total differentiation

[henry]

To find the slope of the IS curve the product market equilibrium is totally differentiated and we get

dy= c'(dy-t'(dy) + i'(dr)

my question is what is total differentiation? and why there is no "d" before c, t, and i and to what respect they are differentiated if a " ' " on them represent differentiation

[Bellatrix]

Dear Henry,

Totally differentiate means, that we are taking differentials. Here are the elaborate calculations

We start with the IS equation which is-
Y = C + I + G
Taking differentials
dY = dC + dI + dG  

This measures a change in all variables, read: not rate of change. That is why we haven’t divided it by any other variable.

Now we know that C is a function of the disposable income, let’s say Y1. Where Y1 = Y – T. also I is a function of r. Ab yeh dekho :

dI= (dI/ dr) . dr = I’.dr

Doing this we get this equation:

dY = C’. dY1 + I’dr + dG (since G is an exogenous variable)

Let’s concentrate on dY1 now

dY1= d(Y- T)= dY –dT = dY – T’dY  (as Taxes are a function of our income)

Now the equation becomes:

dY = C’.(dY –T’dY) + I’dr + dG

Looks familiar now?  This is the IS equation written in differentials.  


Here is a link for understanding differentials:

http://www.economics.utoronto.ca/osborne/MathTutorial/DIFF.HTM

Ok now let us assume that C = C(Y1, M/P) and G= G(Y)
Where M and P have their usual meanings. M measures money supply, P is the price level.
Now tell me the IS equation?

[henry]

   

   i think it would be

     dY = c'( dY-T'dY+ p'dM) + I'dr + G' dY

[Bellatrix]

Think again on what d(M/P) will be

[henry]

I think the equation would be

   dY = c'( dY-T'dY+ M'dp) + I'dr + G' dY

[Bellatrix]

It is still incorrect.
d(M/P) = (P.dM – M.dP)/ P^2  
What does M’ or P’ mean? I never said that they are also a function of some other variable. (They may be a function of time, but that has not been specified in the model)
Now form the IS equation again.

[henry]

how about this

    dY = c'[dY-T'dY{ 1/p^2(dM-MdP}) + I'dr+G'dY

   I hope this to be correct else fix it up and tell me what I am missing again and again

[Bellatrix]

dY= (delC/delY1).dY1 + (delC/ del (M/P))d(M/P) + I’dr  + G’dY

This can be expanded further. But this is the calculation at play behind the C’s :) You were basically missing a plus sign in the last equation that you were getting. However, the C's are not the same so everything cant be taken inside the backet.That's why I rewrote the equation again.

 and I hope things may be clearer now.