# Thread: Thrre sector model and a production fuction question

1. ## Thrre sector model and a production fuction question

hi all, i have been struggling with these problems for a couple of days and would appreciate any help?

2. Originally Posted by bobby87
hi all, i have been struggling with these problems for a couple of days and would appreciate any help?
What problems?

-Dan

3. Hi sorry dan the problems are there now
many thanks

4. 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

5. 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