Solow Growth Model Michael Bar February 28, 208 Contents Introduction 2. Some facts about modern growth........................ 3.2 Questions..................................... 4 2 The Solow Model 5 3 Qualitative analysis 5 3. Steady state analysis............................... 9 3.. Technological improvement........................ 0 3..2 Higher growth rate of population.................... 0 3..3 Higher saving rate............................ 3..4 Summary of analytical results...................... 2 3.2 Golden Rule saving rate............................. 2 4 Quantitative analysis 4 4. Functional forms................................. 4 4.2 Assigning numerical value to.......................... 6 4.3 Equilibrium in the calibrated model with growth................ 8 4.3. Balanced Growth Path (BGP)...................... 8 4.4 Optimal saving rate................................ 2 5 Conclusions 22 San Francisco State University, department of economics.
Introduction The two main branches of macroeconomics are (a) Growth and (b) Business cycles, where growth refers to the general increase in real GDP per capita, while business cycles are deviations from the growth trend. Usually, macroeconomic courses start with studying of growth for two reasons:. The relative importance of growth for the well being of people is much greater than that of business cycles. Take for example the recession of 2007-2009, with the lowest level reached in 2009. There were only 5 years in the entire history prior to 2009, in which the real GDP per capita (standard of living) was higher than in 2009 (these year are 2004-2008). At the lowest point of this recession, the standard of living was still higher than all of the previous history of this country, with the exception of only 5 years (see the next graph). Therefore, volatility of output around the growth trend dwarfs in comparison to the trend itself. In addition, consider the di erence between % growth in standard of living v.s. 2% growth. In the % percent case real GDP per capita will double approximately every 70 years, while in the 2% case, the doubling time is about 35 years, i.e. after 70 years the standard of living quadruples. In other words, small di erences in growth rates, when compounded over a generation or more, have great consequences for standard of living. 2. The second reason why we start with growth is that growth models are the main workhorse in the study of business cycles, with added stochastic components. So it is easier to start with growth. 2
. Some facts about modern growth. Per capita output grows over time, and its growth does not tend to diminish in developed countries. 2. Physical capital per worker grows over time. Capital/output ratio is approximately constant. 3. The rate of return to capital is nearly constant over time. 4. The shares of labor and physical capital in national income are nearly constant over time. 5. The standard of living di ers substantially across countries (see the next gure). 6. The growth rate of output per worker di ers substantially across countries (see the next gure). 7. There is no systematic relationship between standards of living and growth rates across countries (poor countries do not tend to grow faster and catch up). In other words, there is no absolute convergence. Facts -4 describe time-series observations. Facts 5-7 describe cross-sectional observations. 3
We will later evaluate theories with respect to these facts. For example, we will check whether the predictions of our models are consistent with these facts..2 Questions. What factors determine which countries prospered? 2. Can we point to speci c economic policies? 3. Are there speci c country characteristics that determine economic fate? 4. Is prosperity just a result of luck? "Once one starts to think about [these questions], it is hard to think about anything else". Robert Lucas (988). 4
2 The Solow Model. Output is produced with production function Y t = F (K t ; L t ), where Y t is aggregate (real) output, K t is the stock of physical capital, and L t is labor services. The function F (; ) is assumed to exhibit constant returns to scale (CRS), with the following assumptions: (a) : F (0; L) = F (K; 0) = 0 (b) : F K (K; L) > 0; F L (K; L) > 0; F KK (K; L) < 0; F LL (K; L) < 0 (c) : lim K!0 F K (K; L) = lim L!0 F L (K; L) =, lim K! F K (K; L) = lim L! F L (K; L) = 0 The rst assumption (a) means that both inputs are essential for production. The second assumption (b) means that the marginal products of both inputs are positive and diminishing, and the last assumption (c) means that the marginal product of inputs is very high when the input is scarce and very low when it is abundant. The conditions in (c) are sometimes called Inada conditions, after the Japanese economist Ken-Ichi Inada. A production function which is CRS and possesses properties (a) - (c) is called a neoclassical production function. 