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.

1. Derive the marginal product of labour and the marginal product of physical capital schedules at time t.(Please specify what information you need for this question if needed)

2. Show that in equilibrium, the aggregate output/income at time t is the sum of the aggregate labour income: ?_t?_t and the aggregate physical capital income: ?_t?_t.(please specify what information is needed for this if needed)

Answer 1