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ECONOMIC GROWTH COURSE. Main questions: Why are some countries so poor while other countries are rich in terms of income per capita? Why are some countries growing faster than others? REVISED READING LIST FOR ECONOMIC GROWTH COURSE: FALL 2012 2012-11-07. READINGS: Auerbach, A., and L. Kotlikoff (AK), “Macroeconomics-An integrated Approach”, 2nd ed., MIT Press. Chapters: 1-3, 5, 6, 12, and 15. David Weil, “Economic Growth”, 3nd edition, Pearson. All (or most) chapters. Journal articles available on this course page. TOPICS: 0. Review of National Income Accounting and PPP-adjusted GDP AK, basics of ch. 5. Weil, ch. 1, appendix. 1.Production Theory and Growth Accounting: AK: ch. 1. 2. Factor (capital- and labor) markets: AK, ch. 3: pages 53-60. 3. How to calculate growth rates: Weil, ch. 1, pages 30-31. 4. The Solow Model Background: Weil, ch. 1-2 Weil, ch. 3: The Solow Model without population growth and technological progress. Weil, ch. 4.2: The Solow Model with population growth but without technological progress. Weil, skim appendix of ch. 8: Incorporating technological progress in the Solow model Extra-credit exercise: Simulation exercise in Excel of the Solow model without continuous technological progress. See exercise-handout. Deadline will be announced later. 5. On Endogenous growth models Required reading: Handout (see below) 6. Empirics of Economic Growth Required reading: Journal articles available on the course-web: J. Persson, Convergence across the Swedish Counties, 1911-1993, European Economic Review 41 (1997), 1835-1852. R.J Barro, Human Capital and Growth in cross-country growth regressions, Swedish Economic Policy Review, vol. 6, no. 2, autumn 1999. RJ Barro, 2012, … Students not previously exposed to econometrics should read Introduction to econometrics (pdf-file) on the course web. Students should do econometric exercise and write econometric report. This exercise is mandatory and should be completed in groups of 2-4 people. 7. The overlapping generation model (OLG) for a closed economy with no government A. Optimal consumption, and saving by a young person: AK: chapter 2. B. More in-depth analysis of the choices of the household: Regarding saving the household may receive income in the last period of life as well, and may choose the number of hours it want to work in both periods of life. Required reading: Handout (see below). Extra-credit exercise: Saving and labor-supply-exercise to be handed in for extra credit. C. Completing the closed-economy OLG-model: AK, Ch. 3 D. Allowing for population growth in this model: AK, appendix to ch.3. 8. Income distribution and Economic Growth. R J Barro, Inequality and Growth in a Panel of Countries, Journal of Economic Growth, 5: 532 (March 2000); (chapter 13 in Weil) 9. Factor mobility in the OLG-model: An analysis of globalization: AK, ch. 12,basics of ch. 5. 10. A government sector in the OLG-model: AK, chapter 6; 11. Precautionary saving in the OLG-model: AK, ch.15 (pp. 411-423): Handout below; 12. Political economy models, chapter from other textbook (Nicholson). 13. Fertility Choice and The Income level. Reading: handout below. Weil, ch. 4-5. During the latter part of the course we study more applied topics: The topics chosen are based on the chapters in Weil. The topics are covered by in-class student presentations. Example of topics: 14. Economic growth and openness. Chapter 11 in Weil. 15. Economic growth and the government. Chapter 12 in Weil. 16. Economic growth and Culture. Chapter 14 in Weil. 17. Economic growth and Geography. Chapter 15 in Weil. 18. Resources and the Environment at the Global Level. Ch. 16 in Weil. Etc. Review: Read by yourselves: NATIONAL INCOME ACCOUNTING MEASURING PRODUCTION AND INCOME TOPIC 0. Review of National Income Accounting and PPP-adjusted GDP Readings: AK, basics of ch. 5. Weil, ch. 1, appendix. Main lesson: The value of the production of a country (GDP) approximately equals the income its citizens. Gross domestic product (GDP) is the market value of all final goods and services produced within an economy during a year. GDP can be measured in 3 ways: Method 1. through expenditures on final goods and services. Method 2. through income (wages and capital income). Method 3. through production. GDP is calculated by adding up value added in all sectors of production. Value added = total revenue – expenditures on inputs other than capital and labor. These inputs are intermediate goods (insatsvaror). Value added = förädlingsvärde Comparing GDP over time Nominal GDP = GDP in current prices real GDP = GDP in constant prices The value of final goods and services measured at current prices is called nominal GDP. It can change over time either because more goods and services are produced or because there is a change in the prices of these goods and services. We calculate real GDP to see whether the country is producing more or less goods and services over time. Compute GDP in current prices according to method 1. Assume two goods in the economy: Q1 , Q 2 1 1 2 2 Nominal GDP in 2000: GDP2000 P2000 Q2000 P2000 Q2000 Real GDP 2000 (in 2000 year prices) = nominal GDP 2000. 1 1 2 2 Real GDP 2001 (in 2000 year prices) = P2000 Q2001 P2000 Q2001 If real GDP 2001 > real GDP 2000, then the economy produced more goods and services in 2001. Often in news papers and statistical reports you get data on GDP in current prices and data on a price index. How do we calculate GDP in constant prices? Real GDP in 2005 (in 2000 year prices) = nominal GDP in 2005/ price index in 2005 Year Nominal GDP= Price index = Real GDP GDP in current GDP-deflator: P in 2000 year Prices: P*Y prices: (P*Y)/P=Y 2000 2500 billion kr 1.00 2500 2001 2600 billion kr 1.02 2600/1.02=2549 2002 2700 billion kr 1.04 2700/1.04=2596 2003 2800 1.07 2800/1.07=2616 2004 2900 1.10 2900/1.10=2636 2005 3000 1.12 3000/1.12=2678 2006 3100 1.14 3100/1.14=2719 Note that Q can not be directly observed as there are many goods and services in the economy. Nominal GDP increases by (3100-2500)/2500=0.24; that is by 24 percent The price level increases by (1.14-1)/1=0.14; that is by 14 percent between 2000 and 2006. The real GDP increases by (2719-2500)/2500=0.088; that is, by 8.8 percent. In macromodels: real GDP (=GDP in constant prices) is denoted by Y. This means that P is assumed to equal 1 in the base year as 1*Y=Y Nominal GDP is denoted by P*Y. It may be useful to think that there is one good in the economy; e.g. potatoes. Then Y is quantity of potatoes in terms of kilogrammes, and P is the price of a kg potatoes. More rules for computing GDP: 2) used goods are not includes in the calculation of GDP. 3) If newly produced final goods is stored, it is inventory investment which is part of private investment. When the goods are finally sold, they are considered used goods. 