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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.