The stylized Kaldor facts are named after Nicholas Kaldor who uncovered them in 1957:

Transcription

1 Chapter V Growth Professor Thomas Chaney 1 Basic stylized facts of growth The Kaldor facts The stylized Kaldor facts are named after Nicholas Kaldor who uncovered them in 1957: 1. Labor productivity has grown at a sustained rate. 2. Capital per worker has also grown at a sustained rate. 3. The real interest rate, or return on capital, has been stable. 4. The ratio of capital to output has also been stable. 5. Capital and Labor have captured stable shares of national income. 6. Among the fast growing countries of the world, there is an appreciable variation in the rate of growth of the order of 2-5 percent. Fact 1 was meant to reject both what is called the classical growth theory (such as Malthus) and the Marxist growth theory, where the world converges to a steady state with zero growth. Fact 2 suggests that capital accumulation is part of the story, as in Marxist theories. However, facts 3-5 reject the marxist theory of growth, where the collusive behavior of capitalists (capital owners) generates a gradual reduction of the share of labor (until the revolution hits, at which points things are really not very clear in Marx s work). 61

2 CHAPTER V. GROWTH 62 The Solow neoclassical growth model Bob Solow offered in 1956 a neoclassical growth model, that can adequately deal with facts 3-5. By introducing exogenous technological progress, he can also explain facts 1 and 2. Solow recognizes therefore the central role of technological progress in generating sustained growth. However, he also points out very explicitly that in a competitive equilibrium and within a neoclassical framework, there is no money left on the table to reward innovators. So there is no explanation for endogenous technological progress. This is a very important insight. Consider a production function that combines capital, K, and labor, L, ( workers and machines ) to produce some final output, Y, and assume that this production function is characterized by constant returns to scale, Y = F (L, K) s.t. 8 >0, F( L, K) = F (L, K) Now, take the derivative of the constant returns to scale condition with respect to, F (L, In a competitive equilibrium, factors are paid their marginal product. The reason is simple: if say wages are below the marginal product of labor (the contribution of one additional worker to output), then a competitive firm should hire at least one more worker (the extra output [benefit] exceeds the extra cost); conversely, if the wage is above the marginal product of labor, then a competitive firm should fire at least one worker (the benefit from paying fewer workers exceeds the cost of producing a bit less). That implies that, F (L, K) =wl + rk where w is the wage rate and r the cost of capital. In other words, the value of output (F (L, K)) is just enough to pay workers (wl) and machines (rk). There is no money left on the table. In particular, there is no money left to reward innovators. But Solow also shows us that capital accumulation alone (building more machines) can generate growth only for a while. Eventually, the only way to grow is to innovate (build better machines). He therefore points that ultimately, growth must be driven by technological progress. But at the same time, he shows that within a purely competitive equilibrium, the market cannot provide private incentives for innovations. There are two solutions to this puzzle: either innovation is not motivated by private incentives (this is to a large extent the case for fundamental research); or

3 CHAPTER V. GROWTH 63 we do not live in a perfectly competitive world, so that innovators can extract some rents and be rewarded for their effort in generating innovations. Both seem to be true, but the second one is much harder to model rigorously: as one moves away from the assumption of a perfectly competitive world, many of the elegant results of economic theory disappear, and things become very messy. Paul Romer is one of the first to address this question headfirst. He shows how a monopolistic market, i.e. a market with monopoly power but also with very many competitors so that each is in some sense atomistic, can give private incentives to innovate. This work builds on the technical innovations of Joe Stiglitz and Avinash Dixit that devised a simple way to formally treat such a case of monopolistic competition. This was the advent of what became known as endogenous growth theories, in the sense that those theories had an explicit story for where innovations come from, and why they are made (instead of simply assuming that they happen ). The new Kaldor facts Chad Jones and Paul Romer list a series of 6 new stylized facts about growth as of In a sense, they are all related to Kaldor s fact number 6, which remains largely unexplained even after 50 years of research on growth. Here are the new facts: 1. Increases in the extent of the market. Increased flows of goods, ideas, finance and people - via globalization, as well as urbanization - have increased the extent of the market for all workers and consumers. 2. Accelerating growth. For thousands of years, growth in both population and per capita GDP has accelerated, rising from virtually zero to the relatively rapid rates observed in the last century. 3. Variation in modern growth rates. The variation in the rate of growth of per capita GDP increases with the distance from the technology frontier. 4. Large income and total factor productivity (TFP) differences. Differences in measured inputs explain less than half of the enormous cross-country differences in per capita GDP. 5. Increase in human capital per worker. Human capital per worker is rising dramatically throughout the world.

