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

3. 8.1 Differentials 8.1.1 Differentials and derivatives
8.1.2 Differentials and point elasticity

4. 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

7. Differential: dy & dx as finite changes (p. 180)
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.

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

9. 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)
y3+x3-2xy = 0
F(y, x) = 0

10. Overview of Taxonomy – 1st Derivatives & Total Differentials
Differentiation
Form
Function
Specific (parameters)
General (no parameters)
Explicit (causation)
Implicit (no causation)

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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)

18. 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

19. 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 equation dy = f1dx1 + f2dx2

20. 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)

21. 8.3 Rules of Differentials (I-VII)6.
7. d(uvw) = vwdu + uwdv + uvdw

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

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

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

25. 8.3 Example 3, p. 188: Find the total differential (dz) of the function

26. 8.3 Example 3 (revisited using the quotient rule for total differentiation)

27. 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

28. 8.4.1 Finding the total derivative from the differential

29. 8.4.3 Another variation on the theme

30. 8.4.3 Another variation on the theme

31. 8.5 Derivatives of Implicit Functions
8.5.3 Extension to the simultaneous-equation case

32. 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

33. 8.5.1 Implicit functions
Implicit function theorem: given F(y, x1 …, xm) = 0
a) if F has continuous partial derivatives Fy, 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)

34. 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

35. 8.5.1 Implicit functions
Implicit function rule

36. 8.5.1 Deriving the implicit function rule (p. 197)

37. 8.5.1 Deriving the implicit function rule (p. 197)

38. 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 Exercise 8.5-5a on p. 198)
x3 – 2x2y + 3xy = 0
If the function F has continuous partial derivatives Fy, F1, …, Fm
∂F/∂y =-2x2+6xy ∂F/∂x =3x2-4xy+3y2

39. Implicit function problem Exercise 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

40. 8.5.2 Derivatives of implicit functions
Example
If F(z, x, y) = x2z2 + xy2 - z3 + 4yz = 0, then

41. 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 )

42. 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

43. 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

44. 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

45. 8.5.3 Extension to the simultaneous-equation case

46. 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)

47. 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.

48. 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

49. 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

50. 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

51. 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

52. General Function Comparative Statics: A Market Model (8.6.1)

53. General Function Comparative Statics: A Market Model

54. General Function Comparative Statics: A Market Model

55. General Function Comparative Statics: A Market Model

56. General Function Comparative Statics: A Market Model

57. General Function Comparative Statics: A Market Model

58. General Function Comparative Statics: A Market Model

59. Market model comparative static solutions by Cramer’s rule

60. Market model comparative static solutions by matrix inversion

61. 8.7 Limitations of Comparative Statics
Comparative statics answers the question: how does the equilibrium change w/ a change in a parameter.
The adjustment process is ignored
New equilibrium may be unstable
Before dynamic, optimization