Chapter 3 Lecture Notes
1. Elasticity:
a. Intuition: Responsiveness of dependent variable Y to a change in independent
variable X
%Δ𝑌
b. Mathematical Definition: 𝜀𝑌𝑋 = %Δ𝑋 ≈
1 𝑑𝑌
𝑌 𝑑𝑡
1 𝑑𝑋
𝑋 𝑑𝑡
=
𝑑𝑌
𝑌
𝑑𝑋
𝑋
𝑋 𝑑𝑌
= 𝑌 𝑑𝑋
2. Elastic vs Inelastic
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Inelastic
Elastic
3. Price Elasticity of Demand
𝜀𝑄𝑃 =
%ΔQ 𝑃 𝑑𝑄
≈
%Δ𝑃 𝑄 𝑑𝑃
The quantity that buyers will buy depends upon the price charged for the product. An
increase in the price will tend to decrease the quantity bought. That is, the LAW OF
DEMAND implies 𝜀𝑄𝑃 < 0 will tend to hold. The elasticity question is how much demand
will decrease when price increases, or how SENSITIVE demand is to price.
4. Price Elasticity of Demand and Revenue: What happens to revenue when a firm increases
the price of the product it produces? (More is earned on each sale, but there will be less
sales.)
𝑅 = 𝑃𝑄
𝑑𝑅
𝑑𝑄
𝑑𝑃
=𝑃
+𝑄
𝑑𝑃
𝑑𝑃
𝑑𝑃
𝑑𝑅
𝑃 𝑑𝑄
=𝑄
+𝑄
𝑑𝑃
𝑄 𝑑𝑃
𝑑𝑅
= 𝑄[𝜀𝑄𝑃 + 1]
𝑑𝑃
So,
𝑑𝑅
𝑑𝑃
= 0 ⟺ 𝜀𝑄𝑃 = −1,
𝑑𝑅
𝑑𝑃
𝑑𝑅
> 0 ⟺ 𝜀𝑄𝑃 > −1 , and 𝑑𝑃 < 0 ⟺ 𝜀𝑄𝑃 < −1
That is, to maximize revenue, the price should be set so the price elasticity of demand is equal to
negative one, or is unitary elastic. If inelastic, the producer can increase revenue by increasing
the price. (The quantity decrease is not enough to offset the greater revenue per unit. If elastic,
the producer can increase revenue by decreasing the price. (The quantity increase is more than
enough to offset the decrease in revenue per unit.)
5. Price Elasticity of Demand and Profit:
Π=𝑅−𝐶
Π = 𝑃𝑄 − 𝐶
1
dΠ
𝑑𝑄
𝑑𝑃 𝑑𝐶
=𝑃
+𝑄
−
dP
𝑑𝑃
𝑑𝑃 𝑑𝑃
dΠ
𝑃 𝑑𝑄
𝑑𝐶
=𝑄
+𝑄−
dP
𝑄 𝑑𝑃
𝑑𝑃
dΠ
𝑑𝐶
= 𝑄[𝜀𝑄𝑃 + 1] −
dP
𝑑𝑃
dΠ
𝑑𝐶
𝑑𝐶 1
If profits are maximized, then dP = 0, which implies 𝑄[𝜀𝑄𝑃 + 1] = 𝑑𝑃, or [𝜀𝑄𝑃 + 1] = 𝑑𝑃 𝑄.
𝑑𝐶
𝑑𝐶 𝑑𝑄
𝑑𝐶
𝑑𝑄
𝑑𝐶
Since 𝑑𝑃 = 𝑑𝑄 𝑑𝑃 , and since 𝑑𝑄 > 0 and 𝑑𝑃 < 0 , we know 𝑑𝑃 < 0, which implies 𝜀𝑄𝑃 + 1 <
0, or 𝜀𝑄𝑃 < −1. That is, if a firm is maximizing its profit, then the demand facing the firm
will be ELASTIC. This implies, if the firm is operating efficiently, it should be true that a
decrease in price will increase sales revenue, but it does not make sense to do so because the
increased quantity produced will also increase costs by enough to make the change not
worthwhile.
