th t r CHENG Fangyuan ITSC IDfcheng Student ID20369816 Assignment 1 1. Since Y Y er*h , so rln 𝑌𝑡ℎ 𝑌𝑡 /h 1 instantaneous rate 1.000172 rln 𝑌2011 𝑌1981 /30 ln 7.0716 5.2385 /30 0.01000172 2 discrete rate 1.005190 according to the discrete model, Rexpr-1 r ln 𝑌2011 𝑌1981 /30 ln 7.0716 5.2385 /30 0.01000172 Re0.01000172-1 0.01005190 3 central rate 0.9927349 𝑌𝑡ℎ -𝑌𝑡 h*𝑌𝑡ℎ𝑌𝑡 2 7.0716-5.2385 30*7.07165.2385/2 0.009927349 These three rates are not much different. Y Y e th T Doubling time r*h th T , so hln 𝑌𝑡ℎ 𝑌𝑡 /r 𝑙𝑛2 h 69.3 0.01000172 The doubling time implied by the 1981 2011 growth rate is 70 years. Times that HK’s population reaches a size of 10 million hln 𝑌𝑡ℎ 𝑌𝑡 /r ln 10 7.0716 / 0.01000172 34.64 In 2046 HKs population will reach 10 million. Growth rate to reach 10million rln 𝑌𝑡ℎ 𝑌𝑡 /h ln 𝑌2030 𝑌2011 /19 10 ln /19 7.0716 0.018236754 1.82 is high enough for HK population to grown to 10million by 2030. 2. According to the instantaneous growth rate model Y Y er*h , the population growth rate r ln 𝑌𝑡ℎ 𝑌𝑡 /h, the population doubling time h ln 𝑌𝑡ℎ 𝑌𝑡 𝑙𝑛2 /r r 662 1950-1960 r ln 𝑙𝑛2 /100.018171750, h 𝑙𝑛2 38.14 552 r 0.018171750 830 1960-1970 r ln /100.022616014, h 𝑙𝑛2 30.6485 662 0.022616014 987 1970-1980 r ln /100.017324434, h 𝑙𝑛2 48.39 830 0.017324434 1143 1980-1990 r ln /100.014674162, h 𝑙𝑛2 47.24 987 0.014674162 1250 1990-2000 r ln /100.008948717, h 𝑙𝑛2 77.46 1143 0.008948717 Decades Growth Rate Doubling Timeyears 1950-1960 1.82 39 1960-1970 2.26 31 1970-1980 1.73 49 1980-1990 1.47 48 1990-2000 0.89 78 2000-2010 0.62 112 Y Y e Y Y *e 1330 2000-2010 r ln /100.006203539, h 𝑙𝑛2 111.73 1250 0.006203539 1 According to continuous time model, r ln 𝑌𝑡ℎ 𝑌𝑡 /h 830 1960-1970 r ln 662 /100.022616014 r*h th t 0.0022616014*50 2010 1960 662* e0.0022616014*50 2050.97 The population will reach 2050.97 million in 2010. 2 h ln 𝑌𝑡ℎ 𝑌𝑡 /r 1500 ln 1330 /0.006203539 19.39 2015h 2015h 1. 2028 2025 In 2030 the population will reach 1.5billion. 3.1 India Y 1.26*e0.012*h China Y 1.37*e0.005*h 𝑒0.012*ℎ 𝑒0.005*ℎ 1.37 1.26 ln 1.37 h 26 0.007 h 11.957 In 2027, India will overtake China as the most populous country. 2 Assuming n is the time that Chinas GDO overtake Americas GDP, according to the GDP growth model, China10.866*10.069n USA 17.947*10.026n 10.866*10.069n 17.947*10.026n 10.069𝑛 17.947 10.026 10.866 1.0419103n 17.94710.866 17.947 n log 10.866 n 12.22 In 2028, China will overtake USA as the largest economy. China Population size Y Y er*h 1.37*e0.005*13 1.462 billion 13 13 GDP G2028G2015*10.069 10.866*10.069 25.869 trillion GDP per capita India 𝐺2028 𝑌2028 25.869*1012 1.462*109 17694.169 r*h 0.012*13 Population size Y2028Y2025 e 1.26*e 1.473 billion 13 13 GDP G2028G2015*10.078 2.074*10.078 5.506 trillion GDP per capita USA 𝐺2028 𝑌2028 5.506*1012 1.473*109 3737.950 r*h 0.007*13 Population size Y2028Y2025 e 0.32*e 0.350 billion 13 13 GDP G2028G2015*10.026 17.947*10.026 25.056 trillion GDP per capita 𝐺2028 𝑌2028 25.056*1012 0.350*109 71588.571