Question

In: Economics

Let us consider a Solow growth model in which the aggregate production function at time t...

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

at time t for country i is given by:

Yit =Ai(Kit)1-βi(ei Lit) βi

Where Yit is the aggregate real GDP in country i, Kit   is the aggregate physical capital in country i, Lit     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) kit+1=yi yit +(1- ?i) Kit

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 ?it 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)

Solutions

Expert Solution

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

(1) Yit =Ai(Kit)1-βi(ei Lit) βi

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 Lit.

We get,

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

where yti, represents real GDP per worker units and kti 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) kit+1= ?iyit +(1- ?i) Kit

In solow model, steady state condition implies that

kit+1 = Kit

put the above value in (3),

(1+ni) kit= ?iyit +(1- ?i) kit

this implies,

(ni + ?i) kit = ?iyit

(ni + ?i) kit = ?i Ai (kti)1-βi

(kit)1-1+βi = ?i Ai / (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 kti from (4) into (2), we get

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


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