# Difference equation (economic growth model)

• Jan 8th 2010, 08:26 AM
beanbag
Difference equation (economic growth model)
$
\Delta A / A = B (aL)^\gamma A^\theta / A
$

$\Delta L / L = n$

$0< \theta < 1$

$n, B , \gamma > 0$

I need to solve for the procentual growth in A in terms of the parameters. This should according to my textbook equal

$\Delta A / A = \gamma n / (1-\theta)$

I guess it IS NOT a constant but CONVERGES to the previous constant. The only thing is, I can't solve it, work it out properly...
Can anybody solve this for me or put me in the right direction ?

I hope this was placed in the right subforum. Difference equations aren't differential equations, but I don't know in what subfield of Math difference equations would fit. (I'm no math student)
This might also be pre university level math, I don't know. I have never studied difference equations in high school and took a fairly math intensive curriculum.

• Jan 10th 2010, 11:32 AM
Henryt999
What does the a mean?
What does a mean in your equation?
• Jan 10th 2010, 12:37 PM
beanbag
Level of technology in the economy!
• Jan 10th 2010, 12:47 PM
Henryt999
Thanks
Thank you but I menth if it had a specifik value or could be substituted somehow.
But probably not, it is not directly related to any other factor in the equation??? a is then a constant? how was it menth to vanish?
• Jan 10th 2010, 12:56 PM
beanbag
This is all the model specifies.

"B" is what is called a "shift factor" in the literature and is constant. It augments technological growth.
"L" is the population size which grows every year by factor "n".
"a" is the fraction of the population who works in the research and development sector.

I have found an intuitive solution to this problem though, which I will post here the day after tomorrow since I don't have the time right now to go into it, but I would have liked a clean mathematical approach...

But to answer your question: there is no way to substitute any of the other variables or parameters into "A" or vice versa...
A is not a constant... the procentual growth of A however, becomes a constant dynamicaly.
• Jan 11th 2010, 06:26 AM
HallsofIvy
If $\Delta A / A = B (aL)^\gamma A^\theta / A$, doesn't it follow, then, that $\Delta A = B (aL)^\gamma A^\theta$? That is, that $\Delta A$ is simply a constant times A to a constant power.
• Jan 12th 2010, 09:51 AM
beanbag
The solution intuitively:

Given the above equations.

$
\frac{\Delta (\Delta A / A)}{\Delta A / A} = \Delta(Ln (RHS))
= \Delta Ln(B) + \gamma \Delta Ln(a) + \gamma \Delta Ln(L) - (1-\theta) \Delta Ln(A)
$

$
= 0 + 0 + \gamma n - (1-\theta) \Delta A/A
$

while

$\Delta A/A > \frac{\gamma n}{(1-\theta)}$

it's procentual change will be negative, and while it's smaller, it's procentual change will be positive. So it is dynamicaly pushed towards that equilibrium value...

That's the intuition.