Start with a Constant Returns to Scale (CRTS) production function: Y = f (K,L). CRTS implies that by multiplying each input by some factor “z”, output changes by a multiple of that same factor: zY = f ( zK, zL)
In this case, let z = 1/L. That means:
Y * 1/L = f (K * 1/L, L * 1/L)
Y/L = f (K/L, 1)
define y = Y/L and k = K/L, so that the production function can now be written as
y = f (k),
where y is output per worker and k is capital per worker.
A graphical depiction of the production relation is:
The production function shows the production of goods. We now look at the demand for goods. The demand for goods, in this simple model, consists of consumption plus investment:
y = c + i
where y = Y/L; c = C/L; and i = I/L.
Investment, as always, creates additions to the capital stock.
The consumption function in this simple model is: C = (1 – s) Y,
which can be rewritten as c = (1 – s) y, where “s” is the savings rate and 0 < s < 1.
Going back to the demand for goods, y = c + i, we can rewrite this as
y = (1 – s) y + i
y = y – sy + i
so, y – y – sy = i
which means that sy = i: savings equals investment.
We can now put our knowledge to use by looking at a simple model of growth.
Investment adds to the capital stock (investment is created through savings):
i = sy = s f(k)
The higher the level of output, the greater the amount of investment:
Assume that a certain amount of capital stock is consumed each period: depreciation takes away from the capital stock. Let “d“be the depreciation rate. That means that each period d*k is the amount of capital that is “consumed” (i.e., used up):
We can now look at the effect of both investment and depreciation on the capital stock:
Dk = i – dk, which is stating that the stock of capital increases due to additions (created by investment) and decreases due to subtractions (caused by depreciation). This can be rewritten as Dk =s* f(k) – dk.
The steady state level of capital stock is the stock of capital at which investment and depreciation just offset each other: Dk = 0:
if k < k* then i > dk , so k increases towards k*
if k > k* then i <dk , so k decreases towards k*
Once the economy gets to k*, the capital stock does not change.
The Golden Rule level of capital accumulation is the steady state with the highest level of consumption. The idea behind the Golden Rule is that if the government could move the economy to a new steady state, where would they move? The answer is that they would choose the steady state at which consumption is maximized. To alter the steady state, the government must change the savings rate.
Since y = c + i,
then c = y – i
which can be rewritten as c = f(k) – s f(k)
which, in the steady state, means c = f(k) – dk. This indicates that to maximize consumption, we want to have the greatest difference between y and depreciation.
Since we want to maximize c = f(k) – dk, we take the first derivative and set it equal to zero:
Since we are looking at incremental changes in k, dk = 1, which leaves us with
the result that at the Golden Rule, the marginal product of capital must equal the rate of depreciation: MPK =d.
Introducing Population Growth
Let “n” represent growth in the labor force. As this growth occurs, k = K/L declines (due to the increase in L) and y = Y/L also decines (also due to the increase in L).
Thus, as L grows, the change in k is now:
Dk = s*f(k) – d*k – n*k,
where n*k represents the decrease in the capital stock per unit of labor from having more labor. The steady state condition is now that s*f(k) = (d+n) * k:
In the steady state, there’s no change in k so there’s no change in y. That means that output per worker and capital per worker are both constant. Since, however, the labor force is growing at the rate n (i.e., L increases at the rate “n”), Y (not y) is also increasing at the rate “n”. Similarly, K (not k) is increasing at the rate n.
Introducing Technological Progress
We shall assume that technological progress occurs because of increased efficiency of labor. That idea can be incorporated into the production function by simply assuming that each period, labor is able to produce more output than the previous period:
Y = f (K, L*E)
where E represents the efficiency of labor. We will assume that E grows at the rate “g”. Still assuming constant returns to scale, the production function can now be written as:
y = Y / L*E = f ( K/L*E , L/L*E ) = f (k), where k = K/L*E
We are now looking at output per efficiency unit of labor and capital per efficiency unit of labor.
Since k = K / L *E, we can see how k changes over time:
where, the sign of the first term on the right, kdis negative because capital is being consumed by depreciation (dK/K <0).
The steady state condition is modified to reflect the technological progress:
Dk = s*f(k) – (d+g+n)*k,
when Dk = 0 (i.e., at the steady state), s*f(k) = (d+g+n)*k.
At the steady state, y and k are constant. Since y = Y/L*E, and L grows at the rate n while E grows at the rate g, then Y must grow at the rate n+g. Similarly since k = K/L*E, K must grow at the rate of n+g.
The Golden Rule level of capital accumulation with this more complicated model is found by maximizing consumption at a steady state, which yields the following relation:
which simply indicates that the marginal product of capital net of depreciation must equal the sum of population and technological progress.
Let Y = K1/3(LE)2/3
with s = .25, n = .01, d=.1, and g = .015
The production function, because it exhibits CRTS, can be rewritten as
To find the steady state, recall that DΔk = 0, so s*f(k) = (d+n+g) k
which can be rewritten as:
s/ (d+n+g) = k / f(k)
Since f(k) = k1/3, this can be rewritten as:
With this value for k*, we can find y* = (k*)1/3 = 1.41, and c* = y* - s y* = 1.06.
To find the Golden Rule level of capital accumulation, recall that at the GR,
Since Y = K1/3(LE)2/3 then
Since, at the Golden Rule, the above calculated MPK must equal (d+n+g),
Since k** = 4.35,
y** = k1/3 = 1.63
c** = y** - .125k** = 1.088
s** = 1 – (c**/y**) = .333
The graph depicting this would be: