Calculate equilibrium Y and equilibrium r

The goods market equilibrium is given by the equation:

$$Y = C + I + G$$

We are given:
$C = 1000 + 0.75 Y^d$
$I = 1000 - 100r$
$G = 100$
$T = 0.2Y$

First, calculate disposable income ($Y^d$):

$$Y^d = Y - T = Y - 0.2Y = 0.8Y$$

Substitute $Y^d$ into the consumption function:

$$C = 1000 + 0.75(0.8Y) = 1000 + 0.6Y$$

Now substitute C, I, and G into the goods market equilibrium equation:

$$Y = (1000 + 0.6Y) + (1000 - 100r) + 100$$ $$Y = 2100 + 0.6Y - 100r$$

Rearrange to solve for Y:

$$Y - 0.6Y = 2100 - 100r$$ $$0.4Y = 2100 - 100r$$ $$Y = \frac{2100}{0.4} - \frac{100}{0.4}r$$ $$Y = 5250 - 250r \quad \text{(IS curve)}$$

The money market equilibrium is given by the equation:

$$M_s = L_1 + L_2$$

We are given:
$L_1 = Y$
$L_2 = 1 - 1000r$
$M_s = 2751$ (Assuming the 'I' is a typo and $M_s = 2751$)

Substitute $L_1$, $L_2$, and $M_s$ into the money market equilibrium equation:

$$2751 = Y + (1 - 1000r)$$ $$2751 = Y + 1 - 1000r$$

Rearrange to solve for Y:

$$Y = 2751 - 1 + 1000r$$ $$Y = 2750 + 1000r \quad \text{(LM curve)}$$

Now we have a system of two equations:

1. $Y = 5250 - 250r$ (IS curve)
2. $Y = 2750 + 1000r$ (LM curve)

Set the two expressions for Y equal to each other to find r:

$$5250 - 250r = 2750 + 1000r$$ $$5250 - 2750 = 1000r + 250r$$ $$2500 = 1250r$$ $$r = \frac{2500}{1250}$$ $$r = 2$$

Substitute the equilibrium value of r back into either the IS or LM equation to find Y. Using the IS curve:

$$Y = 5250 - 250(2)$$ $$Y = 5250 - 500$$ $$Y = 4750$$

The equilibrium values are:
Equilibrium interest rate $r = \boxed{\text{2}}$
Equilibrium output $Y = \boxed{\text{4750}}$