Using the case of Solow growth model, show that the rate at which k
approaches (converge to) π
β
is Ξ» = π + πΏ β πΌ πΏ + π = (1 β πΌ)(πΏ + π). (Max 2
The Solow growth model is given by the equation:
πΜ = sy β (n + Ξ΄)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 Ξ΄ 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 = sπ¦* β (n + Ξ΄)k*
At the steady state, the change in capital stock per worker is equal to zero, so we have:
sπ¦* = (n + Ξ΄)k*
Since output per worker is a stable function of capital per worker, we can write π¦* = f(k*) where f is the production function. Taking the derivative of both sides with respect to k*, we have:
π¦β² = πβ²(k*) = (1 - Ξ±)π*
Substituting this back into the steady-state equation, we get:
s(1 - Ξ±)π* = (n + Ξ΄)k*
Dividing by k* on both sides gives us:
s(1 - Ξ±) = (n + Ξ΄)/k*
Rearranging this equation gives us the desired result:
k* = (n + Ξ΄) / (s(1 - Ξ±)) = π
Therefore, the rate at which k converges to k* is given by Ξ» = π + πΏ β πΌ πΏ + π = (1 β πΌ)(πΏ + π).