1. Demand

2. PROPORTIES OF DEMAND FUNCTIONS
Comparative statics analysis of ordinary demand functions: the study of how ordinary demands x1*(p1,p2,M) and x2*(p1,p2,M) change as prices p1, p2 and income M change.

3. OWN-PRICE CHANGES
How does x1*(p1,p2,M) change as p1 changes, holding p2 and M constant?
Suppose only p1 increases, from p1 = 1 to p1 = 2 and then to p1= 3.

4. OWN-PRICE CHANGES x2
Fixed p2 and M
p1x1 + p2x2 = M
p1 = 1 x1

5. OWN-PRICE CHANGES
Fixed p2 and M x2
p1x1 + p2x2 = M
p1 = 1
p1= 2 x1

6. OWN-PRICE CHANGES Fixed p2 and M x2 p1x1 + p2x2 = M p1 = 1 p1= 3 p1= 2

7. Own-Price Changes p1 Fixed p2 and M x2 p1 = 1 x1* x1 P1= 1 x1*(p1=1)

8. Own-Price Changes p1 Fixed p2 and M x2 p1 = 2 x1* x1 P1=1 x1*(p1=1)

9. Own-Price Changes p1 Fixed p2 and M x2 p1 = 2 x1* x1 P1=2 P1=1

10. Own-Price Changes p1 Fixed p2 and M x2 p1 = 3 x1* x1 P1=3 P1=2 P1=1

11. Own-Price Changes p1 Ordinary demand curve for product 1
Fixed p2 and M x2
P1=3
P1=2
P1=1
x1*(p1=2)
x1*(p1=1)
x1*
x1*(p1=3)
x1*(p1=3)
x1*(p1=1)
x1*(p1=2) x1

12. Own-Price Changes p1 Ordinary demand curve for product 1
Fixed p2 and M x2
P1=3
P1=2
P1=1
P1 price offer curve
x1*(p1=2)
x1*(p1=1)
x1*
x1*(p1=3)
x1*(p1=3)
x1*(p1=1)
x1*(p1=2) x1

13. OWN-PRICE CHANGES
The curve containing all the utility-maximizing bundles traced out (in x1, x2 space) as p1 changes, with p2 and M constant, is the price offer curve.
The plot of the x1co-ordinates of the price offer curve against p1 is the ordinary demand curve for product 1. [Q = F(P) or X = F(P)]

14. Own-Price Changes
Taking quantity demanded as given and then asking what must be price describes the inverse demand function of a good. [P = F(Q) or P = F(X)]

15. OWN-PRICE CHANGES A Cobb-Douglas example:
is the ordinary demand function and
is the inverse demand function.

16. OWN-PRICE CHANGES A perfect complements example:
is the ordinary demand function and
is the inverse demand function.

17. OWN-PRICE CHANGES
What does a p1 price-offer curve look like for Cobb-Douglas preferences?
Take Then the ordinary demand functions for goods 1 and 2 are

18. OWN-PRICE CHANGES
and
Notice that x2* does not vary with p1 so the price offer curve is flat and the ordinary demand curve for product 1 is a rectangular hyperbola.

19. Own-Price Changes
Fixed p2 and M x2
X2*=bM/
(a+b)p2
X1*=aM/
(a+b)p1 x1

20. Own-Price Changes p1 Fixed p2 and M x2 x1* x1 X2*=bM/ (a+b)p2 X1*=aM/

21. Own-Price Changes p1 Ordinary demand curve for product 1 is
X1*=aM/(a+b)p1
Fixed p2 and M x2
X2*=bM/
(a+b)p2
x1*
X1*=aM/
(a+b)p1 x1

22. OWN-PRICE CHANGES PERFECT COMPLEMENTS
What does a p1 price-offer curve look like for a perfect-complements utility function?
The ordinary demand functions for products 1 and 2 are:

23. OWN-PRICE CHANGES PERFECT COMPLEMENTS
With p2 and M fixed, higher p1 causes smaller x1* and x2*. As As

24. OWN-PRICE CHANGES PERFECT COMPLEMENTSx2
Fixed p2 and M x1

25. M/p2 p1 Fixed p2 and M x2 p1 = p11 x1* x1
OWN-PRICE CHANGES PERFECT COMPLEMENTS p1
Fixed p2 and M x2
p1 = p11
M/p2
p11
x1* x1

26. M/p2 2 p1 Fixed p2 and M x2 p1 = p12 x1* x1
OWN-PRICE CHANGES PERFECT COMPLEMENTS p1
Fixed p2 and M x2
p1 = p12
M/p2
p12
p11
x1* 2 x1

