Course: Microeconomics Text: Varian’s Intermediate Microeconomics 1

 In Chapter 5, we talk about the optimal choice under the budget constraint.  Here, we consider how the changes of exogenous variables affect decisions.  Recall endogenous variables: quantities of goods x 1, x 2 exogenous variables: prices and income. 2

 Consider the two-good case.  Denote the ordinary demand functions as x 1 *(p 1,p 2,m), x 2 *(p 1,p 2,m).  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 ’ Fixed p 2 and m. p 1 x 1 + p 2 x 2 = m

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

6 x1x1 x2x2 p 1 = p 1 ’’ p 1 = p 1 ’’’ Fixed p 2 and y. 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 y. 7

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

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. 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 Fixed p 2 and m. 10

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. 11

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. 12

 The curve containing all the utility-maximizing bundles traced out as p 1 changes, with p 2 and y 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. 13

 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.

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

 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.

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

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

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

Fixed p 2 and y. x1x1 x2x2

p1p1 x1*x1* Perfect Complement x1x1 x2x2 p1’p1’ p 1 = p 1 ’ ’ ’ y/p 2

p1p1 x1*x1* Fixed p 2 and y. Perfect Complement x1x1 x2x2 p1’p1’ p 1 ’’ p 1 = p 1 ’’ y/p 2

p1p1 x1*x1* Fixed p 2 and y. Perfect Complement x1x1 x2x2 p1’p1’ p 1 ’’ p 1 ’’’ p 1 = p 1 ’’’ ’’’ y/p 2

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

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

and

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

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

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

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

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

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

 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?”  Taking quantity demanded as given and then asking what must be price describes the inverse demand function of a commodity.

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

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

 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.

 Substitutes: if the price is higher for one good, you turn to buy other goods to give you satisfaction. E.g. drinks: bubble tea vs fruit juice; entertainment: movie vs magazine.  Complements: things tend to be used together so when one of the prices increases, the demand for the other will decrease. E.g.: camera and memory cards 38

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

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

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.

 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. 42

p1p1 x1*x1* p1’p1’ p 1 ’’ p 1 ’’’ Generally, an 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.

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

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

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

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’’’

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

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

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

 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.

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

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

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

x2x2 x1x1

x2x2 x1x1 Income offer curve

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

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

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

Fixed p 1 and p 2. Income Changes x1x1 x2x2

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

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.

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.

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

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.

 Quasi-linear preferences  For example,

Income Changes; Quasilinear Utility x2x2 x1x1 x1x1 ~

x2x2 x1x1 x1x1 ~ x1*x1* x2*x2* y y x1x1 ~ Engel curve for good 2 Engel curve for good 1

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

 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) We sometimes call this no money illusion. 70

 How are 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: 71

Then the ordinary demand functions for commodities 1 and 2 are

 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. 73

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.

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

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

 Demand Elasticity measures the percentage change of demand as a result of one percent change in exogenous variable.  Own price elasticity of demand: 77

 Cross Price Elasticity of demand:  Income Elasticity of demand: 78

 Ordinary Good: Giffen Good:  Gross Substitutes: Gross Complements:  Normal Good: Inferior Good: 79

 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 80