2. Capital evolves according to K t+ = ( ) K t +I t, where 0 < < is the depreciation rate and I t is aggregate investment. 3. People save a fraction s of their income. This fraction is exogenous. Thus, the total saving and total investment in this (closed) economy is S t = I t = sy t 4. The population of workers grows at a constant rate of n, which is exogenous in this model. Thus, L t+ = ( + n) L t. 5. No government. 3 Qualitative analysis Now we derive the predictions of the model about output per worker, consumption and investment. Our main object of interest is output per worker, denoted by y t Y t L t : Another object of interest is consumption per worker: c t = ( s) y t We call a variable endogenous if it is determined within the model and exogenous if it is determined outside the model. For example, in the model of a market (supply and demand diagram), the price and quantity traded of the good are endogenous variables, while other variables that determine the location of the supply and demand curve, such as income and prices of other goods, are assumed exogenous. 5
which depends on the consumption rate s and output per worker. So we start by examining how output per worker evolves over time in this model. First, we notice that output per worker can be represented as a function of capital per worker: y t = Y t L t = F (K t; L t ) L t = F Kt L t ; The last equality follows from the CRS assumption. Let k t Kt L t denote capital per worker K and f (k t ) F t L t ; be the output per worker as a function of capital per worker 2. Thus, output per worker can be written as y t = f (k t ) Notice that f (0) = 0, f 0 (k) > 0 and f 00 (k) < 0, from the properties of F (K; L). The key is therefore to gure out how capital per worker, k t, evolves over time. To derive the law of motion of capital per worker, we simply divide the law of motion of aggregate capital by L t+ and use the de nition of investment: K t+ = ( ) K t + sf (K t ; L t ) K t+ = ( ) K t + sf (K t; L t ) L t+ ( + n) L t ( + n) L t ( ) k t+ = ( + n) k s t + ( + n) f (k t) () The last equation is the law of motion of capital per worker. If the function f () does not change over time, i.e., there is no change in productivity (say due to technological progress), then we can predict the behavior of this economy in the short run and in the long run pretty easily. Equation () gives us the future capital as a function of current capital k t+ (k t ). This is called a di erence equation 3. First observe that k t+ (0) = 0, i.e. if the economy does not have any physical capital, which is essential in production, there is no way to build new one. First, note that k t+ (k t ) is increasing and strictly concave: Moreover, k 0 t+ (k t ) = k 00 t+ (k t ) = lim k k0 t+ (k t ) = t!0 lim k k0 t+ (k t ) = t! ( ) ( + n) + s ( + n) f 0 (k t ) > 0 s ( + n) f 00 (k t ) < 0 ( ) ( + n) + s ( ) ( + n) + s ( + n) lim f 0 (k t ) = k t!0 {z } = ( + n) lim f 0 (k t ) = k t! {z } =0 ( ) ( + n) < 2 This is called intensive form of production function. 3 A di erence equation is a discrete time analog of the di erential equation in continuos time. 6
Thus, the law of motion of capital per worker starts at the origin, has very a steep slope for small k t, the slope decreases as k t becomes large, and eventually the slope becomes less than. These properties follow from the Inada conditions in assumption (c) above. The graph of the law of motion of capital is illustrated in the next gure. Figure : Law of motion of capital per worker From the above graph, it is clear that starting from any level of capital per worker k 0, it will converge to the steady state level of k ss. The next gure illustrates the convergence of capital per worker to a steady state. Figure 2: Convergence to steady state 7
