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Unformatted text preview: Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Econ 101: Intermediate Microeconomics
Consumer Theory II: Individual Demand Analysis Part 2 Jing Li
Department of Economics University of Pennsylvania February 12, 2009 Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Multiple consequences of a price change Given: m, p2 , p1 → p1 ; Consequences of the price change:
The exchange rate between the two goods are different, i.e., p1 p1 p2 → p2 ; m m The purchasing power has changed, i.e., p1 → p .
1 Change in optimal choice reﬂects both factors. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Multiple consequences of a price change Given: m, p2 , p1 → p1 ; Consequences of the price change:
The exchange rate between the two goods are different, i.e., p1 p1 p2 → p2 ; m m The purchasing power has changed, i.e., p1 → p .
1 Change in optimal choice reﬂects both factors. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Multiple consequences of a price change Given: m, p2 , p1 → p1 ; Consequences of the price change:
The exchange rate between the two goods are different, i.e., p1 p1 p2 → p2 ; m m The purchasing power has changed, i.e., p1 → p .
1 Change in optimal choice reﬂects both factors. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Multiple consequences of a price change Given: m, p2 , p1 → p1 ; Consequences of the price change:
The exchange rate between the two goods are different, i.e., p1 p1 p2 → p2 ; m m The purchasing power has changed, i.e., p1 → p .
1 Change in optimal choice reﬂects both factors. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Multiple consequences of a price change Given: m, p2 , p1 → p1 ; Consequences of the price change:
The exchange rate between the two goods are different, i.e., p1 p1 p2 → p2 ; m m The purchasing power has changed, i.e., p1 → p .
1 Change in optimal choice reﬂects both factors. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Multiple consequences of a price change Given: m, p2 , p1 → p1 ; Consequences of the price change:
The exchange rate between the two goods are different, i.e., p1 p1 p2 → p2 ; m m The purchasing power has changed, i.e., p1 → p .
1 Change in optimal choice reﬂects both factors. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Graphic illustration Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Decomposing the change in demand
1 Substitution effect. When changing the price (p1 p1 ), we change the income, too, (m → m ), so the consumer has the “same level of purchasing power”, meaning he has just enough income to buy the initial optimal choice at the new price. m
∗ ∗ = p1 x1 (p1 , p2 , m) + p2 x2 (p1 , p2 , m) ∗ = m + (p1 − p1 )x1 (p1 , p2 , m). s s.e. ≡ ∆x1 = x1 (p1 , p2 , m ) − x1 (p1 , p2 , m);
2 Income effect. Credit/debit the change in income back: m → m.
n i.e. ≡ ∆x1 = x1 (p1 , p2 , m) − x1 (p1 , p2 , m ). Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Decomposing the change in demand
1 Substitution effect. When changing the price (p1 → p1 ), we change the income, too, (m → m ), so the consumer has the “same level of purchasing power”, meaning he has just enough income to buy the initial optimal choice at the new price. m
∗ ∗ = p1 x1 (p1 , p2 , m) + p2 x2 (p1 , p2 , m) ∗ = m + (p1 − p1 )x1 (p1 , p2 , m). s s.e. ≡ ∆x1 = x1 (p1 , p2 , m ) − x1 (p1 , p2 , m);
2 Income effect. Credit/debit the change in income back: m → m.
n i.e. ≡ ∆x1 = x1 (p1 , p2 , m) − x1 (p1 , p2 , m ). Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Decomposing the change in demand
1 Substitution effect. When changing the price (p1 → p1 ), we change the income, too, (m → m ), so the consumer has the “same level of purchasing power”, meaning he has just enough income to buy the initial optimal choice at the new price. m
∗ ∗ = p1 x1 (p1 , p2 , m) + p2 x2 (p1 , p2 , m) ∗ = m + (p1 − p1 )x1 (p1 , p2 , m). s s.e. ≡ ∆x1 = x1 (p1 , p2 , m ) − x1 (p1 , p2 , m);