4) Some goods are not sold in the market place and do therefore not have market prices. We must use their imputed value as an estimate of their value. For example, home ownership and government services. Price indexes provide an overall measure of the price level in the economy. We have to choose a base year. CPI = Consumer price index. In Swedish: KPI = konsumentprisindex Assume CPI (2000) = 100. Assume 2 goods in the economy: CPI (2001) 1 1 2 2 P2001 Q2000 P2001 Q2000 1 1 2 2 P2000 Q2000 P2000 Q2000 When we use the original basket of consumption to calculate the price index, it is a Laspereys index. If CPI (2001) > CPI (2000) then the overall price level has increased. Alternatively: CPI (2001) 1 1 2 2 P2001 Q2001 P2001 Q2001 1 1 2 2 P2000 Q2001 P2000 Q2001 When we use the current basket of consumption to calculate the price index, it is a Paasche index. CPI versus the GDP deflator (BNP deflatorn) The GDP deflator measures the development of the prices of all goods and services produced. The CPI measures the development of prices only of the goods and services bought by consumers, which includes imported goods. GDP-deflator (2001) = Nom. GPD in 2001/real GDP in 2001 in 2000 year prices GPD deflator P(2001) * Y (2001) / P(2000) * Y (2001) The GDP-deflator is a Paasche index. In macroeconomic models real GDP is denoted by Y (instead by Q that is used for quantity in microeconomics) and the price level is denoted by P. Measuring GDP through expenditures on final goods and services (method 1) in the real world: that is, with a public sector and with international trade. Availability of newly produced goods And services GDP = 3000 billions kronor Imports = 1500 billions kronor Sum: 4500 billions kronor Use of newly produced goods and services Private consumption = 1500 Private investment = 400 Public consumption = 700 Public investment =300 Exports = 1600 Sum: 4500 In other words, GDP+imports = private consumption+private investment + Public consumption and public investment + exports Rearranging: GDP = private consumption+private investment + Public consumption and public investment + exports - imports Expressing this NATIONAL INCOME IDENTITY in real terms (in constant prices): Real GDP (Y) = real private consumption (C)+real private investment (I) + Real Public consumption (GC) and public investment (GI) + exports – imports (NX) Thus, Real GDP (Y) = C+ I+ G+NX Where G = GC+ GI. Note that P*Y= GDP in current prices. C = real expenditures of households on final goods and services. Goods are sometimes categorized into nondurable and durable goods. I = real expenditures on new machines, new buildings, and inventory build-up by firms. I is gross private investment. G: real government spending on (/purchases of) final goods and services. Often G is split up into government consumption (GC) and government investment (GI). Examples of GC are expenditures on teachers’ and doctors’ salaries. Examples of GI is expenditures on new roads and government buildings. Note: Government transfers to households such as unemployment benefits etc. are not included in G: Total government expenditures = G + government transfers to households (unemployments benefits, transfers to poor, to kids, to retired people) and to firms. NX = real net exports = trade balance = value of exports of final goods and services – value of imports of final goods and services. Thus, NX is net expenditure from abroad on our goods and services. One way to think about real variables. Assume that only one good is produced in the economy; e.g. corn. Then production, Y, is measured in tonnes of corn. Some of this corn is used for private consumption (C), some for private investment (it is planted in the ground to yield production next year) (I), some for government consumption and investment (G), and some of the production is shipped abroad (Export) and some corn is imported (Imports). (NX=exports-imports). Ex.: Y=C+I+G+NX: 100 = 50 + 20 + 20 + 10 Other measures of income: Gross national product (GNP) = GDP – (wages and capital income of Swedish workers and firms operating abroad – wages and capital income of foreign workers and firms operating in Sweden). By Swedish worker I mean a Swedish resident. Gross National Product (GNP) = GDP + net factor income from abroad (NFI) GNP counts all final output produced by domestically owned inputs (workers and capital), no matter where those inputs are situated in the world. GDP counts all final output produced within the country regardless of who owns the inputs (foreign or domestic citizens) involved. GNP is an income measure for the residents in a country while GDP is a production measure. In a closed economy: GNP = GDP In Stockholm: GDP > GNP as many workers compute to Stockholm but live in other regions e.g. Uppsala. That is, they contribute to GDP in Stockholm but their incomes are not part of the Stockholm GNP. GNP (Stockholm) = GDP (Stockholm) – wages of workers that live in other regions than Stockholm. Another concept is disposable GNP (DGNP) DGNP = GNP + (transfers from foreign countries – transfers to foreign countries) = GNP + net transfers from abroad (NFTr). For Sweden: DGNP < GNP as net transfers from abroad are negative. In a closed economy macro model: Y = C+I+G Then Y=GDP=GNP=DGDP. DGNP is the income used by government (T) and households, Y-T, which is household disposable income. (Households earn both labor and capital income.) Thus, T + (Y-T) = Y = C+I+G+NX. T = net taxes = tax payments (indirect and direct taxes plus social security contributions) – transfers to households and firms. Typical simplifying assumptions in an open economy macro model: The basic identity in national income accounting: Y = C+I+G+NX Often it is assumed that Net Foreign Income (NFI) =net transfers from abroad (NFTr)=0 Then Y=GDP=GNP=DGDP. Comparing Standard-of-Living across Countries TOPIC 0. PPP-adjusted GDP Readings: Weil, ch. 1, appendix. 2 methods to determine which country has the highest material standard: the amount of goods and services available: 1. The Exchange Rate Method uses the current exchange rate to convert GDP in domestic currency to GDP expressed in dollars: Formula for calculating GDP in domestic currency when assuming 2 goods in the economy: GDP (kr) = Pnt(kr)*Qnt + Pt(kr)*Qt Pnt = price of the non-tradable good, e.g. hair-cuts, housing services, etc Qnt = quantity of the non-tradable good, Pt= price tradable good, Qt=quantity of the tradable good GDP in dollars: GDP ($) = e ($/kronor)*GDP(kr) = e ($/kronor)*Pnt(kr)*Qnt + e ($/kronor)*Pt(kr)*Qt e ($/kronor) is the nominal exchange rate: In 2007: e ($/kronor) =1/6. 