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Growth SAMs Hats Poverty and Development Bill Gibson UVM 8 Feb 2010 Bill Gibson University of Vermont Growth SAMs Hats Robert Lucas Rates of growth of real per-capita income are...diverse, even over sustained periods. Indian incomes double every 50 years and Korean every 10. An Indian will, on average, be twice as well off as his grandfather; a Korean 32 times... I do not see how one can look at figures like these without seeing them as representing possibilities. Is there some action India could take that would lead the economy to grow like Indonesia or Egypt’s If so, what exactly? If not, then what is it about the “nature of India” that makes it so? The consequences for human welfare involved in questions like these are simply staggering. Once one starts to think about them, it is hard to think of anything else. Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception 1580-1820 Netherlands was the fastest growing economy 0.2 percent Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception 1580-1820 Netherlands was the fastest growing economy 0.2 percent UK from 1820-90 only grew 1.2 percent Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception 1580-1820 Netherlands was the fastest growing economy 0.2 percent UK from 1820-90 only grew 1.2 percent US 1890-1990 grew at 2.2 percent Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception 1580-1820 Netherlands was the fastest growing economy 0.2 percent UK from 1820-90 only grew 1.2 percent US 1890-1990 grew at 2.2 percent China now exceeding 10 percent Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception 1580-1820 Netherlands was the fastest growing economy 0.2 percent UK from 1820-90 only grew 1.2 percent US 1890-1990 grew at 2.2 percent China now exceeding 10 percent On average GDP per capita in 1913 1.8 times 1870 Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception 1580-1820 Netherlands was the fastest growing economy 0.2 percent UK from 1820-90 only grew 1.2 percent US 1890-1990 grew at 2.2 percent China now exceeding 10 percent On average GDP per capita in 1913 1.8 times 1870 By 1978 6.7 times Bill Gibson University of Vermont Growth SAMs Hats Growth accelerating Throughout human history growth in income per capita was the exception 1580-1820 Netherlands was the fastest growing economy 0.2 percent UK from 1820-90 only grew 1.2 percent US 1890-1990 grew at 2.2 percent China now exceeding 10 percent On average GDP per capita in 1913 1.8 times 1870 By 1978 6.7 times GDP per capital now growing at an accelerated pace Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Replaces complex analysis with simple differential equation Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Replaces complex analysis with simple differential equation Very good empirically: see Mankiw, Romer and Weil, Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Replaces complex analysis with simple differential equation Very good empirically: see Mankiw, Romer and Weil, A Contribution to the Empirics of Economic Growth, QJE , 1992 Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Replaces complex analysis with simple differential equation Very good empirically: see Mankiw, Romer and Weil, A Contribution to the Empirics of Economic Growth, QJE , 1992 Listen interview with Lucas on Econ Talk. Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Replaces complex analysis with simple differential equation Very good empirically: see Mankiw, Romer and Weil, A Contribution to the Empirics of Economic Growth, QJE , 1992 Listen interview with Lucas on Econ Talk. Says capital accumulation and technology important Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Replaces complex analysis with simple differential equation Very good empirically: see Mankiw, Romer and Weil, A Contribution to the Empirics of Economic Growth, QJE , 1992 Listen interview with Lucas on Econ Talk. Says capital accumulation and technology important Won Solow Noble Prize Bill Gibson University of Vermont Growth SAMs Hats Growth models No poverty traps Replaces complex analysis with simple differential equation Very good empirically: see Mankiw, Romer and Weil, A Contribution to the Empirics of