The following problem uses a modification of the Cobb-Douglas utility function to model consumption when one of the goods causes addiction. Consider the following utility function over two goods: U(x,y)=r(y-A) if y 2 A, and U(x, y) = 0 if y < A. Good Y is an addictive good with addiction level A> 0: consuming USA units leaves the consumer very unhappy (U= 0), and only consumption above that level increases the utility obtained. (a) (3 pts) Draw a rough indifference curve map for these preferences. Are these preferences convex? Are they strictly convex? (b) (2 pts) Find the marginal utilities with respect to both X and Y. Do these preferences satisfy monotonic- ity? (e) (2 pts) How does the utility of consuming any given bundle (r. y) change if the level of addiction A increases? (i.e. find the sign of 4). Interpret why your result makes sense. (d) (3 pts) Assuming an interior solution, find ordinary demand for both goods (r.) under a standard budget constraint pr+Pay Sm. Explain the optimal consumption behavior. (e) (3 pts) How does the demand for both goods change when the level of addiction for good Y increases? (i.e. find the signs of and ). Why do your results make sense? (f) (2 pts) What must be the minimum value of income m in order for the solution from part (d) to make sense? What is the optimal bundle (r.) if m falls below this value?