4 CHAPTER V. GROWTH 64 cr C..I 90.01_ ~~~ Population (Billions) Figure V.1: Population growth versus population 6. Long-run stability of relative wages. The rising quantity of human capital, relative to unskilled labor, has not been matched by a sustained decline in its relative price. 2 A simple model of (semi) endogenous growth: Kremer 1991 Michael Kremer offers an elegant model to explain a series of stylized facts about growth in the very long run (over many thousands of years). He offers both empirical evidence, as well as a theoretical explanation, for fact 2 of Jones and Romer. His model is one of endogenous technological progress, in the sense that the flow of ideas generated depends on several variables that are evolving over time, and in turn influences those variables. It is however not a model of a market economy, so Kremer does not try and explain how the private incentives to innovate may be shaped by market forces. Since he is interested in growth over the very long run, not assuming a market economy is probably a wise choice: in prehistorical times, or even in pre-homo sapiens time, markets were not very developed (even though this is more of a conjecture than a fact). The main empirical fact he wants to explain is that the human population, at least over the first 1 Million years (from 1 Million B.C. to 1940), increased at a rate roughly proportional to the size of the population. That corresponds to a growth that is substantially faster than exponential growth, which would correspond to a constant growth rate. In other words, population growth accelerated continuously until the mid-20th century.

5 CHAPTER V. GROWTH 65 Population as a measure of world GDP. Kremer focuses on the size of the human population, and not on the size of the world income, for two reasons. The first is driven by data availability: going back far into the past, while archeologists have a relatively good idea of the size of the human population at a given point in time, they have very scant evidence (if at all) on the standard of living back then. The second and more fundamental reason is a Malthusian type argument: for most of history, almost all humans were barely surviving. That is, the standards of living were just enough for humans to survive and for women to bear children. Direct evidence on actual data on the standards of living (we can do that with historical data going back all the way to Antiquity) suggests that this is indeed the case. We also have indirect evidence from the study of health of early humans that they did not have a very comfortable life. But we don t know exactly how uncomfortable. We know that they produced enough to survive (as did people until the 18th Century), but not much more. If it is indeed the case that until very late (18th, 19th or even mid-20th Century), the vast majority of humans were simply earning a subsistence wage (call it ȳ), then the total human GDP was simply P ȳ, this subsistence wage (ȳ) multiplied by the number of humans (P ). Aslongasȳ is constant, aggregate GDP growth is simply equal to population growth. A simple model of accelerating population growth. Kremer builds a very simple model of economic growth. It is based on three simple assumptions. First, he assumes the existence of an aggregate production function, which satisfies some simple properties. Second, he assumes a Malthusian type of force. Third, he assumes that humans come up with new ideas, at least some times. Writing down these three assumptions formally as three equations, one gets a very stylized representation of a growing world economy. 1. Aggregate production function. The first ingredient of this model is to have a production function. A production function defines how much output of a particular good can be produced for any combination of inputs. For instance, for a firm where workers combines iron ore and coal in order to produce steel, the production function would be a mathematical function of 3 variables. This function tells us, for a given of workers, that are combining a given number of tons of iron ore and of tons of coal (inputs = what goes in), how many tons of steel can be produced (output = what comes out). Note that we are only defining the best possible way for that particular firm to combine inputs into output. While you can always do worse (throw away some steel for instance), the production function tells

