Derive Harrod-Domar growth equation?

Question

Use algebraic notations.

Answer 1

I had already given you the Harrod Domar equation of the most simple version in response to a later question you posted. It is growth rate of  GDP equals the product of the savings rate (s) and the incremental capital output ratio(v). gdp growth rate g = s/vI reproduce the same once again below:

ICOR is the abbreviation of Increnental capital output ratio. It is the ratio given by incremental output (Y) resulting from incremental addition to the stock of capital (K). ICOR indicates the amount of additional capital required to generate additional GDP or National Income (Y). To represent symbolically, v = ICOR = DK/DY where D stands for incremental change.

Now we also know that DK is nothing but Investment (I) during a period because it is through investment that the capital stock to produce additional output is made available. So , DK = I.  We also know that in the ex post sense in a closed economy Investment equals Savings(S). So, I = S.

but Saving depend on the level of income ( a percentage of income is saved) and this is represented by the equation S= s.Y where s is the saving rate.

Noe let us see what we have:

v= DK/DY = I/DY= S/DY= sY/DY

by transposing we get  DY/Y= s/v

DY/Y is nothing but the rate of growth of GDP or Income. The final equation says that this growth rate equals s/ v or savings rate divided by the ICOR. The higher the ICOR, the lower is the rate of growth of GDP or the economy. The lower the ICOR the higher will be the economy's growth rate measured by the rate of growth of GDP or income. this is so simple intuitively: given the savings rate, the lower the additional capital required to produce an additional unit of GDP, higher will be the additional GDP generated from the given investment and hence higher trhe growth rate. This is a simplistic version of the Harrod Domar growth model.

Answer 2

Follow the link....

http://www.sonoma.edu/users/b/benito/eco...

It will prompt you to save a powerpoint file...

See the file.....It has 21 slides....all the derivation is done in this slides using Algebraic variables....

Answer 3

There´re a lot of Harrod Domar equations and I like the one that considers the growth through a long period of time. This presentation is rather simple and right.

When we say sY=I=µ(K`-K)

K`-K measures how the capital grows in absolute terms but if you write down

sY=I= µK, will mean that investment is measured  in terms of the capital in (t-1).

If K=100 machines and µ=10%, new investment will be 10 machines and

sY=10

K`=110

K=100

If you use

sY=I=µ(K`-K)

It´ll mean that µ=1 and that will not help advance.

Let´s get the expression

sY=I= µK

and let´s operate without using derivatives

sY/Y=µK/Y

s/µ=K/Y

That´s a relationship between capital and output.

Of course that´s not a Harrod Domar equation but HD model has been widely criticized.

This one is my personal Harrod Domar model:

sdY/dt=dI/dt

Investment and output grows through time

dI=µdK

K represents our assets in capital in (t-1)

sdY/dt=µdK/dt

(dY/dt) / (dK/dt)= µ/s

As for your question

Y=µK

sdY/Y=dI/Y

dY/Y=dI/Y /s

dI/Y=(K`-K)/Y

dY=µ(K`-K)

K`-K=dY/µ

dY/Y=µ/s that´s the harrod domar rate of growth