You cannot always use nice recipes like the Lagrangian method to solve for optimal consumption. This problem explores a case where the utility function is not \well- behaving". Consider Joe who was born with 5 legs; 2 left legs and 3 right legs. His parents give him an allowance of I to buy shoes. To get extra utility from having new shoes, he needs 2 new left shoes and 3 new right shoes. 1. Write down Joe's utility function. Denote the number of left shoes he buys by L and right shoes by R. Draw the indi erence curve where the utility level is 2. 2. Are Joe's preferences convex? Show on your graph from the previous part. Are they weakly or strongly monotonic (or not monotonic at all)? 3. By looking at the utility function and your graph, nd the ratio of L to R that Joe will always consume regardless of prices. Explain. 4. Find the demand for L and R. (Hint: Use your result from part 3 and the budget constraint.) 5. Now assume that the price of right shoes increases. What will be the substitu- tion e ect from this price change? Explain.