When Phillips was working on macroeconomic control in the 1950s it was generally thought that the best approach to managing macroeconomic fluctuations was to change either government expenditure or taxes, not interest rates. (a) (2 pt) Show that the optimal path condition for minimizing policy-makers' loss function is Y, - YP YP - = -BY(m -RT) Hints: (i) This part of the derivation is very similar to the optimal rate rule derivation, and (1i) ut - y? = (Y, - YP) JYP.(b) (2 pts) Show the IS curve can be written as G. Y = Yi+ 1 - mpc Cyrt-1 where Y, is Y, with the G: term removed. (c) (3 pts) Use your results from parts 8(a) and 8(b) to derive an equation for G, showing how policy-makers would vary government spending to minimize their loss-function. Your answer should express G, in terms of the coefficients of the IS curve from part 8(b) as well as potential output and the inflation gap.(d) (3 pts) Given you derivation in part 8(a), is it possible to determine whether a government is using an optimal rate rule or an optimal government-expenditure rule by looking only at the optimal path (i.e., the output gap and inflation gap as a function of time)? Briefly explain