Economic Growth, QJE , 1992 Listen interview with Lucas on Econ Talk. Says capital accumulation and technology important Won Solow Noble Prize Gets us to ask the right questions Bill Gibson University of Vermont Growth SAMs Hats Harrod-Domar model Growth is due to abstaining from consumption: set aside Bill Gibson University of Vermont Growth SAMs Hats Harrod-Domar model Growth is due to abstaining from consumption: set aside Based on social accounting matrix (see figure 3.1) Bill Gibson University of Vermont Growth SAMs Hats Harrod-Domar model Growth is due to abstaining from consumption: set aside Based on social accounting matrix (see figure 3.1) Y (t ) = C (t ) + I (t ) (eqn 3.1) Bill Gibson University of Vermont Growth SAMs Hats Harrod-Domar model Growth is due to abstaining from consumption: set aside Based on social accounting matrix (see figure 3.1) Y (t ) = C (t ) + I (t ) (eqn 3.1) S (t ) = I (t ) Bill Gibson University of Vermont Growth SAMs Hats Harrod-Domar model Growth is due to abstaining from consumption: set aside Based on social accounting matrix (see figure 3.1) Y (t ) = C (t ) + I (t ) (eqn 3.1) S (t ) = I (t ) K (t + 1) = (1 − δ)K (t ) + I (t ) note time shift Bill Gibson University of Vermont Growth SAMs Hats Harrod-Domar model Growth is due to abstaining from consumption: set aside Based on social accounting matrix (see figure 3.1) Y (t ) = C (t ) + I (t ) (eqn 3.1) S (t ) = I (t ) K (t + 1) = (1 − δ)K (t ) + I (t ) note time shift s/θ = g + δ Bill Gibson University of Vermont Growth SAMs Hats Harrod-Domar model Growth is due to abstaining from consumption: set aside Based on social accounting matrix (see figure 3.1) Y (t ) = C (t ) + I (t ) (eqn 3.1) S (t ) = I (t ) K (t + 1) = (1 − δ)K (t ) + I (t ) note time shift s/θ = g + δ This is the Harrod-Domar Equation (1939, 1946) Bill Gibson University of Vermont Growth SAMs Hats Growth models Farmer’s pond Bill Gibson University of Vermont Growth SAMs Hats Growth models Farmer’s pond Question arises: Bill Gibson University of Vermont Growth SAMs Hats Growth models Farmer’s pond Question arises: Where does farmer build fence? Bill Gibson University of Vermont Growth SAMs Hats Growth models Farmer’s pond Question arises: Where does farmer build fence? Creek flows into pond then evaporation Bill Gibson University of Vermont Growth SAMs Hats Growth models Farmer’s pond Question arises: Where does farmer build fence? Creek flows into pond then evaporation How big will pond get? Bill Gibson University of Vermont Growth SAMs Hats Social Accounting Matrices SAMs are always nominal, measured in dollars (not physical units) Bill Gibson University of Vermont Growth SAMs Hats Social Accounting Matrices SAMs are always nominal, measured in dollars (not physical units) The SAM for the base year is therefore both real and nominal Bill Gibson University of Vermont Growth SAMs Hats Social Accounting Matrices SAMs are always nominal, measured in dollars (not physical units) The SAM for the base year is therefore both real and nominal SAMs have at most 4 agents: firms, households, government and foreign Bill Gibson University of Vermont Growth SAMs Hats Social Accounting Matrices SAMs are always nominal, measured in dollars (not physical units) The SAM for the base year is therefore both real and nominal SAMs have at most 4 agents: firms, households, government and foreign Each agent has a row and column Bill Gibson University of Vermont Growth SAMs Hats Social Accounting Matrices SAMs are always nominal, measured in dollars (not physical units) The SAM for the base year is therefore both real and nominal SAMs have at most 4 agents: firms, households, government and foreign Each agent has a row and column When income (rows) = expenditure (columns) for all agents of the SAM, then total savings is equal to total investment Bill Gibson University of Vermont Growth SAMs Hats Social Accounting Matrices SAMs are always nominal, measured in dollars (not physical units) The SAM for the base year is therefore both real and nominal SAMs have at most 4 agents: firms, households, government and foreign Each agent has a row and column When income (rows) = expenditure (columns) for all agents of the SAM, then total savings is equal to total