please can I get help with questions circled

1' Suppose the economy is characterized by the following New Keynesian model: 1 . It : Et-TtH ' E [1: ' ENTHL] +715: (1) 7r" : IBETtJrl ] fer: + F: (2) where 1?; denotes the outputgap it the nominal shortterm interest rate.' 11': ination rateS at, a demand shock: ct a costwpush shock and 0 0., K. > 0. Assume that the shocks follow the following proceses: /_\\ D:- v 'U. U: punt1 + 5: 6t : pet71+g A .J_\\ V where the {'s are White noise processes. The central bank sets the nominal interest rate, it to minimize: 1 00 iEt Z ,6\" [773+] + Arid] (5) :0 where )\\ is the weight the central bank assigns to the stabilization of the output~gap at the expense of the stabilization of ination (1) Briey explain Where equations (1) and (2) come from? [3 marks] In the New Keynesian mo( el.' explain why price dispersion is costly for consumers and producers. [3 marks] (iii) Derive and interpret the rstvorder conditions for the optimal policy under discretion [4 marks] Assuming that 71': : "get: nd A, [6 marks] Show that the central ban cannot perfectly stabilize ination and the outputgap if there is a costpush shock q [1 mark] Prove that the Taylor principle is satisfied in the model. [3 marks] [TOTAL MARKS : 20]