Question

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= ??_t^1-? L_t^?

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 ≡ Y_t/N_t denote the output/income per capita at time t and let ?_t ≡k_t/N_t stand for the physical capital per capita at time t.

Question :Write down the production function in per capita units

Answer 1

· Consider a production economy endowed with a single perfectly competitive

Firm renting at every time = t

Households at the real rental rates = ?_t, ?_t

Equilibrium at time

T: ?_t = ???_t and ?_t= ???_t

Where ???_t = marginal product of labour

???_t = marginal product of physical capital

Aggregate output/income ?_t:

                                                   ?_t= ??_t^1-? L_t^?

?_t = the physical capital

?_t = the number of workers

· Let us assume that a constant fraction y∈ (0,1)

Total household population of size = ?_t

                                       ?_t = ??_t

· 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

Aggregate saving in equilibrium: ?_t = ?_t

Output/income per capital: ?_t ≡ Y_t/N_t

Physical capital per capita at time t : ?_t ≡k_t/N_t