2. Purchasing Power Parity (PPP) Method controls for the fact that prices differ across countries. The method replaces domestic prices with average prices across countries (in $). GDP in dollars according to the PPP-method: GDP ($) = e ($/kronor)*GDP(kr) = Average Pnt ($)*Qnt + Average Pt ($)*Qt We want to use the same prices because we want our measure of income to reflect the quantity of goods and services that are available in one country during a year. Using different prices for different countries distorts the picture when we want a measure of material standard. If the law of one price holds for tradable goods, then average Pt ($) = e ($/kronor)*Pt(kr) That is, the law of one price says that the price of a tradable good expressed in the same currency should be the same in all countries. We expect the law of one price to hold if transportation costs and differences in VAT are zero across countries because the market mechanism tends to equalize prices across countries: If there are price differences (in the same currency) then it is profitable to buy the good where the price is low, and ship it to the country where the price is high. This leads to higher demand in the country in which the price was initially low which tends to put an upward pressure on the price in this country. Moreover, a higher supply in the country where the price was initially high tends to put a downward pressure on the price in this country. Thus, the market mechanism tends to equalize prices across countries for tradable goods. The law of one price does not hold for non-tradable goods and services: In a poor country, the price of a non-tradable good tends to be lower relative the price of a non-tradable good in a richer country because the poorer country has lower labor costs. Ex: e($/Etiopian currency)* Pnt(Etiopian currency) < Average Pnt($) PPP-adjusted GDP ($) for Etiopia > Not PPP-adjusted GDP ($) for Etiopia= GDP ($) according to the exchange rate method. Thus, the exchange rate method understates GDP ($) in poor countries. Because domestic prices (in $) are lower in poor countries. In news reports you hear that in this country they earn 400 per capita and year. I have often thought: How can they survive? The explanation is that this income figure is not PPPadjusted. Purchasing Power Parity means that prices expressed in the same currency are the same across countries and regions. PPP does empirically not hold for non-tradable goods; they tend to be lower in poor countries. Application on Regions within a country: If nominal income per capita in the Stockholm region is twice as high as the nominal income per capita in the Karlstad region, the purchasing power of income per capita in the Stockholm region might be less than twice as high due to a higher price level; e.g. on non-tradable goods and services such as housing. Lecture: Production Theory: Factor Markets: Capital and Labor Markets TOPIC 1.Production Theory and Growth Accounting: Readings: AK: ch. 1. TOPIC 2. Factor (capital- and labor) markets: Readings: AK, ch. 3: pages 53-60. Assume an aggregate production function of Cobb-Douglas type: real GDPt Yt F ( At , Kt , Lt ) At G( Kt , Lt ) At Kt L1t , where 0 < Real GDP = GDP in constant prices. Ex.: 0.5 , easy to calculate with… Y A K 0.5 L0.5 < 1. If there is one good in the economy, then Y is the quantity of this good. In kilograms, or liters depending on what the good is. K = aggregate physical capital (machines and buildings) Should be measured by machine-hours and by hours buildings are used per year. But in reality K is measured by the real dollar value of the aggregate physical capital. It is then implicitly assumed that the real dollar value of K is proportional to the number of machine hours and hours buildings are used. L = aggregate labor input is measured by total hours worked (or by number of workers if every worker works the same number of hours). A = totalfactorproductivity, captures the effect on Y of all factors apart from K and L that impact Y. For example, technological progress (innovations) or increased education of workers increases Y at given levels of K and L. Higher energy prices should decrease the use of energy and thereby also A and Y at given levels of K and L. Empirically is estimated to be around 1/3, which is the typical value of the share of capital income of national income. The aggregate production function: Assume that K and A are constant: A=1, K=9, and =0.5. Y A0 K00.5 L0.5 1 90.5 L0.5 3 L0.5 L Y 3 L0.5 MPL Y Y1 Y0 Y/L L L 1 L0 0 0 0 1 3 3 3 2 4.2 4.2-3=1.2 4.2/2=2.1 3 5.2 1 5.2/3=1.7 5 6.7 (6.7-5.2)/2=0.75 6.7/5=1.34 8 8.5 (8.5-6.7)/3=0.6 8.5/8=1.1 From the table we see that: When L increases, Y increases but at a diminishing rate; because the capital-labor ratio decreases when L increases. Each worker gets less capital to work with. *MPL and APL(=Y/L) falls when L increases and K and A are constant. MPL is below APL (=labor productivity=Y/L). Plotting the production function for A=1, K=9 and alpha = 0.5: 14 12 10 8 Y Serie1 6 4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 L MPL is the slope of the production function. 1.6 1.4 1.2 MPL 1 Serie1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 L If the stock of physical capital increases: Y A0 K00.5 L0.5 1160.5 L0.5 4 L0.5 Parameters: A=1, K=16, A=1, K=16, A=1, K=16, =0.5. =0.5. =0.5. 0.5 L Y Y1 Y0 Y/L Y 4 L MPL L L 1 L0 0 0 1 4 4 4 2 5.6 1.6 5.6/2=2.8 3 6.9 1.3 6.9/3=2.3 5 8.9 1 8.9/5=1.8 8 11.3 0.8 11.3/8=1.4 * MPL and APL (=Y/L) increases when K (or A) increases and L is constant. This is shown by comparing the 2 tables above. If A Y and MPL and APL at a given L (and K): Assume that A=1 and A=2 , K=9, and =0.5. A=1 (curve below) and A=2 (curve above) 30 25 Y 20 Serie1 15 Serie2 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 L A=1 (curve below), A=2 (curve above) 3.5 3 MPL 2.5 2 Serie1 1.5 Serie2 1 0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 L Summary: MPL (= Y ) and Y/L decrease when L increases and K and A are constant. L MPL and Y/L increase when K (or A) increases and L is constant. By symmetry, the same holds for MPK and K: that is, MPK (= Y ) and Y/K decrease when K increases and L and A are constant. K MPK and APK(=Y/K) increase when L (or A) increases and K is constant. Review of exponents Definition: x n x x ... x . E.g. 22 2 2 4 n terms m x n Ex.: x2 x3 x5 Rule 1: x n x m Rule 2: Rule 3: Rule 4: Y L x n 1 xn xn m xm x0 1 A K 0.5 L0.5 L Or more generally, Y A K L1 L Ex.: xn L x 4 and x2 x x 1 x4 x 2 x 23 x 1 1 x x3 Ex.: 20 1 Ex.: A K 0.5 L0.5 L1 A K 0.5 L0.51 A K 0.5 L0.5 A K A K L L A K L L 1 1 K A L Production per worker, Y/L, increases if technology improves (A) or if every worker gets more capital (K/L) Deriving the mathematical expression for MPL: Review of the derivative: (1) If y x 2 (2) If y b x 2 where b is a constant (3) More generally, y b x n MPL then then then dy 2 x 21 2 x dx dy 2 b x 21 2 b x dx dy n b x n 1 dx dY (1 ) A0 K0 (1 ) A0 K 0 L dL L Thus, MPL = (1-alpha)*(Y/L). That is, MPL is always lower than Y/L 2. Y A K L1 exhibits constant returns to scale. If you e.g. increase each factor of production by 10 percent, then production also increases by 10 percent. If production increases by more, increasing returns to scale (IRS), and if production increases by less, decreasing returns to scale (DRS). CRS imply constant long-run average costs, IRS imply decreasing long-run average costs and DRS imply increasing long-run average costs when production increases. THE DEMAND OF K AND L IN THE LONG RUN BY FIRMS TOPIC 2. Factor (capital- and labor) markets: Readings: AK, ch. 3: pages 