investment When solving a SAM problem, make sure that total investment is equal to total savings first Bill Gibson University of Vermont Growth SAMs Hats Social Accounting Matrices SAMs are always nominal, measured in dollars (not physical units) The SAM for the base year is therefore both real and nominal SAMs have at most 4 agents: firms, households, government and foreign Each agent has a row and column When income (rows) = expenditure (columns) for all agents of the SAM, then total savings is equal to total investment When solving a SAM problem, make sure that total investment is equal to total savings first Note that savings is included in expenditure! Bill Gibson University of Vermont Growth SAMs Hats SAMs Gross value of production is the row sum for the firms Bill Gibson University of Vermont Growth SAMs Hats SAMs Gross value of production is the row sum for the firms The GDP is GVP less intermediate goods, plus government wages Bill Gibson University of Vermont Growth SAMs Hats SAMs Gross value of production is the row sum for the firms The GDP is GVP less intermediate goods, plus government wages PI is sum of household row/column Bill Gibson University of Vermont Growth SAMs Hats SAMs Gross value of production is the row sum for the firms The GDP is GVP less intermediate goods, plus government wages PI is sum of household row/column Government deficit is negative of government savings Bill Gibson University of Vermont Growth SAMs Hats SAMs Gross value of production is the row sum for the firms The GDP is GVP less intermediate goods, plus government wages PI is sum of household row/column Government deficit is negative of government savings The foreign deficit is foreign savings (total imports less total exports) Bill Gibson University of Vermont Growth SAMs Hats SAMs Gross value of production is the row sum for the firms The GDP is GVP less intermediate goods, plus government wages PI is sum of household row/column Government deficit is negative of government savings The foreign deficit is foreign savings (total imports less total exports) SAM balanced when income is equal to expenditure for all agents Bill Gibson University of Vermont Growth SAMs Hats SAMs Gross value of production is the row sum for the firms The GDP is GVP less intermediate goods, plus government wages PI is sum of household row/column Government deficit is negative of government savings The foreign deficit is foreign savings (total imports less total exports) SAM balanced when income is equal to expenditure for all agents sum of savings equal investment Bill Gibson University of Vermont Growth SAMs Hats A simple SAM 2-agent SAM Firms HH Savings Total check S-I GDP VA error Firms 77 223 300 Households 159 Invest 64 64 223 64 0 223 223 0 Bill Gibson University of Vermont Total 300 223 64 error Growth 0SAMs Hats A 2-agent SAM with wages and profits 2-agent SAM Firms HH Labor Capital Savings Total check S-I GDP VA error Firms HH 80 Invest 20 20 100 20 100 60 40 100 0 100 100 0 Bill Gibson University of Vermont Total 100 100 60 40 20 Growth SAMs Hats 3-agent SAM 3-agent SAM Firms HH Labor Capital Savings Govt Total check S-I GDP VA error Firms 21 308 185 123 329 HH 198 72 68 338 Invest 44 -28 44 0 338 338 0 Bill Gibson Govt 66 30 30 University of Vermont 68 Total 329 338 215 123 44 68 Growth SAMs Hats 4-agent SAM 4-agent SAM Firms HH Labor Capital Savings Taxes Imports Total check S-I GVP GDP VA Firms 9 304 182 122 313 HH 210 Invest 80 Govt 6 6 4 2 50 Exports 8 62 16 23 62 16 310 80 Bill Gibson University of Vermont 0 313 308 308 8 Total 313 310 186 124 80 62 16 Growth SAMs Hats NIPA-SAM NIPA-SAM Firms HH Labor Capital Transfers Saving Depreciation Retained earnings Government Direct tax Indirect tax Social Security Foreign Total Firms HHs Inv Gov Foreign Total 260 1321 793 528 1042 282 204 281 206 75 462 61 219 182 19 2343 23 555 368 177 114 77 -123 2343 1689 970 642 77 282 206 75 800 399 219 182 305 800 305 101 338 338 286 1689 Bill Gibson 282 University of Vermont Growth SAMs Hats NIPA-SAM The GDP is The GNP is Net national product is National income is Personal income is Personal disposable income is The government deficit is The foreign deficit is Capital stock Capital stock next period 2260 