 Previously, we examined a consumer’s optimal choice under his budget constraint.  In this chapter, we will perform comparative static analysis of ordinary demand functions. Namely, how do ordinary demands x 1 *(p 1,p 2,m) and x 2 *(p 1,p 2,m) change as prices p 1, p 2 and income m change? 2

 How does x 1 *(p 1,p 2,m) change as p 1 changes, holding p 2 and m constant?  Suppose only p 1 increases, from p 1 ’ to p 1 ’’ and then to p 1 ’’’. 3

4 x1x1 x2x2 p 1 = p 1 ’’ p 1 = p 1 ’’’ Fixed p 2 and m. p 1 = p 1 ’ p 1 x 1 + p 2 x 2 = m

x 1 *(p 1 ’) Own-Price Changes p 1 = p 1 ’ Fixed p 2 and m. 5

x 1 *(p 1 ’) p1p1 p1’p1’ x1*x1* Own-Price Changes Fixed p 2 and m. p 1 = p 1 ’ 6

x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ x1*x1* Own-Price Changes Fixed p 2 and m. 7

x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Fixed p 2 and m. 8

x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 Fixed p 2 and m. 9

x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x 1 *(p 1 ’) x 1 *(p 1 ’’’) x 1 *(p 1 ’’) p1’p1’ p 1 ’’ p 1 ’’’ x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 p 1 price offer curve Fixed p 2 and m. 10

 The curve containing all the utility-maximizing bundles traced out as p 1 changes, with p 2 and m constant, is the p 1 - price offer curve.  The plot of the x 1 -coordinate of the p 1 - price offer curve against p 1 is the ordinary demand curve for commodity 1. 11

 A good is called an ordinary good if the quantity demanded of it always increases as its own price decreases and vice versa (negatively related), holding all other factors, such as prices, income and preference constant. 12

Fixed p 2 and m. x1x1 x2x2 p 1 price offer curve x1*x1* Downward-sloping demand curve Good 1 is ordinary  p1p1 13

 If the quantity demanded of a good decreases as its own-price decreases and vice versa (i.e. positively related) holding all other factors constant, then the good is called a Giffen Good.  Note: we need to hold other factors constant. Thus, if the price change is also associated with change in income or preference, then even if there’s a positive relation between price and quantity, it is not characterized as Giffen good. 14

Fixed p 2 and m. x1x1 x2x2 p 1 price offer curve x1*x1* Demand curve has a positively sloped part Good 1 is Giffen  p1p1 15

 What does a p 1 price-offer curve look like for a perfect-complements utility function? 16

With p 2 and m fixed, higher p 1 causes smaller x 1 * and x 2 *. As 17

Fixed p 2 and m. x1x1 x2x2 18

p1p1 x1*x1* Fixed p 2 and m. Perfect Complements x1x1 x2x2 p1’p1’ p 1 = p 1 ’ ’ ’ m/p 2 19

p1p1 x1*x1* Fixed p 2 and m. Perfect Complements x1x1 x2x2 p1’p1’ p 1 ’’ p 1 = p 1 ’’ m/p 2 20

p1p1 x1*x1* Fixed p 2 and m. Perfect Complements x1x1 x2x2 p1’p1’ p 1 ’’ p 1 ’’’ p 1 = p 1 ’’’ m/p 2 21

p1p1 x1*x1* Ordinary demand curve for commodity 1 is Fixed p 2 and m. Perfect Complements x1x1 x2x2 p1’p1’ p 1 ’’ p 1 ’’’ m/p 2 22

 What does a p 1 price-offer curve look like for a perfect-substitutes utility function? 23

and 24

Fixed p 2 and Perfect Substitutes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and m. p1’p1’ p 1 = p 1 ’ < p 2 25

Fixed p 2 and m. Perfect Substitutes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and. p1’p1’ p 1 = p 1 ’’ = p 2 26

