This and the next couple posts are going to revolve around the Solow growth model. It’s important for one to be familiar with this model (also called the Solow-Swan model after the economists who developed this model in 1956, though independently) for this post to make sense. For those of you who are unfamiliar or want a quick refresh, here are some basics.
Solow Model Basics
1) What determines the output of an economy (I’m referring to the supply side factors)? Labour (L), Capital (K) and Technology right? So the Solow growth model provides the framework for understanding how changes in these factors (labour, capital and technology) determine the output (Y) of an economy and its growth over time.
2) The Solow model assumes that the production function of this economy (any economy) has constant returns to scale i.e. if you increase both labour and capital by a factor of “x”, the output will also rise by a factor of “x”.
Y = F (K, L) such that xY = F (xK, xL)
The reason this assumption is made is because this allows L, K and Y to be expressed on a per worker basis. On a per worker basis, the production function becomes:
Y/L = F (K/L, L/L) or y = f (k, 1) or y = f (k)
This implies that the output per worker (y) is a function of the capital per worker (k).
3) With a constant savings rate (s) (Savings/Output), a constant rate of depreciation (γ), population growing at a rate “n” per year, and technology improving at a rate “g” every year (technology improvement is represented as an increase in the efficiency of workers at a rate of “g” every year):
The steady-state level of capital (k*) is where “k” (capital per worker) is constant, which means that “y” (output per worker) is constant.
This happens when investment per worker (or the increase in capital per worker) = the amount of capital required to account for depreciation and the increase in labour force (g + n).
Δk = s f (k) - (γ + n + g) k
where investment = savings rate x output = s f (k)
If Δk is 0 i.e. k is constant, then:
s f (k*) = (γ + n + g) k*
or
k* = s f (k*)/ (γ + n + g)
This is the mathematical expression for the Steady-state level of Capital (k*). See Figure 1 below.
Figure 1: Steady-state Level of Capital
If the depreciation rate (γ), the population growth rate (n) and the rate of technology improvement (g) are constant, then for a given savings rate (s), there is just one steady-state level of capital (k*).
4) If capital per worker (k) lies below the steady-state level of capital (k*), then as you can see in the chart above, investment per worker [s f (k)] will be higher than the amount needed to account for depreciation, population and technology growth [(γ + n + g) k]. As a result, “k” will grow till the steady state level is reached.
Alternatively, if k > k*, then investment will be less than the amount needed to keep capital per worker constant and “k” will fall till it is equal to k*.
Thus, an economy will tend to naturally move towards its steady-state level of capital, which represents the long-run equilibrium of the economy. At the steady state level, output per effective worker will remain constant, while total output will grow at a rate of n + g.
5) If the rate of savings (s) in the economy rises, so will the steady-state level of capital. In Figure 1 above, an increase in “s” will manifest as a steeper investment curve. As a result, the investment curve [s f (k)] and the break-even level of capital curve [(γ + n + g) k] will interest at a capital level that is higher than k*.
6) The Golden Rule level of Capital (k*gold) is that steady-state level of capital that maximizes consumption. While there can be different steady-states for the economy depending on what the savings rate is, only one particular savings rate will get the economy to its Golden Rule level of Capital. See Figure 2 below.
Figure 2: Golden Rule Level of Capital
See the bottom panel first. You’ll see that the economy is at its steady state level of capital k*. What is the level of consumption per worker at k* level of capital? It is equal to the distance between the output curve f(k) and investment curve, s1 f(k). Note: we assume that output = consumption + investment. Since at the steady state, the investment curve always interests the break-even level of capital curve, [(γ + n + g) k], the level of consumption at k* is = the distance between the output curve and break-even level of capital curve.
Is k* the Golden Rule level of Capital? No. The Golden Rule level of Capital is at k*gold because this is where the distance between the output curve and the break-even level of capital is the maximum. This is visually apparent in the figure 2. That said, mathematically, this happens when the slope of the output curve is equal to slope of the break-even level of capital curve.
And what is the slope of the output curve? It is = Δf (k)/ Δk which is nothing but the Marginal Product of Capital (the units of output added when you increase the amount of capital used by 1 unit).
So, the Golden Rule level of Capital is where:
MPK = γ + n + g (MPK = Marginal Product of Capital)
Now look at the top panel of Figure 2. Here, the Steady-state level of capital is = Golden Rule level of capital = k*gold. At k*gold level of capital, consumption per worker is the maximum.
Why is the Golden Rule level of Capital achieved in the top panel and not in the bottom panel (we assume that f(k), γ, n and g are the same for both)? This is because in the top panel, the savings rate (s2) is precisely the right rate needed to achieve the Golden Rule level. At any other savings rate (like s1 in the bottom panel), the Golden Rule level will not be achieved.
Is India Operating below its Golden Rule Level of Capital?
So most developing countries are operating below their Golden Rule Level of Capital. India is too. But can we demonstrate this numerically using the Solow model? Yes, we can. I will use the technique that Mankiw uses in Macroeconomics, edition 5, Chapter 8. He uses the case of the US for illustration. I will use India.
So, in order to figure out if the Indian economy is operating at its Golden Rule level of Capital, we need to check if MPK = (γ + n + g).
For n + g, we’ll use the average rate of growth in GDP (output). For estimating MPK and γ, we’ll use some related statistics. See table below.
CORRECTION: In the table above, 66.0% is share of labour in National Income in 2009-10, and not the share of Capital. The share of capital is 34.0%.
From the table above, we get these relationships for India:
1) k = 1.86 y (where k = per capita capital stock, y = per capital GDP)
2) γ k = 0.11 y (where γ k = depreciation per capita)
3) MPK x k = 0.34 y
(Note: MPK x k gives us the total income earnt by capital, since capital earns it Marginal Product)
4) n + g = 0.07
(As I mentioned above, we use the average real GDP growth rate as an estimate for n + g)
For γ, substitute 1) in 2).
γ x 1.86 y = 0.11 y
γ = 0.11/1.86 = 0.059 = 5.9%
For MPK, substitute 1) in 3).
MPK x 1.86 y = 0.34 y
MPK = 0.34/1.86
MPK = 0.1827
We know that n + g = 0.07 and γ = 0.059.
γ + n + g = 0.129
Thus, for India, MPK > γ + n + g. This means that India is operating at a level of capital (k), which is much lower than the Golden Rule level of Capital.
If you look at Figure 2, this means that India is like the economy depicted in the bottom panel. At k*, it is operating in the region where the slope of the output curve is greater than the slope of break-even level of capital curve i.e. in the region that lies below the Golden Rule level of Capital (k*gold).
If India is to operate at its Golden Rule level of Capital, the rate of savings will need to be raised.
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