2241 2035 1816 1689 1351 123 101 6780 6856 VA private public error 2260 2083 177 0 Bill Gibson University of Vermont Growth SAMs Hats Foreign savings We call this foreign savings because it is owed to foreigners since we didn’t pay for all of our imports with exports Bill Gibson University of Vermont Growth SAMs Hats Foreign savings We call this foreign savings because it is owed to foreigners since we didn’t pay for all of our imports with exports Foreigners are therefore “saving” in our country Bill Gibson University of Vermont Growth SAMs Hats Foreign savings We call this foreign savings because it is owed to foreigners since we didn’t pay for all of our imports with exports Foreigners are therefore “saving” in our country the sum of firm savings, households saving, government savings and foreign savings is equal to investment Bill Gibson University of Vermont Growth SAMs Hats Foreign savings We call this foreign savings because it is owed to foreigners since we didn’t pay for all of our imports with exports Foreigners are therefore “saving” in our country the sum of firm savings, households saving, government savings and foreign savings is equal to investment The government deficit is the negative of government savings Bill Gibson University of Vermont Growth SAMs Hats Foreign savings We call this foreign savings because it is owed to foreigners since we didn’t pay for all of our imports with exports Foreigners are therefore “saving” in our country the sum of firm savings, households saving, government savings and foreign savings is equal to investment The government deficit is the negative of government savings When firms operate they have some payments that are contractual, intermediates, wages, indirect taxes, profit taxes, foreign factor payments Bill Gibson University of Vermont Growth SAMs Hats Foreign savings We call this foreign savings because it is owed to foreigners since we didn’t pay for all of our imports with exports Foreigners are therefore “saving” in our country the sum of firm savings, households saving, government savings and foreign savings is equal to investment The government deficit is the negative of government savings When firms operate they have some payments that are contractual, intermediates, wages, indirect taxes, profit taxes, foreign factor payments Twin towers: foreign and govt deficits related Bill Gibson University of Vermont Growth SAMs Hats Growth Models Pond is like capital stock Bill Gibson University of Vermont Growth SAMs Hats Growth Models Pond is like capital stock Output related to capital stock Bill Gibson University of Vermont Growth SAMs Hats Growth Models Pond is like capital stock Output related to capital stock Investment related to output Bill Gibson University of Vermont Growth SAMs Hats Growth Models Pond is like capital stock Output related to capital stock Investment related to output Plowback ratio is s Bill Gibson University of Vermont Growth SAMs Hats Growth Models Pond is like capital stock Output related to capital stock Investment related to output Plowback ratio is s Can still predict how large capital stock will get Bill Gibson University of Vermont Growth SAMs Hats Nature of the growth model Determined by relationship between K and plowback Bill Gibson University of Vermont Growth SAMs Hats Nature of the growth model Determined by relationship between K and plowback First case: fixed capital-output ratio (and α) Bill Gibson University of Vermont Growth SAMs Hats Nature of the growth model Determined by relationship between K and plowback First case: fixed capital-output ratio (and α) Capital-limited economy Bill Gibson University of Vermont Growth SAMs Hats Variable capital-output ratio Could be stochastic around a mean of θ Bill Gibson University of Vermont Growth SAMs Hats Variable capital-output ratio Could be stochastic around a mean of θ Could be a convergent sequence Bill Gibson University of Vermont Growth SAMs Hats Variable capital-output ratio Could be stochastic around a mean of θ Could be a convergent sequence Could also depend on labor as in standard model Bill Gibson University of Vermont Growth SAMs Hats Variable capital-output ratio Could be stochastic around a mean of θ Could be a convergent