You own a piece of raw land in an up$-$and$-$coming area in Gotham City. The costs to construct a building increase disproportionately with the size of the building. A building of q square feet costs $0\mathrm{.}1*$ q$2$ to build. After you construct a building on the lot, it will last forever but you are committed to it: You cannot put another building on the lot. Buildings currently rent at $$100$ per square foot per month. Rents in this area are expected to increase in five years. There is a $50\%$ chance that they will rise to $$200$ per square foot per month and stay there forever, and a $50\%$ chance that they will stay at $$100$ per square foot per month forever. The cost of capital is fixed at $12\%$ per year. a$.$ Should you construct a building on the lot right away? If so$,$ how large should the build$-$ing be$?$b$.$ If you choose to delay the decision, how large a building will you construct in each possible state in five years?