This problem illustrates a consumer's decision to be homeless in the presence of a minimum housing-consumption constraint, imposed through misguided government regulation. Let c denote "bread" consumption and q denote housing consumption in square feet of floor space. Suppose that a unit of bread costs $1 and that q rents for $1 per square foot. The consumer's budget constraint is then c + q = y, where y is income, which equals $1,000 per month.
(a) Plot the budget line, putting q on the vertical axis and c on the horizontal axis. What is the budget line's slope?
(b) Suppose that minimum housing-consumption constraint says that q must be 500 square feet or larger. Show the portion of the budget line that is inaccessible to the consumer under this constraint. Assuming the consumer rents the smallest possible dwelling, with q = 500, what is the resulting level of bread consumption?
Assume that the consumer's utility function is given by U(c, q) = c + α ln(q + 1), where ln is the natural log function (available on your calculator). Using calculus, it can be shown that the slope of the indifference curve at a given point (c, q) in the consumption space is equal to -(q + 1)/α.
(c) Assume that α = 101. Supposing for a moment that the minimum housing-consumption constraint were absent, how large a dwelling would the consumer rent? The answer is found by setting the indifference-curve slope expression equal to the slope of the budget line from (a) and solving for q. Note that this solution gives the tangency point between an indifference curve and the budget line. Is the chosen q smaller than 500? Illustrate the solution graphically.
Compute the associated c value from the budget constraint, and substitute c and q into the utility function to compute the consumer's utility level.
(d) Now reintroduce the housing-consumption constraint, and consider the consumer's choices. The consumer could choose either to be homeless, setting q = 0, or to consume the smallest possible dwelling, setting q = 500. Compute the utility level associated with each option, and indicate which one the consumer chooses. Compute the utility loss relative to the case with no housing-consumption constraint. Illustrate the solution graphically, showing the indifference curves passing through the two possible consumption points.(e) Now assume that α = 61. Repeat (c) for this case.
(f) Repeat (d).
(g) Give an intuitive explanation for why the outcomes in the two cases are different.