2 Income effect. Credit/debit the change in income back: m → m.
n i.e. ≡ ∆x1 = x1 (p1 , p2 , m) − x1 (p1 , p2 , m ). Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Decomposing the change in demand
1 Substitution effect. When changing the price (p1 → p1 ), we change the income, too, (m → m ), so the consumer has the “same level of purchasing power”, meaning he has just enough income to buy the initial optimal choice at the new price. m
∗ ∗ = p1 x1 (p1 , p2 , m) + p2 x2 (p1 , p2 , m) ∗ = m + (p1 − p1 )x1 (p1 , p2 , m). s s.e. ≡ ∆x1 = x1 (p1 , p2 , m ) − x1 (p1 , p2 , m);
2 Income effect. Credit/debit the change in income back: m → m.
n i.e. ≡ ∆x1 = x1 (p1 , p2 , m) − x1 (p1 , p2 , m ). Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Slutsky Identity x1 (p1 , p2 , m) − x1 (p1 , p2 , m) = x1 (p1 , p2 , m) − x1 (p1 , p2 , m )
∆x1
n ∆x1 + x1 (p1 , p2 , m ) − x1 (p1 , p2 , m)
s ∆x1 ∆x1 = n ∆ x1 + s ∆ x1 total change in demand = income effect + substitution effect Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Calculating s.e. and i.e. of a price change Example 1. x1 (p1 , m) = 1m 3 p1 , m = 60, p1 = 2, p1 = 5. Example 2. the case of perfect substitutes Example 3. the case of perfect complements Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Example: perfect complements Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Example: perfect substitutes Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Sign of substitution effect (p1 > p1 ) We must have x1 (p1 , p2 , m ) < x1 (p1 , p2 , m).
The optimal choice given the compensated budget line must lie to the left of the initial consumption. (graphic proof)
s Substitution effect is always negative: ∆x1 < 0 Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Sign of income effect (p1 > p1 ) Could be positive or negative. ∆m = (p1 − p1 )¯1 > 0; x Therefore m + ∆m > m; Normal good: Inferior good:
∂ x1 ∂m ∂ x1 ∂m n > 0 ⇒ ∆x1 < 0 ; n < 0 ⇒ ∆x1 > 0 Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Sign of income effect (p1 > p1 ) Could be positive or negative. ∆m = (p1 − p1 )¯1 > 0; x Therefore m + ∆m > m; Normal good: Inferior good:
∂ x1 ∂m ∂ x1 ∂m n > 0 ⇒ ∆x1 < 0 ; n < 0 ⇒ ∆x1 > 0 Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Sign of income effect (p1 > p1 ) Could be positive or negative. ∆m = (p1 − p1 )¯1 > 0; x Therefore m + ∆m > m; Normal good: Inferior good:
∂ x1 ∂m ∂ x1 ∂m n > 0 ⇒ ∆x1 < 0 ; n < 0 ⇒ ∆x1 > 0 Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 )
Total change in demand:
n s ∆x1 = ∆x1 + ∆x1 (?) (?) (−) For normal good:
n s ∆x1 = ∆x1 + ∆x1 (−) (−) (−) All signs are positive if p1 < p1 ; The law of demand: demand for normal good moves to the opposite direction of its price change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 )
Total change in demand:
n s ∆x1 = ∆x1 + ∆x1 (?) (?) (−) For normal good:
n s ∆x1 = ∆x1 + ∆x1 (−) (−) (−) All signs are positive if p1 < p1 ; The law of demand: demand for normal good moves to the opposite direction of its price change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 )
Total change in demand:
n s ∆x1 = ∆x1 + ∆x1 (?) (?) (−) For normal good:
n s ∆x1 = ∆x1 + ∆x1 (−) (−) (−) All signs are positive if p1 < p1 ; The law of demand: demand for normal good moves to the opposite direction of its price change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 )
Total change in demand:
n s ∆x1 = ∆x1 + ∆x1 (?) (?) (−) For normal good:
n s ∆x1 = ∆x1 + ∆x1 (−) (−) (−) All signs are positive if p1 < p1 ; The law of demand: demand for normal good moves to the opposite direction of its price change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 ) For inferior good:
n s ∆x1 = ∆x1 + ∆x1 (?) (+) (−) Sign is ambiguous.
