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Ch. 8 Comparative-Static Analysis of General-Function Models

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

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Ch. 8 Comparative-Static Analysis of General-Function Models

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  1. Ch. 8Comparative-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, T0) + I0 + G0 T0 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. Differential: dy & dx as finite changes (p. 180) • fi·nite • Mathematics. • Being neither infinite nor infinitesimal. • Having a positive or negative numerical value; not zero. • Possible to reach or exceed by counting. Used of a number. • Having a limited number of elements. Used of a set.

  5. f(x) y=f(x) f(x0+x) y f’(x) f(x0) x x0 x0+x x Difference Quotient, Derivative & Differential B x D A f’(x0)x C

  6. Overview of Taxonomy - Equations: forms and functions

  7. Overview of Taxonomy – 1st Derivatives & Total Differentials

  8. 8.1.1 Differentials and derivatives • From partial differentiation to total differentiation • From partial derivative to total derivative using total differentials • Total derivatives measure the total change in y from the direct and indirect affects of a change in xi

  9. 8.1.1 Differentials and derivatives • The symbols dy and dx are called the differentials of y and x respectively • A differential describes the change in y that results for a specific and not necessarily small change in x from any starting value of x in the domain of the function y = f(x). • The derivative (dy/dx) is the quotient of two differentials (dy) and (dx) • f '(x)dx is a first-order approximation of dy

  10. 8.1.1 Differentials and derivatives • “differentiation” • The process of finding the differential (dy) • (dy/dx) is the converter of (dx) into (dy) as dx 0 • The process of finding the derivative (dy/dx) or • Differentiation with respect to x

  11. 8.1.2 Differentials and point elasticity • Let Qd = f(P) (explicit-function general-form demand equation) • Find the elasticity of demand with respect to price

  12. 8.2 Total Differentials • Extending the concept of differential to smooth continuous functions w/ two or more variables • Let y = f (x1, x2) Find total differential dy

  13. 8.2 Total Differentials (revisited) • Differentiation of U wrt x1 • U/ x1 is the marginal utility of the good x1 • dx1 is the change in consumption of good x1

  14. 8.2 Total Differentials (revisited) Total Differentiation: Let Utility function U = U (x1, x2, …, xn) To find total derivative divide through by the differential dx1 ( partial total derivative)

  15. 8.2 Total Differentials • Let Utility function U = U (x1, x2, …, xn) • Differentiation of U wrt x1..n • U/ xi is the marginal utility of the good xi • dxi is the change in consumption of good xi • dU equals the sum of the marginal changes in the consumption of each good and service in the consumption function

  16. 8.3 Rules of differentials, the straightforward way Find dy given function y=f(x1,x2) • Find partial derivatives f1 and f2 of x1 and x2 • Substitute f1 and f2 into the equationdy = f1dx1 + f2dx2

  17. 8.3 Rules of Differentials (same as rules of derivatives) Let k is a constant function; u = u(x1); v = v(x2) • 1.dk = 0 (constant-function rule) • 2.d(cun) = cnun-1du (power-function rule) • 3.d(u  v) = du  dv (sum-difference rule) • 4.d(uv) = vdu + udv (product rule) • 5. (quotient rule)

  18. 8.3 Rules of Differentials (I-VII) 6. 7. d(uvw) = vwdu + uwdv + uvdw

  19. Rules of Derivatives & Differentials for a Function of One Variable

  20. Rules of Derivatives & Differentials for a Function of One Variable

  21. Rules of Derivatives & Differentials for a Function of One Variable

  22. 8.3 Example 3, p. 188: Find the total differential (dz) of the function

  23. 8.3 Example 3 (revisited using the quotient rule for total differentiation)

  24. 8.4 Total Derivatives • 8.4.1 Finding the total derivative • 8.4.2 A variation on the theme • 8.4.3 Another variation on the theme • 8.4.4 Some general remarks

  25. 8.4.1 Finding the total derivative from the differential

  26. 8.4.3 Another variation on the theme

  27. 8.4.3 Another variation on the theme

  28. 8.5 Derivatives of Implicit Functions • 8.5.1 Implicit functions • 8.5.2 Derivatives of implicit functions • 8.5.3 Extension to the simultaneous-equation case

  29. 8.5.1 Implicit functions • Explicit function: y = f(x)  F(y, x)=0 but reverse may not be true, a relation? • Definition of a function: each x  unique y (p. 16) • Transform a relation into a function by restricting the range of y0, F(y,x)=y2+x2 -9 =0

  30. 8.5.1 Implicit functions • Implicit function theorem: given F(y, x1 …, xm) = 0 a) if F has continuous partial derivativesFy, F1, …, Fm and Fy 0 and b) if at point (y0, x10, …, xm0), we can construct a neighborhood (N) of (x1 …, xm), e.g., by limiting the range of y, y = f(x1 …, xm), i.e., each vector of x’s  unique y then i) y is an implicitly defined function y = f(x1 …, xm) and ii) still satisfies F(y, x1 … xm) for every m-tuple in the N such that F  0 (p. 195) dfn: use  when two side of an equation are equal for any values of x and y dfn: use = when two side of an equation are equal for certain values of x and y (p.197)

