Comparative Statics:Neoclassical ModelMethod of Comparative StaticsComparative Statics, continuedExample 1Example 1, ContinuedExample 1, ContinuedExample 1, ContinuedExample 1, ContinuedExample 1, ContinuedExample 1, ContinuedExample 1, ContinuedExample 2Example 2, ContinuedComparative Comparative StaticsStatics::Neoclassical ModelNeoclassical ModelGraduate Macroeconomics IECON 309 – Cunningham2Method of Comparative Method of Comparative StaticsStaticsGiven, an economic system of the form:.Such a system can be interpreted as the set of certain equilibrium relations or the set of certain optimization conditions. If equilibrium, then and are the excess demands for and prices of the ithcommodity.To consider “shifts” of the system due to changes in exogenous variables, rewrite the system making the exogenous variables explicit:nixxxfni,...,3,2,1,0),...,,(21==ifix,...).,;,...,,(21βαnixxxf3Comparative Comparative StaticsStatics, continued, continuedAssume all second partials exist and are continuous. Then are the “shift parameters” or exogenous variables. If the fiare well-behaved, then we can writeComparative statics is concerned with the effect of a change in one or more of the shift parameters on the equilibrium values of the xi. In effect the equations are linearized around the current equilibrium values of the exogenous variables by taking total differentials. The implied assumption is that the changes will be small and in some neighborhood of the original equilibrium....,,βα.,...,2,1,...),,( nixxii==βα4Example 1Example 1Question: According to the neoclassical model, what happens to output, wages, and employment when the capital stock is increased?Solution: construct as simple a formal model as possible that can examine this question. (Parsimony)⎟⎠⎞⎜⎝⎛===PwNNFPwKNFYSSN),(5Example 1, ContinuedExample 1, ContinuedMake the usual assumptions:00,00,02222==<=>=<=>=KNNKKKKNNNFFdKFdFdKdFFdNFdFdNdFFTake total differentials:()()PwNKNNPwKNdNdNdKFdNFddKFdNFdYPw=+=+=6Example 1, ContinuedExample 1, ContinuedReorganize:()()0000=++=−+=−+dNdNdKFdNFddKFdNFdYPwNKNNPwKNPwIn matrix form:()⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−0101001dKFdKFdNddYNFFNKKPwNNNPwJacobian (J) EndogenousVariablesExogenousVariables7Example 1, ContinuedExample 1, ContinuedFind the determinant of the Jacobian.01det)()(>−==+−PwNFJJNNApply Cramer’s Rule, replacing column 1 with the vector of exogenous variables (RHS).PwPwNFNFdKFFdKFdYNNNNNKNK−−−−=110108Example 1, ContinuedExample 1, ContinuedExpand by minor determinants, third row:PwPwNFdKFdKFFdKFFdKFNdYNNNKKNNNKNK−+−−=110[]PwPwNFdKFdKFFdKFFNdYNNKNNKNKN−+−=19Example 1, ContinuedExample 1, Continued)()()()()()(1+−+−++−+−=PwPwPwNFFFFNFFNdKdYNNKNNKNKN0> 0Ceteris paribus, an increase in the capital stock results in an increase in output, under normal assumptions.10Example 1, ContinuedExample 1, ContinuedWhat happens to real wages? Replace column 2 to find out.())()(10001+=+−−=dKFFdKFFdKFdNKNNNKNKPw()0)(=+=NKPwFdKd(Positive only if FNK > 0.)Cet. par., an increase in the capital stock has no effect on real wages unless some of the productivity of the additional capital accrues to labor. (The increase in K would have to cause increase labor marg. productivity.)11Example 1, ContinuedExample 1, Continued)()(001001+=+−=dKFNNdKFdKFdNNKNKKpwPw0)(=+=NKFNdKdNpw(Positive only if FNK > 0.)Cet. par., changes in the capital stock have no impact on employment unless it improves the marginal productivity of labor.12Example 2Example 2Question: in the neoclassical model, for a given level of output, what is the effect of a change in the money supply?0,222==−=−=−=⎟⎠⎞⎜⎝⎛==dMdYbutdMdYkPdMMdPPkdYPMdPPdMkdYPMdPPdMPMdkYPMkPYMby neutrality.13Example 2, ContinuedExample 2, ContinuedPdPMdMPdMdPMdMMdPP===− 0Cet. par., the inflation rate is equal to the growth rate of the money
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