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.)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