Applied Microeconomics

1
Applied Microeconomics

• Demand

2
Outline

• Demand functions and inverse demand functions
• Elasticity
• Total revenue and marginal revenue
• Marginal revenue and elasticity
• Aggregating demand and elasticity

3
Readings

• Perloff Chapter 2-3
• Kreps Chapter 4
• Zandt Chapter 3

4
Approximation

• More buyers or divisible goods smoothens the
  jagged demand curve we derived in the last
  lecture
• This enables us to treat the demand curve as a
  continuous function

5
Demand functions

• Consider the market for a divisible good
• The demand function D(p) says how much of a given
  product would be purchased at each price p per
  unit, holding other variables fixed
• The inverse demand function P(x) gives the price
  at which q units of the product would be sold,
  holding other variables fixed
• Example D(p)10-2p gives P(x)(10-x)/2

6
Demand and Inverse Demand Functions
7
Do Demand Functions Slope Downward?

• It is common to assume that demand functions are
  downward sloping
• Convenient since this makes the demand function
  invertible
• Generally true
• Exist examples of demand functions that slope
  upwards for some range of prices (Giffen goods)
  when buyers are unsure about product quality and
  believes price signals something about quality

8
Demand Facing Firm and Demand Facing Industry

• Important to distinguish between the demand
  facing an entire industry or the demand facing a
  single firm within the industry
• The demand facing an entire industry is usually
  less responsive to changes in prices than demand
  facing a single firm
• Example A 10 price increase on all laptops vs.
  10 increase on Dell laptops

9
Price Taking Firms and Firms with Market Power

• In the case of a competitive market, such as that
  of crude oil carriers, the demand facing a firm
  is zero if it sets its price above the market
  price - the firm is a price taker
• When, on the other hand, the firm can choose its
  output, selling an amount determined by its
  demand function, it is said to have market power

10
Other Variables Affecting Demand

• A demand function records how the quantity of a
  certain good changes as a function of the own
  price of the good
• This means that all other variables affecting
  demand are held fixed or ceteris paribus
• What is held fixed?
• Prices of other goods
• Income
• Advertisement expenditure etc.
• Sometimes complicated in practice

11
Example Demand for Coke

• Estimated 1992 demand for CokeD(pcoke)26.17-3.98
  pcoke2.25ppepsi2.60acoke-0.62apepsi9.58s0.99y
  
• Where
• aj is advertising expenses
• s1 if summer, s0 otherwise
• y is real income

12
Example Demand for Coke

• The following factors each causes an outward
  shift in the demand for Coke
• Pepsi price crease
• Pepsi ad. exp. crease
• Coke ad. exp. crease
• Real income crease

13
Classification of Goods

• If demand increases as the price of another good
  increases, the goods are substitutes
• If demand decreases as the price of another good
  increases, the goods are complements
• If demand increases as income increases, the good
  is a normal good
• If demand decreases as income increases, the good
  is an inferior good

14
Do Firms Know Their Demand Functions?

• We will generally assume that firms know their
  own demand functions perfectly
• This is not entirely true, but firms do find out
  about the shape of the demand functions in a
  neighborhood of current price using various
  techniques
• Made easier by technologies such as supermarket
  scanners and internet

15
Estimating Demand Functions

• Procedure
• Write down model (equation) for product demand
  with unknown coefficients.
• Fit line or curve to data points using
  statistical techniques (regression).
• Some sources of data
• Consumer surveys
• Consumer focus groups
• Market experiments
• Historical (real) data cross-section,
  time-series, or both (panel)

16
Estimating Demand Functions

• Commonly estimated equations
• Linear D(p)A-BpCy
• Log ln(D(p))A-Bln(p)Cln(y)
• You can try this out by doing problem 4.15 in
  Kreps (data on the web and answer in Student
  Companion)

17
Price Sensitivity

• Which is the most price sensitive of the demand
  functions in each diagram?
• For which market would you set the higher price?

