According to the Solow growth model, only exogenous technological
progress can generate persistently rising living standards. Expalin
this statement, using appropriate algerbra and diagrams.

Hi bbshawn,

I'm going to assume that you have a fairly reasonable understanding
already of the Solow model.  If not you may wish to check out some of
the resources I've listed at the end of this answer.  Probably the
best resource I found to explain the model is here:
http://www.digitaleconomist.com/cap_4020.html where there are a couple
of interactive graphs that allow you to see the effect of different
savings rates, depreciation rates, etc. on output.

Output in the economy, Y, is a function of the amount of captial (K)
and labor (L) employed.  Capital is assumed to wear out over time, or
depreciate.  So over time the level of capital will naturally fall,
and so K/L will fall (assuming L remains the same).  The rate at which
this happens is the depreciation rate, d.

So in order to maintain a constant level of capital to labor some of
the economy's output must be used to invest in new equipment instead
of for final consumption.  Representing investment as I, this means
that for K/L to remain constant,

I = d * K 

or the depreciation rate multiplied by the level of capital.

Naturally the more capital is employed in the economy the more wears
out in a year, so with greater levels of K, greater amouts of new
investment are required to offset depreciation.

As the population increases K/L falls due to the increase in L.  So to
offset this investment must account for the population growth rate n. 
So

I = (n + d) * K

Now, dividing both sides by L to produce per-capital values, this is

i = (n + d) * k

where i = I/L and k = K/L.  

Investment needed to purchase new capital comes from savings, which,
in a closed economy without foreign lending and borrowing, is final
output (Y) less consumption (C) and government expenditure (G):

S = Y - C - G

representing the proportion of of output not devoted to C and G as s,
this is:

S = sY

If savings is equal to investment, then S = I and so

sY = I

and in per-capita terms

sy = i  

where y = Y/L and so substituting in for i

sy = (n + d) * k

The steady state level of capital is defined by this equation, and if
you're graphing these values is found at the intersection of the sy
line and the (n + d) * k line.

If sy < (n + d) * k then the change in capital per worker delta(k) <
0.  This means that the savings rate is too low to keep up with
depreciation and the amount of capital in the economy is decreasing.
So, if the population remains the same then K/L is falling.

Likewise if delta(k) > 0 then the amount of capital is increasing, so
K/L is rising.  With more capital per worker output per worker
increases, and so there is growth in the living standard.  This growth
lasts only until the steady state is reached again and output per
worker stabilizes.

So the Solow model says that growth in the capital to labor ratio K/L
causes increases in the standard of living.  In the above paragraph
that growth came from an increase in K, but it could also come from a
decrease in L, which would happen if the population fell.  Then there
would be fewer workers, so more capital per worker, and greater output
per worker.

Since the economy will always tend towards a steady state, an increase
in savings (investment) will only cause a brief growth in the standard
of living.  Eventually the steady state is reached and the growth
ceases.  Likewise the standard of living for each worker will increase
as the population falls, but once it stabilizes the economy will
converge at the steady state and the standard of living will again
stabilize.

Persistent increases in the standard of living must, therefore, come
from somewhere other than increases in capital or decreases in
population.

Technology is measured by the variable A.  A describes a worker's
efficiency.  So L workers with efficiency A can produce as much as A *
L workers with efficiency of 1.

A is assumed to grow at the rate g.

Incorporating A into the model, Y is a function of K and AL instead of
just K and L.  Or:

Y = F(K, AL)

Now, you may remember that the Solow model expresses output as a
cobb-douglas function where Y = F(K, L) = K^alpha * L^(1-alpha).  In
this case, expressing the above as a cobb-douglas function yields:

Y = F(K, AL) = K^alpha * L^(1-alpha)

Dividing by L to produce per-capita values again, 

y = F(K/L, A)

So expressing that as a cobb-douglas function you get

y = (K/L)^alpha * A^(1-alpha)

This shows us that output per worker is dependent on the K/L ratio,
which remains constant in a steady state, and A, the level of
technology.  Since A grows at rate g, y which expresses the standard
of living, will always be increasing and cannot converge to a steady
state.  It is this that causes long-term growth in the living
standard.

