I am solving the following question:
Suppose that we live in a two good world, books (x) and movies (y), with utility function given by $u(x,y)=min(x+2y,2x+y)$. Prices of books and movies are 25 and 10 respectively. Income is 1000. Consider the following two schemes:
- The person gets one movie free on each book they purchase.
- In addition to buying books and movies separately you also get an option to buy a book with four movies combo pack for 50 only.
Q1. What is the optimal consumption in scheme 1?
A. (40,40) B. (0,100) C. (10,75) D. None of the above
Q2. What is the optimal consumption in scheme 2? A. (0,100) B. (10,75) C. (10,80) D. None of the above
For the first part I figured the following: If I buy $x$ books, I only have to pay for $y-x$ movies, so therefore my budget constraint will look like this:
$$25x + 10(y-x) = 1000$$ $$\implies 15x + 10y = 1000$$
Now solving this budget constraint with the given utility function, I get the optimal consumption as $(40,40)$.
On to the next part, I apply the same logic as above. If I buy $z$ units of the combo for 50, I only have to pay for $x-z$ books and $y-4z$ movies. So therefore my budget constraint will look like this:
$$25(x-z) + 10(y-4z) + 50z = 1000$$ $$\implies 25x + 10y - 15z = 1000$$
My question here is two-fold:
Is my reasoning correct in the above two scenarios?
How do I solve for the optimal consumption in the second scheme, where I have the option to buy a combo?