Suppose the economy starts with k 0 < k ss capital per worker. The law of motion of capital per worker, k t+ (k t ) can be used to obtain k (k 0 ). The 45 0 line helps us to locate k on the horizontal axis, so that in period the economy starts with k. Again, the law of motion of capital per worker determines k 2 (k ), and the process continues forever. Similarly, Suppose the economy starts above the steady state, k 0 > k ss. The law of motion helps to nd k (k 0 ), k 2 (k ),... Thus far we have proved mathematically, and illustrated graphically, an important proposition about the Solow model. Proposition If the aggregate production function F (K; L) is CRS, satis es conditions (a)-(c), and the production function does not change over time (no technological progress), then starting with any positive level of initial capital per worker k 0 > 0, the economy will converge to a steady state. Formally, for any k 0 > 0, we have lim t! k t = k ss, such that k t+ (k ss ) = k ss. Since the capital per worker converges to the steady state, output per worker and consumption per worker will also converge to steady state levels: lim y t = y ss = f (k ss ), and lim c t = ( s) y ss = ( s) f (k ss ) t! t! Alternatively, if there was technological progress, then we could write f (k t ; t) to indicate the dependence of the production function on time. In this case technological improvement would shift the law of motion of capital per worker. The next gure illustrates a case of technological improvement. Figure 3: Technological improvement 8
3. Steady state analysis In the previous section we showed that in the Solow model with no change in productivity, starting with any positive level of capital per worker, the economy will converge to the steady state. In other words, the prediction of the model is that in the long run, all variables converge to steady state, which is similar to the prediction of the supply and demand model of a market that the price and quantity will converge to the equilibrium. So in a sense, the steady state (k ss ; y ss ; c ss ) is the model s prediction about the economy in the long run. Our next task is to nd the steady state and investigate how it depends on the exogenous parameters. We start by nding the steady state level of capital per worker using the de nition of the steady state, that k t+ (k ss ) = k ss. Thus, substituting k t = k t+ = k ss in the law of motion of capital per worker and rearranging, gives s ( + n) f (k t) k t+ = ( ) ( + n) k t + k ss = ( ) ( + n) k ss + k ss ( + n) = ( ) k ss + sf (k ss ) s ( + n) f (k ss) k ss (n + ) = sf (k ss ) (2) Equation (2) characterizes the steady state level of capital and also provides economic intuition about the steady state. The left hand side, k ss (n + ), represents the decline in capital per worker due to depreciation and increasing number of workers (" ow out"). The right hand side represents the increase in capital per worker due to investment per worker (" ow in"). At the steady state the " ow out" is exactly o set by the " ow in". We cannot solve for k ss without knowing the function f (), but we can nevertheless illustrate the steady state graphically and perform qualitative analysis. The next gure illustrates the ows (" ow in" and " ow out" in the Solow model). Figure 4: Steady state with " ows" 9
Figure 4 should not be confused with Figure. Both gures show the steady state, but in a di erent way. The curve in Figure is law of motion of capital per worker k t+ (k t ) and the straight line is 45 0 which helps us see all the points where k t+ = k t. The curved lines in Figure 4 on the other hand, are f (k t ) and sf (k t ) (production and investment), and the straight line represents the depletion of capital due to depreciation and growth in the number of workers, k t (n + ). Figure 4 illustrates not just the steady state capital per worker, but output and consumption as well: (k ss ; y ss ; c ss ). Thus, Figure 4 is more useful for analyzing the steady state, while Figure s main purpose is to demonstrate convergence to steady state (see Figure 2). 3.. Technological improvement Here we illustrate the e ect of a once-and-for-all increase in productivity, so that f (k t ) becomes greater for any level of k t. The production function f (k t ) has therefore shifted up. As a result, the saving function shifts up as well. All the steady state variables went up: (k ss "; y ss "; c ss "). Figure 5: Technological improvement The intuition is simple. With higher technology, the return on investment in capital is higher, so investment and capital go up. As a result, output per worker goes up, and since consumption is a constant fraction of output, ( s), it also goes up. 3..2 Higher growth rate of population Higher population growth rate, n ", increase the slope of the " ow out" line. Intuitively, higher growth rate of population results in higher depletion of capital per worker. In a 0
