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.(Already answered )

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.(already answered)

3.Write down the production function in per capita units.(Already answered)

4.Write-down the equilibrium law of motion of the physical capital per capita if all the labour income is saved:S_t =w_tL_t .Derive the steady state physical capital per capita and the steady-state output/income per capita(Already answered)

Question: Write-down the equilibrium law of motion of the physical capital per capita if all the physical capital income is saved: S_t=r_tK_t.

Derive the steady-state physical capital per capita and the steady-state output/income per capita.(please specify what information is needed)

Answer 1

Equilibrium law of motion of the physical capital per capita:

The law of motion of physical capital is given by:

K_t+1=(1-?)K_t+I_t

We know that I_t=S_t Here, we have S_t=r_tK_t

therefore, K_t+1=(1-?)K_t+r_tK_t

K_t+1=(1-?+r_t)K_t

In per capita terms, this can be written as:

(1+n)k_t+1 = (1-?+r_t)k_t  

k_t+1= (1-?-n+r_t)k_t . This gives the equilibrium law of motion of physical capital per capita.

To get the steady state physical capital per capita:

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

^{Converting into per capita terms;}

?_t/L_t= (??_t^1-? L_t^?)/L_t

y=Z(K/L)^1-? == y=Z.k^1-?

^{We know that at steady state per capita capital stock has to be constant i.e.}

Δk= s.f(k)-?k (Δk=0)

s.f(k)-?k =0

k/f(k)=s/?

k/Z.k^1-? =s/?

k^? =(s/?).Z

k= ((s/?).Z)^1/?

Since S_t=r_tK_t , s= r_t(K_t /?_t)

k=((r_tk_tZ)/?)^1/?

Solving this for k, we get:

k=((r_tZ)/?)^1/(?-1)

This gives the steady state level of capital.

To get the steady state level of output, we put this value in the putput function:

y=Z.k^1-?

y=Z.(((r_tZ)/?)^1/(?-1) )^1-?

y=Z.(((r_tZ)/?)^-1/(1-?) )^1-?

y=Z.((r_tZ)/?)^-1 )

y=Z.(?/r_tZ)

y=(?/r_t)

This gives the steady state level of output.