6 CHAPTER V. GROWTH 66 us the maximum amount of output that can be produced for a given combination of inputs. In the same way as one can define a production function for an individual firm, one can do so for an entire country. The concept of an aggregate production function is somewhat harder to grasp. Consider for instance a simplified view of the economy, where there is a single output (call it units of GDP, number of real euros, instead of the sum of all cars, pants, apples, oranges... etc produced in a country), and only two inputs, labor and land. This production function tells us, for a country with that many million of workers, and that many thousands of km 2,howlargeis the GDP. It s a summary of everything that is happening inside the country. Let s turn to Michael Kremer s model now. He assumes that there is one type of output (that could be number of calories of food produced, as for most of mankind s history, that was the main consumption), called Y. It is produced by combining workers (measured by the size of the population, p), land (measured by T, which we shall assume is constant over time, and normalized to 1), and ideas (summarized by a number, A). We will assume the following simple production function, Y = AP T 1, with 0 < <1 (V.1) This type of production function, linear in logs (take the log of Y and check that it is a linear function of the log of P,thelogofT and the log of A) is called a Cobb-Douglas function, named after Charles Cobb and Paul Douglas. While this particular mathematical function is very specific, it is (i) very convenient [easy to manipulate], but also (ii) it describes the actual process of aggregate production in the data quite well. It has several important properties, that happen to hold approximately in the data. First, this function exhibits constant returns to scale with respect to land and labor. This means that if there are twice as many humans and twice as much land, output will be exactly twice as large. This seems like a natural assumption, given that if given new land and more humans, one can always replicate everything. Second, and this is a direct corollary of the CRS assumption above, this function exhibits decreasing returns to scale with respect to labor. This means that with the same amount of land, if one doubles up the number of humans, output will grow, but by less than a factor of two. A consequence of this property is that for a given level of technology A, output per capita (how much there is to eat for the average human) will be a decreasing function of the size of the

7 CHAPTER V. GROWTH 67 population, y Y P = A T < 0 In other words, if we have a given amount of land to harvest, the more we are, the less we will be able to eat per person, unless we find a smarter way to use this given amount of land. This assumption will naturally give rise to a Malthusian trap, as we will see below. Third, and also as a corollary to the CRS assumption above, this production function exhibits increasing returns to scale with respect to labor (P ), land(t ) and ideas (A). This means that if we double up the number of ideas, humans and land, we will more than double up total output. This assumption will naturally lead to a sustained economic growth, as we will see below. 2. Malthusian trap. The second ingredient of the model is to assume a link between how well off humans are, and how many babies they make. It is a very stylized representation of both some biological evidence (when a man and a woman are healthy, they are able to bear and raise children; when they are ill nourished, less so), and some behavioral evidence (men and women, when they can, tend to like having babies). Formally, this means that there is a subsistence level of income per capita, ȳ, such that above this threshold, population grows, and below, it shrinks. In other words, when humans are well fed and in good health (y >ȳ), there natural tendency to want to have babies means that couples have on average more than two viable kids over their lifetime, so that population grows; when humans are ill nourished or sick (y <ȳ), they tend to die young (before they can have kids), they tend to be less fertile, and their children are not very likely to survive, couples have less than two viable kids on average, and population shrinks. Formally, we have, 8 >< > 0 if y>ȳ P =0 if y =ȳ >: < 0 if y<ȳ where I use the physics convention For a variable that is a function of time, putting a dot over that variable means how much that variable grows (or shrinks) from one period to the next. This biological/behavioral assumption, combined with the assumption of decreasing returns to scale with respect to labor, implies that the size of the population will tend to be determined by this subsistence level ȳ, P = ȳ 1 1 A (V.2)