53-60. In macro models households/individuals supply L and K. Firms demand L and K. Assumptions: There exist many identical firms that produce an identical good. Perfect competition is assumed Firms are price-takers and they are assumed to maximize profits. The problem of the representative firm is to choose K and L and thereby output (Y) so that profits are maximized. Assuming two factors of production (K, L): Profits ($) = Total revenue ($) – capital cost ($) – labor costs ($) Profits ($) = P Y – R K – W L where Y A K L1 , P=price of the good, Y = quantity of the good, R= rental price of one unit of capital per period of time, W= nominal wage per worker per period of time. Problem of the firm: Choose K and L (and thereby output) to maximize profits: Profits ($) = P A K L1 – R K – W L Assuming perfect competition implies that the individual firm cannot influence prices: P, R and W are exogenous from the point of view of the firm. We also assume that A and are exogenous from the point of view of the firm as they are assumed to be given by the technology. To maximize profits, the firm should choose K and L so that: R P MPK and W P MPL or R MPK P and W MPL P R Y A MPK P K K / L 1 W Y K and MPL (1 ) (1 ) A P L L These 2 conditions for profit-maximization should hold simultaneously: If we combine them we find the profit-maximizing firm’s capital-labor ratio, K/L, which is independent of the level of production. In other words, when assuming a Cobb-Douglas production function the optimal (=cost-minimizing) capital-labor ratio is identical for a low level and for a high level of Y. Divide by W/P on both sides of the first condition: R / P (Y / K ) (Y / K ) Y 1 L L W /P (W / P ) (1 ) (Y / L) K (1 ) Y (1 ) K R L W (1 ) K K R L W (1 ) K L demand W (1 ) R Thus, a higher Wage relative to the Rental price of capital makes it optimal to use more capital relative to labor at a given level of production. You may remember the concept of isoquants. Determining Equilibrium Factor Prices: They are found where Supply = Demand Assume that the supply of K and L is fixed: K , L demand W / P (1 ) K W K K L L (1 ) R R/P L Thus, a higher capital labor ratio increases the equilbrium real wage relative to the equilibrium real rental rate. Example: Assume: Y A K L1 , and A=1, = 0.5 , K =9, L =9: L supply increases from 9 to 16 2.5 MPK, R/P 2 1.5 Serie1 1 Serie2 0.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 K (Kbar = 9) L supply increases from 9 to 16 1.6 1.4 MPL, W/P 1.2 1 0.8 Serie1 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 L (Lbar = 9 or 16) R / P MPK 0.5 ( K / L )0.5 0.5 (9 / 9) 0.5 0.5 W / P MPL 0.5 ( K / L )0.5 0.5 (9 / 9)0.5 0.5 Assume now that L =16 due to labor immigration R / P MPK 0.5 ( K / L )0.5 0.5 (9 /16)0.5 0.6667 W / P MPL 0.5 ( K / L )0.5 0.5 (9 /16)0.5 0.375 15 16 17 Thus, If L (due to labor immigration) and K constant R / P MPK and W/P=MPL . Domestic workers loose in terms of W/P from labor immigration, whereas domestic capital owners gain as R / P MPK . 2 other examples: (2) If L (due to black death) and K constant R / P MPK , W/P=MPL Workers gain and capital owners loose income. (3) If K and L constant R / P MPK and W/P Workers gain and capital owners loose. Conclusion: ( K / L ) W/P , R / P MPK Do Problem 1 on the Labor Market in the document: Exercises based on lecture notes1.doc The distribution of income (Y) between workers and capital-owners. Assume no government sector. Pr ofits($) P Y R K W L Assuming perfect competition in the rental market for capital: Pr ofits($) P Y (r ) P K W L , where r=real return to capital Note: PY=GDP=GNP, and (r ) P K = nominal capital income Real Profits = where Pr ofits($) W Y (r ) K L P P (r ) K = real capital income, W/P= real wage Assuming perfect competition in the goods market means that the profits are zero and that firms maximizes profits by choosing K and L so that W/P=MPL and R / P (r ) MPK . Y MPK K MPL L Y Y Y K (1 ) L Y Y (1 ) Y 0 K L Y Y (1 ) Y Thus, the share of GDP (net of taxes) that goes to capital owners is . The share of GDP (net of taxes) that goes to workers is 1- . We have data on labor and capital income. Thereby, we get an estimate of , which is around 1/3 for both developing and developed countries. Why? In poor (rich) countries: K is low (high), r is high (low), L is high (low) and W/P is low (high) In poor countries r is high and W/P is low because of low K/L. If 0 : Y=r*K+(W/P)*L, Y, K, (W/P) are in real terms; eg. In kilo of potatoes or in constant dollars. r is real interest rate. For example in r=0,03. This means that rK is in kg potatoes. rK is capital income in kg of potatoes and (W/P)*L is labor income in kg of potatoes. Equation in nominal terms (in kronor): PY=r*PK+W*L, P is the price of a kg of potatoes in kronor. r*PK= capital income in kr, and WL= labor income are in kronor. Real wage and labor productivity: When perfect competition: MPL (1 )*Y / L W / P If the share of income (Y) that goes to workers: (1-alpha) is constant, Then the real wage grows at the same rate that labor productivity, Y/L, grows. What nominal wage demands, keep the income distribution between capital owners and workers unchanged? Trade union strategy: “We need to forecast future development of labor productivity and inflation, when we make our wage demands”: W/P = MPL=(1-alfa)* (Y/L), when assuming perfect competition and a cobb-Douglas pf. This means:ΔW/W - ΔP/P = Δ(1-alfa)/(1-alfa) + Δ(Y/L)/(Y/L) If the distribution of income between firm owners and workers are constant: Δ(1-alfa)/(1-alfa) = 0. Thus: ΔW/W = Δ(Y/L)/(Y/L) + ΔP/P Do Problem 6 on The Supply Side of the Economy: Aggregate Production and Factor Markets in the document: Exercises based on lecture notes1.doc LECTURE: TOPIC 3. How to calculate growth rates: Readings: Weil, ch. 1, pages 30-31. How to calculate growth rates: Weil, ch. 1. Math: Growth rate = Percentage Change y y y , e.g. r1 = 0.02, that is, 2 % . 1 0 r1 y y0 where y 0 = income per capita year 0, y1 = income per capita year 1. r1 = growth rate/percentage change between year 0 and year1. y1 y0 r1 y0 y1 r1 y0 y0 y0 (1 r1 ) Analogously: y2 y1 (1 r2 ) , y3 y2 (1 r3 ) y3 y0 (1 r1 ) (1 r2 ) (1 r3 ) y1 y2 At a constant yearly percentage change (growth rate) income year 3 is: y3 y0 (1 r ) (1 r ) (1 r ) y0 (1 r )3 where r = constant yearly growth rate/percentage change. After t years and a constant growth rate income per capita equals: yt y0 (1 r )t , where t = number of years. Exercise: If GDP per capita (in 1995 prices) in 1995 and in 2000 was 194 and 222 thousands, what is the average annual growth rate during this 5-year period? Graphical representation of the exponential function: yt y0 (1 r )t . Let y0 1 and r = 0.03: yt (1 0.03)t 4.5 4 3.5 3 2.5 y Serie1 2 1.5 1 0.5 0 0 10 20 30 40 time If r increases, steeper slope. If y0 increases, the curve shifts upwards. Students do not have to know logarithms: 50 Alternative graphical representation of the function: yt y0 (1 r )t t ln( yt ) = ln( y 0 ) + ln( (1 r ) ) (1 r )t ) ln( y 0 ) + t ln(1 r ) ln yt ln y0 ln(1 r) t ln( yt ) = ln( y0 ln( yt ) = int ercept This is the equation for a straight line: y = a + b x If r is a small number < 0.1 ln(1 r) r slope coefficient ln yt ln y0 r t The logarithm function (r=0.03) 1.6 1.4 1.2 ln y 1 0.8 Serie1 0.6 0.4 0.2 0 0 10 20 30 40 time Formula: yt y0 (1 r ) How many years does it take to double y at different growth rates? t 2 y0 y0 (1 r )t 2 (1 r )t t ln(2) = ln( (1 r ) ) ln(2) = ln(1 r ) t r t ln(2)/r t t ln(2)/r If r = 0.05 t 14 years. If r = 0.015 t 46 years. 