A firm produces and sells two commodities. The demand function for one commodity is = 24 1/4 x . The demand function for the other commodity is = 42 1/2py . Note, the quantities are measured in tons. The total cost of producing and selling tons of the first commodity and tons of the second is given by (,) = 2^2 + 2 + ^2.

a.) Solve the firms profit maximisation problem. What is the demand elasticity of price for each commodity at the maximum point?

b.Suppose the firms production causes pollution and the local authorities decide to apply a tax on its production, T=( + ), where is a parameter. Given the tax, the firm maximises its profit and the authorities maximise their tax revenue. What is the maximum profit the firm can obtain? What is the demand elasticity of price for each commodity at the maximum point?

c. Suppose the firms production activity causes so much pollution that the authorities limit its output to 8.2 tons in total instead of a tax on the production. Use the Lagrange Multiplier Method to solve the profit maximisation problem for the firm. What is the demand elasticity of price for each commodity at the maximum point?

notes. 1.can you please give detailed solving and explanation.

2. can you also provide kkt conditions for c

3. for part a I got x=6 and y=12