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Lecture 7 Economic Growth It’s amazing how much we have achieved But huge difference across countries Country comparisons GDP http://www.google.com/publicdata?ds=wb-wdi&met=ny_gdp_mktp_cd&idim=country:USA&dl=en&hl=en&q=gdp GDP growth rates http://www.google.com/publicdata?ds=wb-wdi&met=ny_gdp_mktp_kd_zg&idim=country:USA&dl=en&hl=en&q=gdp+growth+of+us Growth and differences Nigeria is only 1/43 of the US. We study Why so much growth Why so much difference Robert Solow: 1924 - Won Nobel Prize in economics in 1979 for his contribution in the growth theory. Basic idea Previously we know output is mainly determined by Capital stock Labor We focus on capital stock. Basic idea Solow considers How capital stock increases How capital stock decreases Equilibrium is reached when: Increase of the capital stock = decrease of the capital stock The accumulation of capital stock Per capita production function Y F K , L K L 1 The per worker production function is: 1 Y K L y L L 1 K L K 1 k L L L Per capita production function. The marginal production of capital, MPK: Y K 1 1 MPK K L K L 1 k 1 y k MPK is obtained by taking the first derivatives from the aggregate production function, or from the per capita production function. Per capita production function Per capita production function At per capita level: y = c + i Per capita consumption is: c = (1 – s) y. Rearrange terms, we get: i = s*y=s*f(k). Output/investment graph Evolvement of capital stock Capital stock: Increases if investment. Decrease if depreciation. K t 1 K t 1 I t Each period, Amount of increase: It Amount of depreciation: δK If labor force does not change At per capita level k t 1 k t 1 it Rearrange this: kt it kt 1 sf (kt ) kt 1 Equilibrium: At the equilibrium, we must have: kt sf kt kt 1 0 kt kt 1 k * At the steady state level k*, we have: sf k k * * Equilibrium Discussions: Why steady state? If k > k*: * * From the graph, sf k k Depreciation > investment k level would decrease. If k < k*: * * From the graph, sf k k Depreciation < investment k level would increase. Discussions: an increase in saving rate Saving rate and per capita output A key prediction of the Solow model is that higher saving would be the cause of higher per capita output. Investment and per capita output Investment and per capita output Discussions: A higher level of saving would lead to a higher level of per capita output The most important growth policy is the policy of raising the saving rate. China experience: GDP growth rates http://www.google.com/publicdata?ds=wbwdi&met=ny_gdp_mktp_kd_zg&idim=country:CHN&dl=en&hl=en&q=china+gdp+grow th+rates Example: China’s gross national saving as a percentage of GDP Public policies that affect saving rates Public policies that may raise savings rates: Tax benefits for IRA, Roth IRA, 401K, 403B, and 529 raise private saving rate. Reducing budge deficit would raise the public saving and hence the total saving. Reducing trade deficit would raise the total both public and private saving. Reducing capital gains tax. Establishing social security and Medicare would reduce demand for precautionary saving. China’s saving rates Government Enterprises Households 28 China’s problem: saving is too high Various measures of reducing savings are apparently not successful. Expand the enrollment of higher education and raise the tuition for higher education. However, it creates wrong incentives – some parents now would save for higher education while others pay for higher educations. New enrollment is six times as much in 2010 than in 1999 New College Enrollments in China Enrollments (in 1,000) 7000 6000 5000 4000 3000 2000 1000 0 1985 1990 1995 2000 Year 2005 2010 Reducing saving in China Establishing social safety network Nationwide health insurance 1998 – urban employees 2003 – rural residents 2007 – urban residents (nonemployees) It helped reducing save rate but not much (increasing consumption by roughly 10%). Reducing saving rate in China This is important for US because of the large trade deficit between US and China. So far, nothing worked. Compromise between saving and consumption A higher saving rate higher per capita output in the future but a lower consumption rate. In the extreme