How to solve this rbc problem

Consider the following economy where households have logarithmic utility function: U = Jo e-pt Inc(t) dt (1) A household has a fixed amount of labor (L) which the household supplies inelastically to the market for ruing wage, we- Normalize the household population to 1. A household's budget constraint is given by: a(t) = ra(t) + w(t)L - C() (2) (a) Set up household's optimization problem and find the optimal growth rate of consumption. (10 points) Assume that the output in this economy is produced according to Y = Ly "[ x/dj (3) Where x, represents intermediate good of variety ). The variable Ly represents the part of the total labor supply (L) that is used in the final good production. The rest of the labor supply, (L - L, ) are used to innovate new varieties of intermediate goods. The number of varieties evolve according to = = 6(L - Ly) (4) The profit for the final good sector (competitive) firms and an intermediate good (monopolistic) producer of variety j are respectively given by n = ly" , x/ dj - why - fo pixidj (5) 1; = XPj - Xi (6) Here, p; represents the price of the intermediate good of variety j. (b) Set up the final good producers' optimization problem and derive the expressions governing the demands for intermediate goods and the demand for labor. ( 5 points) (c) Assume that intermediate good producers are operating in a monopolistic environment. Find the profit of each intermediate good producer. (10 points) (d) Show that the costs of innovation for a new intermediate good is = (5 points) (e) Show that w, = (1 - a)@1-#N (10 points) (f) Use the free entry (no-arbitrage) condition to establish that r = ably (10 points) We know that model such as this will yield = = = ". Assume this at the outset (no need to show this). (8) Show that the growth rate for the decentralized economy is given by