sense, growth in the number of workers is similar to depreciation, since with more workers capital per worker decreases. Thus, the net rate of return on investment in capital is smaller (because it "depreciates" faster), and steady state capital per worker falls. Figure 6: Higher growth rate of population As a result of the fall in steady state capital per worker, output per worker falls and consumption per worker fall as well. 3..3 Higher saving rate Now we increase the saving rate, i.e. s ". The steady state capital per worker goes up, because higher saving rate means that higher fraction of output is saved (=invested) and not consumed. With higher steady state capital per worker, output per worker goes up as well. Notice however that when saving rate goes up, the steady state consumption per worker does not necessarily go up. In the next gure it actually goes down. To understand why this is happening, look carefully at the de nition of steady state consumption per worker: z } { c ss = ( s) f (k {z } ss (s)) # Notice that steady state capital per worker has increased when saving rate went up, and so did the output per worker. But at the same time, a smaller fraction of the new output is being consumed. Therefore, when s ", consumption per worker does not necessarily goe up. Figure 7: Higher saving rate "
3..4 Summary of analytical results () : Technology " ) k ss "; y ss "; c ss " (2) : n " ) k ss #; y ss #; c ss # (3) : s " ) k ss "; y ss "; c ss? 3.2 Golden Rule saving rate In the last section we have seen that the steady state depends, among other things, on the saving (or investment) rate 4 s. Now we want to nd out the optimal saving rate, i.e. the saving rate that would maximize the steady state level of consumption per worker. Since we do not have an explicit solution for k ss as a function of s, we restrict our attention to nding the optimal level of steady state capital per worker. Formally, we want to solve max c ss k ss = ( s) f (k ss ) = f (k ss ) sf (k ss ) s:t: k ss (n + ) = sf (k ss ) (the steady state condition) Substituting the constraint into the objective max k ss c ss = f (k ss ) k ss (n + ) 4 In closed economies saving rate equals to the investment rate. 2
F.O.C. f 0 (k GR ) (n + ) = 0 f 0 (k GR ) = n + In words, the capital per worker has to be such that the marginal product of capital is equal to the rate of capital depletion (n + ). Intuitively, suppose that k ss > k GR, i.e. the economy saves too much. Is this optimal? The return on an extra unit of capital is the marginal product of capital f 0 (k GR ). The loss from the extra unit of capital is the depreciation rate + the rate of growth in the population of workers. Thus, if k ss > k GR, we have f 0 (k ss ) (n + ) < 0, so the net bene t from the extra unit of capital is negative. Similarly, if k ss < k GR, we have f 0 (k ss ) (n + ) > 0, so the net bene t from extra unit is positive, and the level of steady state capital per worker needs to increase. The golden rule saving rate can be illustrated graphically. Observe that in Figure 7, the steady state consumption per worker is equal to the vertical distance between the output per worker curve, f (k t ), and the " ow out line", k t (n + ). In order to nd the optimal saving rate, we need to maximize the vertical distance between the two lines. The vertical distance in maximized when the slopes of the two lines are the same. When the slopes are di erent, this means that the lines are getting closer together or farther away from each other. The next gure illustrates the golden rule saving rate. Figure 8: Golden rule The dotted line is parallel to the " ow out" line, k t (n + ), so the slope of f (k GR ) is the same as the slope of the " ow out" line, i.e. f 0 (k GR ) = n +. 3
4 Quantitative analysis The theoretical model described in the previous section is not suited for quantitative investigation. For example, we want to answer questions such as "if the saving rate goes up by %, what is the percentage change in the steady state capital per worker, output per worker, and consumption per worker?". In order to answer this and other similar questions, we need to calibrate the model. Calibration consists of two steps:. Assigning functional forms to the functions used in the model. 2. Assigning numerical values to the parameters of the model. What is the best way to calibrate an economic model is an ongoing debate among economists. We will not attempt to resolve the debate, but rather discuss a few possible ways of calibrating the model, and mention some of their advantages in disadvantages. 4. Functional forms In the Solow model the only function that we need to calibrate is the production function. Economists usually assume that the production function has the Cobb-Douglas form: Y = AK L, 0 First, make sure that you know how to check that Cobb-Douglas production function is neoclassical, i.e., it is CRS and satis es the assumptions (a) - (c). In addition to all of these properties, the Cobb-Douglas has the property that the input factor shares are constant (independent of the prices of inputs). Suppose that the rm is competitive, and solves the following pro t maximization problem. max = F (K; L) rk wl K;L where r is the rental rate of capital and w is the real wage rate. The rst order conditions for pro t maximization are F K (K; L) = r F L (K; L) = w So a competitive rm pays each factor its marginal product. The total payments to K and L are rk = F K (K; L) K wl = F L (K; L) L Factor shares are the fractions of total output that are paid to each factor. The factor shares of capital and labor are therefore Capital share: Labor share: 4 F K (K; L) K F (K; L) F L (K; L) L F (K; L)