50 GROWTH ACCOUNTING (TILLVÄXTBOKFÖRING) TOPIC 1. Growth Accounting: Readings: AK: ch. 1. Expressing levels into growth rates; that is, into percentage changes: y x z . y x z Rule 1. If y(t) = x(t)*z(t), then Where does this approximate formula come from? dy = (dy/dx)*dx + (dy/dz)*dz= z*dx + x*dz. dy/y=(1/y)*(z*dx+x*dz)=dx/x+dz/z Formula can also be derived by taking logs of the equation. Ex1.: Total Revenue (TR) = Price(P)*Quantity(Q). If P increases by 10 % and (Q thereby decreases by 5%, then TR increases by 5 %. Ex.1: real wage = MPL = (1-alpha)*Y/L: The percentage change of real wage = percentage change of (1-alpha) + percentage change of Y/L. Note: (1-alpha) = share of labor income Rule 2. If y(t) = x(t)/z(t), then y x z . y x z Ex.1: GDP per capita (y) = GNP(Y)/Population(Pop) If GNP (Y) increases by 5 % and the population increases by 3 % , then GNP per capita increases by 2 %. Ex. 2: Debt-income-ratio (y)= nominal debt(D)/nominal GDP. In case no amortizations are done, the growth rate of nominal debt is the interest rate. Thus: Percentage change of y = interest rate – percentage change of nominal GDP. Ex3: Real wage(w) = W/P. y x a . y x Rule 3. If y(t) x(t)a , then Rule 4. If Yt At Kt Lt1 , then Y A K L (1 ) Y A K L (1) (2) (3) thus the growth rate of Y equals: (1) The growth rate of totalfactorproductivity. (2) The contribution of physical capital. (3) The contribution of labor. Question addressed by so-called growth accounting growth accounting: How big share of the growth rate of the GDP can be attributed to changes in capital, to changes in the labor input and to changes in total factor productivity? For developed countries we have good data on We have not direct data on A A Y K , Y K and L L , as A captures the influence on Y of many different factors on Y. E.g. taxes, climate for business, educational level of work force, infrastructure, social capital etc. Under perfect competion, , is the share of national income that is capital income, and (1 ) is the share of national income that is labor income. We have data on labor income and national income. Thereby, we get an estimate of . Example: Year 2005 2006 Y 100 103 A ? ? K 300 306 L 1000 1010 Y K L 0.03, 0.02 and 0.01 Y K L A 0.03 = + 0.3*0.02 + 0.7*0.01 A A = 0.03 – 0.006 – 0.007 = 0.017. A 0.017/0.03 = 0.57 : 57 percent of the growth rate of Y can be attributed to an increase in A. 0.006/0.03 = 0.2: 20 percent can be attributed to an increase in K. 0.007/0.03 = 0.23: 23 procent can be attributed to an increase in L. We have not explained why K, L and A changes over time. We have only been engaged in accounting. The neoclassical growth model explains why K and thereby Y increase. (A and L are exogenously given in this model; that is, they are determined outside the model.) The growth rate of output equals the growth rate of aggregate demand, which can be split up as follows: Y C C I I G G NX NX Y Y C Y I Y G Y NX THE SOLOW GROWTH MODEL: TOPIC 4. The Solow Model Readings: Background: Weil, ch. 1-2 Weil, ch. 3: The Solow Model without population growth and technological progress. Weil, ch. 4.2: The Model with population growth but without technological progress. Weil, skim appendix of ch. 8: Incorporating technological progress in the Solow model Aim to explain the development over time of K and Y, k=K/L, and y=Y/L. In the model the growth rate of the technological level, A / A g , and the growth rate of the numbers of workers, L / L n , are exogenous variables. In the model everyone is a worker. Thus, the number of workers = population. Thus, they are determined outside the model. As A / A , and L / L are assumed to be exogenous variables, the model is about the accumulation of physical capital, and its effects on k, Y, and y. To simplify we start by assuming: A / A L / L 0 . Thus, the level of A and L are assumed to be constant over time. To explain this model it is best to start to work with K and Y. Instead of expressing variables in terms of per worker, which this handout unfortunately does. Assumption (A1): The production function: where 0 < Yt A Kt L1 , < 1. Expressing production in terms of per worker (labor productivity): Yt At Kt Lt1 Lt Lt At Kt Lt At Kt Lt Labor productivity depends on: * Totalfaktorproductivity, A. If A Y/L * Physical capital per worker, K/L. If K/L Y/L Note: (1- )* Labor productivity (Y/L) = MPL (= W/P) y A k Thus, the level of A and L are assumed to be constant over time. As L is assumed to be constant we can assume L=1. Small letters indicate that variables are expressed in terms of per worker. K At t Lt Labor productivity y=Y/L 9 8 7 6 5 4 3 2 1 0 Y/L=A(K/L) A=2 A=2 Serie1 Serie2 A=1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k=K/L In figure: A=1,2, and =0.5 . Slope of the curve above is MPK=dY/dK. Note: L is assumed to be a constant; for example, L=1 K=k More complicated proof which is optional: dy A k 1 , which is MPK: dk 1 dY K ( 1) 1 1 1 MPK= A K L A K L A ] dK L [ If you increase saving, K increases, the rate of return is dY/dK-d, which falls. Assumption (A2): A constant share of income is saved: S=s*Y, 0<s<1. (= a constant share of production is invested) and a constant share of income is consumed: C=(1-s)*Y. Goods market equilibrium condition: Y C I G NX We assume a closed economy without a government sector: G=NX=0 S=Y-C=I National saving equals gross investment. It is easy to augment the model so it includes a government sector as well as exports and imports we do this on the C-level. (A2): S s Y , where s is the share of income that is saved. Note1: Y=GDP=GNP Note2: s is not saving per worker even though s is a small letter. s Y I s Y I L L s y i s S Y Y CI C I ci L L L L y c i (1 s) y s y (1 s) A k s A k Moreover, y=Y/L,i production and investment per worker 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 y=Ak i=sy=sAk 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Serie1 Serie2 16 17 k=K/L In figure: s=0.3, A=1, and =0.5. The vertical distance between the curves for production per worker, and investment per worker is consumption per worker. Assumption (A3): K I K where K is net investment, I = gross investment, and K = depreciation of capital per period. is the depreciation rate, which is between 0 and 1; e.g. 0.05. (That is, 5 %). If I K , then K 0 If I K , then K 0 ; K 0 . k i k If I K , then Expressing (A3) in terms of per worker: K I I L I /L i K K K L K/L k k K L L Using k = K/L , where =0 by assumption. L k K L k K k i k i k ] k K k k Derivation optional:[ Depreciation of capital per worker d*k 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Serie1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k In figure: 0.1 The whole model: (A1): y A k , (A2): i s y constant share of income. (A3): k i k , the production function investment = saving (equilibrium condition) where saving is The time path of the capital stock per worker The whole model can be reduced to one equation: Inserting (A1) and (A2) into (A3): k i k s y k s A k k The long-run equilibrium (steady state) value for k, That is, when gross investment equals depreciation s y k s A k k Solving for k in equilibrium: s A k k k * , occurs when k 0 . s A k 1 1 1 A 1 s s A 1 * k k * What is the long-run equilibrium value of y, y ? 