Question

Suppose that you have a standard Solow model with a Cobb-Douglas production function. The central equation of the model can be written:

kₜ₊₁=sAkᵗα+(1−δ)kₜ.
Analytically solve for an expression for the steady state capital stock per worker.

Answer 1

To find the steady state capital stock per worker in the Solow model, we set kt+1 equal to kt and solve for k*, resulting in k* = (sA/δ)1/(1-α) as the expression for steady state capital stock.

Step-by-step explanation:

To find the steady state capital stock per worker in the standard Solow model with a Cobb-Douglas production function, we need to set the capital accumulation equation to a point where the capital stock does not change, that is when kt+1 equals kt.

The central equation provided is kt+1 = sAktα + (1-δ)kt, where s is the savings rate, A is total factor productivity, α is the output elasticity of capital, and δ is the depreciation rate.

At the steady state, we set kt+1 = kt = k* (steady state capital stock per worker) and solve for k*:

k* = sAk*α + (1-δ)k*

To isolate k*, we rearrange the equation:

k* - (1-δ)k* = sAk*α

k* (δ) = sAk*α

k*1-α = sA / δ

Finally, we find the steady state capital stock per worker:

k* = (sA / δ)1/(1-α)

This gives us the expression for k* in the Solow model's steady state.