s n Substitution effect dominates income effect (∆x1  > ∆x1 ): (), regular inferior good, downward sloping demand curve; n s Income effect dominates substitution effect (∆x1  > ∆x1 ): (+), Giffen good. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 ) For inferior good:
n s ∆x1 = ∆x1 + ∆x1 (?) (+) (−) Sign is ambiguous.
s n Substitution effect dominates income effect (∆x1  > ∆x1 ): (), regular inferior good, downward sloping demand curve; n s Income effect dominates substitution effect (∆x1  > ∆x1 ): (+), Giffen good. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 ) For inferior good:
n s ∆x1 = ∆x1 + ∆x1 (?) (+) (−) Sign is ambiguous.
s n Substitution effect dominates income effect (∆x1  > ∆x1 ): (), regular inferior good, downward sloping demand curve; n s Income effect dominates substitution effect (∆x1  > ∆x1 ): (+), Giffen good. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 ) For inferior good:
n s ∆x1 = ∆x1 + ∆x1 (?) (+) (−) Sign is ambiguous.
s n Substitution effect dominates income effect (∆x1  > ∆x1 ): (), regular inferior good, downward sloping demand curve; n s Income effect dominates substitution effect (∆x1  > ∆x1 ): (+), Giffen good. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 ) For inferior good:
n s ∆x1 = ∆x1 + ∆x1 (?) (+) (−) Sign is ambiguous.
s n Substitution effect dominates income effect (∆x1  > ∆x1 ): (), regular inferior good, downward sloping demand curve; n s Income effect dominates substitution effect (∆x1  > ∆x1 ): (+), Giffen good. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Total change in demand (p1 > p1 ) For inferior good:
n s ∆x1 = ∆x1 + ∆x1 (?) (+) (−) Sign is ambiguous.
s n Substitution effect dominates income effect (∆x1  > ∆x1 ): (), regular inferior good, downward sloping demand curve; n s Income effect dominates substitution effect (∆x1  > ∆x1 ): (+), Giffen good. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Inferior goods Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Exercises p1 < p1 ? graph, analysis...... perfect substitutes perfect complements Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Standard of compensation? To separate substitution effect, we have to compensate the consumer for the change in purchasing power caused by the change in relative price. Two compensation standards:
Maintain the same consumption; Slutsky: the consumer can just afford the original optimal bundle. Maintain the same level of satisfaction/utility. Hicks: the consumer stays at the same indifference curve. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Standard of compensation? To separate substitution effect, we have to compensate the consumer for the change in purchasing power caused by the change in relative price. Two compensation standards:
Maintain the same consumption; Slutsky: the consumer can just afford the original optimal bundle. Maintain the same level of satisfaction/utility. Hicks: the consumer stays at the same indifference curve. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Standard of compensation? To separate substitution effect, we have to compensate the consumer for the change in purchasing power caused by the change in relative price. Two compensation standards:
Maintain the same consumption; Slutsky: the consumer can just afford the original optimal bundle. Maintain the same level of satisfaction/utility. Hicks: the consumer stays at the same indifference curve. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Standard of compensation? To separate substitution effect, we have to compensate the consumer for the change in purchasing power caused by the change in relative price. Two compensation standards:
Maintain the same consumption; Slutsky: the consumer can just afford the original optimal bundle. Maintain the same level of satisfaction/utility. Hicks: the consumer stays at the same indifference curve. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Standard of compensation? To separate substitution effect, we have to compensate the consumer for the change in purchasing power caused by the change in relative price. Two compensation standards:
