3.2 Introducing Government Purchases in the Basic New Keynesian Model Assume that the government purchases quantity G, (1) of good i, for all i e [0, 1]. Let G. = S. G.()1- di denote an index of public consumption, which the government seeks to maximize for any level of expenditures f P.() G,(i) di. Assume government expenditures are financed by means of lump-sum taxes. a) Derive an expression for total demand facing firm i. b) Derive a log-linear aggregate goods market clearing condition that is valid around a steady state with a constant public consumption share S = . c) Derive the corresponding expression for average real marginal cost as a function of aggregate output, government purchases, and technology and provide some intuition for the effect of government purchases. d) How is the equilibrium relationship linking interest rates to current and expected output affected by the presence of government purchases