sequence Could also depend on labor as in standard model ∆K + δK = sY = sf (K , L) Bill Gibson University of Vermont Growth SAMs Hats Variable capital-output ratio Could be stochastic around a mean of θ Could be a convergent sequence Could also depend on labor as in standard model ∆K + δK = sY = sf (K , L) Solow model has variable capital-output with diminishing returns Bill Gibson University of Vermont Growth SAMs Hats Figure 3.2 Bill Gibson University of Vermont Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable Bill Gibson University of Vermont Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable GDP is 100 in 2000 and grows to 106 in 2001, t Bill Gibson University of Vermont Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable GDP is 100 in 2000 and grows to 106 in 2001, t Total percentage change (106-100)/100 = 0.06 or 6 percent Bill Gibson University of Vermont Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable GDP is 100 in 2000 and grows to 106 in 2001, t Total percentage change (106-100)/100 = 0.06 or 6 percent Made up of “dots” and levels, ẋ = ∆x /∆t where x is the level Bill Gibson University of Vermont Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable GDP is 100 in 2000 and grows to 106 in 2001, t Total percentage change (106-100)/100 = 0.06 or 6 percent Made up of “dots” and levels, ẋ = ∆x /∆t where x is the level Hats are related to logarithms and they follow the same rules Bill Gibson University of Vermont Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable GDP is 100 in 2000 and grows to 106 in 2001, t Total percentage change (106-100)/100 = 0.06 or 6 percent Made up of “dots” and levels, ẋ = ∆x /∆t where x is the level Hats are related to logarithms and they follow the same rules If have y = ln (x ) then the derivative of y is Bill Gibson University of Vermont dy dt = dx /dt x = x̂ Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable GDP is 100 in 2000 and grows to 106 in 2001, t Total percentage change (106-100)/100 = 0.06 or 6 percent Made up of “dots” and levels, ẋ = ∆x /∆t where x is the level Hats are related to logarithms and they follow the same rules If have y = ln (x ) then the derivative of y is Elasticity dln(x ) dln(t ) = dx /dt x /t = x̂ /t̂ Bill Gibson University of Vermont dy dt = dx /dt x = x̂ Growth SAMs Hats Hats Growth rates or percent changes of the underlying level variable GDP is 100 in 2000 and grows to 106 in 2001, t Total percentage change (106-100)/100 = 0.06 or 6 percent Made up of “dots” and levels, ẋ = ∆x /∆t where x is the level Hats are related to logarithms and they follow the same rules If have y = ln (x ) then the derivative of y is Elasticity dln(x ) dln(t ) = dx /dt x /t dy dt = dx /dt x = x̂ /t̂ Know your hat rules...saves a lot of time and effort Bill Gibson University of Vermont = x̂ Growth SAMs Hats Rule 1 Multiplication Let x = yz where x, y and z are levels of the three variables x̂ = ŷ + ẑ Bill Gibson University of Vermont Growth SAMs Hats Rule 1 Multiplication Let x = yz where x, y and z are levels of the three variables x̂ = ŷ + ẑ If levels are multiplied, then the hats are added Bill Gibson University of Vermont Growth SAMs Hats Rule 1 Multiplication Let x = yz where x, y and z are levels of the three variables x̂ = ŷ + ẑ If levels are multiplied, then the hats are added Example Nominal GDP grows at 6% but inflation is 4%. What is the approximate growth rate of real GDP? Bill Gibson University of Vermont Growth SAMs Hats Rule 1 Multiplication Let x = yz where x, y and z are levels of the three variables x̂ = ŷ + ẑ If levels are multiplied, then the hats are added Example Nominal GDP grows at 6% but inflation is 4%. What is the approximate growth rate of real GDP? Answer: 2 % Bill Gibson University of Vermont Growth SAMs Hats Rule 2 Division Let x = y /z x̂ = ŷ − ẑ Bill Gibson University of Vermont Growth SAMs Hats Rule 2 Division Let x = y /z x̂ = ŷ − ẑ Levels are divided hats are subtracted Bill Gibson University of Vermont Growth SAMs Hats Rule 2 Division Let x = y /z x̂ = ŷ − ẑ Levels are divided hats are subtracted Example Output per worker is defined as ρ = X /L where X is GDP and L is employment. We know that productivity usually grows at around 1%. If employment grows