Maintain the same consumption; Slutsky: the consumer can just afford the original optimal bundle. Maintain the same level of satisfaction/utility. Hicks: the consumer stays at the same indifference curve. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Standard of compensation? To separate substitution effect, we have to compensate the consumer for the change in purchasing power caused by the change in relative price. Two compensation standards:
Maintain the same consumption; Slutsky: the consumer can just afford the original optimal bundle. Maintain the same level of satisfaction/utility. Hicks: the consumer stays at the same indifference curve. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Hicks substitution effect Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Understanding Hicks substitution effect Hicks substitution effect must be negative; (general argument; an argument by convexity) Hicks s.e. is theoretically more relevant in many situations; But, choice is observable while preferences are not; For inﬁnitesimal changes in price, the two substitution effects are the same. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Understanding Hicks substitution effect Hicks substitution effect must be negative; (general argument; an argument by convexity) Hicks s.e. is theoretically more relevant in many situations; But, choice is observable while preferences are not; For inﬁnitesimal changes in price, the two substitution effects are the same. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Understanding Hicks substitution effect Hicks substitution effect must be negative; (general argument; an argument by convexity) Hicks s.e. is theoretically more relevant in many situations; But, choice is observable while preferences are not; For inﬁnitesimal changes in price, the two substitution effects are the same. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Understanding Hicks substitution effect Hicks substitution effect must be negative; (general argument; an argument by convexity) Hicks s.e. is theoretically more relevant in many situations; But, choice is observable while preferences are not; For inﬁnitesimal changes in price, the two substitution effects are the same. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Understanding Hicks substitution effect Hicks substitution effect must be negative; (general argument; an argument by convexity) Hicks s.e. is theoretically more relevant in many situations; But, choice is observable while preferences are not; For inﬁnitesimal changes in price, the two substitution effects are the same. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Demand curves
Notation: Initial bundle (¯1 , ¯2 ), initial prices ¯1 , ¯2 , initial xx pp ¯ income m. Marshallian demand curve: x1 (p1 , ¯2 , m) p¯
How demand changes with respect to price, given income; Reﬂect both substitution effect and income effect; Slope could be positive.
s Slutsky demand curve: x1 ≡ x1 (p1 , ¯2 , p1¯1 + ¯2¯2 ) p x px How demand changes with respect to price, given purchasing power; Downward sloping.
h Hicksian demand curve: x1 ≡ x1 (p1 , ¯2 , u(¯1 , ¯2 )) p xx How demand changes with respect to price, given welfare; Downward sloping.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Slutsky equation
Let m(p1 ) = p1¯1 + ¯2¯2 . x px
s x1 ≡ x1 (p1 , p2 , m(p1 )) s ∂ x1 ∂ p1 s ∂ x1 = ∂ x1 ∂ x1 dm + ∂ p1 ∂ m dp1
=¯1 x ∂ p1 − ∂ x1 ¯1 x ∂m = ∂ x1 ∂ p1
total effect substitution effect income effect Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Sign of the Slutsky equation Normal goods: s ∂ x1 ∂ x1 ∂ x1 ¯1 = − . x ∂ p1 ∂ m ∂ p1 (−) (−) (−) Inferior goods: s ∂ x1 ∂ x1 ∂ x1 ¯1 = − . x ∂ p1 ∂ m ∂ p1 (−) (+) (?) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Understanding the Slutsky equation Rate of change formulation; Alternative way to see it: ∆x1 ∆x1 ∆p1 (∆m = m − m = −∆p1¯1 x ∆x1 ∆p1
s n ∆x1 + ∆x1 s n ∆x1 ∆x1 = + ∆p1 ∆p1 ∆m ⇒ ∆p1 = − ) ¯1 x s n ∆x1 ∆x1 ¯1 = − x ∆p1 ∆m = Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Substitution Effect and Income Effect Signs of Substitution Effect and Income Effect Hicks Substitution Effect Slutsky Equation Understanding the Slutsky equation Rate of change formulation; Alternative way to see it: ∆x1 ∆x1 ∆p1 (∆m = m − m = −∆p1¯1 x ∆x1 ∆p1
s n ∆x1 + ∆x1 s n ∆x1 ∆x1 = + ∆p1 ∆p1 ∆m ⇒ ∆p1 = − ) ¯1 x s n ∆x1 ∆x1 ¯1 = − x ∆p1 ∆m = Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Description Good 2 is money, p2 = 1; Constant marginal utility for money; Good 1 is some good that consists of a “small” part of one’s consumption; Money “substitutes” good 1 to some degree; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Description Good 2 is money, p2 = 1; Constant marginal utility for money; Good 1 is some good that consists of a “small” part of one’s consumption; Money “substitutes” good 1 to some degree; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Description Good 2 is money, p2 = 1; Constant marginal utility for money; Good 1 is some good that consists of a “small” part of one’s consumption; Money “substitutes” good 1 to some degree; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Description Good 2 is money, p2 = 1; Constant marginal utility for money; Good 1 is some good that consists of a “small” part of one’s consumption; Money “substitutes” good 1 to some degree; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Indifference curves Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Utility function General form of utility function for quasilinear preferences: u(x1 , x2 ) = v(x1 ) + x2 . MU1 MRS = − MU2 = −v (x1 ); Monotonicity? Convexity? Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Utility function General form of utility function for quasilinear preferences: u(x1 , x2 ) = v(x1 ) + x2 . MU1 MRS = − MU2 = −v (x1 ); Monotonicity? Convexity? Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Utility function General form of utility function for quasilinear preferences: u(x1 , x2 ) = v(x1 ) + x2 . MU1 MRS = − MU2 = −v (x1 ); Monotonicity? Convexity? Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Choice Example: u(x1 , x2 ) = 10 ln x1 + x2 p1 = 2, m = 20. Optimal choice? p1 = 4? m = 25? Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Demand function
The constrained optimization problem is: max
x 1 ,x 2 v(x1 ) + x2 p1 x1 + x2 ≤ m. subject to: Assuming concavity of v, the optimal solution satisﬁes: v (x1 ) = p1 p1 x1 + x2 = m No income effect:
∂ x1 ∂m ⇒ x1 (p1 ) does not depend on m. = 0. (Engel curve?)
∂ x1 ∂ p1 All changes come from substitution effect (S.E.): (Note: x2 (p1 , m) = m − p1 x1 (p1 ).)
Jing Li Intermediate Microeconomics = s ∂ x1 ∂ p1 . Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Demand function
The constrained optimization problem is: max
x 1 ,x 2 v(x1 ) + x2 p1 x1 + x2 ≤ m. subject to: Assuming concavity of v, the optimal solution satisﬁes: v (x1 ) = p1 p1 x1 + x2 = m No income effect:
∂ x1 ∂m ⇒ x1 (p1 ) does not depend on m. = 0. (Engel curve?)
∂ x1 ∂ p1 All changes come from substitution effect (S.E.): (Note: x2 (p1 , m) = m − p1 x1 (p1 ).)
Jing Li Intermediate Microeconomics = s ∂ x1 ∂ p1 . Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Demand function
The constrained optimization problem is: max
x 1 ,x 2 v(x1 ) + x2 p1 x1 + x2 ≤ m. subject to: Assuming concavity of v, the optimal solution satisﬁes: v (x1 ) = p1 p1 x1 + x2 = m No income effect:
∂ x1 ∂m ⇒ x1 (p1 ) does not depend on m. = 0. (Engel curve?)
∂ x1 ∂ p1 All changes come from substitution effect (S.E.): (Note: x2 (p1 , m) = m − p1 x1 (p1 ).)
Jing Li Intermediate Microeconomics = s ∂ x1 ∂ p1 . Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Demand function
The constrained optimization problem is: max
x 1 ,x 2 v(x1 ) + x2 p1 x1 + x2 ≤ m. subject to: Assuming concavity of v, the optimal solution satisﬁes: v (x1 ) = p1 p1 x1 + x2 = m No income effect:
∂ x1 ∂m ⇒ x1 (p1 ) does not depend on m. = 0. (Engel curve?)
∂ x1 ∂ p1 All changes come from substitution effect (S.E.): (Note: x2 (p1 , m) = m − p1 x1 (p1 ).)