6. Linear Demand Curve: 𝑄 = 𝑎 − 𝑏𝑃
dQ
It follows that dP = −b, and 𝜀𝑄𝑃 =
dQ 𝑃
dP 𝑄
𝑃
= −𝑏 𝑄.
We can show where the unitary elasticity point is on the demand curve by setting 𝜀𝑄𝑃 =
𝑃
𝑏𝑃
−1. This implies −𝑏 𝑄 = −1, or
𝑎−𝑏𝑃
𝑎
= 1, which implies 𝑃 = 2𝑏. And, substituting this
𝑎
result back into the demand curve equation, we find 𝑄 = 2.
𝑎
Similarly, it follows that setting
𝑎
𝑎
𝑎
𝜀𝑄𝑃 < −1 implies 𝑃 > 2𝑏 𝑄 < 2, while 0 > 𝜀𝑄𝑃 > −1 implies 𝑃 < 2𝑏 𝑄 > 2. Plotting
the linear demand curve, we there for know that the unitary elastic point is the midpoint of
the curve, the elastic portion is associated with a relatively high price, and the inelastic
portion is associated with a relatively low price, as shown in the figure below.
P
𝑎
𝑏
𝑎
2𝑏
Elastic 𝜀𝑄𝑃 < −1
Unitary Elasticity 𝜀𝑄𝑃 = −1
Inelastic -1<𝜀𝑄𝑃 < 0
𝑎
2
a
Q
2
7. Market Demand Curves, their slope, and their point price elasticity of demand
P
P
Perfectly Inelastic
Perfectly Elastic
Q
Q
P
More Inelastic
Point Elasticity
More Elastic
Q
8. Primary factors affecting OWN PRICE ELASTICITY OF DEMAND
a. Available substitutes (More substitutes implies more elastic)
b. Time (More time to adjust implies more elastic)
c. Expenditure share (Smaller share of total expenditure implies less elastic)
9. Price Elasticity Estimates (Market Demand)
a. Elastic
i. Motorcycles and bicycles
ii. Cereal
iii. Recreation (especially elastic long term)
b. Inelastic
i. Transportation
ii. Food (but elastic long term)
iii. Alcohol and tobacco (very inelastic short term)
10. Marginal Revenue and Price Facing the Firm
𝑑𝑅
𝑀𝑅 = 𝑑𝑄 = 𝑃 [
𝜀𝑄𝑃 +1
𝑀𝑅
𝜀𝑄𝑃
𝑃
], so
=[
𝜀𝑄𝑃 +1
𝜀𝑄𝑃
] < 1, so MR<P
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Intuition: To sell the next unit, the firm must lower the price. Therefore, even though an extra
unit is sold the revenue obtain from selling the next unit is not as great as what the price had
been.
Perfect Competition: 𝜀𝑄𝑃 = −∞, meaning demand is infinitely sensitive to price, or it means
demand is perfectly elastic. In this case, MR=P. The firm is called a price taker, meaning the
firm cannot increase its price or it will lose all customers, and it need not lower its price because
the firm can sell all it wants at the market price. In this case, the market entirely sets the price.
11. Cross Price Elasticity: 𝜀𝑄𝑌𝑃𝑋
a. How will the demand for good Y change when the price of good X increases?
b. If goods X and Y are substitutes, then 𝜀𝑄𝑌𝑃𝑋 > 0
i. Examples: Food and Recreation\
c. If goods X and Y are complements, then 𝜀𝑄𝑌𝑃𝑋 < 0
i. Examples: Transportation and Recreation, Clothing and food
d. Using Cross price elasticity in a restaurant
i. If items on the menu are complements, then decreasing the price of one item
may not only lead to an increase in its purchase, but it may also lead to an
increase in the purchases of the complement.
ii. So, cross elasticities should be considered, not just own elasticities when
developing pricing policies for multiple product delivery.