  31. 8.5.1 Implicit functions • If the function F(y, x1, x2, . . ., xn) = k is an implicit function of y = f(x1, x2, . . ., xn), then • where Fy = F/y; Fx1 = F/x1 • Implicit function rule • F(y, x) = 0; F(y, x1, x2 … xn) = 0, set dx2 to n = 0

  32. 8.5.1 Implicit functions • Implicit function rule

  33. 8.5.1 Deriving the implicit function rule (p. 197)

  34. 8.5.1 Deriving the implicit function rule (p. 197)

  35. Implicit function problem:Exercise 8.5-5a, p. 198 • Given the equation F(y, x) = 0 shown below, is it an implicit function y = f(x) defined around the point (y = 3, x = 1)? (see Exercise8.5-5a on p. 198) • x3 – 2x2y + 3xy2 - 22 = 0 • If the function F has continuous partial derivatives Fy, F1, …, Fm • ∂F/∂y =-2x2+6xy ∂F/∂x =3x2-4xy+3y2

  36. Implicit function problemExercise 8.5-5a, p. 198 • If at a point (y0, x10, …, xm0) satisfying the equation F (y, x1 …, xm) = 0, Fy is nonzero (y = 3, x = 1) • This implicit function defines a continuous function f with continuous partial derivatives • If your answer is affirmative, find dy/dx by the implicit-function rule, and evaluate it at point (y = 3, x = 1) • ∂F/∂y =-2x2+6xy ∂F/∂x =3x2-4xy+3y2 • dy/dx = - Fx/Fy =- (3x2-4xy+3y2 )/-2x2+6xy • dy/dx = -(3*12-4*1*3+3*32 )/(-2*12+6*1*3)=-18/16=-9/8

  37. 8.5.2 Derivatives of implicit functions • Example If F(z, x, y) = x2z2 + xy2 - z3 + 4yz = 0, then

  38. 8.5 Implicit production function • F (Q, K, L) Implicit production function • K/L = -(FL/FK) MRTS: Slope of the isoquant • Q/L = -(FL/FQ) MPPL • Q/K = -(FK/FQ) MPPK (pp. 198-99)

  39. Overview of the Problem –8.6.1 Market model • Assume the demand and supply functions for a commodity are general form explicit functions Qd = D(P, Y0) (Dp < 0; DY0 > 0) Qs = S(P, T0) (Sp > 0; ST0 < 0) • where Q is quantity, P is price, (endogenous variables) Y0 is income, T0 is the tax (exogenous variables)no parameters, all derivatives are continuous • Find P/Y0, P/T0 Q/Y0, Q/T0

  40. Overview of the Procedure -8.6.1 Market model • Given Qd = D(P, Y0) (Dp < 0; DY0 > 0) Qs = S(P, T0) (Sp > 0; ST0 < 0) • Find P/Y0, P/T0, Q/Y0, Q/T0 Solution: • Either take total differential or apply implicit function rule • Use the partial derivatives as parameters • Set up structural form equations as Ax = d, • Invert A matrix or use Cramer’s rule to solve for x/d

  41. 8.5.3 Extension to the simultaneous-equation case • Find total differential of each implicit function • Let all the differentials dxi = 0 except dx1 and divide each term by dx1 (note: dx1 is a choice ) • Rewrite the system of partial total derivatives of the implicit functions in matrix notation

  42. 8.5.3 Extension to the simultaneous-equation case

  43. 8.5.3 Extension to the simultaneous-equation case • Rewrite the system of partial total derivatives of the implicit functions in matrix notation (Ax=d)

  44. 7.6 Note on Jacobian Determinants • Use Jacobian determinants to test the existence of functional dependence between the functions /J/ • Not limited to linear functions as /A/ (special case of /J/ • If /J/ = 0 then the non-linear or linear functions are dependent and a solution does not exist.

  45. 8.5.3 Extension to the simultaneous-equation case • Solve the comparative statics of endogenous variables in terms of exogenous variables using Cramer’s rule

  46. 8.6 Comparative Statics of General-Function Models • 8.6.1 Market model • 8.6.2 Simultaneous-equation approach • 8.6.3 Use of total derivatives • 8.6.4 National income model • 8.6.5 Summary of the procedure

  47. Overview of the Problem –8.6.1 Market model • Assume the demand and supply functions for a commodity are general form explicit functions Qd = D(P, Y0) (Dp < 0; DY0 > 0) Qs = S(P, T0) (Sp > 0; ST0 < 0) • where Q is quantity, P is price, (endogenous variables) Y0 is income, T0 is the tax (exogenous variables)no parameters, all derivatives are continuous • Find P/Y0, P/T0 Q/Y0, Q/T0

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