1. Suppose there is an exchange economy composed of 2 individuals with identical preferences and they are each endowed with a positive quantity of two goods. Their preferences are described by Cobb-Douglas utility as given by the equation Ui(ri, r;) = ri"r, ". Endowments are (el, e;) = (15,9) and (e], e}) = (5, 11). (a) What are the equilibrium prices and allocation for this exchange economy? Show this in an edgeworth box diagram. (b) What does the contract curve for this economy look like? Add the contract curve to your edgeworth diagram. (c) Give a graph depicting the Utility Possibilities Set for this exchange economy. (Hint: You'll find helpful hints by looking at the appendix at the end of the assignment.) (d) Suppose that the ethical values of society can be summarized by a social welfare function given by Is it necessary for a government (dictator) to intervene in the market in order to achieve the ethical objectives of society? If so, find a set of taxes and transfers that the government can use so that markets will achieve it's ethical goals. (Think 2nd welfare theorem) If not, why? (e) Suppose that the ethical values of society can be summarized by a social welfare function given by W(U], U?) = min {U'(x], x]), U?(x], x]) ) . Is it necessary for a government (dictator) to intervene in the market in order to achieve the ethical objectives of society? If so, find a set of taxes and transfers that the government can use so that markets will achieve it's ethical goals. (Think 2nd welfare theorem) If not, why?Suppose a 2 person 2 good exchange economy has total endowments a = e, + er and ez = e, + e; and individuals have identical CD utilities given by the equation U'(ri, 1)) = z z where 1> a > 0. The following explains and details the steps for deriving an equation to describe the UPF. First, write an equation that shows that my can be written as it depends on a quantity of ry and the utility of person A. This is a manipulation of an equation for an Indifference Curve for person A. UA = 1 2 Al-a rearrange Now, we can write an equation for the utility of person B. UB = (21 - ri)" (62 - 17) Substituting to get UB = (21 - (U4)= (24) ")" (22 - 24) 1-0 We know that a point is on the UPF if for any allotment of utility to person A, person B's utility is as big as possible. We have written an equation for person B's utility as it depends on only one variable, ry, as we hold Ud constant. So we can maximize US by taking the derivative wit ry. Then finding the critical points where that derivative is equal to zero. Doing this leads to the following expression for ?" This can now be substituted back into US. Which reduces to This expression gives the equation for the PPF for this simple exchange economy