8 CHAPTER V. GROWTH 68 If population is strictly above this P, then because of decreasing returns, output per capita will be strictly below the subsistence level ȳ, and population will shrink back down towards P (remember, more humans means less to eat for each of us). If population is strictly below this P, then output per capita will be strictly above the subsistence level ȳ, and population will grow back up towards P. A population size exactly equal to P is the only stable state. We call that a Malthusian trap, following the work of Thomas Malthus. It s a trap in the sense that humans desire to have children, combined with the limitation of our capacity to produce, forces us to earn just a subsistence income. The above equation also shows that the only way to escape such a Malthusian trap is to have new ideas, meaning increase A. It does not strictly speaking allow us to escape the trap (we still only earn a subsistence income), but at least, it allows us as a specie to grow. This is the next and final ingredient of the model. 3. Building on the shoulders of giants The final ingredient of Kremer s model has to do with the process through which new ideas are generated. Kremer assumes simply that any human has some chance of coming up with a great idea. Such a great idea will allow all humans to produce more efficiently. Think for instance of the neolithic revolution, by which someone somewhere around Mesopotamia roughly 12,000 years ago figured out how to grow wheat. This technological innovation may have taken a little while to be adopted by every human on the planet, but eventually, it did. We will make two assumptions for the process through which ideas are generated. First, the average number of ideas per person per period is some constant number, g. There will of course be a tremendous amount of randomness in the process of coming up with ideas (most of us never get to have a single idea that benefits mankind in our whole life); but with a large enough population, the law of large number means that the total number of ideas will simply be proportional to the size of the human population. The second assumption is that new ideas are improvements above and beyond what we already know. In other words, the more ideas we already have, or equivalently, the more efficient we are at producing, the better the new ideas will be. We can now formally express the link between the size of the human population (P ) and how the stock of ideas (A) grows, Solving the model. A A = gp (V.3)

9 CHAPTER V. GROWTH 69 We now have all the ingredients we need to solve the model! We have three equations, (V.1), (V.2) and(v.3), with three unknowns, P, Y and A. Equation(V.1) tells us how ideas and workers together can produce food. Equation (V.2) tells us how the available amount of food in turn determines the size of the population. And Equation (V.3) tells us how new ideas are invented by people. This is a relatively simple dynamic system: humans and ideas give food (V.1), food allows us to make babies (V.2), and those babies will themselves have new ideas (V.3), so that the next generation will have more people and more ideas, and they will be able to produce more food (V.1), which in turn will allow them to have babies of their own... etc. While this simple story in words makes sense, having equations allows to characterize precisely how this system will evolve. That means solving this system of 3 equations with 3 unknowns. First a little bit of math: for a variable X t that is a function of time, we Ẋ X X t ln X In words, the derivative of the log of X with respect to time of that variable Ẋ ln X is equal to the growth rate. This result is very useful, and important to remember. We can now take the log of Equation (V.2), and then take the derivative with respect to time, and get, ln P =lnȳ ln A P P = 1 A 1 A Plugging this expression into Equation (V.3), we solve the model and get the main testable prediction that Kremer brings to the data, P P = g 1 P In words, the growth rate of the population is proportional to the size of the population. This is exactly what we see in the data, as shown on Figure V.1. At least, this is true for most of human history, i.e. until the mid-20 th century. One can also say that the model is not really rejected by what happens in the data after The model actually tells us that as long as mankind is stuck in a Malthusian trap, we should expect the relationship between population growth and the size of the population, which implies a faster than exponential growth rate of the population. It also implicitly tells us that if mankind escapes from this Malthusian trap (for instance, people decide to have fewer babies, even though

10 CHAPTER V. GROWTH 70 they could very well have more, as is the case in most countries in the world today), then we should not expect the same relation between population growth and population size. But of course, one then has to write another model, where mankind can escape from the Malthusian trap is was stuck in for one million years. This is the second part of the Kremer paper, which is technically harder, and which I will not go over. 3 A simple model of exogenous growth: Solow 1956 We now turn to a more conventional, and influential, model of economic growth, the neoclassical growth model developed by Bob Solow in Solow shows that capital accumulation alone cannot generate sustained growth in the long run, but it can explain the convergence of income between initially different countries. I will go over this model rather quickly. Solow with neither population growth and nor technological progress Set-up. We represent the aggregate production of a country at time t, i.e., it s GDP, as Y t. This output is produced by combining capital and labor according to the constant returns to scale production function, Y t = F (K t,l) To start, I assume that the population is fixed, L is constant over time. Every period, a constant fraction of the stock of capital is lost to depreciation. This depreciation rate is meant to capture both the attrition of time, as machines eventually break down, natural or man-made disasters that wipe out part of a country s capital as bombs and hurricanes destroy buildings and machines, but also the obsolescence of old machines (you guys probably never even heard of Windows 95). In addition, agents can invest into building new capital. To do so, they use part of their production to produce new machines. For simplicity, I abstract from the complex decision of how much of one s income to save, and assume instead that everyone saves a constant fraction s to build new/replace old machines. The law of motion for capital is therefore, K = sf (K, L) K (V.4) where K represents the change in the stock of capital from one year to the next, and where I omit time subscripts for concision. The first term represents investment into new machines, and the second term represents the loss of existing machines to depreciation.

11 CHAPTER V. GROWTH 71 Model in per capita terms. Ultimately, we care not so much about the total production of a country, but rather about how much each person in that country earns. I will therefore express all variables in per capita term. I denote all per capita variables with lower case letters: x X L. To express the model in per capita terms, I use the property of a constant returns to scale (homogeneous of degree one) function: F (K, L) =LF K L, 1 and introduce the notation f (k) =F (k, 1), where f (k) = Y L summarized by a very simply difference equation, represents GDP per capita. The Solow growth model can be k = sf (k) k (V.5) Steady state. If the function f (k) is increasing, convex, and sufficiently steep around zero, then this dynamic system converges to a steady state such that, sf (k )= k At the steady state level of capital per worker k, investment (sf (k )) is just enough to replace depreciated capital ( k ), and the economy neither grows nor shrinks from one period to the next ( k, y = 0). If the stock of capital per worker k is below k, then the economy grows towards its steady state ( k, y>0), if it is above, it shrinks ( k, y<0). This result comes from the assumption of decreasing returns with respect to capital in the aggregate, for which there is strong empirical evidence. The very first few machines are tremendously productive, and they easily overcome the (small) depreciation of existing machines. This is why starting from a small level of capital per worker; the economy grows at a fast pace. Once most workers are equipped with capital, additional investment only brings about little increase in production, which can barely replace the (large) depreciation of existing capital, if at all; the economy grows at a slow pace, or even shrinks. This result also shows that in the long run, and in a world of decreasing returns to scale with respect to capital, economic growth in the long run cannot come only from capital accumulation. For a little while, the accumulation of capital brings the economy closer to its steady state. But once the steady state is reached, growth stops. Solow with population growth but no technical progress I now introduce a constant (exogenous) population growth rate, n, L L = n

12 CHAPTER V. GROWTH 72 The model remains almost identical, except that population growth now has to be accounted for. First, I rewrite the increase in the stock of capital per workers as K L = k + kn.1 Plugging in Equation (V.4), I get the following law of motion for capital per worker (k) in the presence of population growth, k = sf (k) ( + n) k (V.6) This expression very much resembles the law of motion in the absence of population growth in Equation (V.6), except that the depreciation is augmented by the population growth n. The intuition for this result is simple. When population grows, new workers are born without capital. Their parents therefore have to share their existing capital with them, so that everything is as if population growth eats up the existing capital stock. Capital accumulation must not only replace depreciated machines, but also provide new machines for newborn workers. This system also reaches a steady state level of capital per worker k at which growth stops, sf (k )=( + n) k Once again, due to the assumption of decreasing returns with respect to capital, capital accumulation can generate growth for some time, but ultimately, output per worker stops growing. The economy as a whole does grow in the steady state, but only at the speed of population growth. The world becomes richer because there are more workers, but the income of each individual worker remains constant. The primary lesson from Solow s growth model is that ultimately, growth of income per capita cannot come from either capital accumulation, or from population growth. It must come from some other (mysterious) source... Solow does not offer any explicit answer as to where this mysterious source might lie, but he does give us several tools to analyze this mystery. That is why he got the Nobel prize after all. First, he tells us what the world would look like if such a mysterious source of growth does exist, and he tells us what this source should be (basically, technical progress). He also tells us how to measure such a dark and mysterious source of growth (growth accounting). Finally, he explains why capital accumulation in his simple model can explain why some countries that are K To derive this result, I first use the quotient rule (derivative of a ratio of functions) to get L t = (@K t /@t)l t K t (@L t /@t) L 2 t = (@K t/@t) L t K t (@L t /@t) L t L t. I then use the following approximation, the previous equation as k K k L.Withtheconstantpopulationgrowthrate, L L bit, I get the proposed K = k + kn. L L L = n, andrearranginga

