Difference in economic may lead to vast differences in standards of living, especially for human welfare. A small difference in growth might result to a big difference to income over long periods of time, such as contrast Argentina with Austria. Many other differences, for example nutrition, literacy, etc, are also associated with real income differences. Therefore, welfare consequences of long-run growth swamp any possible effects of the short-run fluctuations that macroeconomics traditionally focus on (Dome, 1996). The Solow Growth Model is one of the models which economists have traditionally used to study, and it is also regarded as the starting point for almost all analysis of growth.
Quote from Dome: the principal conclusion of the Solow model is that the accumulation of physical capital cannot account for either the vast growth over time in output per person or the vast geographic differences in output per person. Briefly, Solow Growth Model owns self-correcting mechanism which extends the Harrod-Domar Model and assumes that capital directly contributes to production, and other potential sources of differences are exogenous. Therefore, it is hard to explain the enormous differences between real income and capital inputs.
There are four main variables in Solow Growth Model: output (Y), capital (K), Labour (L), and technology (A). The production function could be structured as: Y(t)=F (K(t), A(t)L(t)) where t denotes time
In the production function, time t only affects K, L and A, and then indirectly influence the output. In other world, the change of output over time is only caused by the change of inputs in the production function. If there is an increase in technology, the productivity of capital K will increase (the capital-augmenting Y=F(AK,L)). When the labour L benefit from technology progress (AL), then the effectiveness of labour will increase. ( labour-augmenting, Y=FK,AL). It is called Hicks-neural when the technology progress forms a production function likeY=AF(K,L).
In order to make the analysis works well, there are 6 assumptions in total. But the model’s critical assumption concerning the production function is that It has constant returns to scale in is two arguments, capital and effective labour (Dome, 1996). Basically, it means that if both K and L increase c times in quantity (c is constant) with fixed A, the output will increase c times as well. Mathematically, F (cK,cAL)=cF(K,AL)=cY. However, the constant returns to scale relay on two sub-assumptions. First it supposes that the economy is sufficient large in order to use extra inputs essentially. Second, except K, L and A, other influential factors are ignored such as natural resources. However, these two situations are all difficult to exist in reality.
The other assumptions under Solow Model are: 1: K and L are substitute which indicates that the quantity of L and K moves in opposite way in order to keep output level. 2: Law of diminishing marginal returns holds. This means that as additional inputs increase, the output increases in a deceasing rate. In other world, the first order condition of production function is positive (dYdK>0), and the second order condition is negative (d2YdK2<0).so does the labour.3: Production Function is homogeneous of degree 1. This implies that if all inputs are increased by 3 times, the output will increase 3 times. 4: Capital depreciates at a fixed rate, δ. This assumption is also associated with dynamic of capital. Under this assumption, capital becomes less valuable at a fixed rate over time.5: Labour grows at a constant rate, n. 6: Individuals save a constant fraction of their income, because a change in saving rate will affect both consumption and output under Solow Model.
By taking advantages of the assumption of constant return to scale, we can analyze in per capital rather than in overall. Setting c=1L and multiply in both side of production function, we get