2. (50 points) Consider an enriched version of the two-period consumption-savings framework from Chapter 3 and 4, in which the representative individual not only makes decisions about consumption and savings but also housing purchases. For this particular application, it is useful to interpret period 1 as the *young period" of the individual's life, and interpret period 2 as the "old period" of the individual's life. In the young period of an individual's life, utility depends only on period-1 consumption cy. In the old period of an individual's life, utility depends both on period-2 consumption cz , as well as his/her "quantity" of housing denoted h ). From the perspective of the beginning of period 1, the individual's lifetime utility function is Ing + Inc2 + Ink, where In(.) stands for the natural log function; the term In indicates that people directly obtain happiness from their housing. Due to the "time-to-build" nature of housing (i.e., it takes time to build a housing unit), the representative individual has to incur expenses in his/her young period to purchase housing for his/her old period. The real price in period 1 (i.e., measured in terms of period-1 consumption) of a " unit" of housing (again, think of a unit of housing as square footage) is pu, and the real price in period 2 (i.e., measured in terms of period-2 consumption) of a unit of housing is pa . In addition to housing decisions, the representative individual makes stock purchase decisions. The individual begins period 1 with zero stock holdings (an = 0), and ends period 2 with zero stock holdings (as = 0). How many shares of stock the individual ends period 1 with, and hence begins period 2 with, is to be optimally chosen. The real price in period 1 (i.e., measured in terms of period-1 consumption) of each share of stock is a1, and the real price in period 2 (i.e., measured in terms of period-2 consumption) of each share of stock is sy . For simplicity, suppose that stock never pays any dividends (i.e., dividends = 0 always). Because housing is a big-ticket item, the representative individual has to accumulate financial assets (stock) while young to overcome the informational asymmetry problem and be able to purchase housing. Suppose that the financing constraint that governs the purchase of housing is pih_ (technically an inequality constraint, but we will assume it always holds with strict equality). In the financing constraint, Ry > 0 is a government-controlled "leverage ratio" for housing. Note well the subscripts on variables that appear in the financing constraint. Finally, the real quantities of income in the young period and the old period are y and y2, over which the individual has no choice. a. Let p the Lagrange multiplier on the financing constraint and dj and Ag are, respectively, the Lagrange multipliers on the period-1 and period-2 budget constraints. Write the sequential Lagrangian for the representative individual's problem lifetime utility maximization problem and in no more than two brief sentences/phrases, qualitatively describe what an informational asymmetry is, and why it can be a serious problem in financial transactions. b. In no more than three brief sentences/phrases, qualitatively describe the role that the leverage ratio RA plays in the "housing finance" market. In particular, briefly describe/ discuss what higher leverage ratios imply for the individual's ability to finance a house purchase (i.e., "obtain a mortgage"). c. Based on the sequential Lagrangian presented above, compute the two first-order conditions: with respect to aj and h. d. Based on the first-order condition with respect to h computed in part c, solve for the period-1 real price of housing pf (that is, your final expression should be of the form pf = ... where the term on the right hand side is for you to determine). e. Based on the expression forpf computed in part d, and assuming that the Lagrange multiplier a > 0 (recall also that R" > 0), answer the following: is the period-1 price of housing larger than or smaller than what it would be if financing constraints for housing were not at all an issue? Or is it impossible to determine? Carefully explain the logic of your argument/analysis, and provide brief economic interpretation of your conclusion