hi all, i have been struggling with these problems for a couple of days and would appreciate any help?
The first part of 2 isn't as much hard as it is long.
$\displaystyle Y= C + I$
So
$\displaystyle Y = (aY_d + b) + (cr + d)$
$\displaystyle Y = aY_d + b + cr + d$
Then
$\displaystyle Y =a(Y - T) + b + cr + d$
$\displaystyle Y = aY - aT + b + cr + d$
Then
$\displaystyle Y = aY - a(tY + T^*) + b + cr + d$
$\displaystyle Y = aY - atY - aT^* + b + cr + d$
Now
$\displaystyle Y - aY + atY = -aT^* + b + cr + d$
$\displaystyle Y(1 - a + at) = -aT^* + b + cr + d$
$\displaystyle Y = \frac{-aT^* + b + cr + d}{1 - a + at}$
$\displaystyle Y = \frac{-aT^* + b + cr + d}{1 - a(1 - t)}$
-Dan
For the partial derivatives:
$\displaystyle Y = \frac{-aT^* + b + cr + d}{1 - a(1 - t)}$
$\displaystyle \frac{\partial Y}{\partial c} = \frac{r}{1 - a(1 - t)}$
since all other terms are considered to be constant.
Also
$\displaystyle \frac{\partial Y}{\partial a} = \frac{(-T^*)(1 - a(1 - t)) - (-aT^* + b + cr + d)(-(1 - t))}{(1 - a(1 - t))^2}$ <-- By the quotient rule
And I'll let you clean this up yourself.
-Dan