Hello! Kindly provide detailed proof and analysis. Thanks!

1. Suppose Emily's preferences over 1' and y can be represented by: U (3:, y) = 2:1: + By. Given that income is $120, and p1 = $5 while pg = $6, nd optimal bundle for Emily to consume. 2. Rippin' Good Cookies is based in Ripon, WI. Suppose the cookie factory uses two inputs to produce cookies according to a production function given by f {2:1, 3:2) = 23:?\" + 3:32. Cookies sell at $200 per crate. Input 1 costs $10 per unit and input 2 costs $50. (a) If input 2 is xed at 3'2 = 5, nd the prot maximizing level of 3:1. (b) Suppose Rippin' Good Cookies is free to vary its usage of all inputs. H a t is the prot maximizing level of $1? 3. Consider an industry with market demand characterized by: P = 6300 36}. Suppose the market is currently served by a large collection of rms all with constant marginal costs: M0 = 4200. (a) What will be the initial price in this market? (b) Suppose one rm is able to innovate and drive their own costs down to F = 600. What price and quantity would this rm want to set to maximize prots as a single-price monopolist if it were to have the market to itself? (c) Suppose one rm is able to innovate and drive their own costs down to F = 00. What price and quantity will this rm actually set and what are the rm's prots? Is this a large or small innovation? (:1) HGiven that there's a large collection of rms in the market and given t at one rrn has innovated to reduce its costs, what will be the output and prots of this innovating rm when it cheeses to practice rst degree price discriminatiOn after having driven its costs down to M0 = 3600