please answer the question

1. Consider the Melitz model of trade. [5 marks each = 20 marks] (a) For the case of exactly TWO varieties, the utility maximization problem is given by 1 mum+sv 11192 subject to Pith +P29'2 = R Form the Lagrangian, solve the Optimization problem and verify that demand is given by: R P(w) cr where each variable is dened exactly like in the Melitz (2003) model. Describe what the aggregate Price index P stands for. (b) Given that prot maximization for each rm producing variety to is given by: 1 P 90 = _ ( ) .090 where (,0 is rm productivity, show that 1 We) = 4010) - f 0 (c) Hence, use the ZCP condition \"((9') = 0 to show that average prot in the industry is: f = \"(grl 1) and is hence independent of labor supply L. ((1) Verify that MOP)