Chiang, Ch. 7 Rules of Differentiation and Their Use in Comparative Statics

1. Chiang, Ch. 7Rules of Differentiation and Their Use in Comparative Statics • 7.1 Rules of Differentiation for a Function of One Variable • 7.2 Rules of Differentiation Involving Two or More Functions of the Same Variable • 7.3 Rules of Differentiation Involving Functions of Different Variables • 7.4 Partial Differentiation • 7.5 Applications to Comparative-Static Analysis • 7.6 Note on Jacobian Determinants

2. 7.1 Rules of Differentiation for a Function of One Variable • 7.1.1 Constant-Function Rule • 7.1.2 Power-Function Rule • 7.1.3 Power-Function Rule Generalized

3. A Review of the 9 Rules of Differentiation for a Function of One Variable

4. y=f(x) Secant slope:rise/run =f(x0+x)-f(x0)/(x0+x-x0) f(x0) x x0 x0+x Difference Quotient f(x) f(x0+x)

5. y=f(x) Secant slope:rise/run =f(x)-f(N)/(x-N) f(N) x N x Difference Quotient f(x) f(x)

6. 6.2 Rate of Change and the DerivativeThe difference quotient & the derivative (p. 144)

7. 7.1.1 Constant-Function Rule

8. 7.1.1 Power-Function Rule

11. 7.1.1 Power-Function Rule

12. 7.1.2 Power-Function Rule

13. 7.2 Rules of Differentiation Involving Two or More Functions of the Same Variable • 7.2.1 Sum-difference rule • 7.2.2 Product rule • 7.2.3 Finding marginal-revenue function from average-revenue function • 7.2.4 Quotient rule • 7.2.5 Relationship between marginal-cost and average-cost functions

14. 7.2.1 Sum or difference rule

15. 7.2.2 Product rule

16. 7.2.3 Finding marginal-revenue function from average-revenue function using the product rule

17. A Review of the 9 Rules of Differentiation for a Function of One Variable

18. 7.2.4 Quotient rule

19. C MC AC Q 7.2.5 Relationship between marginal-cost and average-cost functions

20. 7.3 Rules of Differentiation Involving Functions of Different Variables • 7.3.1 Chain rule • 7.3.2 Inverse-function rule

21. A Review of the 9 Rules of Differentiation Involving Functions of Different Variables

22. 7.3.1 Chain rule

23. 7.3.1 Chain rule

24. 7.3.1 Chain rule and its relation to total differential

25. A Review of the 9 Rules of Differentiation Involving Functions of Different Variables

26. Inverse-function rule

27. 7.3.2 Inverse-function rule

28. 7.3.2 Inverse-function rule • This property of one to one mapping is unique to the class of functions known as monotonic functions: • Recall the definition of a function, p. 17, function: one y for each xmonotonic function: one x for each y (inverse function) • if x1 > x2 f(x1) > f(x2) monotonically increasing Qs = b0 + b1P supply function (where b1 > 0) P = -b0/b1 + (1/b1)Qs inverse supply function • if x1 > x2 f(x1) < f(x2) monotonically decreasing Qd = a0 - a1P demand function (where a1 > 0) P = a0/a1 - (1/a1)Qd inverse demand function

29. 7.4 Partial Differentiation • 7.4.1 Partial derivatives • 7.4.2 Techniques of partial differentiation • 7.4.3 Geometric interpretation of partial derivatives

30. 7.4.1 Partial derivatives

31. 7.4.1 Partial derivatives

32. 7.5 Applications to Comparative-Static Analysis • 7.5.1 Market model • 7.5.2 National-income model • 7.5.3 Input-output model

33. 7.4.2 Techniques of partial differentiation: Market Model

34. 7.4.2 Techniques of partial differentiation: Market Model

35. 7.4.2 Techniques of partial differentiation: Market Model

36. 7.4.3 Geometric interpretation of partial derivatives • Market model

37. 7.4.3 Geometric interpretation of partial derivatives • Market model

38. D1 Q S1 S0 D P Q S D P 7.4.3 Geometric interpretation of partial derivatives

39. S1 Q Q S0 S0 Q0 Q1 D0 D P D1 P 7.5.1 Market model

40. 7.5.2 National-income model

41. 7.5.3 Input-output model

42. 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.