In the Cobb-Douglas case, the factor shares are Capital share: Labor share: AK L K = AK L ( ) AK L L = ( ) AK L Thus, if the production function is Cobb-Douglas, the factor shares are constant (independent of factor prices) and equal to the exponents and ( ). In fact, the Cobb-Douglas production function is the only production function with this property. So what if the factor shares are constant? If the factor shares in the data are constant, then the right functional form for the aggregate production function would be the Cobb- Douglas. What we need to check then, is whether the factor shares are indeed constant in the data. We will therefore look at National Income and Product Accounts (NIPA), measure the total income, and decompose it into payments to labor and payments to capital (everything else). We then plot the graphs of these empirical factor shares, and see if they are indeed constant. The next gure plots the labor share,, in the U.S. during 929-2006. We see that indeed the labor share uctuates mostly between 62% and 70%, with an average of 66%. But how do we come up with the numerical values for the factor shares? We provide a detailed answer to this question in the next section. 5
4.2 Assigning numerical value to We have seen in the previous section that the theory predicts that a fraction of total output is paid to capital and a fraction is paid to labor. We now look at the National Income and Product Accounts, and try to decompose the GDP into payments to labor and payments to capital (everything else). Recall that the GDP can be computed using the income approach as follows NI = W + Int + Rent + p + B + NF C GDP = NI + Dep NF I + Sd The next two table contain data on the ingredients of National income and its relation to GDP. 6
For simplicity, we ignore the Net Factor Income and Statistical Discrepancy, and treat the di erence between GDP and NI as Depreciation (consumption of xed capital). Thus, we will work with NI = W + Int + Rent + p + B + NF C GDP = NI + Dep where Dep = GDP N I Now we need to classify all the above magnitudes into payments to labor and everything else. Obviously, compensation of employees is a labor income. But it is a convention to treat proprietors income and nonfactor charges as part labor and part capital income. In particular, we assume that a fraction of p and NF C is paid to labor, and the rest is paid to capital. To nd the labor share then, we have to solve the following equation ( ) = W + ( ) ( p + NF C) GDP ( ) GDP = W + ( ) ( p + NF C) W ( ) = GDP p NF C Using data from the above tables, for the year 2006. gives: W = 7448:3, GDP = 394:7, p = 006:7, NF C = 993:9. This gives ( ) = 7448:3 394:7 006:7 993:9 = 0:665 7
4.3 Equilibrium in the calibrated model with growth We repeat the description of the model from section 2, but this time the production function is Cobb-Douglas, and we allow for growth in productivity.. Output is produced with production function Y t = A t Kt L t, where Y t is aggregate (real) output, A t is the total factor productivity (TFP), K t is the stock of physical capital, and L t is labor services. We further assume that A t+ = ( + A ) A t i.e. the TFP grows at constant, and exogenous, rate A. This assumption is equivalent to A t = A 0 ( + A ) t. 2. Capital evolves according to K t+ = ( ) K t + I t, where is the depreciation rate and I t is aggregate investment. 