Fixed p 2 and Perfect Substitutes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and m. p1’p1’ p 1 = p 1 ’’ = p 2 p 2 = p 1 ’’ 27

Fixed p 2 and m. Perfect Substitutes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and p1’p1’ p 1 ’’’ p 2 = p 1 ’’ 28

Fixed p 2 and m. Perfect Substitutes x2x2 x1x1 p1p1 x1*x1* Fixed p 2 and p1’p1’ p 2 = p 1 ’’ p 1 ’’’ p 1 price offer curve Ordinary demand curve for commodity 1 29

 Usually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?”  But we could also ask the inverse question “At what price for commodity 1 would a given quantity of commodity 1 be demanded?” 30

p1p1 x1*x1* p1’p1’ Given p 1 ’, what quantity is demanded of commodity 1? Answer: x 1 ’ units. x1’x1’ 31

p1p1 x1*x1* p1’p1’ x1’x1’ Given p 1 ’, what quantity is demanded of commodity 1? Answer: x 1 ’ units. The inverse question is: Given x 1 ’ units are demanded, what is the price of commodity 1? Answer: p 1 ’ 32

 If an increase in p 2, holding p 1 constant,  increases demand for commodity 1 then commodity 1 is a gross substitute for commodity 2.  reduces demand for commodity 1 then commodity 1 is a gross complement for commodity 2. 33

A perfect-complements example : so Therefore commodity 2 is a gross complement for commodity 1. 34

p1p1 x1*x1* p1’p1’ p 1 ’’ p 1 ’’’ Increase the price of good 2 from p 2 ’ to p 2 ’’ and 35

p1p1 x1*x1* p1’p1’ p 1 ’’ p 1 ’’’ Increase the price of good 2 from p 2 ’ to p 2 ’’ and the demand curve for good 1 shifts inwards -- good 1 is a complement for good 2. 36

 For perfect substitutes, how will the quantity demanded x 1 change when p 2 increases?  Note: it only changes the point where x 1 starts to be positive. 37

p1p1 x1*x1* p1’p1’ p 1 ’’ p 1 ’’’ Generally, if increase the price of good 2 from p 2 ’ to p 2 ’’ and the demand curve for good 1 shifts outwards -- good 1 is a substitute for good 2. 38

 How does the value of x 1 *(p 1,p 2,m) change as m changes, holding both p 1 and p 2 constant? 39

Income Changes Fixed p 1 and p 2. m’ < m’’ < m’’’ 40

Income Changes Fixed p 1 and p 2. x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ m’ < m’’ < m’’’ 41

Income Changes Fixed p 1 and p 2. x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve m’ < m’’ < m’’’ 42

 A plot of income against quantity demanded is called an Engel curve. 43

Income Changes Fixed p 1 and p 2. m’ < m’’ < m’’’ x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x1*x1* x2*x2* y y x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ m’ m’’ m’’’ m’ m’’ m’’’ Engel curve; good 2 Engel curve; good 1 44

p1p1 x1*x1* p1’p1’ p 1 ’’ p 1 ’’’ When income increases, the curve shifts outward for each given price, if the good is a normal good. 45

 A good for which quantity demanded rises with income is called normal.  Therefore a normal good’s Engel curve is positively sloped.  Generally, most goods are normal goods. 46

 A good for which quantity demanded falls as income increases is called inferior.  Therefore an inferior good’s Engel curve is negatively sloped.  E.g.: low-quality goods. 47

Income Changes; Goods 1 & 2 Normal x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ Income offer curve x1*x1* x2*x2* y y x 1 ’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ y’ y’’ y’’’ Engel curve; good 2 Engel curve; good 1 48

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior x2x2 x1x1 Income offer curve 49

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior x2x2 x1x1 x1*x1* y Engel curve for good 1 50

 Now we compute the equations of Engel curves for the perfectly-complementary case.  The ordinary demand equations are 51

Rearranging these equations, we get : Engel curve for good 1 Engel curve for good 2 52