Jing Li Intermediate Microeconomics = s ∂ x1 ∂ p1 . Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Demand function
The constrained optimization problem is: max
x 1 ,x 2 v(x1 ) + x2 p1 x1 + x2 ≤ m. subject to: Assuming concavity of v, the optimal solution satisﬁes: v (x1 ) = p1 p1 x1 + x2 = m No income effect:
∂ x1 ∂m ⇒ x1 (p1 ) does not depend on m. = 0. (Engel curve?)
∂ x1 ∂ p1 All changes come from substitution effect (S.E.): (Note: x2 (p1 , m) = m − p1 x1 (p1 ).)
Jing Li Intermediate Microeconomics = s ∂ x1 ∂ p1 . Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Preferences Choice Demand Analysis Demand function
The constrained optimization problem is: max
x 1 ,x 2 v(x1 ) + x2 p1 x1 + x2 ≤ m. subject to: Assuming concavity of v, the optimal solution satisﬁes: v (x1 ) = p1 p1 x1 + x2 = m No income effect:
∂ x1 ∂m ⇒ x1 (p1 ) does not depend on m. = 0. (Engel curve?)
∂ x1 ∂ p1 All changes come from substitution effect (S.E.): (Note: x2 (p1 , m) = m − p1 x1 (p1 ).)
Jing Li Intermediate Microeconomics = s ∂ x1 ∂ p1 . Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Endowment as money Instead of having money as intermediary, the consumer owns an initial endowment of commodities (ω1 , ω2 ); Budget line: we have a point and a slope; Consumer sells as well as buys at the market prices; Prices affect income as well as consumption costs. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Endowment as money Instead of having money as intermediary, the consumer owns an initial endowment of commodities (ω1 , ω2 ); Budget line: we have a point and a slope; Consumer sells as well as buys at the market prices; Prices affect income as well as consumption costs. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Endowment as money Instead of having money as intermediary, the consumer owns an initial endowment of commodities (ω1 , ω2 ); Budget line: we have a point and a slope; Consumer sells as well as buys at the market prices; Prices affect income as well as consumption costs. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Endowment as money Instead of having money as intermediary, the consumer owns an initial endowment of commodities (ω1 , ω2 ); Budget line: we have a point and a slope; Consumer sells as well as buys at the market prices; Prices affect income as well as consumption costs. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Graphic illustration of choice problem with endowment Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choice problem with endowment Choice problem with money: < p1 , p2 , m, u >; Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Connection: m = p1 ω1 + p2 ω2 . The constrained optimization problem is: max
x 1 ,x 2 u(x1 , x2 ) p1 x1 + p2 x2 ≤ p1 ω1 + p2 ω2 . subject to: Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choice problem with endowment Choice problem with money: < p1 , p2 , m, u >; Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Connection: m = p1 ω1 + p2 ω2 . The constrained optimization problem is: max
x 1 ,x 2 u(x1 , x2 ) p1 x1 + p2 x2 ≤ p1 ω1 + p2 ω2 . subject to: Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choice problem with endowment Choice problem with money: < p1 , p2 , m, u >; Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Connection: m = p1 ω1 + p2 ω2 . The constrained optimization problem is: max
x 1 ,x 2 u(x1 , x2 ) p1 x1 + p2 x2 ≤ p1 ω1 + p2 ω2 . subject to: Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choice problem with endowment Choice problem with money: < p1 , p2 , m, u >; Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Connection: m = p1 ω1 + p2 ω2 . The constrained optimization problem is: max
x 1 ,x 2 u(x1 , x2 ) p1 x1 + p2 x2 ≤ p1 ω1 + p2 ω2 . subject to: Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Demand function with endowment Lagrangian method again; The optimal solution (x1 , x2 ) satisﬁes: MRS = p1 p2 p1 x1 + p2 x2 = p1 ω1 + p2 ω2
g g Solve for gross demand functions x1 (p1 , p2 ), x2 (p1 , p2 ); g g Net demand functions: x1 − ω1 , x2 − ω2 ; Net demand > 0: buying Net demand < 0: selling Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Demand function with endowment Lagrangian method again; The optimal solution (x1 , x2 ) satisﬁes: MRS = p1 p2 p1 x1 + p2 x2 = p1 ω1 + p2 ω2