12. Income Elasticity
a. How elastic is demand when income changes?
b. Normal Good: Elasticity is positive (Steak)
c. Inferior Good: Elasticity is negative (Top Ramen)
13. Estimating Elasticities
a. Linear Demand Function: 𝑄𝑥 = 𝛼0 + 𝛼1 𝑃𝑥 + 𝛼2 𝑃𝑌 + 𝛼3 𝑌
i. Use linear regression to estimate 𝛼1 , 𝛼2 , 𝑎𝑛𝑑 𝛼3
dQ
𝑃 dQ
𝑃
ii. Note that dP x = α1 , so 𝜀𝑄𝑥 𝑃𝑥 = 𝑄𝑥 dPx = α1 𝑄𝑥
x
𝑥
x
𝑥
iii. Similarly, we can derive the cross elasticity 𝜀
𝑄𝑥 𝑃𝑦 =α2
𝑃𝑦
and the income
𝑄𝑥
elasticity 𝜀𝑄
𝑌
𝑥 𝑌=α3 𝑄
𝑥
b. Cobb-Douglas (Log-Linear) Demand Function: 𝑄𝑥 = 𝛼0 𝑃𝑥 𝛼1 𝑃𝑌 𝛼2 𝑌 𝛼3
i. There is a nonlinear relationship between demand and the other factors
ii. To estimate the parameters, this model must be linearized, which can be done
by taking the natural log. Doing so, one obtains:
iii. 𝑙𝑛(𝑄𝑥 ) = 𝑙𝑛(𝛼0 ) + 𝛼1 𝑙𝑛(𝑃𝑥 ) + 𝛼2 𝑙𝑛(𝑃𝑌 ) + 𝛼3 𝑙𝑛(𝑌)
1 dQ
iv. Taking the derivative with respect to 𝑃𝑥 , we obtain Q dP x = α1 , which implies
x
x
𝜀𝑄𝑥 𝑃𝑥 = α1 . So we can estimate the elasticity α1 by regressing 𝑙𝑛(𝑄𝑥 ) on
𝑙𝑛(𝑃𝑥 ), 𝑙𝑛(𝑃𝑌 ), and 𝑙𝑛(𝑌).
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14. Deriving Elasticities using Excel
a. Use BeerData for Wisconsin in class (quantity of beer, price of beer, income of
consumers)
b. Estimate a linear model of the demand for beer, as it depends upon the beer price and
consumer income
c. Find price elasticity of demand for beer in Wisconsin, week by week, and income
elasticity of demand
d. Find the averages for the elasticities estimated
e. Estimate a nonlinear model and obtain the elasticity of demand for beer in Wisconsin,
and income elasticity of demand
f. Show professional presentation of results
g. Interpret results
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Chapter 3 Questions
CH 3, Question 1: Use the BeerData for Ohio to do the following:
a. Estimate a linear model of the demand for beer, as it depends upon the beer price and
consumer income. Present your estimated model professionally.
b. Find price elasticity of demand for beer in Ohio, week by week. (Nothing to turn in
for this.)
c. Find the income elasticity of demand, week by week. (Nothing to turn in for this.)
d. Find the averages of the elasticities for parts (b) and (c) and report these.
e. Estimate a LogLinear model to obtain the price elasticity of demand for beer in Ohio,
and the income elasticity of demand. Report your results in a professional manner,
and identify the elasticities.
f. Interpret your results from both the linear and nonlinear models.
CH3, Question 2: Use the MilkData to do the following:
a. Estimate a linear model of the demand for milk, as it depends upon the milk price and
dollars spent on advertising. Present your estimated model professionally.
g. Find price elasticity of demand for milk, week by week. (Nothing to turn in for this.)
h. Find the advertising elasticity of demand, week by week . (Nothing to turn in for
this.)
b. Find the averages of the elasticities for parts (b) and (c) and report these.
c. Estimate a LogLinear model to obtain the price elasticity of demand for milk, and the
income elasticity of demand. Report your results in a professional manner, and
identify the elasticities.
d. Interpret your results from both the linear and nonlinear models.
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