case, a saving rate 100% no current consumption. The Golden Rule level of capital stock “Golden rule” the steady level consumption is the highest. At steady state, we have: sf k k * * The Golden Rule level of capital stock The steady state level of consumption: c 1 s y f k sf k f k k * * * * * * Maximizing c* to get the Golden rule level of consumption: c* * f ' k 0 * k The Golden Rule level of capital stock The Golden rule of capital stock is given by: MPK = δ The Golden Rule level of capital stock A numerical example Production function: y k 1 / 2 Depreciation rate: δ = 0.1 At the optimum: MPK 0.5k 1 / 2 k * 0.25 2 k* = 25 A numerical example k In the steady state: sf k s f k k * * s = 0.5 k * 1/ 2 * * Summarize: Two unknowns, saving rate s, and optimal level of capital stock k. Two equations: Golden rule equation: f ' kG 0 G G sf k k Steady state equations: Population growth Assume population grows at n, ΔL/L = n. The evolvement of capital stock remains at: ΔK = I – δK The evolvement of per-labor capital stock is more complicated: Evolvement of per-labor capital stock L K K k K 2 L L L I K L K L L L i k nk sf k n k Population growth The steady state is determined by: Δk=0 Therefore, at the steady state, sf k * n k * Population growth Population growth Prediction: a higher population growth rate, a lower level of per capita capital stock and output. Population growth Population growth Discussions Causality: here it is suggested that a higher population growth rate a lower per capita output. It is possible that the reverse causality is true: a higher per capita output a lower population growth Discussions Reasons for reverse causality: In poor countries, children sometimes serve as the saving for retirement. A higher income would reduce such demand. Richer people would enjoy leisure more and hence less likely to have more children. US data: income and number of children Technology To introduce technology growth, we introduce a concept of efficiency labor, E. A higher E means that labor becomes more effective. Production function now becomes: Y F K , EL K EL 1 Technology We now work with per-efficiency laborer capital stock: K Y k , y EL EL Define: Let the growth rate of E be g: E g E Technology The evolvement of aggregate capital stock remains the same: ΔK = I – δK The evolvement of the little k: E L E L K K k K EL2 EL EL I K K E L E L EL EL EL sY K K E L EL EL E L s y k k g n sf k n g k Technology K K 1 k K EL EL EL 1 EL Technology At the steady state Δk =0 sf k * n g k * The steady state The Golden rule c =y–i = y – sy (output – investment) (output – saving) (at steady state, sy = (δ + n + g) k) c = y – (δ + n + g) k = f(k) - (δ + n + g) k The Golden rule The first order condition: f’(k) - (δ + n + g) = 0 The Golden Rule level capital stock: n g f' k G Or: MPK = δ + n + g Summary Symbol capital per effective worker output per effective worker output per worker capital per worker total output total capital k = K/(E*L) y = Y/(E*L) = f(k) Y/L = y * E K/L = k * E Y = y * (E * L) K = k * (E * L) steady-state growth rate 0 0 g g g+n g+n Discussions: To calculate the Golden Rule saving rate, two equations and two unknowns: The Golden rule equation: f ' kG n g Steady state equation: sf k G n g k G A numerical example In the US, we have: k = 2.5y δk = 0.1y MPK*k = 0.3y depreciation income for the owners of capital stock What is the Golden-rule saving rate? A numerical example The depreciation rate: δ = 0.1y/k=0.1/2.5y = 0.04 MPK = 0.3y/k = 0.3y/2.5y = 0.12 The Golden rule level: MPK G n g Given n = 0.01, δ = 0.04, and g < 0.07 We have: MPK G MPK 0.12 A numerical example Current MPK is too high, suggesting We should invest more Our saving rate is probably too low. A numerical example: Now to obtain the Golden rule saving rate: The Golden-rule: MPK = δ + n + g 0.3y/k = δ + n + g k(δ + n + g) = 0.3y A numerical example At the steady state: sy = k(δ + n + g) the golden rule saving rate is at: s = k(δ + n + g) / y Therefore: s G 0 .3 A numerical example The optimal (the Golden rule) saving rate for US is roughly 30% Current US saving rate is: http://www.bea.gov/briefrm/saving.htm US net national savings rate US net private savings rate Current US saving Why increase? US government savings Discussions Convergence: the Solow model suggests convergence to the steady state where the growth rate would tend to be the same. Evidence from states within US support this. Evidence across countries not necessarily true. Discussions Solow model can only explain a small portion of the variations across countries. Consider US and Mexico. Per capita income: US/Mexico = 4 Numerical example (1) U.S. and Mexico both have the Cobb-Douglas production function Y= K1/2L1/2. (2) Suppose technology growth is zero in both countries. (3) Other information: US Mexico 0.01 0.025 n 0.04 0.04 δ 0.22 0.16 s per capita income: yus/ymexico= 4 What is the ratio of the two countries according to the Solow model? A numerical example At the steady state: sy = (n+δ)k 1/ 2 sk n k sk 1/ 2 s / n y = s/(n+δ) US-Mexico yUS y Mexico sUS 0.22 nUS 0.01 0.04 1.79 s Mexico 0.16 0.025 0.04 nMexico US-Mexico Therefore, according to the Solow model, the ratio between US and Mexico is 1.79, much smaller than the actual GDP per capita, which is 4. So the Solow model can only explains a small portion of the ratio. What is the potential problem? US-Mexico A potentially different efficiency E. A different level of E Consider the Solow model with technology growth sy = (n+δ+g)k yUS y Mexico YUS EUS LUS YUS / LUS E Mexico E Mexico 1.79 4 YMexico YMexico / LMexico EUS EUS E Mexico LMexico A different level of E A Mexico worker is 45% of efficiency of a US worker’s level. E Mexico 1.79 .4475 EUS 4 Endogenous growth theory Basic idea: investment, especially investment in R&D, would lead to higher productivity. Suppose E = B* K/L Y K E L 1 K B K / L L 1 B1 K AK Endogenous growth model Consider the evolvement of capital stock: ΔK = sY – δK = sAK – δK = (sA – δ)K Increase of capital stock: sAK Decrease of capital stock: δK Endogenous growth model Increase in k = sk Decrease in k = δk No steady state equilibrium If sA > δ No steady state, capital stock will continue to rise forever. The DOTCOM Bubble in 1990s The DOTCOM Bubble The spectacular rise and fall of the NASDAQ (techheavy): In 1995 – NASDAQ at 900 March 10, 2000 – NASDAQ rose to 5,048 Oct 4, 2002 – NASDAQ down to 815 Oct 4, 2010 – NASDAQ 1,975 Individual stock – example: Microstrategy http://www.google.com/finance?q=mstr Michael Saylor – lost 6 billion dollars in one single day. http://www.slate.com/id/77774/ Broadcast.com and Facebook.com Broadcast.com – in 1999, $50 million revenue, 330 employees. Facebook.com – in 2010, $1 billion revenues, 1500 employees. Broadcast.com was sold to yahoo.com at the peak of the internet bubble at US$ 5.9 billion. One third of employees are millionaires on paper. If evaluation based on broadcast.com, Facebook.com would be worth 118 billion. Currently Facebook.com is evaluated at US$ 11 billion. Example Whole market becomes crazy during the internet boom. Market evaluation of internet companies is completely wrong. Could worth 25 billion (if in 1999) Mark Zuckerberg But only worth 7 billion today Actually worth 2.5 billion Mark Cuban Would only worth 200 million if sold today Broadcast.com and Facebook.com Mark Cuban purchased NBA Dallas Mavericks for $285 million. He wouldn’t be able to do that if based on the current evaluation. No steady-state equilibrium Capital stock keeps rising no steady state equilibrium. During the DOTCOM bubble in 1990’s, it is widely believed that growth is unlimited. The key person is Paul Romer, a Stanford economist, one of TIME Magazine’s 25 most influential economists in 1997. http://www.time.com/time/magazin e/article/0,9171,98620610,00.html No longer – burst of the DOTCOM Bubble Summary Amount of per effective capital stock increase due to investment: sk Amount of per effective capital stock decrease due to depreciation, population growth, and technology growth. n g k Equilibrium condition: s k n g k The Solow model Steady state equilibrium: decrease of k = increase of k Decrease of k: (δ+g+n)k Increase of k: sy k* Summary Golden rule: the steady state equilibrium where the consumption is maximized. Golden rule condition: MPK k 1 n g Summary Policy implications: The most important long-run economic policy is to encourage both public and private savings. This is particularly important for the United States since our saving rate is too low.