Class Notes. Intermediate Macroeconomics. Li Gan. Lecture 7: Economic Growth. It is amazing how much we have achieved.

Transcription

1 Class Notes Intermediate Macroeconomics Li Gan Lecture 7: Economic Growth It is amazing how much we have achieved. It is also to know how much difference across countries. Nigeria is only 1/43 of the US. 1

2 Obvious we are interested in understanding why some countries have grown faster than others. Previously we argue that output is mainly determined by capital stock and labor. Here we would discuss how capital and labor determine the output. Robert Solow ( ) The model to explain the economic growth is the Solow model. Robert Solow won Nobel Prize in economics in The accumulation of capital stock 1. Per capita production function. Consider the following aggregate production function: Y = F α 1 α ( K, L) = K L Then per capita (or per worker) production function: α 1 α α 1 α Y K L K L = = = α 1 α K y = = k L L L L L The marginal product of capital, MPK is given by: α α MPK Y = K = α K α 1 α 1 1 α α 1 L K = α L = αk y = k Therefore, MPK can be obtained by taking the first derivatives from the aggregate production function, or from the per capita production function. Two points: (i) the per worker production function increases with capital per worker, but the marginal is decreasing. 2

3 (ii) Note at per capita level: y = c + i where the per worker consumption c = (1 i) y. Rearrange terms, we get: i = s*y=s*f(k). 2. Evolvement of capital stock: Capital stock increases because of investment, and decreases because of depreciation. So consider the following evolvement pattern for capital stock: ( 1 δ ) Kt I t K + t = 1 Each period, because of depreciation, δk amount of capital stock disappears. So on (1- δ)k remains, but there is also an investment I. So the total capital stock will be at: (1- δ)k + I. When labor force does not change, L is constant, dividing both sides of the previous equation by L: ( 1 δ ) kt it k + t = 1 Rearrange this: Δk = i δ k t t t 1 Given i = sf(k), we have: ( kt ) kt 1 Δk = sf δ t 3

4 At steady state equilibrium, we must have: Δk t = 0, or k t = k t-1. Therefore, at the steady state level k *, we have: sf * * ( k ) = δk Discussions: (1) Why steady state? Suppose k > k * * *, then from the graph, sf ( k ) < δk, i.e., the amount of depreciation is higher than the investment k level would decrease. Suppose k < k * * *, then from the graph, sf ( k ) > δk, i.e., the amount of depreciation is lower than the investment k level would increase. (2) What if the saving rate increases? If saving rate increases, then we would have a higher level of k *, and therefore a higher level of per capita output. It turns out the most important growth policy is the policy of raising the saving rate. Examples of public policies that may raise saving rates: Tax benefits for IRA, Roth IRA, 401K, 403B, and 529 (college saving account raise private savings. Reducing budget deficit would raise the public saving and hence the total saving. Reducing capital gains tax. Examples of public policies that may reduce saving rate: 4

5 Social security and Medicare would reduce demand for precautionary saving. (3) What if depreciation rate decreases? A higher depreciation rate would lower the level of k * and output. Example: The China Experience China has been having about 10% growth rates for last two decades. One of the most important characteristics of Chinese economy is its high saving rates. See the following graph and table. 5

6 Gross national saving has been rising since 2000, and it stands at roughly 50% right now. Various measures of reducing save rate have not been successful. (1) Expand the enrollment of higher education and raise the tuition for higher education. New College Enrollments in China 7000 (in 1,000) Enrollments Year

7 College enrollments have increased dramatically over the last decade. Right now the enrollment is about 7 times as much the enrollment in Before 1999, college education is almost free. Right now the college tuition on average is about 5000 RMB (or US$720) per year. However, expanding college education creates wrong incentives some parents now would save for higher education while others pay for higher educations. (2) Establishing social safety network Nationwide health insurance a urban employees b rural residents c urban residents (non-employees) It helped reducing save rate but not much (increasing consumption by roughly 10%). T he Golden Rule Level of Capital Stock Compromise between the saving and consumption: Obviously a higher saving rate would lead to a higher per capita output, but a higher saving also means a lower level of consumption. In the extreme case, for example, a saving rate of 100% would lead to a higher level of per capita output but no consumption. Therefore, because there is a compromise of saving and consumption, it is possible to find the optimal level of capital stock that maximizes the steady-state level of consumption. At steady state, sf * * ( k ) = δk The steady state level of consumption is given by: c * = * * * * * ( 1 s) y = f ( k ) sf ( k ) = f ( k ) δk The Golden Rule level of capital stock is the level of capital stock that maximizes the steady-state consumption level. 7

