1. MARKET DEMAND AND ELASTICITY
Chapter 7
MARKET DEMAND AND ELASTICITY
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

2. Market Demand Curves
Assume that there are only two goods (X and Y) and two individuals (1 and 2)
The first person’s demand for X is
X1 = dX1(PX,PY,I1)
The second person’s demand for X is
X2 = dX2(PX,PY,I2)

3. Market Demand Curves Features of these demand curves:
Both individuals are assumed to face the same prices
Each buyer is assumed to be a price taker
must accept the prices prevailing in the market
Each person’s demand depends on his or her own income

4. total X = dX1(PX,PY,I1) + dX2(PX,PY,I2)
Market Demand Curves
The total demand for X is the sum of the amounts demanded by the two buyers
The demand function will depend on PX, PY, I1, and I2
total X = X1 + X2
total X = dX1(PX,PY,I1) + dX2(PX,PY,I2)
total X = DX(PX,PY,I1,I2)

5. Market Demand Curves
To construct the market demand curve, PX is allowed to vary while PY, I1, and I2 are held constant
If each individual’s demand for X is downward sloping, the market demand curve will also be downward sloping

6. Market Demand Curves To derive the market demand curve, we sum the
quantities demanded at every price PX PX
Individual 1’s
demand curve
Individual 2’s
demand curve PX
Market demand
curve X* DX
X1* + X2* = X*
X1*
X2*
PX*
dX1
dX2 X X X

7. Shifts in the Market Demand Curve
The market demand summarizes the ceteris paribus relationship between X and PX
Changes in PX result in movements along the curve (change in quantity demanded)
Changes in other determinants of the demand for X cause the demand curve to shift to a new position (change in demand)

8. Shifts in Market Demand
Suppose that individual 1’s demand for oranges is given by
X1 = 10 – 2PX + 0.1I PY
and individual 2’s demand is
X2 = 17 – PX I PY
The market demand curve is
X = X1 + X2 = 27 – 3PX + 0.1I I2 + PY

9. Shifts in Market Demand
To graph the demand curve, we must assume values for PY, I1, and I2
If PY = 4, I1 = 40, and I2 = 20, the market demand curve becomes
X = 27 – 3PX = 36 – 3PX

10. Shifts in Market Demand
If PY rises to 6, the market demand curve shifts outward to
X = 27 – 3PX = 38 – 3PX
Note that X and Y are substitutes
If I1 fell to 30 while I2 rose to 30, the market demand would shift inward to
X = 27 – 3PX = 35.5 – 3PX
Note that X is a normal good for both buyers

11. Generalizations
Suppose that there are n goods (Xi, i = 1,n) with prices Pi, i = 1,n.
Assume that there are m individuals in the economy
The j th’s demand for the i th good will depend on all prices and on Ij
Xij = dij(P1,…,Pn, Ij)

12. Generalizations
The market demand function for Xi is the sum of each individual’s demand for that good
The market demand function depends on the prices of all goods and the incomes and preferences of all buyers

13. Elasticity
Suppose that a particular variable (B) depends on another variable (A)
B = f(A…)
We define the elasticity of B with respect to A as
The elasticity shows how B responds (ceteris paribus) to a 1 percent change in A

14. Price Elasticity of Demand
The most important elasticity is the price elasticity of demand
measures the change in quantity demanded caused by a change in the price of the good
eQ,P will generally be negative
except in cases of Giffen’s paradox

15. Distinguishing Values of eQ,P
Value of eQ,P at a Point
Classification of Elasticity at This Point
eQ,P < -1
Elastic
eQ,P = -1
Unit Elastic
eQ,P > -1
Inelastic

16. Price Elasticity and Total Expenditure
Total expenditure on any good is equal to
total expenditure = PQ
Using elasticity, we can determine how total expenditure changes when the price of a good changes

17. Price Elasticity and Total Expenditure
Differentiating total expenditure with respect to P yields
Dividing both sides by Q, we get

18. Price Elasticity and Total Expenditure
Note that the sign of PQ/P depends on whether eQ,P is greater or less than -1
If eQ,P > -1, demand is inelastic and price and total expenditures move in the same direction
If eQ,P < -1, demand is elastic and price and total expenditures move in opposite directions

19. Price Elasticity and Total Expenditure
Responses of PQ
Demand
Price Increase
Price Decrease
Elastic
Falls
Rises
Unit Elastic
No Change
Inelastic

20. Income Elasticity of Demand
The income elasticity of demand (eQ,I) measures the relationship between income changes and quantity changes
Normal goods  eQ,I > 0
Luxury goods  eQ,I > 1
Inferior goods  eQ,I < 0

21. Cross-Price Elasticity of Demand
The cross-price elasticity of demand (eQ,P’) measures the relationship between changes in the price of one good and and quantity changes in another
Gross substitutes  eQ,P’ > 0
Gross complements  eQ,P’ < 0

