**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) **

**or **

**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**

**if k > k ^{* }then i <dk
, so 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: MP _{K} =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.**

** **

**Example:**

**Let Y = K ^{1/3}(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) = k ^{1/3}, 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, **

**MP _{K} =(d+n+g).**

**Since Y = K ^{1/3}(LE)^{2/3} then**

_{ }

**Since, at the Golden Rule, the above calculated
MP _{K} must equal (d+n+g),**

_{ }

**Since k** = 4.35,**

**y** = k ^{1/3} = 1.63**

**c** = y** - .125k** = 1.088**

**s** = 1 – (c**/y**) = .333**

**The graph depicting this would be:**

** **