13 CHAPTER V. GROWTH 73 initially poor do seem to catch up to richer countries (think of Germany or Japan after WWII). Let look at each of those contributions in turn. Solow with population growth and technical progress The easiest way to introduce technical progress is to assume that over time, workers become more and more efficient. In other words, what could be produced with 10 workers can now be produced with fewer. This corresponds to the notion of labor augmenting technological progress. Formerly, we introduce L and A, L = AL where A is a term that captures the level of advancement of labor productivity, and L measures the number of efficiency units of labor. IfA goes from 1 to 2, it means that the same number of workers can now produce what twice as many workers could. An increase in A corresponds to labor augmenting technological progress. I assume with Solow that for some mysterious reason, A grows at a constant rate g over time, A A = g This form of technical progress may corresponds to either learning by doing (over time, workers get better at performing the same task), technological innovations (workers learn about a more efficient way of performing the same task), better management practices (the same number of workers perform more tasks as they waste less of their time)... etc. We can now rewrite the basic Solow model to incorporate technical progress. Output is produced combining capital and effective workers, Y = F K, L The stock of effective workers grows at the constant rate L L = g + n.2 The model is exactly isomorphic to the Solow model with population growth but without technical progress, except 2 To derive this result, I use the following approximation, ln A +lnl, Igettheproposed X t/@t X t ln X Then, noting that ln L = L ln L ln ln A A + L L = g + n

14 CHAPTER V. GROWTH 74 that every variable has to be expressed in per effective unit of labor, instead of per worker. The law of motion for the stock of capital per unit of effective worker is given by, k = sf k ( + g + n) k (V.7) The stock of capital per effective worker converges towards a steady state k defined by the condition, sf k =( + g + n) k (V.8) But while k does converge to a steady state, output per worker keeps growing forever. So instead of getting stuck in a steady state, the economy converges to a steady state growth path. Along that path, all variables grow at the rate of technical progress g. To see this, note that along this path, we have y t = Yt L t = F(Kt, L t) L t = L k tf( k ) L t = A t f. Output per unit of effective worker, f k, is fixed, but A t grows at a constant (exogenous) rate g. Along the steady state growth path, we have, Y Y = g + n, y y = g and ỹ ỹ =0 We now turn to the question of measuring the various sources of economic growth in the data. Growth accounting and Solow residuals While Solow does not offer an explicit theory of what the ultimate source of technical progress is, he offers a simple way to quantify in the data the relative contributions of factor accumulation and technical progress to economic growth. It is easiest to understand his method growth growth accounting using a particular functional form for the production function F, namely the Cobb- Douglas production function, Y t = A t F (K t,l t )=A t K t L 1 t where is a parameter that governs the relative importance of capital versus labor in the production process, and A t is called total factor productivity (note that this A t is not the same as A t in the previous section). Taking the logarithm of this expression, differencing over 2 consecutive years, and using the approximation we can write, X t g X for the growth rate g X of a variable X, g Y = g A + g K +(1 ) g L Going to the data, it is easy to measure the growth rate of aggregate output (GDP growth), as well as the growth rate of the stock of capital, and the growth rate of the population. Putting

15 CHAPTER V. GROWTH 75 together data for many years (t) and many countries (c), we can use econometric techniques to estimate the following equation, g Y (c, t) =b g K (c, t)+ \ (1 )g L (c, t)+bu (c, t) The coefficients b and \ (1 ) are precisely estimated, they approximately sum up to 1, with b 1 3. The error term bu (c, t) is called the Solow residual. It is a measure of the contribution of technical progress to economic growth. More precisely, it is a measure of all the growth that cannot be explained either by capital accumulation or population growth. In that sense, this is not a test of the Solow growth model. Rather, under the assumption that the Solow model is true, it is a measure of the contribution of technical progress to economic growth. Convergence and conditional convergence in the Solow model One of the key early contributions of Solow is to show that while capital accumulation alone cannot sustain growth in the long run, it can explain why some countries (or regions within a country) that are initially poorer catch up to their richer neighbors. The simplest version of this prediction of the Solow model only receives very mild support from the data, and is blatantly rejected by the data in many cases. A more nuanced version of this prediction however finds strong support from the data. Combining Equations (V.7) for the law of motion of k, the stock of capital per unit of effective worker, and Equation (V.8) for the definition of the steady state level of k, we get,! k k =( + g + n) ỹ/ k 1 ỹ / k Because of decreasing returns with respect to capital, the ratio of output per unit of capital, (V.9) decreases with k. The above equation therefore implies that the lower the initial stock of capital of a country (the lower k), the faster it will grow. As a country catches up to its steady state level of output per unit of capital ỹ, growth slows down. In other words, countries that are k initially poorer will grow at a faster pace, and converge towards other wealthier countries. We should observe convergence in income per capita over time. A naive interpretation of this prediction from the Solow model is that the level of output per worker across countries should converge: initially poorer countries will grow at a faster pace than richer ones, closing the gap between them. If one looks at individual regions within a country, US States within the US for instance, there is strong evidence of convergence. In the case of US, ỹ k