g g Solve for gross demand functions x1 (p1 , p2 ), x2 (p1 , p2 ); g g Net demand functions: x1 − ω1 , x2 − ω2 ; Net demand > 0: buying Net demand < 0: selling Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Demand function with endowment Lagrangian method again; The optimal solution (x1 , x2 ) satisﬁes: MRS = p1 p2 p1 x1 + p2 x2 = p1 ω1 + p2 ω2
g g Solve for gross demand functions x1 (p1 , p2 ), x2 (p1 , p2 ); g g Net demand functions: x1 − ω1 , x2 − ω2 ; Net demand > 0: buying Net demand < 0: selling Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Demand function with endowment Lagrangian method again; The optimal solution (x1 , x2 ) satisﬁes: MRS = p1 p2 p1 x1 + p2 x2 = p1 ω1 + p2 ω2
g g Solve for gross demand functions x1 (p1 , p2 ), x2 (p1 , p2 ); g g Net demand functions: x1 − ω1 , x2 − ω2 ; Net demand > 0: buying Net demand < 0: selling Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Demand function with endowment Lagrangian method again; The optimal solution (x1 , x2 ) satisﬁes: MRS = p1 p2 p1 x1 + p2 x2 = p1 ω1 + p2 ω2
g g Solve for gross demand functions x1 (p1 , p2 ), x2 (p1 , p2 ); g g Net demand functions: x1 − ω1 , x2 − ω2 ; Net demand > 0: buying Net demand < 0: selling Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Exogenous variables in endowment choice problems Endowment choice problem: < p1 , p2 , (ω1 , ω2 ), u > Exogenous variables: (ω1 , ω2 ), p1 , p2
Change in endowment: (ω1 , ω2 ) → (ω1 , ω2 ); Change in relative price: p1 → p1 . Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in endowment (ω1 , ω2 ) → (ω1 , ω2 ); Just a change in income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 ; Parallel shift of budget line, same slope, passing through the new endowment point. (graph) Change in gross demand/net demand. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in endowment (ω1 , ω2 ) → (ω1 , ω2 ); Just a change in income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 ; Parallel shift of budget line, same slope, passing through the new endowment point. (graph) Change in gross demand/net demand. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in endowment (ω1 , ω2 ) → (ω1 , ω2 ); Just a change in income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 ; Parallel shift of budget line, same slope, passing through the new endowment point. (graph) Change in gross demand/net demand. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in endowment (ω1 , ω2 ) → (ω1 , ω2 ); Just a change in income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 ; Parallel shift of budget line, same slope, passing through the new endowment point. (graph) Change in gross demand/net demand. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in relative price p1 → p1 ; New budget line passes through the original endowment point, but with a new slope; Different relative price
p1 p2 → p1 p2 ; Different income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 . Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in relative price p1 → p1 ; New budget line passes through the original endowment point, but with a new slope; Different relative price
p1 p2 → p1 p2 ; Different income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 . Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in relative price p1 → p1 ; New budget line passes through the original endowment point, but with a new slope; Different relative price
p1 p2 → p1 p2 ; Different income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 . Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Change in relative price p1 → p1 ; New budget line passes through the original endowment point, but with a new slope; Different relative price
p1 p2 → p1 p2 ; Different income: p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2 . Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Graphic illustration: change in relative price Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Graphic illustration: Slutsky decomposition Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky decomposition Different relative price: p1 p2 → p1 p2 Regular Slutsky decomposition: substitution effect and income effect; Different income p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2
Endowment income effect: an additional shift of the budget line; Slutsky identity with endowment:
