Based on the Lagrange function above, the first-order conditions with respect to C ; C , a, , and DUR Guy " are, respectively u'(c)-1,=0. (1) (2) - As, +Q R . 5, + By (s,it dal ) = 0. (3) and Bv'l C+1 DUR - Ap." DUR + B(1-6)MAP =0. (4) a. (9 points) Using the first-order conditions stated above, provide (in an easy-to-read, organized manner) the important algebraic steps that lead to the period- relative price of durable consumption goods. The final solution should be expressed as DUR Ptin which R" > 1 is a government-controlled "leverage ratio" for financing expensive durable goods. (Note: there are no typos on the time subscripts of the variables in the financing constraint.) Letting / stand for the one-period-ahead "impatience" parameter, 1, stand for the multiplier on the period- budget constraint, and o the multiplier on the period- financing constraint, the sequential Lagrangian for the representative consumer's optimization problem starting from period t 15 u ( c ) +v ( c, " ) + B u ( c, . ) + v ( c [ ]) + B u(C+2 ) + v(6142 ) DER ) + 83 u(c, 3 ) + v (c 45 7 ) +... the (s , + d, ) at y - G - pi DUR CA - (1-5)cPUR ) - s,a, to Roof .s,a, - P, DUR _DUR DUR DUR DUR DUR DUR DUR C+ +3 -(1-8)(12 )-5,42042] DUR _ DUR +.....DOUR demand function D GOV for RINITIAL DUR Figure 2. Demand function for durable consumption goods given initial value of leverage ratio.(NOTE: FOR THE GRADING PROCESS, IT WILL BE DIFFICULT, IF NOT IMPOSSIBLE, TO CARRY ERROR(S) IN PART A INTO THE ENSUING PART B.) b. (6 points) Assume that financing constraints always hold (that is, purchasing durable goods always requires obtaining a loan). If the government increases the leverage ratio RU from some initial value RGOF 'nwizz : qualitatively show in which direction the demand function for durable goods plotted below shifts? Briefly, but carefully, explain the logic of your argument/analysis and provide brief economic interpretation of your conclusion. (Note: economic interpretation is not simply a verbal restatement of the mathematics.)Problem 2. Financing Constraints and Durable Consumption {15 points}. Consider the infiniteperiod consumptionsavings amework of Chapter 3. Suppose there are two assets, one of which is stocks and the other is durable consumption goods. The periodr utility function of the representative consumer is ncl) + Wiring!) , in which c, denotes \"non-durable\" consumption goods in period 2* and c?\" denotes \"durable" consumption goods in period 3.1 The periodr budget constraint is LIL]! BL]! DER '5: +2\": (CHE _(1_5)C: l+53c ='1"Ta"lti+'I'l'-\""iI-I!":1.'l413"]!F in which pf)\" denotes the periodr relative price of durable goods, 5 denotes the I' periodr real price of stock, 111', denotes the periodr real stock dividend, )3 denotes periodr real income, and the exogenous parameter :2? denotes the oneperiod depreciation rate of durable goods. The depreciation parameter lies between zero and one {i.e., :1 :3 e, I). Because durable goods are much more expensive than nondurable goods, loans are required in order to purchase durable goods. Due to informational asymmetries, there are nancing constraints involved in these loans. The period: financing constraint is LILR LIL}! GEE-i =3 . P: CHE 5 {1' ft? in which the term in ellipses ("...") on the right-hand side is for you to determine. Note: The final expression must contain only the period- relative price of durable consumption goods on the left-hand side, and the right-hand side may not include any Lagrange multipliers. A conclusion, even if correct, will be awarded zero points if there is no or very little clearly organized algebraic support behind it