16 CHAPTER V. GROWTH 76 the income gap between rich and poor States closes by about 2% per year, so that half the gap is eliminated in 35 years. There is equally strong evidence of convergence between European regions over the period , with the income gap also closing at about 2% per year. There is also strong evidence of convergence between OECD countries (Western Europe, North America, Japan and Oceania mainly). This evidence however is very problematic, and may very well be due to what is called selection bias. To be a member of the OECD, a country has to be rich at the time when they join the organization, around 1960 for most members. So by definition, countries that were initially poor must have grown faster than their initially richer counterparts for them to be equally rich by Not surprisingly, we will find that for this very selected sample of countries, poorer countries grew faster than richer ones. If one looks at all the countries in the world, and plots their initial income say around 1960 versus their growth rate over the period for instance, there is clear pattern of an upward sloping relationship. In other words, it does not seem that initially poor countries grew faster than initially rich ones. This is however not evidence against the notion of convergence in the Solow model. What the theory predicts is not necessarily that all countries should converge to the same level of capital per worker k, but rather that they should converge to their own level of k. This steady state level k in the Solow model depends on various parameters that may very well (and actually do) vary across countries. For instance, a higher savings rate, a lower population growth all increase the steady state k. In other words, convergence in the Solow model should be conditional on a country s characteristics, and not absolute. Robert Barro and Xavier Sala-i- Martin show evidence that conditional convergence does seem to hold, even when all countries are included. The Golden rule savings rate The last point I will make about the Solow model concerns the distinction between income and actual consumption. So far, we have only described the growth of aggregate income (Y ). While aggregate income is often a good proxy for the level of physical well being in a country, it is not exactly what people care about. In particular, only a fraction of this income is actually spent on consumption, while the rest is invested in order to increase future production, but not consumed by any actual person. People usually care about what they consume. Trivially, in the Solow model, if one were to increase the savings rate to 100%, one would achieve the highest possible growth rate, but nothing would be left to consume, and everyone would starve to death. This is obviously not

17 CHAPTER V. GROWTH 77 a very desirable outcome. On the other extreme, if the savings rate falls to zero, then depreciation would gradually eat up any existing stock of capital, and eventually, output would drop to zero, and everyone again would starve to death. Not a very desirable outcome either. Within the Solow model, we can answer a simple question: what is the savings rate that would maximize not the steady state level of output per effective worker ỹ, but rather the steady state level of consumption per effective worker, c? This optimal savings rate is called the golden rule savings rate. By construction, total output is divided between consumption and investment, so that C = Y I and c =ỹ ĩ. In the steady state, we have ĩ =( + g + n) k so that c = f k ( + g + n) k. The optimal savings rate will therefore correspond to a steady state level k that solves, max f k k ( + g + n) k To solve this (unconstrained) maximization program, we take the first order condition, and find that a solution must be such that f k 0 = + g + n. The savings rate such that this condition holds in the steady state is the golden rule savings rate. At this point, the marginal product of capital per effective worker f k 0 exactly equals the effective depreciation rate of capital per effective worker. Given the decreasing returns with respect to capital, this point is where the distance between what is being produced and what is being saved (which in the steady state exactly equals the [effective] depreciation of capital) is the largest. In other words, it corresponds to the maximum level of consumption.