I would like some help solving this problem and understand the steps. thank you!

Consider the following 2-period model of a closed economy with a representative house- hold and competitive firms. The household has logarithmic utility from consumption (Ct in period t) and a quadratic disutility of work (Lt ): log (C1 )-(L1)^2 +Beta[log (C2 )-(L2)^2]; 0 < Beta< 1 The household's saving (S) in Period 1 is S = w1* L1-C1 + T1 and consumption in Period 2 equals C2 = (1 + r) S + w2* L2 + T2 where wt is the real wage in period t, and r is the real interest rate. T1 and T2 are lump- sum transfers (equal to firm profits) that have no effect on the household's optimality conditions. Competitive firms have a fixed capital stock and operate the production function: Yt=At*(Lt)^.5, At > 0

Firms maximize profits (Yt-wt *Lt) At each period. In equilibrium we have Ct = Yt in each period. (a) What are the equilibrium values of the real wage and output? (b) What is the equilibrium value of the real interest rate? (c) Consider the equilibrium you get with A1 = A2 and compare it to the one you get with A1 < A2 . Give some intuition for how (if at all) this makes a difference for the equilibrium real interest rate. (d) What is the social planner's optimization problem for this economy? (e) Show that the planner's solution has the same output levels as the competitive equilibrium.