n and knowledge g L t n Lt Ȧt g At - Definition growth rateproportional rate of change growth rate of X Ẋt Xt, where Ẋt dxt dt d ln Xt dt Ẋt Xt - Assumption: * L and A each grow exponentially Lt L0 e nt At A0 e gt ln At ln A0 gt d ln At dt Ȧt At - So far we have assumed that both employment and knowledge are exogenously determined. This is not the case for the third factor of production, capital k t kt st gross investment saving St s Yt where s: constant, exogenous. - Summing up L t n Lt Ȧt g At K t s Yt Kt - But * Only a single good * Government is absent * Full employment * Just three inputs in production * Parameters u, g, s and are constant. Dynamics of the model g
Dynamics of k k K A L ln k ln K ln A ln L Ȧ A d ln k dt K sy K k k K K Ȧ A L L K K s Y K sy g; L L k k n s Y K g n k s Y K K AL g nk k sy g nk - sfk * Actual investment per unit of labour * s denotes the fraction of output that is invested - n g k * Break-even investment * Investment needed to prevent k from falling * k : Capital is depreciating * n gk : The (effective) labor force is growing f(k) Break-even investment ( n+ g+δ) k sf(k) Actual Investment O - Long-run equilibriumk k 0 s fk g nk - Small values of k : f0 0 * at k 0 actual investment and break-even investmetn are equal. lim f k k 0 at k 0thesfk line is steeper than the g nk line. So, for small values of the actural investment is larger than break-even k
investment. - Large values of k : lim k f k 0 * So, "at some point", the slope of the actual investment line will be lower than the slope of the break-even line. f k 0 * The two lines intersect only once; * k is a unique equilibrium - What happens when k k? * k k sfk g nk k 0 * k k sfk g nk * The equilibrium k is stable. k 0 Balanced growth path - A balanced growth path means that each variable of the model will be growing at a constant rate. This is exactly the case in the Solow growth model. * k k y y * Unique * Stable - K (Capital stock) K A L k ln K ln A ln L ln k K K Ȧ L A L g n So, K is growing by g n. *
g 0 K K L L i.e. the capital-labor ratio is rising over time. - YGDP Y A L y A L fk Ẏ Ȧ L g n Y A L The growth rate of GDP is determined by exogenous parameters. Y - y GDP per capita) L y A fk ẏ y Ȧ A g GDP per capita will be growing by g. - k K L (capital per worker) k A k k Ȧ g k A - Empirical relevance? * Yes! Growth rates of labor, capital and output have each been roughly constant. Growth rates of capital and output have been about equal. K Y ratio almost constant and larger than growth rate of labor. Growth rate of labor productivity has been increasing. * Some problems with these facts? However, they will be taken up later. The Impact of a Change in the Saving Rate Empirical regularities - A negative correlation between the world saving rate and the world real interest rate, observed since 1920s. - A positive cross-country correlation is found between national saving rates and per capita income levels. But this association tends to level off at high income levels. - A strong positive association is observed between saving rates and per capita income growth rates. - Saving and investment rates show a very robust positive correlation across countries. - There is evidence of a strong negative association between foreign aid and national saving rates. Changing the saving rate s - The effect on k
k adjusts gradually towards k s (and NOT immediately). k t 0 t t 0 t 0 t k t * k 1 * k 0 t 0 t t - The effect on Growth rate of Y/L y Y A fk L ln y ln A ln fk ẏ y Ȧ A ḟk fk 0 k
y y Growth rate of Y/L g t t 0 t * Change in saving rate has a LEVEL effect but NOT a GROWTH effect. * In the Solow model, it is only changes in the rate of technological progress that has a growth effect! The Effect on Consumption - Why look at consumption? A: Welfare - What happens to consumption? C Y S c C y S fk sfk A L A L c 1 s fk s fk n gk c 1 sfk fk s fk c f k k f k n g n g k k 0
c 0iff k n g c The effect on c* is ambiguous?? t0 t t - f k n g * More than enoughly additional, output is created to maintain k at its higher level. In fact, there is something to "left over" is used as consumption. f( k) ( δ + n+ g) k s f() k < + + * f ( k ) δ n g More savings would lower consumption. * k k - f k n g * Additional output is not enough to maintain capital at the new high level, and consumption must fall. - f k g n
* "Golden Rule" * max c s Quantitative Implications Quantitative implications - The model s quantitative predictions * What are the eventual effects of a change in the saving rate? * How rapidly those effects occur? - Approach * For most models, obtaining exact quantitative results requires specifying functional forms and values of the parameters. * But In many cases, it is possible to learn a great deal by considering approximations around the long-run equilibrium. The effect on output in the long run - The long-run effect of a rise in saving on output y f k k s, n, g, 1 where y fk : the level of output per unit of effective labor on the balanced growth path. - To find y /, we need to find k /. - Note that k is defined by the condition that k 0; Thus sfk s, n, g, n g k s, n, g, fk sf k k k n g k fk n g sf k - Substituting 2 into 1 yields y f k fk n g sf k 3 - Two changes help in interpreting (3) * Covert it to an elasticity by multiplying both sides by s/y * Use the fact that sfk n g k - Making these changes gives us 2