by 2%, what is the growth rate of GDP? Bill Gibson University of Vermont Growth SAMs Hats Rule 2 Division Let x = y /z x̂ = ŷ − ẑ Levels are divided hats are subtracted Example Output per worker is defined as ρ = X /L where X is GDP and L is employment. We know that productivity usually grows at around 1%. If employment grows by 2%, what is the growth rate of GDP? Answer: 3% Bill Gibson University of Vermont Growth SAMs Hats Rule 3 Multiplication by a constant Let x = ay with a constant x̂ = ŷ If level is multiplied by a constant, it drops out Bill Gibson University of Vermont Growth SAMs Hats Rule 3 Multiplication by a constant Let x = ay with a constant x̂ = ŷ If level is multiplied by a constant, it drops out If level is multiplied by a constant, it drops out Bill Gibson University of Vermont Growth SAMs Hats Rule 3 Multiplication by a constant Let x = ay with a constant x̂ = ŷ If level is multiplied by a constant, it drops out If level is multiplied by a constant, it drops out Example Consumption is a constant fraction of GDP, 70%. GDP grows at 6%. What is the approximate growth rate of consumption? Bill Gibson University of Vermont Growth SAMs Hats Rule 3 Multiplication by a constant Let x = ay with a constant x̂ = ŷ If level is multiplied by a constant, it drops out If level is multiplied by a constant, it drops out Example Consumption is a constant fraction of GDP, 70%. GDP grows at 6%. What is the approximate growth rate of consumption? Answer: 6% Bill Gibson University of Vermont Growth SAMs Hats Rule 4 Exponents Let x = ya with levels of the two variable x and y and a constant Bill Gibson University of Vermont Growth SAMs Hats Rule 4 Exponents Let x = ya with levels of the two variable x and y and a constant The fourth rule says x̂ = aŷ Bill Gibson University of Vermont Growth SAMs Hats Rule 4 Exponents Let x = ya with levels of the two variable x and y and a constant The fourth rule says x̂ = aŷ If levels are raised to a constant exponent constant does not disappear Bill Gibson University of Vermont Growth SAMs Hats Rule 4 Exponents Let x = ya with levels of the two variable x and y and a constant The fourth rule says x̂ = aŷ If levels are raised to a constant exponent constant does not disappear Example Let Y = L0.5 If the growth rate of labor is 3% how fast is GDP growing? Bill Gibson University of Vermont Growth SAMs Hats Rule 4 Exponents Let x = ya with levels of the two variable x and y and a constant The fourth rule says x̂ = aŷ If levels are raised to a constant exponent constant does not disappear Example Let Y = L0.5 If the growth rate of labor is 3% how fast is GDP growing? Answer: 1.5 % Bill Gibson University of Vermont Growth SAMs Hats Rule 5 Special case Of the exponent rule when y = e, the base of the natural logarithm system: y = e gt Bill Gibson University of Vermont Growth SAMs Hats Rule 5 Special case Of the exponent rule when y = e, the base of the natural logarithm system: y = e gt The fifth rule says ŷ = g Bill Gibson University of Vermont Growth SAMs Hats Rule 5 Special case Of the exponent rule when y = e, the base of the natural logarithm system: y = e gt The fifth rule says ŷ = g The growth rate is the exponent when the base is e Bill Gibson University of Vermont Growth SAMs Hats Rule 5 Special case Of the exponent rule when y = e, the base of the natural logarithm system: y = e gt The fifth rule says ŷ = g The growth rate is the exponent when the base is e Example Let Y = Y0 e 0.05t with Y0 as the initial condition. How fast is GDP growing? Bill Gibson University of Vermont Growth SAMs Hats Rule 5 Special case Of the exponent rule when y = e, the base of the natural logarithm system: y = e gt The fifth rule says ŷ = g The growth rate is the exponent when the base is e Example Let Y = Y0 e 0.05t with Y0 as the initial condition. How fast is GDP growing? Answer: 5% Bill Gibson University of Vermont Growth SAMs Hats Rule 6 Weighted average. Let x = ay + bz with a and b constant. Calculate the rate of growth of x is more complicated and requires that we write x̂ = ay bz ŷ + ẑ ay + bz ay + bz Bill Gibson University of Vermont Growth SAMs Hats Rule 6 Weighted average. Let x = ay + bz with a and b constant. Calculate the rate of growth of x