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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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 Fall '11
 Dr.Gwartney
 Economics

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