1 1 1 s y * A (k * ) A A 1 If s or A k * and y* . If the economy is not in its equilibrium, it converges over time towards the equilibrium because if k< k * , then i> k k * , and if k> k * , then i< k k * . See figure below. Showing the equilibrium in the Solow diagram: y, i, dk The SOLOW MODEL 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 y=Ak y=A*k*exp(alfa) Serie1 Serie2 Serie3 dk dk i=sy=sAk 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k In figure: A=1, s=0.3, 0.1 , and =0.5. The transition to equilibrium: a numerical example Starting below the equilibrium: The initial value of k: k(year=1)=4.00. Assume also: A=1, s=0.3, 0.1 , and =0.5. year K i k k y y k 0.5 c (1 0.3) y 0.3 y 1 2 3 4 5 … 4.00 4.2 4.395 4.584 4.768 2.00 2.049 2.096 2.141 2.184 1.4 1.435 1.467 1.499 1.529 0.6 0.615 0.629 0.642 0.655 0.4 0.420 0.440 0.458 0.477 0.2 0.195 0.189 0.184 0.178 0.049 0.047 0.045 0.043 9 3 2.1 0.9 0.9 0 0 The equilibrium values of k and y are calculated by using the formulas: 1 s A 1 k* , k k y 0.05 0.046 0.043 0.040 0.0245 0.0229 0.021 0.020 0 0 y s A 1 y * A (k * ) A How to fill out the Table based on an initial value and assumed parameter values: A=1, s=0.3, 0.1 , and =0.5. Start by filling out the column for k based on the formula: k i k s y k s A k k k2 k1 s A k1 k1 k2 k1 s A k1 k1 (1 ) k1 s A k1 If k(year=1)=4, A=1, s=0.3, 0.1 , and =0.5. k2 0.9 k1 0.3 k10.5 , k3 0.9 k2 0.3 k20.5 , etc. After the values of k has been filled out, all other values of other variables (columns) can be calculated. Graphical description of transition to equilibrium when economy start below and above the equilibrium ln( y* 3 )=1.1: (1) k(t=1)=4, y (t=1)=2 , and ln(y (t=1)=2)=0.69 (2) k(t=1)=14, y (t=1)=3.74 , and ln(y (t=1)=2)=1.32 Transition to equilibrium 1.4 1.2 ln (Y/L) 1 0.8 Serie1 Serie2 0.6 0.4 0.2 0 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 time According to model: The growth rates of k and y are higher the lower k and y are. This explains why the slope of the curves for lny becomes flatter and flatter when lny approaches its equilibrium. Recall that the slope of lny is the growth rate of y. If two economies share the same equilibrium; that is, have the same parameter values on A, s (as well as on and ) but differ with respect to initial values, then the economy with lower k and y experience higher growth rates of k and y than the economy with higher k and y. the model says that y (and k) of these two economies converge over time. In other words, the model says that y over time converge across economies if the economies share the same equilibrium value of y). Main lesson of empirical work on growth: Real per capita income tends over time to converge across economies, which are similar with respect to “institutions”. An economy with an initially relatively low real income per capita has on average a higher growth rate of real income per capita than an economy with an initially relatively high real income per capita if “institutions” are similar. Ex.: EU-countries and regions within countries. Evidence from the OECD-countries (the currently rich countries) Growth rate of GDP p.c. Average annual growth rate of GDP p.c., 1960-2000, and GDP p.c. in 1960 0.05 0.04 0.03 Serie1 0.02 0.01 0 0 2000 4000 6000 8000 10000 12000 14000 16000 Real GDP per capita 1960 Sample includes: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Greece, Iceland, Ireland, Italy, Japan, Netherlands, New Zealand, Norway Portugal, Spain, Sweden, Switzerland, United Kingdom and USA. Initially poor countries grow faster in terms of real GDP per capita during the period 19602000 than initially rich countries. The correlation between the average annual growth rate of real GDP per capita between 1960 and 2000 and real GDP per capita in 1960 = - 0.89 Evidence from the 24 Swedish Regions, 1911-1993 Regions that were relatively poor in terms of real income per capita in 1911, on average had a higher growth rate of real income per capita.Higher growth rates in poor regions caused relative differences in real per capita income to diminish across the Swedish Regions between 1 911 and 1993. The dispersion is lower for real per capita income when it is adjusted for regional differences in cost of living as counties with high unadjusted real per capita incomes tend to have cost of living. Per capita Income adjusted and unadjusted for cost of living The empirical evidence on convergence in real per capita income across the Swedish regions is consistent with the predications of the textbook model: Low real per capita income Little capital (physical + human) per worker, low wages, high rates of return to capital capital per worker production per worker income per capita Also factor mobility tends to contribute to convergence: Low wages and high returns to capital out-migration, and foreign investment capital per worker production per worker Evidence from the countries of the world Growth rate of GDP per capita Average annual growth rate growth rate of GDP p.c., 19602000, and GDP p.c. in 1960 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01 0 -0.02 S 2000 4000 6000 8000 10000 12000 14000 Real GDP per capita in 1960 Sample includes 80 countries. No convergence in real GDP per capita across the countries of the world. The correlation between the average annual growth rate of real GDP per capita between 1960 and 2000 and real GDP per capita in 1960 = + 0.14. Is lack of convergence in GDP per capita for the countries of the world, evidence against the model? NO! The model says that if countries have the same equilibrium, the poorer country should grow faster in terms of y and k than the country that is richer in terms of y and k. But if countries differ with respect to equilibrium, that is, with respect to values on A, s (as well as on and ), the poorer economy need, according to model, not grow faster than the initially richer economy. Africa is poor because it has a low equilibrium. 16000 Example that a rich country can grow faster than a poor country Country A (Poor Country): A=1, s=0.2, 0.1 , and =0.5 : k * Assumed initial values of k and y: 4 and 2. Country A’s growth rate of y=0 Country B (Rich Country): A=1, s=0.3, 0.1 , and Assumed initial values of k and y: 5 and 2.24. Country B’s growth rate of y is positive. 