Download 7. Breaking Up the Reciprocal Slope of the Demand Curve into

7. Breaking Up the Reciprocal Slope of the Demand Curve
into Substitution and Income Ratios
Commodity Space
Demand Curve
y
𝑝𝑥
Price-consumption curve
α
γ
β
𝑝𝑥𝛼
𝛽
𝑝𝑥
α
β
𝛽
𝑥𝛼 𝑥𝛾
𝑥𝛽
x
𝑝𝑦 = 𝑝𝑦0
𝑚 = 𝑚0
𝑝𝑥
𝑥𝛼
𝑥𝛽
x
Start with 𝑝𝑦0 and 𝑚0 fixed, 𝑝𝑥𝛼 specified, and the budget constraint 𝑥𝑝𝑥𝛼 + 𝑦𝑝𝑦0 = 𝑚0
identified by 𝑝𝑥𝛼 in the left-hand diagram. Then utility is maximized subject to this constraint at
𝛽
α. Now lower the price of good x to 𝑝𝑥 keeping the price of good y and income at their original
𝛽
𝛽
values 𝑝𝑦0 and 𝑚0 . The budget constraint becomes 𝑥𝑝𝑥 + 𝑦𝑝𝑦0 = 𝑚0 and is labeled with 𝑝𝑥 in
the left-hand diagram. With the lower price, utility is maximized subject to this latter budget
constraint at β. The demand curve for good x (drawn with the price of good y and income fixed,
respectively, at 𝑝𝑦0 and 𝑚0 ) appears in the right-hand diagram. From the left-hand diagram,
when the price of good x is 𝑝𝑥𝛼 , the utility-maximizing quantity of good x demanded is 𝑥 𝛼 ; and
𝛽
when the price of good x is 𝑝𝑥 , the utility-maximizing quantity of good x demanded is 𝑥 𝛽 .
These points on the demand curve are identified in the right-hand diagram by, respectively, α and
β. Drawing a budget line parallel to the outer budget line and tangent to the inner indifference
curve in the left-hand diagram, constrained utility maximization with respect to this last budget
constraint occurs at γ with x-quantity 𝑥 𝛾 .
The reciprocal of the slope of the demand curve at α in the right-hand diagram is
approximated by the slope of the straight line segment connecting α to β. Letting ∆𝑥 = 𝑥 𝛼 − 𝑥 𝛽
𝛽
and ∆𝑝𝑥 = 𝑝𝑥𝛼 − 𝑝𝑥 the expression for this approximate slope is:
∆𝑥
𝑥𝛼 − 𝑥𝛽 𝑥𝛼 − 𝑥𝛾 + 𝑥𝛾 − 𝑥𝛽
=
=
∆𝑝𝑥
∆𝑝𝑥
∆𝑝𝑥
𝑥𝛼 − 𝑥𝛾 𝑥𝛽 − 𝑥𝛾
=
−
.
∆𝑝𝑥
∆𝑝𝑥
The term (𝑥 𝛼 − 𝑥 𝛾 )⁄∆𝑝𝑥 is the substitution ratio and (𝑥 𝛽 − 𝑥 𝛾 )⁄∆𝑝𝑥 is the income ratio.