Fixed p 1 and p 2. Income Changes x1x1 x2x2 53

Income Changes x1x1 x2x2 y’ < y’’ < y’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ x 1 ’’’ Fixed p 1 and p 2. 54

Income Changes x1x1 x2x2 y’ < y’’ < y’’’ x 1 ’’ x1’x1’ x 2 ’’’ x 2 ’’ x2’x2’ x 1 ’’’ x1*x1* x2*x2* y y x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ y’ y’’ y’’’ Engel curve; good 2 Engel curve; good 1 x 1 ’’’ x 1 ’’ x1’x1’ Fixed p 1 and p 2. 55

Income Changes x1*x1* x2*x2* y y x 2 ’’’ x 2 ’’ x2’x2’ y’ y’’ y’’’ y’ y’’ y’’’ x 1 ’’’ x 1 ’’ x1’x1’ Engel curve; good 2 Engel curve; good 1 Fixed p 1 and p 2. 56

 Another example of computing the equations of Engel curves; the perfectly-substitution case.  The ordinary demand equations are 57

yy x1*x1*x2*x2* 0 Engel curve for good 1 Engel curve for good 2 Suppose p 1 < p 2 It is the opposite when p 1 > p 2. 58

 In every example so far the Engel curves have all been straight lines.  Q: Is this true in general?  A: No. Engel curves are straight lines if the consumer’s preferences are homothetic. 59

 A consumer’s preferences are homothetic if and only if (x 1,x 2 ) (y 1,y 2 )  (kx 1,kx 2 ) (ky 1,ky 2 ) for every k > 0.  That is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin. 60 

 Quasi-linear preferences  For example, 61

Income Changes; Quasilinear Utility x2x2 x1x1 x1x1 ~ 62

Income Changes; Quasilinear Utility x2x2 x1x1 x1x1 ~ x1*x1* x2*x2* y y x1x1 ~ Engel curve for good 2 Engel curve for good 1 63

Income Changes; Good 2 Is Normal, Good 1 Becomes Inferior x2x2 x1x1 x1*x1* x2*x2* y y Engel curve for good 2 Engel curve for good 1 64

 If income and prices all change by the same proportion k, how does the consumer demand change? (e.g. same rate of inflation for prices and income)  Recall that prices and income only affects the budget constraint: p 1 x 1 + p 2 x 2 = m. Now it becomes: kp 1 x 1 + kp 2 x 2 = km. Which clearly gets back to the original one.  Thus, x i (kp 1, kp 2, km) = x i (p 1, p 2, m). 65

 How do the demand curves and Engel curves look like for a Cobb-Douglas utility?  Recall the Cobb-Douglas utility function:  On solving using the MRS=p 1 / p 2 and the budget constraint, you will have: 66

The ordinary demand functions for commodities 1 and 2 are 67

 Own-Price effect: inversely related to its own price.  Cross-Price effect: no cross price effect  Income Effect: Engle curve is a straight line passing through the origin. 68

x 1 *(p 1 ’’’) x 1 *(p 1 ’) x 1 *(p 1 ’’) p1p1 x1*x1* Own-Price Changes Ordinary demand curve for commodity 1 is Fixed p 2 and y. 69

Rearranged to isolate m, these are: Engel curve for good 1 Engel curve for good 2 70

m m x1*x1* x2*x2* Engel curve for good 1 Engel curve for good 2 71

 Demand Elasticity measures the percentage change of demand as a result of one percent change in exogenous variable.  Own price elasticity of demand: 72

 Cross Price Elasticity of demand:  Income Elasticity of demand: 73

 Ordinary Good: Giffen Good:  Gross Substitutes: Gross Complements:  Normal Good: Inferior Good: 74

 The consumer demand function for a good generally depends on prices of all goods and income.  Ordinary: demand decreases with own price Giffen: demand increases with own price  Substitute: demand increases with other price Complement: demand decreases with other price  Normal good: demand increases with income Inferior good: demand decreases with income 75