Derive the IS curve based on the given information about autonomous spending, investment spending, and interest rates.

Step 1: Identify the given points on the investment demand curve.
At interest rate $r = 10\%$, investment spending $I = 3000$.
At interest rate $r = 8\%$, investment spending $I = 4000$.

Step 2: Calculate the change in investment and interest rate.
$\Delta I = 4000 - 3000 = 1000$
$\Delta r = 8 - 10 = -2$

Step 3: Calculate the slope of the investment demand curve.
Formula for slope: $m = \frac{\Delta I}{\Delta r}$
Substitute values: $m = \frac{1000}{-2} = -500$

The slope of the investment demand curve is $-500$.

To derive the equation of the investment line:
$I - 3000 = -500(r - 10)$
$I = 3000 - 500r + 5000$
$I = 8000 - 500r$

Check at $r=8\%$: $I = 8000 - 500(8) = 8000 - 4000 = 4000$ ✓

This linear investment function $I(r) = 8000 - 500r$ can be used to derive the IS curve via $sY = I(r)$, or $r = 16 - \frac{s}{500}Y$ (where $s$ is the marginal propensity to save).

-500