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Solow growth model assumptions of solow growth model and steady state in solow model

The Solow Growth model is an exogenous model of economic Growth  that analyzes changes in the level of output in an economy over time as a result of changes in the population growth rate, the savings rate, and the rate of technological progress.

Assumptions of the Solow growth model:

1. One composite good produced which is either consumed or accumulated in capital form
2. Homogeneous labour
3. Two factors of production— Capital (K) and Labour
4. Constant Return to Scale (CRS) production function
5. Economy exhibits diminishing returns in the labor and capital inputs.
6. Marginal Propensity to Save (s)—assumed constant
7. Exogenously given labour force growth rate (n)
8. Closed economy and Laissez Faire

Structure of the solow growth model:

Suppose that there is no depreciation, Consider aggregate production function.

Where Y is the aggregate output K is the aggregate capital and L is the labour force
We have already assumed that production function exhibit constant returns to scale that means
if L,K increases by λ then Y will increase by λ.
Y=F(K,L) for all λ>1
Let λ=1/L
then Y/L = F(K/L, L/L)
or y=f(k)
where y is output per labour
 k is capital per labour

if we draw the picture of the above equation

In the above diagram output per worker(y) depicts on vertical axis while capital per worker(k) depicts upon horizontal axis.
The slope of the curve at any point represents the marginal product of capital (MPK), the slope in the 
graph decreases because of diminishing MPK. This implies that as an additional unit of capital added to a 
fixed labour supply , the gain in output is positive but less than the extra output generated from the 
addition of the previous unit of capital.
Accumulation of Capital: The change in the capital stock per worker (known as capital deepening) is the 
investment per worker, given by 
i = 𝑆/L (here i is a investment and we know that saving equals to investment )
i = 𝑠𝑌/𝐿 since Y/L =y
i= sy since y = f(k)
i = s f (k) 
where s is the MPS and i = 𝐼/ 𝐿 , and as I = S, we have 𝑖 = 𝑆/𝐿
Graphically given as

For equilibrium and
Equilibrium and steady state

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That is at equilibrium, the investment function and the 45 line (nk function) cut each other at k*.
 If , k < k* then,  sf (k) > nk, so k increases towards k*
 If , k > k* then,  sf (k) < nk, so k decreases towards k* Once the economy gets to k*, the capital stock does not change. 
Also, when k = k* , we get, y = y*, that is, at equilibrium both capital and output per capita level are settled to the constant levels k*  and y* , respectively. 
This in turn results in the constant capital-output ratio at equilibrium. With k (per capita capital) and y (the per capita output) constant at equilibrium, we get the  growth rate of capital (K) and output (Y) equal to the exogenous growth rate of labour (n) at equilibrium.

Conclusion from the Solow growth model Model 

1. There exists a steady state (balanced growth) solution for the model, which is achieved whatever be the initial values of all the variables. The economy eventually reverts to the steady state equilibrium value of y and k (as we saw the adjustment when k ≠ k* ).  
2. The balanced rate of growth of output and capital is the constant exogenous growth of the labour force, which is n. 

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