Chiang_Ch8 - 1 Ch. 8 Comparative-Static Analysis of...

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Unformatted text preview: 1 Ch. 8 Comparative-Static Analysis of General-Function Models 8.1 Differentials 8.2 Total Differentials 8.3 Rules of Differentials (I-VII) 8.4 Total Derivatives 8.5 Derivatives of Implicit Functions 8.6 Comparative Statics of General- Function Models 8.7 Limitations of Comparative Statics 2 8.1 Differentials 8.1.1 Differentials and derivatives 8.1.2 Differentials and point elasticity 3 8.1.1 Differentials and derivatives Problem: What if no explicit reduced-form solution exists because of the general form of the model? Example: What is Y / T when Y = C(Y, T ) + I + G 0 T can affect C direct and indirectly thru Y, violating the partial derivative assumption Solution: Find the derivatives directly from the original equations in the model. Take the total differential The partial derivatives become the parameters 4 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 T G I T Y C T Y C Y T G I T G I T Y C Y G I T Y C Y G I T Y G I T Y C Y T Y C C G I C Y b b Y T b G I bT a Y G I bT a bY Y G I b(Y-T) a Y b(Y-T) a C G I C Y * * * * * * * * * , , , , : problem ? ) 7 Then dependent mutually ) T and arguments (C var exog. , , , ) 6 , 5) exists equil. assuming , , Y 4) data parameter no , 3) , 2) 1) If 1 ) 4 Then t independen mutually vars exog. and parameters 1 ) 4 3) data parameter 2) 1) If = = + + + + = + + = = + + =-- = - + +- = + +- =- + + + = + = + + = 5 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 x x f y x x x f x f x f x f dx x f dy x x x f y x x x f y x f x y x x f x y x y x f y dx d x = - + = = = + = + = = - = = 1 1 1 ) 8 ) 7 : is y of ion approximat series Taylor order - first A ality proportion of factor the is where al differenti ) 6 finite as ) 5 ) 4 ) 3 then as ) 2 derivative lim ) 1 als Differenti 1 . 8 Differential: dy & dx as finite changes (p. 180) 6 finite Mathematics. a.Being neither infinite nor infinitesimal. b.Having a positive or negative numerical value; not zero. c.Possible to reach or exceed by counting. Used of a number. d.Having a limited number of elements. Used of a set. 7 Difference Quotient, Derivative & Differential f (x + x) f (x) f (x ) x x + x y= f (x) x y x f(x) f(x ) x x A C D B 8 Overview of Taxonomy - Equations: forms and functions Primitive Form Function Specific (parameters) General (no parameters) Explicit (causation) y = a+bx y = f (x) Implicit (no causation) y 3 +x 3-2xy = 0 F(y, x) = 9 Overview of Taxonomy 1 st Derivatives & Total Differentials Differentiation Form Function Specific (parameters) General (no parameters) Explicit (causation) Implicit (no causation) b y dx d = ) ( dx ) ( x f dx dy x f dy = = ( 29 ( 29 ( 29 ( 29 x y y x dx dy dx y x dy x y 2 3 2 2 2 2 2 3 2 2 2 2--- = =- +- y x y x F F dx dy dx F F dy- =- = 10...
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Chiang_Ch8 - 1 Ch. 8 Comparative-Static Analysis of...

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