----

Now, that was quite a lot of information to ingest, and I hope it was
fairly clear.  Let me try to summarize it for you just to make sure
it's reasonably straightforward.

Output Y is a function of K and L.  

Y = F(K,L)

Divide by L to get a per-capita output level:

y = F(K/L, 1)

as a cobb-douglas function, this is:

y = (K/L)^alpha * 1^(1-alpha)

So output per worker y is dependent on (K/L) which, in the steady
state, is constant.  Changes in the savings rate (and so investment)
and/or in the population level will cause short-run changes in the
living standard as a new steady state is approached, but will not
cause long-run changes.

When technological change is factored in y is a function of K and AL:

Y = F(K, AL)

y = F(K/L, A)

y = (K/L)^alpha * A^(1-alpha)

Since A grows at rate g, it is impossible for y to converge to a
steady state, and there will be constant growth in the standard of
living.

----

I hope this has helped you out.  If you're interested in some more
in-depth analysis, or just to get some different interpretations of
the model, you can check out some of the links below.

If you have any questions, please feel free to ask for clarification,

Hibiscus

----

Links:

A good one for discussion about the effect of technology in the model:
http://www.econ.ucsb.edu/~bohn/133/133overheads03.pdf

Good powerpoint presentation which summarizes the Solow model:
http://www.mines.edu/Academic/courses/econbus/ebgn412/slides/16

And there's a somewhat more mathematical discussion of the same thing
here:
http://emlab.berkeley.edu/users/bhhall/e124lec3.pdf

Interactive graphs and a brief description of the model:
http://www.digitaleconomist.com/cap_4020.html

Search terms: "solow growth model", "solow residual", solow
"technological growth", solow "effective labor"

---

Request for Answer Clarification by bbshawn-ga on 15 Jun 2003 01:04 PDT

hi,
  do you mind to provide some diagrams to support your answer.
thank you
bbshawn

---

Clarification of Answer by hibiscus-ga on 15 Jun 2003 11:35 PDT

Hi bbshawn,

It's not possible to include graphs directly in the answer as Google
Answers only supports text.

However, I can send you to links with the appropriate diagrams. 
Rather than recreate diagrams that already exist, I'm just going to
recommend you look at the diagrams listed here:
http://online.sfsu.edu/~haase/e100/homework/hw_3/hw_3_key.htm

These are homework questions with answers at San Francisco State
University.  Of particular interest to you will be the graph for
question 4 which shows how an improvement in production technology
affects long-run potential output.  It shows that improvements in
technology shift the production function upward which in turn
increases capital due to increased savings (a function of output).  As
technology advances (by rate g) the effectiveness of capital continues
to increase and the production function will continue to shift upward.

If you wished to demonstrate the effect of persistently rising
technology levels, you might wish to draw the graph of question 4 with
a number of production function curves, each higher than the next, and
each labelled for a different time period.  So, for instance, Y1 would
be higher than Y0, Y2 higher than Y1, etc.  Each would represent a
higher level of technology since technological progress increases at
the rate g.  With the more advanced technology output would be greater
while labor levels would remain the same, which would indicate rising
living standards.

Hopefully this will provide you with enough information but if not,
let me know.

Also, I noticed that the second link at the bottom of my answer was
wrong and doesn't work.  It should be
http://www.mines.edu/Academic/courses/econbus/ebgn412/slides/Chapter4.ppt

Hibiscus

---

Request for Answer Clarification by bbshawn-ga on 15 Jun 2003 14:40 PDT

hi,
  do you know how can i leave common for this question?
thank you 
bbshawn

---

Clarification of Answer by hibiscus-ga on 15 Jun 2003 15:14 PDT

Hi again bbshawn, 

I'm guessing you meant to ask how you could leave a comment?  It
appears that you figured it out.  You do it when you rate the answer.

Let me know if you meant something different.

Thanks for the tip, it's appreciated.

Hibiscus

your answer really helps me to understand the Solow model and it make
the question more clear for me. thanks a lot. i really want to tip you
more, but i'm broke now! sorry!!