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If in an economic model we focus on four variables:

Y(t) = Output after time t

K(t) = Kapital

L(t) = Labor

A(t) = Skill or knowledge of the labor force

at any one time the economy has some output, kapital, and labor with its knowledge so we can say that the output or production has the form that follows (Solow model):

Y(t) =F (K(t),L(t),A(t))

where F(...) is the function that relates these variables to each other and the final output after a time "t."

if we define A(t)*K(t) as the effective labor and call it AK we then get for the output

Y(t) = F ( AK(t) , L(t) )

if we have:

Y = F (cK, cAL) = c F(K, AL) for c>=0

assuming that there are constant returns in the AL and K variable.

so what if we take c to be any value such as c = 1 / AL

so we have.

Y = F(ck, cAL) = (1/AL*K , 1/AL * AL) = (K/AL, 1)

K/AL is defined as the amoutn of effective kapital per unit of effective labor.

we also konw that Y/AL is the production over effective labor which means:

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

ok, lets for the sake of simplicity define Y/AL as "y"

and K/AL as "k" we , and also lets define f(k,1) as f(k)

we would have

y = f(k)

that is we can write the output per effective unit of labor as a function of kapital per effective unit of labor.

The intuition behind why we used y = f(k) is this:

think of dividing the economy into AL small econmies, each with 1 unit of effective labor and K/AL units of capital. Since the production function has constant returns, each of these small economies produces 1/AL as much as is produced in large, undivided economy!!!

This means that the amount of outpuer per unit of effective labor depends only on the quantity of kapital per unit of effective labor. and not on the overall size of the economy!!

but the function y=f(k) describes the output per effective unit of labor for only these AL untis of economy. What if we want the estimation for the entire economy?

then we multiply by AL to get y AL = AL f(k) and since y = Y /AL

Y = ALf(k)

assumptions: if k=0 it means that kapital is zero and so we can assume that y=0


f(0) = 0

f'(k) > 0 because we now that Y = ALf(k)=ALf(K/AL)


dY /dK = AL f'(K/AL)(1/AL) = f'(K/AL)

so f'(k) = f'(K/AL)

and we know that 0 < AL f'(K/AL)(1/AL)

f''(k) = df'(K/AL)/dK= f''(K/AL)(1/AL) = f''(K/AL)/AL

since f'(k) is positive, it is assumed that f''(k) is negative.


f'(k) >0 and f''(k) <0


I have been pursuing a small business that follows more or less the Cobb-Douglas model:

F (K, AL) = K^a * {(AL)^[1-a]} 0>a<1

if my previous assumption about the solow function is right then this function should also be true for a constant return or F(cK, cAL) = cF(K, AL)

I have:

F (K, AL) = K^a * {(AL)^[1-a]}

F (cK, cAL) = (cK)^a * {(cAL)^[1-a]} = c^a * K^a * c^(1-a) * (AL)^[1-a]

since c^ a * c ^(1-a) = c ^ (a+1-a) = c so we have

F(cK,CAL) = c * K^a * {(AL)^[1-a]} = cF(K,AL)

so it is proven that my productoin function meets the condition of Solow model and that I can use it to an appreciable degree.

Now that my function has met the first condition I can go ahead and check it for the intesive form y=f(k)

dividing both inputs by AL (like before I have)

F(K/AL,1) which I again take to be f(k)

I also konw that with this my function or F (K, AL) = K^a * {(AL)^[1-a]} if I put in the values of K and AL i get:

f(k) = F(K/AL, 1) = (K/AL)^a * (1)^[1-a] = (K/AL)^a

f(k) = (K/AL)^a or k^a (sinc k = K/AL)

finding f'(k)

f'(k) = ak^(a-1)


f''(k) = a(a-1)k^(a-2) = - (1-a)ak^(a-2) and since 0<a<1 so 1-a>0 and so is the rest but since we have a "-" sign in the front then


so far the model has met he criteria of the solow model.

this would give the following shape to my enterprize:

user posted image

as u can see above at a certain k the value of f(k) fixes and falls on the equall production and return. Beyond that if you increase the capital per effective force or output per effective force it only goes down.

What is the point I am making?

There is a lot of talk about these tiny asian economies and Japan, and everybody seems to think that these economies would keep on growing forever but in fact as recent evidence shows Japan is in serious serious economic recess, and so is Korea, and all the other nations who have already reached their k(critical). These nations can not go any further and any further attempt with the old standards would result only in loss of kapital, unless of course the entire system is abandoned for the sake of a newever version. Of coruse there is one factor these old economies can use to rescue them and that is advertisement manipulation and falsification of data and records to create a false sense of security for the consumer. Kind of like what Enron did, expect as countries, these entities are not observed by authorities who are impartial. So if japan says it makes this amount of money even if it is a lie, there is no authority who would report to world bank.

I also noticed one thing about the American economy, which seems to make people anxious about its survival but is in fact its very invulnerability!!! Unlike other economies U.S. does not only rely on its Kapital, Effective work force, Labor knowledge and skills, or even its natural resources (counted as knowledge of the labor). It has a new variable which I am surprised to find out is missing from all the economic theories present today.