- Denoting s y y s f k fk fk n g sf k n g k f k fk n g n g k f k /fk k f k /fk 1 k f k /fk K k k f k fk Where K k : the elasticity of output with respect to capital at k k. - We have s y y K k 1 K k 4 - If markets are competitive and there are no externalities, capital earns is marginal product. In this case, the total amount received by capital on the balanced growth path is k f k. - Thus, if capital earns its marginal product, the share of total income that goes to capital on the balanced growth path is K k k f k fk - If K k 1 3 Then s y y 1 2 - Thus, a 10% increase in the saving rate raises output per worker in the long run by about 5% relative to the path it would have followed. Even a 50% increase in s raises y only by 22%. - Thus significant changes in saving have only moderate effects on the level of output on the balanced growth path. - Intuitively, a small value of K k makes the impact of saving on output low for two reasons: * It implies that the actual investment curve, sfk, bends fairly sharply; as a result, an upward shift of the curve moves iits intersection with the break-even investment line relatively little. Thus the impact of a change in s on k is small. * A low value of K k means that the impact of a change in k on y is small. The speed of convergence - Q: How rapidly those effects occur? - For simplicity: * We focus on the behavior of k rather than y. * Our goal is thus to determine how rapidly k approaches k. - Recall that k sfk n g k
Thus we can write k k k When k k, k 0. - A first-order Taylor-series approximation of k k around k k therefore yields k k k k kk k k Denoting k k k 0? Then k t kt k kt k e t k0 k Where k0 : the initial value of k. - It remains to find, this is where the specifics of the model enter the analysis. k k k kk sf k n g n g sf k n g n g k f k fk 1 K k n g - Thus, k converges to its balanced-growth-path value at rate 1 K k n g. - Typically, n g 6%, K 1/3 Then 4% - Therefore k and y move 4% of the remaining distance toward k and y each year. - As the half-life, t, is the solution to e t 0. 5, where is the rate of decrease. Then t ln0. 5/ 0. 69/. - Thus, it takes approxiamately 18 years to get halfway to their balanced-growth-path values. - In our example of 10% increase in the saving rate * Output is 0. 04 0. 5 0. 2% above ist prvious path after 1 year; * is 0. 5 5% 2. 5% above after 18 years; * and asymptotically approaches 5% percent above the previous path. - Thus not only is the overall impact of a substatantial change in the saving rate modest, but it does not occur very quickly. The Solow Model and the Central Questions of kk
Growth Theory What explains the vast differences in wealth across time and space? Two ways - Consider the required differences in capital per worker. - Study that the required differences in capital imply enormous differences in the rate of return on capital. First way - Consider a C-D function Y K A L 1 Y L A 1 K L a - Consider two different countries * 1 (high income) * 2 (low income) - We assume that A is the same in two countries. Only K/L differs. Would that be enough to explain the differences that we observe in the real world? - Analysis * Assume * Suppose, Y 1 L 1 K 1 L 1 A 1 Y 2 L 2 K 2 L 2 A 1 Y 1 L 1 X Y 1/L 1 Y 2 /L 2 X X 10 1 3 Y 2 L 2 K 1/L 1 K 2 /L 2 X K 1/L 1 K 2 /L 2 X 1/ empirically relevent Hence, K 1 /L 1 K 2 /L 2 10 3 1000 * The implied difference in the K/L ratio between country 1 and 2 is far from what is empirically observed. (Empirically 2-3 times). * This implies that within the framework of the Solow model, we have to explain income differences by differences in A. - Why is it possible that 1-countries have higher As than 2-countries? * Knowledge base bigger in 1-countrie than 2-countries * Education
* Better infrastucture, cultural attitudes toward entrepreneurshing and work, etc. Second Way - Rate of return on capitalf k - C-D y Y AL AL K. AL AL 1 k fk f k k 1 k 1 y 1 - Again, consider country 1 and country 2. y 1 Y 1 A L 1 y 2 Y 2 A L 2 Assume Y 1 L 1 X Y 2 L 2 y 1 X y 2 - MP K f k 1 y 1 1 X y 2 1 1 y 2 X 1 f k 1 f k 2 X 1 * Assume X 10 and 1/3 1 2 * Then f k 1 f k 2 10 2 f k 2 100 f k 1 - Again, the Solow model would suggest (or predict) a relation between returns to capital in different regions, which grossly unrealistic. - Differences in physical capital per worker cannot account for the differences in output pre worker that we observe. Effectiveness of labor, A: - Differences in A across regions may coexist with realistic differences in capital per labour, rates of return variation in output per capita. - The Solow model offers an incomplete treatment of "effectiveness of labour". * Its rate of growth is exogenous (not modelling but assuming growth) * Catch all factors other than labor and capital that affect output. * Does "effectiveness of labour" correspond to knowledge. * If so, what are the determinants of knowledge over time? * If so, why do firms in some countries have access to more knowledge than is other countries? * Why is knowledge difficult to transfer to poor countries?