27. M/p2 p1 Fixed p2 and M. x2 p1 = p13 x1* x1
OWN-PRICE CHANGES PERFECT COMPLEMENTS p1
Fixed p2 and M.
p13 x2
p1 = p13
M/p2
p12
p11
x1* x1

28. M/p2 p1 Ordinary demand curve for product 1 is Fixed p2 and M x2 x1*
OWN-PRICE CHANGES PERFECT COMPLEMENTS p1
Ordinary demand curve for product 1 is
Fixed p2 and M
p13 x2
M/p2
p12
p11
x1* x1

29. OWN-PRICE CHANGES PERFECT SUBSITUTES
What does a price-offer curve look like for a perfect-substitutes utility function?
Then the ordinary demand functions for products 1 and 2 are

30. OWN-PRICE CHANGES PERFECT SUBSTITUTES
and

31. OWN-PRICE CHANGES PERFECT SUBSTITUTES
Homework: Draw the diagrams.

32. INCOME CHANGES
How does the value of x1*(p1,p2,M) change as M changes, holding both p1 and p2 constant?

33. INCOME CHANGES
A plot of quantity demanded against income is called an Engel curve.

34. INCOME CHANGES M1 < M2 < M3 Engel curve; good 1
Fixed p1 and p2
M1 < M2 < M3
Engel curve;
good 1
Income offer curve M
x23 M3
x22 M2
x21 M1
x11
x13
x11
x13
x1*
x12
x12

35. INCOME CHANGES Engel curve; good 2 M M3 M2 M1 < M2 < M3 M1
Fixed p1 and p2 M3 M2
M1 < M2 < M3 M1
Income offer curve
x21
x23
x2*
x22
x23
x22
Income offer curve = income expansion path.
x21
x11
x13
x12

36. INCOME CHANGES and HOMOTHETIC PREFERENCES
Demand for each good goes up by the same proportion as income
Income offer curve (also known as the income expansion path) is a straight line through the origin
Engel curve is a straight line through the origin

37. HOMOTHETICITY A consumer’s preferences are homothetic if and only if
(x1, x2) > (y1, y2)  (tx1, tx2) > (ty1, ty2)
for all positive t.

38. HOMOTHETICITY
Linear expansion path through the origin X2 X1

39. Income Effects
A product for which quantity demanded rises with income is called normal.
Therefore a normal product’s Engel curve is positively sloped.

40. Income Effects
A product for which quantity demanded falls as income increases is called inferior.
Therefore an inferior product’s Engel curve is negatively sloped.

41. Income Changes: Products 1 & 2 Normal
Engel curve;
good 2 M M3 M2 M1
Income offer curve
x21
x23
x2* M
x22
x23 M3
Engel curve;
good 1
x22 M2
x21 M1
x11
x13
x11
x13
x1*
x12
x12

42. As income changes… Engel curves M x2 product 2 x2* M product 1 x1* x1
Inferior x1/M<0
product 1
Normal x1/M>0 x1
x1*
Product 2 Is Normal, Product 1 Becomes Inferior

43. ORDINARY PRODUCTS
A product is called ordinary if the quantity demanded always increases as its own price decreases.

44. ORDINARY PRODUCTS Downward sloping demand curve p1
Fixed p2 and M.
Downward sloping demand curve x2 p1
p1 price offer curve Û
Product 1 is ordinary
x1* x1

45. GIFFEN PRODUCTS
If, for some values of its own price, the quantity demanded of a product rises as its own price increases then the product is called a Giffen product.

46. GIFFEN PRODUCTS Demand curve has a positively sloped part p1
Fixed p2 and M x2 p1
p1 price offer
curve Û
Product 1 is Giffen
x1* x1

47. CROSS PRICE EFFECTS If an increase in p2
increases demand for product 1 then product 1 is a gross substitute for product 2.
reduces demand for product 1 then product 1 is a gross complement for product 2.

48. CROSS PRICE EFFECTS A perfect complements example: so
Therefore product 2 is a gross complement for product 1

49. CROSS PRICE EFFECTS p1
Increase the price of product 2 from p21 to p22 and
p13
p12
p11
x1*

50. CROSS PRICE EFFECTS p1
Increase the price of product 2 from p21 to p22 and the demand curve
for product 1 shifts inwards -- product 2 is a complement for product 1.
p13
p12
p11
x1*

51. CROSS PRICE EFFECTS
A Cobb- Douglas example: so

52. CROSS PRICE EFFECTS A Cobb- Douglas example: so
Therefore product 1 is neither a gross complement nor a gross substitute for product 2.

53. SUMMARY I Changes in income Cobb Douglas Perfect Substitutes
Perfect Complements

54. SUMMARY II Cross Price Changes Cobb Douglas Perfect Substitutes
(Be Careful!)
Perfect Complements