Economic Growth

...

In the absence of technological progress, economic growth eventually slows down because:

- Capital accumulation slows down and we cannot move away/beyond k*, and

- Production function is concave, i.e. has diminishing returns.

k* and y* are affected by population growth rates (n), savings rates (s) and depreciation rates (δ). Increasing savings rates can increase k* and y* only temporarily, but not permanently.

k* and y* set maximum possible economic growth rates. The maximum possible growth rate is population growth rate + depreciation rate. Any attempts to stimulate economic growth beyond the limits are doomed to failure.

Investment function

For the analysis, let us begin with the macroeconomic equilibrium condition that aggregate expenditure equal aggregate supply, or investment equals savings:

Ye = Y

I = S

Consider consumption function, C = bY, where b is the marginal propensity to consume.

Now, by definition, savings are S = Y - C = Y - bY or simply S = (1-b)Y. Letting s = (1-b), the marginal propensity to save, then we can express savings as some proportion of total output:

S = sY

So combining this with the macroeconomic equilibrium condition we get:

I = sY

Dividing through by L, the amount of labour in the economy, then:

I/L = s(Y/L)

so, letting i = I/L and y = Y/L, we see that the macroeconomic equilibrium condition becomes:

i = sy

Production function

Now, aggregate supply (output) is given by a production function of the general form:

Y = F(K, L)

which is assumed to vary continuously with K and L.

Dividing through by L [Solow model is capital accumulation model in contrast to Malthus]:

Y/L = F(K/L, 1)

or, letting k = K/L, we can rewrite this as:

y = f(k)

where f (*) is the "per capita" form of the production function. As a result, the macroeconomic equilibrium condition can be rewritten as:

i = sf(k)

This can be thought as representing equilibrium investment per person.

Showing investment and production functions on the graph

Figure below depicts the “per capita” production function y = f(k) and the actual (equilibrium) investment function, i = sf(k). Notice that at any k, we can derive investment per person, i, output per person, y, and, residually, consumption per person (c = C/L = y - i).

[pic 4]

The slope of the “per capita” production function is MPK = df(k)/dk which is the marginal product of capital.

Finally, notice that the capital-output ratio, v = K/Y = k/y, is captured as the slope of a ray from the origin to production function. 1/v is respectively Y/K or capital productivity.

Required rate of capital accumulation

By assumption, we assume population grows exogenously at the rate n, i.e.

gL = (dL/dt)/L = n

If there is no investment, then k = K/L will automatically fall as population grows. So, for k to be constant, there must be investment (i.e. capital must grow) at rate n:

gKr = (dK/dt)/K = n

,where we have attached the superscript "r" to indicate that this is the required growth rate of capital to keep the capital-labour ratio, k, steady. As investment is defined as I = dK/dt, then we can rewrite this as:

Ir = nK

,where Ir is required investment. Dividing through by labour, L, Ir/L = nK/L, or:

ir = nk

,which is the required investment per person to maintain a steady k.

Why are we obsessed with keeping k constant? Well, we are interested in "steady-state" growth which means "proportional" growth in a manner that there are no induced changes in relative prices over time.

It is obvious (from figure above, for instance) that a change in k will change the marginal products of capital and labour.

This will change the shares of labour and capital in GDP and will introduce additional social issues, which are difficult to model.

Putting all together

We can depict the steady-state k in Figure below, by superimposing the required investment function, ir = nk, on top of our old diagram.

[pic 5]

Convergence to steady state k* and y*

It is a simple matter to note that k* is a stable capital-labour ratio. If our initial capital labour ratio is below k* (e.g. at k1), then actual investment is greater than required investment, i > ir, which means that capital is actually growing faster than labour, so k will increase. Conversely, if our initial k is above k* (e.g. at k2), then actual investment is below required, i r, so capital is growing slower than labour, so k will fall. Thus, the steady-state capital-labour ratio, k*, is stable in the sense that any other k will have the tendency to approach it over time.

At k1, the supply of capital is bigger than demand for capital. This makes excess supply of capital and price of capital falls. Relative to labour capital becomes cheaper, and hence substitution from labour to capital starts. Firms adopt more capital-intensive technologies, k↑ and y↑ (capital accumulation accelerates) until k* and y* are reached.

At k2 in contrast, supply of capital is smaller than

...