That is where my contribution to this thread comes in. I have noticed the same worry in the economic trends 10 years ago with Clinton or folks before him. The fact is we have to introduce two new variable into America:

Politial strenght P(t)


Military Power or Acquisitive power Z(t)

I like to define the P(t) as the politial influence in the security counsel, as a NATO boss, and as the prominant force of the globe.

I like to define Z(t) as the power to mobilize its military for monetary gains without losses. It can easily attack Iraq or Iran or buy out Saudi Arabia or Kuwait. This gives it great power.

In the new model which I think makes sense I propose the following modification to the Solow's principal basic model:

Y = F( K(t),L(t),A(t),P(t),Z(t))

just like before we ball A(t)L(t) = AL = effecitive work force and this time we call the ALK=effecitive labour kapital, and lets add PZ as effective global influence.

The purpose is to find the same graph between f(k) and k but this time between f(PZ) and PZ for simplicity sake we take then both as E = PZ

First we have to see if this function is true return for c as is the case in Solow assumptive theory.

we have

Y = F(KAL, E)

we assume this is a consatant return function as it is following Solow assumption so:

Y = F(cKAL, cE) = c F(KAL, E)

lets use as c = 1/KAL


F(1,E/KAL) = Y/KAL

like before lets assume that y = Y/KAL

and k = E /KAL

and that means,

y = f(k)


again checking for f'(x) and f''(x) proves that the values are:

f'(x) > 0 , and f''(x)<0

now before we used the internatinally accepted Cobb-Douglass formula but I like to add the new variable into the set

it used to be:

F (K, AL) = K^a * {(AL)^[1-a]} 0>a<1

this is the new form

F(KAL, E) = E^(a) * K^a * AL^ [1-2a]

again working with this function gives:

F(cKAL, cE) = (cE)^a *(cK)^a*(cAL^[1-2a]) = {c^a*c^a*c^1-2a} E^(a) * K^a * AL^ [1-2a] = c * {E^(a) * K^a * AL^ [1-2a]} = cF(KAL,E)

so it is checked for the return constant and this makes sense at least in terms of its math.

to find the intensive form of f(k) = y

we proceed as follows:

the original formula is:

F(KAL, E) = E^(a) * K^a * AL^ [1-2a]

we divide both inputs by c = 1/KAL

we have:

F(1, E/KAL) = (E/KAL)^a * 1^a * 1^[1-2a] = (E/KAL)^a = k^a

f'(k)= ak^(a-1)

f''(k) = -(1-a)k^(a-2)

since the graph has a y = k^a shape it imitates yet another hyperbolic shape but with one big difference.

The economy of america is not controlled by its kapital per effective labor force but by a combination of its political and military related market influence.

user posted image

therefore for U.S. it will never experience economic slow down, or at leat its critical point of economic balance is not defined because it sets it itself. It is in contorl of its own economic destiny by controling the very variable by which the game is played.

Oh well, regard this as just a play on formula or who knows maybe I am onto something...but regardless it is not questionable that European and Asian economies have a high growth rate but as fast ast they go up they usually fall down. U.S. seems to enjoy a steady yet shaky improvement that is not impressive but it not prone to falling. In short, U.S. economy has a huge stability factor that other economies wont have and cant have, unelss of course if we change the polarity of our world.

Your fellow human (yfh)
I liked reading this, thanks.
lol. you are welcome.
Drude, this post has been very helpful to me. I'm a graduate student trying to get through Macro Econ. We are using the Romer text and your post could have come from that text. However, some of your clarifications have been the key for me to understand the text. I am weak in Calc. I'm 59 and haven't been to school for over 30 years. I am stuck on understanding part of your post which is also part of the Romer text (page 11).

Your post reads (page 3):


F(0) = 0 (I'm good to this point)

f'(k) > 0 because we know that Y = ALf(k) = ALf(k/AL) (I'm now lost. I can't figure how you derive this. I can logically see that we are effectively moving from per capita measure back to the whole economy. But I'm confused as to the logical assumption that f'(k) > 0.)


dY/dK = AL f'(K/AL)(1/AL) = f'(K/AL) (This is where I'm totally lost. I can't figure how to derive this the way you have.)

I'm sure this is a very elementary calc question but any help would be appreciated.
Kind regards
This guy had an alternate solution.
The Governor of the Reserve Bank in Zimbabwe, Gideon Gono, won the Mathematics prize for simplifying the currency for the people and helping them cope with rampant inflation. He simply had the bank print money in denominations that covered all the bases, with note values ranging from one cent to one hundred trillion dollars.

That was actually incorrect on a number of levels.

1. Learn to spell.

2. Kapital? Is that what you win by Mortal Kombat?

3. Reducing an economy to 4 factors is a massive oversimplification and is bound to fail. You exclude factors like position, distance to markets, availability of markets, corporate behavior, competition, etc.

4. The US is not unique in it's political system and I don't see that it has any specific strengths vis a vis other democracies.

5. None of the wars of the last century have been wars of aquisition per se. In order to do that, the US would need to acquire Canada or Mexico as the most logical targets. Anywhere else becomes cost prohibitive. might want to actually study economic theories before posting your own.

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