8 Maximizing c *, we have: * * ( k ) δ = 0 c = f ' * k In other words, the Golden rule of capital stock is given by: MPK = δ A numerical example to find the Golden Rule level of consumption: Production function: 1/ 2 y = k, δ=0.1 t the optimum: = 1/ 2 2 A MPK 0.5k = δ k = 0.25/ δ k = 25 In a steady state, sf * * ( k ) δk =. * * ( k ) * ( k ) 1/ 2 δk s = = δ s = 0.5 f To summarize, two unknowns capital stock, k G ), and two equations: Population growth (saving rate s, and optimal level of per-labor G ( k ) Golden rule equation: f ' δ = 0 G G Steady state equation: sf ( k ) = δk Assume population grows at n, i.e., ΔL/L = n. The evolvement of capital stock remains the same at the aggregate level: 8

9 ΔK = I δk The evolvement of the per-labor capital stock now becomes: K ΔK ΔL Δk = Δ = K 2 L L L I δk ΔL K = L L L = i δ k nk = sf ( k ) ( δ + n)k Note: here we apply the total differentiation formula. The steady state is determined by Δk=0 Therefore, at the steady state, sf ( k * ) = ( δ + n) k * Intuitively, the population growth serves in a similar role as the depreciation at the per capita level. Higher the population growth rate, the per capita capital stock would decrease. The following graph illustrates the point: One of the predictions of the Solow model is that a higher population growth rate would result in a lower level of per capita capital stock and output. The following graph presents some international evidence supporting this prediction. From the graph, the relationship between population growth rate and per capital GDP is clearly negatively correlated. 9

10 Discussions: (1) Note here that the Solow model suggests causality: A higher population growth rate a lower per capita output. However, the causality can be just opposite. Many economists would argue the opposite causality: A higher per capita output a lower population growth rate. There are several plausible stories to argue for the opposite causality. In poor countries, children sometimes serve as the saving for retirement. A higher income would reduce such the demand. Richer people would enjoy leisure more and hence less likely to have more children. The following graph gives the US data. It shows (1) people with higher incomes have lower numbers of kids, and (2) overtime people have less numbers of kids. 10

11 Technology Growth To introduce technology growth, we introduce a concept of efficiency labor, E. A higher E means that labor becomes more effective. The production function becomes: Y = F ( ) α 1 K, EL = K ( EL) α Instead of working with per laborer capital stock, we now work with perefficiency laborer capital stock. Define K Y k =, y =. Let the growth rate of E be: EL EL ΔE g =. E The evolvement of aggregate capital stock remains the same: ΔK = I δk The evolvement of the per-efficiency labor capital stock now becomes: K ΔK ΔE L + E ΔL Δk = Δ = K 2 EL EL ( EL) I δk K ΔE L + E ΔL = EL EL EL sy δk K ΔE ΔL = + EL EL E L = s = sf y δk k( g + n) ( k) ( δ + n + g)k 11

12 The steady state is determined by Δk =0 Therefore, at the steady state, sf ( k * ) = ( δ + n + g) k * The Golden Rule with population and technology growth In this case, the consumption per efficiency laborer is given by: c = y i (output investment) = y sy (output saving) = y (δ + n + g) k (at steady state, sy = (δ + n + g) k) = f(k) - (δ + n + g) k To obtain the optimal level of consumption, the first order condition is given by: f (k) - (δ + n + g) = 0 The Golden Rule level of capital stock is determined by: G f k = δ + n + or, MPK = δ + n + g ( ) g It is necessary to find out the growth rate at the steady state for each of the variables at aggregate level, per capita level, and per efficiency-labor level. Symbol steady-state growth rate capital per effective worker k = K/(E*L) 0 output per effective worker y = Y/(E*L) = f(k) 0 output per worker Y/L = y * E g capital per worker K/L = k * E g 12

13 total output Y = y * (E * L) g+n total capital K = k * (E * L) g+n If we want per worker output to grow when the economy is at the steady state, the only way is to have technology growth. Discussions: (1) To calculate the Golden Rule saving rate, two equations and two unknowns: G The Golden rule equation: f ' ( k ) = δ + n + g Steady state equation: sf ( k G ) = ( δ + n + g) k G Numerical example: In the US, it is typically assumed that (i) the capital stock is 2.5 times one year s GDP; (ii) Depreciation of capital is about 10% of GDP; and (iii) Capital income is about 30% of GDP. What is the Golden-rule saving rate? We have: k = 2.5y δk = 0.1y MPK*k = 0.3y The depreciation rate is given by: δ = 0.1y/2.5y = 0.04 And MPK is given by: MPK = 0.3y/k = 0.3y/2.5y = 0.12 Therefore, current MPK = The Golden rule level: MPK G = δ + n + g Given that n = 0.01, δ = 0.04, and g < 0.07 We have: MPK G < MPK = The current MPK is too high the current steady state capital stock k * : * G k = k What is the Golden Rule saving rate? We have: 0.3y/k = δ + n + g, or k(δ + n + g) = 0.3y Since k(δ + n + g) = sy, we must have: 13

