Suppose you have a job at a start-up and are looking for a house to buy. You consider indoor and outdoor spaces (like a yard or a deck) to be somewhat fungible because, when you host parties, they can be indoors, outdoors, or a combination of the two. In addition, you know that you need a house of a minimum size, say 1000 square feet, but that you could buy something with no outdoor space at all. Consistent with these notions, you write down the following utility function:

u(x₁,x₂) = log(x₁+x₂) s.t. u(x₁,x₂) = inf if x₂ <1

where x1 is the size of the yard in 1000's of sq ft and x2 is the size of the house in 1000's

of sq ft. Answer the following questions:

1. Are x1 and x2 substitutes or complements in utility? Explain how this calculation is consistent with your use of the spaces.
2. Suppose your budget is $210,000 (your start-up is doing well), the price of outdoor space in Cambridge is $10 per sq ft and the price for indoor space is $100 per sq ft. How would you allocate your budget? (hint: it might be useful to think of your "disposable income" as being the amount you can allocate after you pay for the minimum-sized house)
3. It turns out that, while you were house-hunting, every biotech firm in the world decided to locate in Kendall Square, and Cambridge buildings became much more expensive. In particular, indoor space increased to $210 per sq ft. How do you allocate your budget now?
4. Are indoor and outdoor spaces complements or substitutes in demand? How can you interpret this result in light of your answer in part 1?