How to solve Langrangian problem. To find ei^CP that the central planner would optimally allow each firm to emit. Calculate the value of the Lagrange multiplier, ?

Problem 1 There are 10 rms emitting pollution. The cost to rm i of achieving the emission level 6, is C2025) = (Z + 2 81)2 for i = 1,2,.., 10. This implies that rm 1 has cost function 01(61) 2 (3 el)2; rm 2 has cost function 02(62) = (4 2)2, etc. The upper bound on total emissions has been set at 55: 10 E = 2.2, = 55 i=1 1. Please write down the central planner's problem. 2. Solve the Langrangian problem in part 1. Calculate the amount of pol- lution, eEP , that the central planner woud optimally allow each rm to emit. Also, calculate the value of the Lagrange multiplier, A. 3. Now, consider a market for emission licences. Each rm is endowed with the same initial amount of licenses: em = 5.5 for all 2'. Write the cost minimizing problem for rm i. 4. Combining the market equilibrium condition and each rm's rst order condition, fully solve the problem. Calculate the equilibrium price of a unit of emission, p, and the individual emission level, 82;\