18
Price Sensitivity

• Suppose we want to know what happens to demand as
  we increase the price slightly
• Crucial for profit maximization to find out how
  sensitive demand is to changes in price
• The tool for this is elasticity

19
Elasticity

• The (own-price) elasticity of demand at a
  particular price p0 and quantity x0 is the
  change in quantity demanded per 1 change in
  price (?x/x0)/(?p/p0)
• Example Price increase from 100 to 102 causes
  demand decrease from 10 to 9, giving elasticity
  of -10/2-5 at p0100
• We can also calculate the midpoint Arc Elasticity
  which is given by vA(p0,p1)(x1-x0)/0.5(x1x0)/(p
  1-p0)/0.5(p1p0))
• Example gives (1/9.5)/(2/101)-5.32

20
Elasticity

• With a differentiable demand, this can be
  expressed as v(p0) ?D(p0)/?pp0/D(p0)D(p0)p0/
  D(p0)
• We may also express the elasticity using the
  inverse demand function v(x0)1/P(x0)P(x0)/x0
• Note the difference between the functions v(p0)
  and v(x0)

21
Two Extreme Cases
22
Calculating Elasticity

• If we estimate a demand function of the form
  D(p)A-BpCy, the own-price elasticity is
  v(p)?
• If we estimate a demand function of the form
  ln(D(p))A-Bln(p)Cln(y), the own-price
  elasticity is v(p)?

23
Demand and Elasticity
24
Why Elasticity and Not the Derivative?

• Example Suppose a consumer has monthly demand
  for gasoline given by DM(p)50(3-p) and annual
  demand given by DA(p)12DM(p)
• Suppose we want to find the effect on his demand
  of a small change in the price
• The price derivative of the monthly demand is 50
  and of the annual demand is -1250-600
• However, the elasticity of demand is
  v(p0)-p0/3-p0 for both demand functions!

25
Other Elasticities

• Income elasticity measures demand sensitivity to
  changes in income vy(y) ?D/?yy/D
• Cross-price elasticity measures demand
  sensitivity to changes in the price of another
  good vC1(p2) ?D1/?p2p2/D1

26
Marginal Revenue

• The firms total revenue from selling x units of
  a good is given by the function TR(x)xP(x)
• If we take the derivative of this with respect to
  quantity, we obtain the marginal revenue function
  MR(x)TR(x)
• It tells us how much more revenue we get from
  adjusting price so that we sell one more unit of
  the good

27
Total Revenue and Marginal Revenue
28
Elasticity and Marginal Revenue

• MR(x)P(x)xP(x)P(x)(1xP(x)/P(x))
• But recall that xP(x)/P(x)1/v(x)
• This gives, MR(x)P(x)(11/v(x))
• Hence, marginal revenue is a function of price
  and elasticity!

29
Elasticity and Marginal Revenue
30
Demand, Elasticity, and Marginal Revenue
MR(x)0
31
Aggregating Demand Functions

• The individual demand function Di(p) is the
  demand from a single consumer
• The aggregate or market demand is the demand from
  a group of I consumers D(p)
• The relationship between the two is D(p)D1(p)
  D2(p) DI(p)

32
Aggregating Demand Functions Example

• 2 consumers with individual demand functions
  D1(p)10-2p and D2(p)4-p
• gives aggregate demand D(p)14-3p for plt4,
  10-2p for 4plt5, and 0 for 5p

33
Aggregating Elasticity

• Suppose we know the demand functions for three
  segments of the market (Austria, Belgium and
  Cyprus) DTotal(p)DA(p)DB(p)DC(p)
• The elasticities of the segments are given by
• vA(p)pDA(p)/DA(p)
• vB(p)pDB(p)/DB(p)
• vC(p)pDC(p)/DC(p)

34
Aggregating Elasticity

• Then we have thatvA(p)DA(p)vB(p)DB(p)vC(p)DC(p
  )pDtotal(p)
• Hence vTotal(p)pDtotal(p)/Dtotal(p)(vA(p)DA(p
  )vB(p)DB(p)vC(p)DC(p))/Dtotal(p)
• In other words the total elasticity is the
  weighted average of the segments elasticities,
  weighted by their shares of total demand

35
Disaggregating Demand Functions

• Firms often tries to break down total demand into
  segments, charging different segments different
  prices
• Example Student discounts, vouchers, air fares
• This is known as price discrimination
• You will learn more about this in a couple of
  weeks

36
Conclusions

• Demand functions measure quantity that can be
  sold at each price, inverse demand functions
  price that can be charged for each quantity sold
• Elasticity is the change in demand per 1
  change in price
• Aggregate demand is the horizontal sum of
  individual demand
• Aggregate elasticity is the weighted average of
  individual elasticities
• MR(x)P(x)(11/v(x))