Extra assignment i'm working on

In the market of cars, there are two rms operating. The Industry Demand Curve is a function of the outputs being produced by both rms, and is given as: P = 240 (X1+X2), where X1 and X2 are the outputs of Firm 1 and Firm 2 respectively. The Total Cost faced by Firm 1 is T01 = 20X] and by Firm 2 is T02 = 20X2. Each rm maximizm its own prot by choosing its own output, while taking the output of the other rm as given. Write down the prot function of each rm, and nd the level of output for each rm that will maximize its prot. What are the corresponding levels of prices and prots for each rm? , Suppose a person has a utility function U (331, 332) = :r'l'+:c' , which she maximizes subject to her budget constraint, pm + 9'932 = m, where p, q,m are all positive. Use the Lagrangian method to solve the maximization problem, and nd the demand functions for the consumer. Show that the demand functions are homogeneous of degree zero in prices (p,q) and income (m). A consumer has a Cobb-Douglas Utility Function: U = 33053905. The price of good 56 is $2, the price of good y is $4, while the person has an income of $40. Find the solution to the consumer's maximization problem, and verify that you have found the maximum point. , A rm produces a single output, Q, using the production function: Q = K %L%, where K is capital and L is labour. The cost of K per unit is $1 and L per unit is $2. Find the optimal choices of K and L that will minimize the cost of production for the rm when it intends to produce (:2 units of output