4 , y* 2 =0.5 : k* 9 , y* 3 y, i, dk Countries with different saving rates 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 y=k**0.5 i=0.3*y i=0.2*y 0.1*k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k Time path of y of two countries 3.5 3 Y/L 2.5 2 Country B 1.5 Country A 1 0.5 0 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 time The short and long run effects of an increase of L (e.g. due to immigration) y, i, dk The SOLOW MODEL 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 y=A*k*exp(alfa) y=Ak Serie1 dk dk Serie2 Serie3 i=sy=sAk 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k A one-time increase of L:L(t=0)<L(t=1)= L(t=2)= L(t=3)= L(t=4) At time 1: K/L and Y/L, At time 2 and onwards: K/L and Y/L If the economy initially is in its equilibrium, it will over time revert to the initial equilibrium as gross investment exceeds depreciation of capital. K/L Time path K/L 10 9 8 7 6 5 4 3 2 1 0 Serie1 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 time In figure: A=1, s=0.3, 0.1 , and =0.5, K(0)=900, L(0)=100 and L(1)=200. The long run values of k and y are unchanged. However, adjustment takes a long time so migration plays a role for y during a long time according to model. What happens to the long run values of Y and K? Y * A K L1 A k * L Y * A k * L(1) Y * A k * L(0) K * k * L(1) K * A k * L(0) Size of economy increases when L increases. Example: Y * increases from 3*100= 300 to 3*200=600, and K * increases from 9*100=900 to 9*200=1800. In case of a pandemic, L decreases, the results are the opposite. The effect of an increase in A y, i, dk The SOLOW MODEL 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 y=A*k*exp(alfa) y=Ak Serie1 dk dk Serie2 Serie3 i=sy=sAk 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k 0.1 , and =0.5 k* 9 and y* 3 . y* 6.75 New value: A=1.5 k * 20.25 , Old value: A=1, s=0.3, The transition to the new long run equilibrium Transition to new equilibrium 8 7 6 Y/L 5 Serie1 4 Serie2 3 2 1 0 1 7 13 19 25 31 37 43 49 time 55 61 67 73 79 85 91 97 Long-run growth of the number of workers Before we assumed: Lt 1 Lt L(0)=L(1)=L(2)=L(3) Now we assume : Lt 1 (1 n) Lt where n is the constant growth rate of the number of workers; e.g. 0.01. L L L L L L t 1 (1 n) t 1 1 n t 1 t n t 1 n Lt Lt n Lt Lt Lt Lt Lt L(0)<L(1)<L(2)<L(3) Assumption A4. To keep k constant gross investment (I) now needs to compensate not only for depreciation of capital to keep k constant but also for the fact that the number of workers increases over time: (A3): K I K Derivation below optional: [ K I I L I /L i K K K L K/L k k K L L n by assumption. , where L k K L k K k n i ] k i (n ) k n k k K k Using k = K/L The whole model: (A1): y A k , the production function (A2): i s y investment = saving (equilibrium condition) where saving is constant share of income. (A3)+(A4): k i (n ) k , The time path of the capital stock per worker A-level students need not know mathematical derivation below: The whole model can be reduced to one equation: Inserting (A1) and (A2) into (A3)+(A4): k i (n ) k s y (n ) k s A k (n ) k The long-run equilibrium The long-run equilibrium (steady state) value for k, k * , occurs when k That is, when gross investment equals “depreciation” s y (n ) k s A k (n ) k Solving for k in equilibrium: 1 1 1 k 1 1 s A n s A n k k s A k 1 n 1 s A 1 k* n 0. What is the long-run equilibrium value of y, y* ? s A 1 If n y * A (k * ) A n k * and y* . The transition to the equilibrium If the economy initially is in equilibrium and n the economy moves over time to the new lower equilibrium because when i< (n ) k k: y, i,(n+d)k Growth rate of L increases 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 y=k**0.5 i=0.3*y (0+0.1)*k (0.05+0.1)*k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 k In figure: A=1, s=0.3, 0.1 , =0.5 and n=0 and 0.05. (1) when n=0 k * 9 and y* 3 . k * 4 and y * 2 . (2) when n=0.05 Transition to new equilibrium 3.5 3 Y/L 2.5 Serie1 2 Serie2 1.5 Serie3 1 0.5 0 1 7 13 19 25 31 37 43 49 55 Time The growth rates of aggregate variables: K k L K k L k n K k L k 61 67 73 79 85 91 97 Y yL Y y L y n Y y L y In the steady-state: k * y* * 0 k* y K * k * L * 0n n L K* k * * y L Y * * 0n n L Y y K* k* L Y * y* L Factor prices: In the model: C + I = Y = capital income + labor income = MPL*L+MPK*K= (W/P)*L + (r+ )K r= real return on physical capital. Note: There is only one good in the Solow model, which is consumed or invested. If it is invested it is an asset which yields a return. K is the only asset in the economy. There exist no bonds, shares or money in the model. Expressing the equilibrium condition above in terms of per worker: c + i=y=(W/P) + (r+ )k (1) W/P=MPL= (1 ) A k (1 ) y (2) r A k 1 y / k If k W/P and r In poor and rich countries K/L is low and high, respectively. Factor mobility across economies If the value of A is the same in poor and rich countries, the real return on capital is higher in poor countries. As a result, we expect capital to move from rich to poor countries, increasing K/L in poor countries and lowering K/L in rich countries. Thereby, mobility of capital contributes to convergence in K/L between rich and poor economies. We expect L to move the opposite way because W/P is higher in rich countries. Mobility of L increases K/L in poor countries and decreases K/L in rich countries. Thereby, it also contributes to convergence in K/L between rich and poor countries. Why do capital not flow to Africa? In other words, why are not large investments taking place in some African countries? Answer: Because A is low, which means that MPK=r+d is not so high. This can be seen in Solow-diagram. (Allow countries to differ w.r.t. A.) In other words, if A is the same across countries (which it is not). (assume g=0). K will move from richer countries with low r (due to high k) to poorer countries where r is high due to low k. Thereby k and y will tend to equalize across countries. Nowadays, rich EU-countries invest capital or move production to new EU-countries or to CHINA or India. L will move the opposite way, from low-wage countries to high wage countries, which also contributes to equalize real wages, r and k across countries. Workers move from new EU-countries to old EU-countries where real wages are higher. Specific example: assume two countries that are the same with respect to the parameters: A, n, d, and alfa, but one country has a higher saving rate than the other. Assume that these countries are in their respective equilibria. Allowing for factor mobility across countries equalizes the real wage, the real return to capital, k and y across countries. The new equilibrium will be joint for the two countries and is determined by a weighted average of the saving rates in the 2 countries, where the weights are given by the size of the populations in the two countries. Capital to labor (k) ratio is low in developing countries. As a result, one would expect a high real rate of return on investment in those countries. Why then do not a lot of investment (construction of new factories, etc.) take