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

Question

approaches (converge to) 𝑘
∗
is λ = 𝑛 + 𝛿 − 𝛼 𝛿 + 𝑛 = (1 − 𝛼)(𝛿 + 𝑛). (Max 2

Answer 1

The Solow growth model is given by the equation:

𝑘̇ = sy − (n + δ)k

where k represents the capital stock per worker, s is the savings rate, y is output per worker, n is the population growth rate, and δ is the depreciation rate. The steady-state level of capital per worker, denoted as k*, is the level at which capital does not change over time:

0 = s𝑦* − (n + δ)k*

At the steady state, the change in capital stock per worker is equal to zero, so we have:

s𝑦* = (n + δ)k*

Since output per worker is a stable function of capital per worker, we can write 𝑦* = f(k*) where f is the production function. Taking the derivative of both sides with respect to k*, we have:

𝑦′ = 𝑓′(k*) = (1 - α)𝑘*

Substituting this back into the steady-state equation, we get:

s(1 - α)𝑘* = (n + δ)k*

Dividing by k* on both sides gives us:

s(1 - α) = (n + δ)/k*

Rearranging this equation gives us the desired result:

k* = (n + δ) / (s(1 - α)) = 𝜆

Therefore, the rate at which k converges to k* is given by λ = 𝑛 + 𝛿 − 𝛼 𝛿 + 𝑛 = (1 − 𝛼)(𝛿 + 𝑛).