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Appendix A Derivation of Proposition 1
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Appendix A Derivation of Proposition 1 The reaction functions in proposition 1 in chapter 2 follow directly from the actor’s loss functions and the Phillips curve mechanism. The government’s problem is to choose g so as to minimize its loss function, which, if we substitute in the Phillips curve process (equation (4) in chap. 2) that determines y and the right-hand side of (2) for the government’s ideal point for output and assume (without loss of generality) that * 0, becomes Li (yn ( e) g kgovyn)2 a2. (A.1) To find the minimum, we take the partial derivative with respect to g: L 2(yn ( e) g kgovyn), (A.2) g which when set equal to zero and solved for g becomes 1 g 冤yn(kgov 1) ( e)冥 (A.3) as proposed. An analogous process can be used to determine the central bank’s optimal response. Here the problem is to find the level of inflation that minimizes its loss function. Once again, substituting the Phillips curve process (equation (4)) and the right-hand side of (3) for the bank’s ideal point for output and assuming (without loss of generality) that * 0, the bank’s loss function is 177 Lcb (yn ( e) g kcbyn) 2. (A.4) Differentiating with respect to yields L 2(yn ( e) g kcbyn) , (A.5) which when set to zero and solved for becomes 1 冤e yn(kcb 1) g冥 (A.6) as proposed. Since the central banker observes the government’s choice before setting monetary policy, the equilibrium policies in table 2 were deduced in the following manner. Because the government anticipates central banker’s response to be (A.6), this is substituted for in the government’s loss function (A.1). Differentiating with respect to, and solving for g, yields the equilibrium fiscal policies in table 2. These are then substituted into the central banker’s loss function to derive equlibrium monetary policies. 178 Appendix ...