place in many of these countries? Answer: There is a lot of corruption, which makes the actual rate of return much lower; that is, after the investor have paid off a lot of government official, there might not be so much money left. A is low. There might also be a political risk. Investors might risk that some bandits take over the factories, like in Zimbabve. Important Exercise: Derive the equilibrium expressions for the real wage and for the real return on capital; that is, express the real wage and the real return to capital as functions of the exogenous variables: s, A, n, the depreciation rate and alfa. Golden rule is optional reading for A-level students: The golden rule level of capital: The level of capital that maximizes consumption per worker in equilibrium Consumption per worker is the distance between the curve for labor productivity ( y A k ) and the curve for depreciation of capital per worker: (n+d)k. This distance is maximized at the level of k where the slopes of these two curves are the same: dy A k 1 n dk Solving the equation A k 1 n for k yields the answer. MPK A government that wants to maximize consumption per worker should choose the saving rate (s) so that this level of capital is achieved. An economy can save too much. That is, by decreasing the saving rate per capita consumption can increase in the steady state. Adding realism in the model: continuing technological progress There is technological progress if new production techniques arise due to innovations such as the computer, engine, electricity, etc. A dA / dt g A A gt [optional reading: A(t ) A(0) e ] Model assumption: (A5): , where g =rate of technological progress is exogenously assumed. [Optional reading: The model is here formulated in continuous time which means that time changes continuously. Previously the model was in discrete time which means that the time is in periods. If the model were in discrete time: At A0 (1 g )t .] There are exercises on this model. Only technological progress can explain long run increases in the living standard= GDP per capita = Y/L=y Growth rates in the long-run equilibrium: k * y* k * y* w* , Now: 0 * g w k* y* k* y y* k * r* K * Y * * ( * * ) 0 g n, g n , r d ( y / k) , r y k K* Y* Before: Technological progress is exogenous As the rate of technological progress is unexplained by the SOLOW model (that is, exogenous), adding g to the model does not add any more economic insights than the version of the model with g=0. For this reason and because it is simpler we will focus on the version of the model where g=0. Keep however in mind that technological progress makes the model more realistic because in the real world y typically increases over time due to new production techniques; that is, due to innovations. What happens if the economy is off its equilibrium growth path? ln(Y/L) Transition to equilibrium growth path 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Serie1 Serie2 Serie3 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 Time When the economy approaches its equilibrium growth path, the growth rate of y deviates from the long-run growth rate (g). If an economy starts out below (above) the equilibrium growth path, the growth rate of y is higher (lower) than g. Holding constant the equilibrium growth path that is holding constant A(0), s, n, g, d and alfa, a lower y means a higher growth rate of y. What happens to the growth rate and to the equilibrium growth path if the saving rate increases (or institutions improve or population growth )? Transition to higher equilibrium growth path 2.5 ln(Y/L) 2 Serie1 1.5 Serie2 1 Serie3 0.5 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 time If s increases, the equilibrium shifts upwards, and the growth rate of y is higher than the long run growth rate during to the transition to the new equilibrium growth path. READ BY YOURSELVES: Factors that impact GDP per capita in the real world: GDP GDP Hours worked Employment POP 19 64 POP Hours worked Employment POP 19 64 POP POP = Population. If GDP per hour (=labor productivity) increases or the hours worked per employed increases or the number of employed as a share of population increases, then GDP per person increases. In other words, get each worker to produce more or get more people in production, then GDP per person increases. Production per employed (= the first 2 terms on the left hand side of the equation above) is in macromodels is GDP per worker, Y/L. GDP (or GNP) per capita as a measure of the standard of living The income distribution GDP per capita (=average income) can be a poor indicator of the income of the average citizen; that is, of the median income. The median is the person in the middle of the income distribution. Typical income distribution Number of income earners 60 50 40 30 Serie1 20 10 0 Income classes of equal size The income distribution is typically assymmetric: median income < average income the more unequal income distribution the greater difference between median and average income and a larger proportion of the population tends to have an income below the average income. An extreme example: Country Equal. 10 individuals each with an income of 5000. The median and mean income is 5000. Country Unequal. 9 individuals each has an income of 2000. One individual has an income of 32000. Average income is 5000. Median income is 2000. GDP per capita as an indicator of “human development/happiness” We have concluded that average income per capita may be a poor indicator of the income of the average person; that is, of the median person. What is the relationship between income per capita and other indicators of “welfare/happiness”? We want (but cannot) measure is happiness/utility: U = U (y, x1, x2, x3, x4,…) Where y = income per capita, x1=literacy rate, x2=assess to clean water, x3= infant mortality rate, x4=life expectancy, etc. 2 views: 1. The correlation between income per capita and other variables (x1,x2,x3,x4,..) which we believe impact the welfare of people is high. Therefore, it is sufficient to study determinants to income per capita. 2.The correlation is not necessarily high. The UN (UNDP’s) “Human Development Index” has 3 components: 1. Life expectancy. 2. Educational level (e.g. literacy rate). 2. Income per capita. According to this index Sweden’s is a top 5 whereas with respect to income per capita Sweden is only top 20. Problem of Household surveys that ask “Are you happy?” is that the meaning of the word happy may differ across cultures. We are rich now but are we happier? The importance of relative position. Harvard-students were asked what alternative they preferred: a) USD 50000/year whereas others get half. b) USD 100000/year whereas others get the double. Source: The economist, Aug. 9, 2003. Some characteristics of poor countries Large agricultural sector. They have a comparative advantage with respect to labor-intensive production as they have a lot of labor but only a little capital (physical and human). Demography: Young populations, many kids per woman.