4. In its most abstract (though still very simple) form, we can write our model of individual choice as subject to P1331 +P2332 + ' ' ' +Pn33n S I. Let us rst examine the role of the budget constraint in this model. In principle, the agent need not spend all her money, and so we should write the budget constraint as an inequality, or mm + 102562 + + 10,133,, S I, as above. But it is typically much easier to deal with equality constraints. (a) To explore this issue, let us simplify the model as much as possible by assuming there is only one good, 31, with utility function given by U(:E1) = 100331 33% Let p1, the price of good one, be 5. The budget constraint is then 5:51 3 I. Suppose I = 50. What is the utilitymaximizing consumption 3:1 for this agent? Would we be justied in writing the budget constraint as 531 : I in this case? Suppose I = 500. What is the utility-maximizing consumption 3:1 for this agent? Would we be justied in writing the budget constraint as 5331 = I in this case? Which of the assumptions we made in class about utility functions fails in this latter case? Now, let us take a look at the second element of our model of consumer behavior, preferences. Suppose the person in question consumes two goods, with utility function U(3:1, 3:2) : 43:1- 232. For example, 1151 might be candy and :62 might be broccoli, and this agent dislikes broccoli. (b) Draw the indifference curves for this agent. (c) Suppose p1 = 5, p2 = 2, and I = 200. Find the utilitymaximizing consumption bundle for this agent. You should be able to do this without using any calculusit should be clear from your indierence curves. This utility function once again violates one of the assumptions we've made on utility functions. Which one? A violation like this is typically an indication that we have not constructed our model