Economics Dynamics Problems 99

# Economics Dynamics Problems 99 - Continuous dynamic systems...

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Unformatted text preview: Continuous dynamic systems 15. Given the differential equation ˙ Y− 16. s Y=0 v for the Harrod–Domar growth model: ˙ (i) construct a diagram of Y against Y and establish the phase line for this model (ii) establish Y (t) given Y (0) = Y0 . From Domar (1944), assume income Y (t) grows at a constant rate r. In order to maintain full employment the budget deﬁcit, D(t), changes in proportion k to Y (t), i.e., ˙ D(t) = kY (t) Show that D(t) = Y (t) 17. 18. 19. 20. D0 k k − e−rt + Y0 r r At what nominal interest rate will it take to double a real initial investment of A over 25 years, assuming a constant rate of inﬂation of 5% per annum? Table 2A.1 provides annual GDP growth rates for a number of countries based on the period 1960–1990 (Jones 1998, table 1.1). (a) In each case, calculate the number of years required for a doubling of GDP. (b) Interpret the negative numbers in the ‘years to double’ when the growth rate is negative. In 1960 China’s population was 667,073,000 and by 1992 it was 1,162,000,000. (a) What is China’s annual population growth over this period? (b) How many years will it take for China’s population to double? (c) Given China’s population in 1992, and assuming the same annual: growth rate in population, what was the predicted size of China’s population at the beginning of the new millennium (2000)? An individual opens up a retirement pension at age 25 of an amount £5,000. He contributes £2,000 per annum each year up to his retirement at age 65. Interest is 5% compounded continuously. What payment will he receive on his retirement? Table 2A.1 GDP growth rates, selected countries, 1960–1990 ‘Rich’ countries Growth rate France Japan West Germany UK USA 2.7 5.0 2.5 2.0 1.4 Source: Jones (1998, table 1.1). Years to double ‘Poor’ countries Growth rate China India Uganda Zimbabwe 2.4 2.0 −0.2 0.2 Years to double 83 ...
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