In: Economics
Let us consider a production economy endowed with a single perfectly competitive firm renting at every time t both labour and physical capital from households at the real rental rates ?t, ?t, respectively. In equilibrium at time t: ?t = ???t and ?t= ???t where ???t and ???t denote the marginal product of labour and the marginal product of physical capital, respectively. The aggregate output/income ?t at every time t is produced according to the following production function:
?t= ??t1-? Lt?
where Z>0 stands for the total factor productivity parameter, ?t represents the physical capital and ?t denotes the number of workers with ? ∈ (0,1) stands for the labour share of output parameter. Let us assume that a constant fraction y∈ (0,1) of the total household population of size ?t works at every time t:
?t = ??t
The aggregate population of households grows at a constant rate n∈ (0, +∞):
?t+1= (1 + ?)?t
The law of motion for the physical capital from time t to time t+1 can be written as:
?t+1 =?t + (1 −δ)?t
where δ ∈ (0,1) represents the physical capital depreciation rate parameter and ?t denotes the aggregate investment in physical capital at time t which is equal to the aggregate saving in equilibrium:
?t = ?t
Let ?t ≡ Yt/Nt denote the output/income per capita at time t and let ?t ≡kt/Nt stand for the physical capital per capita at time t.
Question:Write-down the equilibrium law of motion of the physical capital per capita if all the physical capital income is saved: St=rtKt. Derive the steady-state physical capital per capita and the steady-state output/income per capita.