Equilibrium interest rate and national savings

This post is an extension from this post. If you are not familiar with how to calculate national, private and public savings you should consult it first. Consider the following example:

C = 250 + 0.75(Y-T)

Y = 5000

I = 1000 - 50r

T = 1000

G = 1000

Suppose that we wish to calculate the equilibrium interest rate; the private savings; the public savings and the national savings. The first thing we should do is calculate consumption from the consumption function, which in this case is C = 250 + 0.75(Y-T).

This is relatively straightforward since we have been told that the tax rate (T) is 1000 and government income (Y) is 5000, therefore:

C = 250 + 0.75(5000-1000) = 3250

Next, we need to use our national accounting identity Y = C + I + G to determine I:

5000 = 3250 + I + 1000

I = 750

Now we substitute in our investment formula from above I = 1000 - 50r such that:

1000 - 50r = 750

50r = 250

r = 5

Next, we wish to calculate our national, private, and public savings. The last is the easiest since the tax rate is the same as government spending, which means that the government is operating a balanced budget and therefore have no savings. Since National savings = public savings + private savings this also means that national savings equal private savings. Finally, since National savings = investment (See here) we know that national savings are:

S = I = 750

To recap:

National savings = 750

Private savings = 750

Public savings = 0.