Solution

1. Tilly's preferences are to choose x and y such that y=2*x. We can combine that with her budget constraint to obtain her demand equations as follows:

   I = Px*x + Py*y
   I = Px*x + Py*2*x
   I = (Px + 2*Py)*x
   x = I/(Px + 2*Py)
   y = 2*x = 2I/(Px + 2*Py)

   We don't need to know Tilly's utility function because she regards the goods as perfect complements: she always consumes 2 popcorn servings per movie.

2. Movies are normal goods. If her income goes up by 50%, Tilly's consumption of movies will rise by 50% (replacing I by 1.5*I causes x to become 1.5 times its previous value). Her income elasticity of demand for movies is thus 1, which makes movies a normal good.
3. The graph appears below:

   

4. When the price of movies falls to $4, her consumption of movies rises to $24/($4+2*$1) = 4 and her consumption of popcorn rises to 8. Her equilibrium looks as follows:

   

   To find the income and substitution effects, imagine going through the reasoning behind compensating variation. Shift Tilly's new budget constraint inward until she is just back to her initial indifference curve:

   

   The substitution effect is the change in her consumption of movies from her original value (X=3) to the value she would consume after the price change and CV shift (X=3). Since the values are equal, the substitution effect is zero. The income effect is the difference between her compensated consumption of movies (X=3) and her actual consumption after the price change (X=4): 4 - 3 = 1.

   What's unusual is that the entire change in consumption of X is due to the income effect. That happens because she regards the goods as perfect complements and is therefore unwilling to substitute one for the other.

5. To achieve her original utility, her consumption of movies must be 3 and her consumption of popcorn must be 6. After the price change, the cost of that bundle will be $4*3 + $1*6 = $18. The CV for the policy is $18 - $24 = -$6. The CV is negative because the policy makes her better off: to compensate for it, we would need to take money away from her.