- In any proposed view of what A is, the questions arise * How does it affect output? * How does it evolve over time? * Why it differs across regions? Empirical Applications Growth Accounting Convergence Saving and Investment Growth Accounting Consider again the production function Yt FKt, AtLt. Q: How much of the observed growth in Y can be attributed to K, L and A? Yt FKt, AtLt Yt Ẏt Kt K t Ẏt Yt Kt Yt Yt Kt Yt Lt L t K t Kt Lt Yt Yt At Ȧt Yt Lt L t Lt At Yt Yt At Ȧt At I * I: Growth in GDP as a result of a higher capital stock. * II: Growth in GDP as a result of an increase in the labour force. * III: Growth in GDP as a result of technological advances. - Reformulation * K t Yt Kt : Elasticity of GDP with respect to capital. * L t * Rt Kt Yt Lt At Yt Yt Lt Yt Yt At : Elasticity of GDP with respect to labor. Ȧt At : Solow residual Interpreted as a measure of the contribution of technologic progress. But it reflects all sources of growth, other than the contribution of capital accumulation via its private return. - Thus Ẏt Yt K t K t Kt II L t L t Lt Rt - Subtracting L t/lt from both sides and using the fact that L t K t 1 (see Problem 1.9) Ẏt L t K t K t L t Yt Lt Kt Lt Rt III * We can find data for Y, K, L and K Extensions - Different types of capital and labour. - Adjust for changes in the quality of inputs. - Imperfect competition. (w and r Applications Yt Lt Yt Kt )
- Denmark (Slide) - OECD (Slide) - Productivity growth slowdown * slower growth in workers skills; * oil price increases in early 1970s; * slowdown in the rate inventive activity; * the effects of government regulations. - "New Economy" * Since mid-1990s, (US) productivity growth has returned to close to its level before the slowdown. * Role of technological progress in computers. Convergence We have introduced the phase diagramme for k : We restrict attention to the segment where k 0 k This could be interpreted such that we do not consider cases "too" far away from the steady state. Indeed, k depends on the distance of k from k, i.e. k hk k and, by assumption, k h k k k 0 So, the greater is k, the smaller is k and for k k, k is even negative. We have already shown that there is a unique relationship between k and y : y fk k vy; v 0 y fk k vy Hence: k k k h y k y k k v y So, 0 y k k 0
Let s now study the growth rate of GDP: Y A L fk where To what extent does Ẏ Y - We know that a "small" y means: * k is small * f k is large * fk is small * k hk k is large - Hence, Ẏ Y g n f k fk hk k k hk k depend on how rich the country is intially? Ẏ Y g n f k fk hk k is large. - This boils down to saying that if a country is poor (y is small), then the Solow model would predict that the country has a higher growth rate. - Of course, the oppostite holds if the country initially is rich (y"large") - According to the Solow model, we could expect convergence to occur. The intuition is that in the Solow model, a possible reason for being poor is that k is far below k. We would therefore expect k 0, i.e. high growth rate in y as well. - Is this empirically relevant? * It is relevant to study the question of convergence, since if convergence really occurs, then global poverty problem would disappear over time. * Unfortunately, most empirical studies show that very rich and very poor countries do not really converge probably in line with most people s observation intuition. * However, if comparisions are limited to relatively similar countries, the support to convergence is much stronger. Problem Set 1.3, 1.5, 1.7, 1.9