Question

PART 3: Long questions (2 questions; 20 points each): 1. Let GDP per capita in year t be denoted by Yt. Suppose GDP per capita grows at a constant rate g per year. Let the initial value of GDP per capita (that is, the value in year 0) be denoted by yo. (a) What is the value of GDP per capita in year 1? That is, write the expression for yı as a function of yo and g. (b) What is the value of GDP per capita in year 4? That is, write the expression for y4 as a function of yo and g. (c) What is the value of GDP per capita in year t? That is, write the expression for yt as a function of yo and g. (d) Suppose you know the value of GDP per capita in year 0 and in year 5. That is, you know the value of yo and Y5. What is the average annual growth rate of GDP per capita between year 0 and year 5? That is, write the expression for g as a function of yo and y5.

Answer 1

a)

Since GDP grows at a constant growth rate,

y₁=y₀*(1+g)

b)

y₄=y₀*(1+g)*(1+g)*(1+g)*(1+g)=y_o*(1+g)⁴

c)

y_t=y₀*(1+g)^t

d)

We know that

y₅=y₀*(1+g)⁵

On rearranging we get

(y₅/y₀)=(1+g)⁵

or

(1+g)=(y₅/y₀)^1/5

or

g=(y₅/y₀)^1/5-1