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In: Economics

Let us consider a production economy endowed with a single perfectly competitive firm renting at every...

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.

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