3. People save a fraction s of their income. This fraction is exogenous 5. Thus, the total saving and total investment in this economy is S t = I t = sy t 4. The population of workers grows at a constant rate of n, which is exogenous in this model. Thus, L t+ = ( + n) L t. 5. No government. In the Cobb-Douglas case, output per worker is given by y t = A t kt and the law of motion of capital per worker is ( ) k t+ = ( + n) k s t + ( + n) A tkt The graph of this law of motion, for xed A t, looks like the one in gure 2. However, when A t is growing, we cannot plot the law of motion since it will be constantly shifting up (see gure 3). The good news is that we are able to analyze the dynamics of this model in a special case of a balanced growth path. 4.3. Balanced Growth Path (BGP) A balanced growth path is a sequence of endogenous variables such that they all grow at constant rate (not necessarily the same for all variables). It turns out that if the growth rate of productivity is constant, then the economy converges to the unique BGP. 5 We call a variable endogenous if it is determined within the model and exogenous if it is determined outside the model. For example, in the model of a market (supply and demand diagram), the price and quantity traded of the good are endogenous variables, while other variables that determine the location of the supply and demand curve, such as income and prices of other goods, are assumed exogenous. 8
Proposition 2 Along a balanced growth path, output per worker, capital per worker, consumption per worker, and saving (=investment) per worker, all grow at the same rate. In other words, suppose that the economy is on a balanced growth path where y t+ = + y yt, k t+ = ( + k ) k t, c t+ = ( + c ) c t, s t+ = ( + s ) s t. Then the proposition claims that y = k = c = s Proof. Divide both sides of the law of motion of capital per worker by k t k t+ ( ) s = + k {z} t ( + n) ( + n) {z } const const This implies that the ratio y t =k t must also be constant, so y = k. Since consumption and saving (or investment) per worker are proportional to output per worker, y t k t c t = ( s) y t s t = sy t we must have c = s = y Let denote the common growth rate by. We can nd this common growth rate by using the fact that on a BGP, the ratio y t =k t must be constant. Therefore A t kt is constant, and A t+ kt+ = A t kt ( + A ) ( + ) = + = ( + A ) This means that when TFP grows at.2% per year, and capital share is 32%, the growth rate of all the endogenous per worker variables in the Solow model is + = ( + 0:02) 0:32 = :077 = :77% In the above discussion we analyzed what happens on a BGP, if such BGP exists. We have not established yet whether such a path exists. Proposition 3 Suppose that the TFP grows at a constant rate A. Then the endogenous variables converge to the unique balanced growth path with the common growth rate = ( + A ). 9
Proof. De ne e ciency variables (also called detrended variables) as the original endogenous variables divided by the cumulative growth factor: k t k t ( + ) t, y t y t ( + ) t, c t c t ( + ) t Observe that at time t = 0, the original and detrended variables are the same. Also notice that when the original variables are on the balanced growth path, the detrended variables are at a steady state (constant). The idea is to rewrite the law of motion of capital per worker in terms of these detrended variables, and prove that this new law of motion has a unique stable steady state. k t+ = kt+ ( + ) t+ = kt+ ( + ) t+ = kt+ ( + ) t+ = kt+ = s ( + n) A tk t ( ) ( + n) k t + ( ) ( + n) k t ( + ) t + ( ) ( + n) k t ( + ) t + ( ) ( + n) k t ( + ) t + s ( + n) A 0 ( + A ) t k t ( + ) t s ( + n) A 0 ( + ) t s ( + n) A 0kt ( + ) t ( ) s ( + n) ( + ) k t + ( + n) ( + ) A 0kt k t ( + ) t The function k t+ (k t ) satis es the properties of the original model (with no growth) in section 3, that guarantee a unique steady state as illustrated in gure 2. In particular, Thus, k 0 t+ (k t ) = ( ) ( + n) ( + ) + s ( + n) ( + ) A 0kt lim k0 kt t+ (kt ) =!0 lim k0 kt t+ (kt ) =! ( ) ( + n) ( + ) + s ( ) ( + n) ( + ) + s ( + n) ( + ) lim A 0k kt!0 t = {z } = ( + n) ( + ) lim A 0k kt! t = {z } =0 ( ) ( + n) ( + ) < 20
Now we can nd the steady state in terms for the detrended variables, the way we did for the original variables in the case of no productivity growth. k t+ = k ss = kss ( + n) ( + ) = ( ) kss + sa 0 kss kss [( + n) ( + ) ( )] = sa 0 kss [ + + n + n + ] = sa 0 k k ss = ( ) s ( + n) ( + ) k t + ( + n) ( + ) A 0kt ( ) s ( + n) ( + ) k ss + ( + n) ( + ) A 0kss ss sa 0 n + + + n y ss = A 0 k ss c ss = ( s) A 0 k ss sa0 = n + + The last approximation is valid for small values of and n. If the economy starts with initial capital stock per worker of k 0 = k ss, then the detrended variables are at the steady state, while the original variables are on the balanced growth path. The steady state analysis that we performed in section 3. carries through to the steady state in detrended variables. Everything else equal, if we compare two countries with di erent initial productivity, then the country with higher A 0 will have higher k ss; y ss and c ss and thus higher capital per worker, output per worker and consumption per worker on the BGP. Everything else equal, if we compare two countries with di erent population growth rates, then the country with higher n should have lower k ss; y ss and c ss and thus lower capital per worker, output per worker and consumption per worker on the BGP. Everything else equal, if we compare two countries with di erent investment rates, then the country with higher s should have higher k ss and y ss, but we can t tell if c ss is higher or lower. Obviously, countries with higher are growing faster by de nition of (the BGP growth rate of endogenous variables). 