# Utility maximization in a 2-good scenario with an option to buy a combo of the two

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:

1. The person gets one movie free on each book they purchase.
2. 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

My working:

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:

1. Is my reasoning correct in the above two scenarios?

2. How do I solve for the optimal consumption in the second scheme, where I have the option to buy a combo?

Thanks!

For Q2, a little graphical intuition would help.

Suppose you spend all your income on the combo, you'd be able to purchase $$80$$ movies and $$20$$ books. This is the point $$(20,80)$$ on the $$(x,y)$$-plane.

If you buy $$1$$ less combo, you'll have $$76$$ movies, $$19$$ books, and $$\50$$ cash. Spending the entire $$\50$$ on movies would move you to $$(19,81)$$; spending all the cash on books would move you to $$(21,76)$$. (Of course you can split the cash between the two goods, but it's easy see on a graph that such a split is not efficient.)

Therefore, trading one combo for movies implies a relative price of 1 book for 1 movie, and trading one combo for books implies a relative price of 4 movies for 1 book. Using the information, we can plot the budget constraint as below:

The green segment, given by $$y=100-x$$ ($$x\in[0,20]$$), represents trade-off between the two goods implied by switching from combos to movies. The red segment, given by $$y=160-4x$$ ($$x\in[20,40]$$), represents the trade-off implied by switching from combos to books. It is clear from the graph that no other ways of using the income is more efficient. (As a reference, the budget line without the combo is plotted using the black dashed line.)

Given this budget constraint, it should be straightforward to solve for the optimal consumption bundle.

• Thank you, @Herr K. This was very helpful. Apr 18 '19 at 20:07
• @Vizag: You're welcome. Glad it helped. Apr 18 '19 at 20:22