The solow-swan model of economic growth
The figure above represents a solow-swan diagram in which there are constant returns to capital K and labour L. As we know, for the returns to capital to be constant, the production function has to be Yt = AKt. This therefore means that labour is assumed to be constant as it is growing at an exogenous rate and it is expressed as n ≡ (dL/dt)/L. to show the constant rate in the figure, labour growth (n) is represented by a straight line. Given that the savings rate, s, is fixed, the investments given as dK/dt is equal to the savings. The rate growth of capital will therefore be given as (dK/dt)/K = sY/K. In the diagram, the capital growth rate is given as a straight line through the origin since Y/K varies along the x axis.
The equilibrium attained at E occurs when rates of growth of labour, capital and output coincide. To the left of the equilibrium point, output is growing at a faster rate than capital therefore, Y/K rises towards (Y/K)E and to the right of E, output is growing slower than the capital thereby prompting the Y/K to fall.
The solow-swan model of economic growth links output to capital and labour inputs in a bid to explain the steady state of an economy. More so, to understand how the steady state comes about, specific equations must be understood. With the assumption that technical progress is constant, the production function will be given as Y = F (K,L), where L represents labour and K capital. Given the condition of constant returns to scale, the production function can be divided by L and the new production function is given as Y/L = F (K/L, 1) = L.f(k).
Y = Y/L refers to output or income per worker and k= K/L is the capital-labour ratio.
Therefore, the production function can further be expressed as y= f (k). In the Swan model, savings (s) is regarded as a constant, usually a fraction, of the income. Savings for each worker is then sy, and since income is equivalent to the output, sy = sf(k). To sustain capital per worker (k), an investment is required and this investment is determined by population growth and the depreciation rate expressed as d. Population growth is assumed to be at a constant rate n, and the stock for capital is then said to grow at a rate expressed as n.k in order to provide enough capital for the growing population.
In the model depreciation is also a constant, d, and if it is calculated per cent of the capital stock, the resulting expression, d.k, becomes the investment needed to replace capital that wears out over time. The depreciation investment d.k, which is per worker, is then added to nk, which is the investment per worker to sustain the capital-labour ratio for the growing population. The resulting equation, (nk + dk) = (n + d) k, becomes the required investment to maintain the capital per worker.
From all this, we can deduce the fundamental equation for the solow-swan model which helps us in determining the steady state. For a steady state to be achieved, sf(k) = (n + d)k where k = 0.
In this figure, the steady state can then be explained. y =f(k) is the production function which depicts that the output per worker increases at a diminishing rate as k increases as per the law of diminishing returns. sf (k) curve shows the savings per worker while the (n + d) k is the investment requirement line which has a positive slope equal to (n + d). In this diagram, the steady state is achieved at point E where the sf (k) curve intersects the (n+d)k, to show that at that point, the savings per worker and investment per worker are equal.
In the question, we are assuming that the sum of the population growth and the depreciation rate, (n + d) k, is less than the savings rate expressed as sf (k). With this we can conclude that such an economy does not converge at a steady state. This situation is explained by an economy that starts thecapital-labour ration at k1. At this point, savings per worker k1B are more that the investment necessary to maintain the capital-labour ratio cosntant, hence convergence is not at a steady state.
References
Agénor. (2004). The economics of adjustment and growth. Cambridge, Mass: Harvard University Press.
Barro, R. & Martin, X. (2004).
Economic growth. Cambridge, Mass: MIT Press.
Ramanathan, R. (1982). Introduction to the Theory of Economic Growth. Berlin, Heidelberg: Springer Berlin Heidelberg.
Solow, R. (1977). Growth theory : an exposition. Oxford: Clarendon Press.