4.4 Optimal saving rate In section 3.2 we derived the condition for Golden Rule saving rate, i.e. the saving rate which maximizes the steady state consumption per worker: f 0 (k GR ) = n + (3) When a Cobb-Douglas production function is assumed, the output per worker is f (k t ) = A t kt. Assuming that there is no growth in productivity, A t = A 8t, the law of motion of capital per worker is: k t+ = + n k t + sak t + n 2
The steady state of capital per worker is found in the same way as in the last section, by plugging k t = k t+ = k ss into the law of motion of capital per worker: k ss = + n k ss + sak ss + n k ss ( + n) = ( ) k ss + sakss k ss (n + ) = sak ss The left hand side is the " ow out" of the stock of capital per worker, due to depreciation and growing number of workers. The right hand side is the " ow in" to the stock of capital per worker due to investment. The solution is: k ss = sa n + Now we nd the optimal capital stock in the steady state, k GR, which maximizes the steady state consumption per worker: max k ss c ss = Ak ss sak ss = Ak ss k ss (n + ) The last step substitutes the steady state condition: " ow out" = " ow in". optimal capital per worker must satisfy: Thus, the Ak GR = n + Where the left hand side is the marginal product of capital f 0 (k GR ) as in (3). Solving for the golden rule capital per worker, gives: k GR = A n + We see that in order to achieve this level of capital per worker in a steady state, the saving rate must be: s GR = 5 Conclusions We started these notes with some motivating facts and questions. Does the Solow model help us understand the facts and answering some of these questions? The Solow model is the starting point for looking at the growth experience of a country and comparing cross-country economic performance. The Solow model demonstrates that without growth in productivity, a country cannot experience sustained growth. In other words, growth that is based primarily on factor accumulation cannot go on forever. So the Solow model tells us that countries that grow, do so because of growth in productivity, and the reason why some countries stagnate is the lack of growth of productivity. We can use the Cobb-Douglas production function to decompose a growth of a country into a part coming from factor accumulation v.s. the part 22
coming from productivity growth. This is called growth accounting, and is carried out as follows. Using the production function, we can relate the growth rates of output with the growth rates of TFP, capital and labor. Y t+ = A t+kt+l t+ Y t A t Kt Lt + ^Y = + ^A + ^K + ^L where hat above the variable denotes the growth rate of the variable: Taking ln s of both sides ln + ^Y ^x x t+ For small growth rates, the above is approximately x t x t = ln + ^A + ln + ^K + ( ) ln + ^L ^Y = ^A + ^K + ( ) ^L So suppose that the growth rate of real GDP is 3%, capital grows at % and labor grows at.5%. Further assume that the capital share is 35%. Thus, 3% = ^A + 0:35 % + ( 0:35) :5% ) ^A = :675% This means that the growth in this country is fueled primarily by growth in productivity. The Solow model can also be used to compare two or more countries at a point in time. Assuming no growth in productivity, the steady state output per worker is sa y ss = Akss = A = A n + s n + We can use this equation to account for cross country di erences in GDP per worker, under the assumption that both countries (i and j) are in the steady state: y i y j = A i A j s i n i + sj n j + This equation can be used to measure how much of the di erence in output per worker between countries i and j can be accounted for by di erences in TFP, investment rate, population growth rate. This cross-country accounting tells us which country speci c characteristics are important for explaining the di erence between the two countries. Once this accounting is performed for many countries, economists learned that the most important factor in accounting for cross-country di erences is productivity. The Solow model does not provide a theory of productivity, but it points out the importance of developing one. 23