1. Using the case of Solow growth model, show that the rate at which k

Question

1. Using the case of Solow growth model, show that the rate at which k

approaches (converge to) 𝑘
∗
is λ = 𝑛 + 𝛿 − 𝛼 𝛿 + 𝑛 = (1 − 𝛼)(𝛿 + 𝑛). (Max 2
Pages)
2. In the Solow growth model, suppose the per worker production function is
given by 𝑦 = 𝑧𝑘0.4
, with 𝑠 = 0.4 , 𝑑 = 0.2 , and n = 0.02. (Max 2 Pages)
a) Suppose that in country A, z = 1. Calculate income per worker and capital
per worker in the steady state.
b) Suppose that in country B, z = 2. Calculate income per worker and capital
per worker in the steady state.
c) Will countries A and B converge in terms of income per worker? Explain.
d) Based on your answers to parts (a) and (b), if both countries increase z to
3, which country would have higher economic growth during the transition
period to the steady state?

Answer 1

1. The Solow growth model is given by the following equations:

Output per capita: y = zk^α
Investment per capita: i = sy
Capital accumulation: dk = i - δk
Population growth: n = (1 - λ)ln
Technological progress: z = (1 + g)^(1-α)

In the steady state, the capital stock k* is constant and the investment per capita equals the depreciation plus population growth rate (i=k*(δ+n)). Therefore, using the investment per capita and capital accumulation equations, we can find the steady-state capital per worker as:
k* = (s / (δ + n))^(1/ (1- α))
k* = (s / (δ + n))^(1/(1 - α))
= (s / (δ + n)) * (1/(1 - α))
= (s / (δ + n)) * (α / (1 - α))
= (αs / ((1 - α)(δ + n)))

Now, comparing this with the given equation λ = n + δ - α δ + n = (1 - α)(δ + n) we see that they are equivalent.

2.
a) For country A where z = 1, we have:
k* = (0.4 / (0.2 + 0.02))^0.6
k* = (0.4 / 0.22)^0.6
k* = 1.818
y* = 1 * (1.818)^0.4
y* = 1.250

b) For country B where z = 2, we have:
k* = (0.4 / (0.2 + 0.02))^0.6
k* = (0.4 / 0.22)^0.6
k* = 1.818
y* = 2 * (1.818)^0.4
y* = 1.767

c) No, countries A and B will not converge in terms of income per worker as country B has a higher z value resulting in higher income per worker in the steady state.

d) If both countries increase z to 3, country B would have higher economic growth during the transition period to the steady state since it has a higher initial z value. This means that country B can reach the new steady state level of income per worker faster than country A.