14 G s = 0. 3 (2) Convergence: The Solow model suggests convergence to the steady state where the growth rate would tend to be the same. The following graph suggests the convergence of growth rates from different states within United States. (3) Solow model can explain only a small portion of the per capital difference across countries. Numeric example: Suppose U.S. and Mexico both have the Cobb-Douglas production function Y= K 1/2 L 1/2. Technology growth is zero in both countries. Given the following information, US Mexico n δ s per capita income: y us /y mexico = 4 14

15 What is the ratio of the two countries according to the Solow model? At the steady state, sy = (n+δ)k 1/ 2 sk = ( n + δ )k 1/ 2 sk = s /( n + δ ) y = s/(n+δ) Therefore, y y US Mexico = n n s s US US Mexico + δ = Mexico + δ = 1.79 However, we know that y us / y mexico = 4. So the Solow model can only explain a small portion of the ratio. It turns out that Solow model can only explain a small potion of the difference across countries, especially across developing and developed countries. (a) One possibility is that the two countries have different level of E. Consider the Solow model with technology growth. At the steady state, we have: sy = (n+δ+g)k In this case, please note that y = Y/(E*L), and k = K/(E*L). Assume g = 0, then we have: y y US Mexico US US US US Mexico = = 1.79 = = 4 YMexico YMexico / LMexico EUS E E Y Mexico US L L Mexico Y / L E E E Mexico US EMexico 1.79 Therefore, = = E 4 US In another word, a US worker is twice as productive as the Mexican worker (at the given level of capital stock). (b) Another possibility is to have different production function. For Mexico, y=k β. 15

16 s k β = (δ+n) k. So k 1-β = s/(δ+n). y = [s/(δ+n)] β/(1-β). y us /y mexico = 4.4/2.46 β/(1-β) = 4. Solve for β, we get β = So in Mexico, the share of capital could potentially be very small. Endogenous Growth Theory (1) One sector endogenous growth model: The basic idea: investment, especially investment in R&D would lead to more productivity. Let E = B*K/L this is to say that labor efficiency depends on the how much capital stock per worker has. A is a scaling factor. Given this, the total output function becomes: Y = K α (E*L) 1-α = K α (B*K/L* L) 1-α = B 1-α K = AK where A = B 1-α, a constant. Again, consider the evolvement of capital stock: ΔK = sy δk = sak δk = (sa δ)k The growth rates of the total output and the total capital stock are: ΔK/K = sa δ ΔY/Y = sa δ In the following graph, sa > δ, there will be no steady state, capital stock will continue to grow. 16

17 Graph for endogenous growth Increase in k = sak Decrease in k = δk This model suggests that no equilibrium can be ever achieved. The growth of a country would accelerate. The DOTCOM Bubble This graph shows the tech-heavy NASDAQ index s spectacular rise and fall during 1990s, from roughly 900 in , to 5048 on March 10, 2000, and then back to about 1000 in Right now NADQAQ index is at (Oct 4, 2010) 1,

18 During the DOTCOM bubble period in 1990 s, this type of new growth model, pioneered by economist Paul Romer from Stanford, has generated a substantial interest. TIME magazine said Paul Romer is one of the 25 most influential persons in the US. After the bust of the DOTCOM bubble, this theory is no longer as influential. (2) A two-sector model: One sector is the manufacturing firms, with exogenous growth as in the Solow model. The other sector is research universities, producing E. The model is characterized by: (i) Output is produced by manufacturing firms. Y = F [ ( ) ] α 1 K, 1 u EL = K (( 1 u) EL) α (ii) However, efficiency of labor is produced by research universities. g ( u) ΔE = E (iii) Evolvement of capital stock ΔK = sy δk where u is the fraction of the labor force in universities (and 1- u is the fraction of labor force in manufacturing). This model has some interesting features: (a) From previous discussion, at the steady state, the per worker output would increase at the rate of g(u). Therefore, this model suggests a way of a persistent economic growth at per worker level. (b) In the Solow model, the growth policy is a policy about the saving rate. Here, in addition to the saving rate, the percentage of people working in research universities also becomes a growth policy factor too. Higher u, the growth rate of E would be higher higher growth rate of per capita output. However, a higher level of u would lead to a lower level of Y. This is similar to the discussion about the Golden rule saving rate. (3) Schumpeter s Creative Destruction 18

19 Joseph Schumpeter suggests a different way of economic growth. (a) When a new product emerges, the firm often would have monopoly power of this product they would have monopoly profit. In fact, it is the monopoly profit that encourages the innovation of the new product. (b) Because of the innovation of the new product, the existing product would be lost in the competition. So they would be destructed. They often resort to the political process to block the innovation. (c) Many examples: Many computer companies fail, and new companies emerge. 19