22. Relationships Among Elasticities
Suppose that there are only two goods (X and Y) so that the budget constraint is given by
PXX + PYY = I
The individual’s demand functions are
X = dX(PX,PY,I)
Y = dY(PX,PY,I)

23. Relationships Among Elasticities
Differentiation of the budget constraint with respect to I yields
Multiplying each item by 1

24. Relationships Among Elasticities
Since (PX · X)/I is the proportion of income spent on X and (PY · Y)/I is the proportion of income spent on Y,
sXeX,I + sYeY,I = 1
For every good that has an income elasticity of demand less than 1, there must be goods that have income elasticities greater than 1

25. Slutsky Equation in Elasticities
The Slutsky equation shows how an individual’s demand for a good responds to a change in price
Multiplying by PX /X yields

26. Slutsky Equation in Elasticities
Multiplying the final term by I/I yields

27. Slutsky Equation in Elasticities
A substitution elasticity shows how the compensated demand for X responds to proportional compensated price changes
it is the price elasticity of demand for movement along the compensated demand curve

28. Slutsky Equation in Elasticities
Thus, the Slutsky relationship can be shown in elasticity form
It shows how the price elasticity of demand can be disaggregated into substitution and income components
Note that the relative size of the income component depends on the proportion of total expenditures devoted to the good (sX)

29. Homogeneity
Remember that demand functions are homogeneous of degree zero in all prices and income
Euler’s theorem for homogenous functions shows that

30. Homogeneity Dividing by X, we get
Using our definitions, this means that
An equal percentage change in all prices and income will leave the quantity of X demanded unchanged

31. Cobb-Douglas Elasticities
The Cobb-Douglas utility function is
U(X,Y) = XY
The demand functions for X and Y are
The elasticities can be calculated

32. Cobb-Douglas Elasticities
Similar calculations show
Note that

33. Cobb-Douglas Elasticities
Homogeneity can be shown for these elasticities
The elasticity version of the Slutsky equation can also be used

34. Cobb-Douglas Elasticities
The price elasticity of demand for this compensated demand function is equal to (minus) the expenditure share of the other good
More generally
where  is the elasticity of substitution

35. Linear Demand Q = a + bP + cI + dP’ where: Q = quantity demanded
P = price of the good
I = income
P’ = price of other goods
a, b, c, d = various demand parameters

36. Linear Demand Q = a + bP + cI + dP’ Assume that:
Q/P = b  0 (no Giffen’s paradox)
Q/I = c  0 (the good is a normal good)
Q/P’ = d ⋛ 0 (depending on whether the other good is a gross substitute or gross complement)

37. Linear Demand
If I and P’ are held constant at I* and P’*, the demand function can be written
Q = a’ + bP
where a’ = a + cI* + dP’*
Note that this implies a linear demand curve
Changes in I or P’ will alter a’ and shift the demand curve

38. Linear Demand
Along a linear demand curve, the slope (Q/P) is constant
the price elasticity of demand will not be constant along the demand curve
As price rises and quantity falls, the elasticity will become a larger negative number (b < 0)

39. Linear Demand Demand becomes more elastic at higher prices
eQ,P < -1
-a’/b
eQ,P = -1
eQ,P > -1 Q a’

40. Constant Elasticity Functions
If one wanted elasticities that were constant over a range of prices, this demand function can be used
Q = aPbIcP’d
where a > 0, b  0, c  0, and d ⋛ 0.
For particular values of I and P’,
Q = a’Pb
where a’ = aIcP’d

41. Constant Elasticity Functions
This equation can also be written as
ln Q = ln a’ + b ln P
Applying the definition of elasticity,
The price elasticity of demand is equal to the exponent on P

42. Important Points to Note:
The market demand curve is negatively sloped on the assumption that most individuals will buy more of a good when the price falls
it is assumed that Giffen’s paradox does not occur
Effects of movements along the demand curve are measured by the price elasticity of demand (eQ,P)
% change in quantity from a 1% change in price

43. Important Points to Note:
Changes in total expenditures on a good caused by changes in price can be predicted from the price elasticity of demand
if demand is inelastic (0 > eQ,P > -1) , price and total expenditures move in the same direction
if demand is elastic (eQ,P < -1) , price and total expenditures move in opposite directions

44. Important Points to Note:
If other factors that enter the demand function (prices of other goods, income, preferences) change, the market demand curve will shift
the income elasticity (eQ,I) measures the effect of changes in income on quantity demanded
the cross-price elasticity (eQ,P’) measures the effect of changes in another good’s price on quantity demanded

45. Important Points to Note:
There are a number of relationships among the various demand elasticities
the Slutsky equation shows the relationship between uncompensated and compensated price elasticities
homogeneity is reflected in the fact that the sum of the elasticities of demand for all of the arguments in the demand function is zero