is more complicated and requires that we write x̂ = ay bz ŷ + ẑ ay + bz ay + bz When a variable is defined as the weighted sum of the two levels y and z with weights a and b, then the growth rate of x is the weighted sum of the growth rates of y and z. Bill Gibson University of Vermont Growth SAMs Hats Rule 6 Weighted average. Let x = ay + bz with a and b constant. Calculate the rate of growth of x is more complicated and requires that we write x̂ = ay bz ŷ + ẑ ay + bz ay + bz When a variable is defined as the weighted sum of the two levels y and z with weights a and b, then the growth rate of x is the weighted sum of the growth rates of y and z. Since the levels of y and z are changing, this rule is often difficult to apply Bill Gibson University of Vermont Growth SAMs Hats Example Let Y = C + I where C = 80 and I = 20. C is growing at 4% and I is growth at 2%. The rate of growth of Y is Answer: 0.8(4)+0.2(2)=3.6 but...next time Bill Gibson University of Vermont Growth SAMs Hats Example Let Y = C + I where C = 80 and I = 20. C is growing at 4% and I is growth at 2%. The rate of growth of Y is Answer: 0.8(4)+0.2(2)=3.6 but...next time Example Let Y = C + I where C = 83.2 and I = 20.4. C is growing at 4% and I is growth at 2%. Y grows at Bill Gibson University of Vermont Growth SAMs Hats Example Let Y = C + I where C = 80 and I = 20. C is growing at 4% and I is growth at 2%. The rate of growth of Y is Answer: 0.8(4)+0.2(2)=3.6 but...next time Example Let Y = C + I where C = 83.2 and I = 20.4. C is growing at 4% and I is growth at 2%. Y grows at Answer: [83.20/(83.2+20.4)]4+[20.4/(83.2+20.40)]2=3.61 ...and so on with weights changing each time Bill Gibson University of Vermont Growth SAMs Hats Rule 7 Average rate of growth xt = (1 + g )t x0 where x is any variable and x0 is the initial value and the final value is xt . Bill Gibson University of Vermont Growth SAMs Hats Rule 7 Average rate of growth xt = (1 + g )t x0 where x is any variable and x0 is the initial value and the final value is xt . We can calculate the average growth rate, g , by solving this equation xt g = ( )1/t − 1 x0 Bill Gibson University of Vermont Growth SAMs Hats Rule 7 Average rate of growth xt = (1 + g )t x0 where x is any variable and x0 is the initial value and the final value is xt . We can calculate the average growth rate, g , by solving this equation xt g = ( )1/t − 1 x0 Rules only apply for small changes; for large changes they are only approximations Bill Gibson University of Vermont Growth SAMs Hats Rule 7 Average rate of growth Example 2005 = 1 2006 = 2 2007 = 3 What is the average rate of growth? Bill Gibson University of Vermont Growth SAMs Hats Rule 7 Average rate of growth Example 2005 = 1 2006 = 2 2007 = 3 What is the average rate of growth? Answer: (last/first) raised to inverse number of growth periods then subtract one = (3/1)1/2 − 1 = 0.73 Be careful about parentheses on these! Bill Gibson University of Vermont Growth SAMs Hats Rule 8: Doubling rule Growth path given by xt = x0 (1 + g )t where x is any variable and x0 is the initial value Bill Gibson University of Vermont Growth SAMs Hats Rule 8: Doubling rule Growth path given by xt = x0 (1 + g )t where x is any variable and x0 is the initial value Solve for t with ( Bill Gibson xt )=2 x0 University of Vermont Growth SAMs Hats Rule 8: Doubling rule Growth path given by xt = x0 (1 + g )t where x is any variable and x0 is the initial value Solve for t with ( xt )=2 x0 Take logs of both sides and not that ln(1 + g ) ≈ (1 + g ) tg = ln 2 = 0.693147 Bill Gibson University of Vermont Growth SAMs Hats Rule 8: Doubling rule Growth path given by xt = x0 (1 + g )t where x is any variable and x0 is the initial value Solve for t with ( xt )=2 x0 Take logs of both sides and not that ln(1 + g ) ≈ (1 + g ) tg = ln 2 = 0.693147 Doubling time t = 0.69/g or “rule of 70” Bill Gibson University of Vermont Growth SAMs Hats Hat rules Example A country is growing at 3.5 percent. Approximately how long will it take for income to double? Bill Gibson University of Vermont Growth SAMs Hats Hat rules Example A country is growing at 3.5 percent. Approximately how long will it take for income to double? Answer: 70/3.5 = 20 years Bill Gibson University of Vermont
