Question

Let us consider a Solow growth model in which the aggregate production function

at time t for country i is given by:

Yⁱ_t =Aⁱ(Kⁱ_t)^1-βi(eⁱ Lⁱ_t) ^βi

Where Yⁱ_t is the aggregate real GDP in country i, Kⁱ_t   is the aggregate physical capital in country i, Lⁱ_t     is the aggregate number of workers in country i, eⁱ is the average working time of a worker parameter in country i, Aⁱ >0 is the total factor productivity parameter in country i and βⁱ ∈ (0,1) is the labor share of output parameter in the country i. The equilibrium law of motions of the physical capita per worker from time t to time t+1 in country i can be written as:

                               (1+nⁱ) kⁱ_t+1=yⁱ yⁱ_t +(1- ?ⁱ) Kⁱ_t

where ?ⁱ ∈ (−1, +∞) represents the growth rate of the population of workers parameter in country i, ?ⁱ ∈ (0,1) denotes the investment rate parameter in country i, ?ⁱ ∈ (0,1) is the depreciation rate parameter in country i and ?ⁱ_t denotes the output per worker in country i.

1. Write-down the production function in per worker units at time t for country i and derive the steady-state physical capital per worker and the steady-state real GDP per worker formulas for country i. (please specify if more info needed and what kind of info)

Answer 1

production function at time t for country i is given by:

(1) Yⁱ_t =Aⁱ(Kⁱ_t)^1-β_i(eⁱ Lⁱ_t) ^β_i

To write-down the production function in per worker units at time t for country i, divide both sides of the production function by eⁱ Lⁱ_t.

We get,

(2)    ,this represents the producion function in per worker units.

where y_tⁱ, represents real GDP per worker units and k_tⁱ represents physical capital per worker units at time t for country i.

Equilibrium law of motions of the physical capita per worker from time t to time t+1 in country i:

(3)    (1+nⁱ) kⁱ_t+1= ?ⁱyⁱ_t +(1- ?ⁱ) Kⁱ_t

In solow model, steady state condition implies that

kⁱ_t+1 = Kⁱ_t

put the above value in (3),

(1+nⁱ) kⁱ_t= ?ⁱyⁱ_t +(1- ?ⁱ) kⁱ_t

this implies,

(nⁱ + ?ⁱ) kⁱ_t = ?ⁱyⁱ_t

(nⁱ + ?ⁱ) kⁱ_t = ?ⁱ A_i (k_tⁱ)^1-β_i

(kⁱ_t)^1-1+β_i = ?ⁱ A_i / (nⁱ + ?ⁱ)

(4) , it is the formula for the steady-state physical capital per worker.

The steady-state physical capital per worker:

put the the value of k_tⁱ from (4) into (2), we get

, this is the formula for the steady state real GDP per worker.