e s n ∆x1 = ∆x1 + ∆x1 + ∆x1 . (C−A) (B−A) (D−B) (C−D) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky decomposition Different relative price: p1 p2 → p1 p2 Regular Slutsky decomposition: substitution effect and income effect; Different income p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2
Endowment income effect: an additional shift of the budget line; Slutsky identity with endowment:
e s n ∆x1 = ∆x1 + ∆x1 + ∆x1 . (C−A) (B−A) (D−B) (C−D) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky decomposition Different relative price: p1 p2 → p1 p2 Regular Slutsky decomposition: substitution effect and income effect; Different income p1 ω1 + p2 ω2 → p1 ω1 + p2 ω2
Endowment income effect: an additional shift of the budget line; Slutsky identity with endowment:
e s n ∆x1 = ∆x1 + ∆x1 + ∆x1 . (C−A) (B−A) (D−B) (C−D) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Example Demand function: x1 (p1 , m) = p1 = 5. m p1 −6 p1 = 2, p2 = 4, (ω1 , ω2 ) = (8, 10) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky Equation with endowment e Demand function with endowment: x1 (p1 , p2 , p1 ω1 + p2 ω2 ); Let p2 = ¯2 be constant; p p Income depends on p1 : m(p1 ) = p1 ω1 + ¯2 ω2 ;
e Demand function with endowment: x1 (p1 , m(p1 )) A function of p1 alone. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky Equation with endowment e Demand function with endowment: x1 (p1 , p2 , p1 ω1 + p2 ω2 ); Let p2 = ¯2 be constant; p p Income depends on p1 : m(p1 ) = p1 ω1 + ¯2 ω2 ;
e Demand function with endowment: x1 (p1 , m(p1 )) A function of p1 alone. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky Equation with endowment e Demand function with endowment: x1 (p1 , p2 , p1 ω1 + p2 ω2 ); Let p2 = ¯2 be constant; p p Income depends on p1 : m(p1 ) = p1 ω1 + ¯2 ω2 ;
e Demand function with endowment: x1 (p1 , m(p1 )) A function of p1 alone. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky Equation with endowment e Demand function with endowment: x1 (p1 , p2 , p1 ω1 + p2 ω2 ); Let p2 = ¯2 be constant; p p Income depends on p1 : m(p1 ) = p1 ω1 + ¯2 ω2 ;
e Demand function with endowment: x1 (p1 , m(p1 )) A function of p1 alone. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky Equation with endowment e Demand function with endowment: x1 (p1 , p2 , p1 ω1 + p2 ω2 ); Let p2 = ¯2 be constant; p p Income depends on p1 : m(p1 ) = p1 ω1 + ¯2 ω2 ;
e Demand function with endowment: x1 (p1 , m(p1 )) A function of p1 alone. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Slutsky Equation with endowment e dx1 (p1 , m(p1 )) dp1 = ∂ x1 (p1 , m) ∂ x1 (p1 , m) dm + ∂ p1 ∂m dp1
standard.... s ∂ x1 ∂ x1 (p1 , m) =ω1 = ∂ p1
s ∂ x1 − ∂m ¯1 + x ∂ x1 (p1 , m) ω1 ∂m
endowment income effect standard S.E. = ∂ p1 − ∂ x1 (p1 , m) (¯1 − ω1 ) x ∂m Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Sign of Slutsky Equation with endowment Endowment income effect and the regular income effect always have the opposite signs; Sign check of the equation.... With ﬁxed endowment, total change in demand resulting from relative price change is always ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Sign of Slutsky Equation with endowment Endowment income effect and the regular income effect always have the opposite signs; Sign check of the equation.... With ﬁxed endowment, total change in demand resulting from relative price change is always ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Sign of Slutsky Equation with endowment Endowment income effect and the regular income effect always have the opposite signs; Sign check of the equation.... With ﬁxed endowment, total change in demand resulting from relative price change is always ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choosing labor supply Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Endowment is (time/leisure, money/other consumption) = (R, C); Additional constraint: R ≤ 16; Labor supply ≡ L = 16 − R; p2 = 1; p1 is wage, write it as w;
w is opportunity cost of leisure. Labor supply problem: < w, 1, (R, C), u > Slope of the budget constraint = w; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choosing labor supply Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Endowment is (time/leisure, money/other consumption) = (R, C); Additional constraint: R ≤ 16; Labor supply ≡ L = 16 − R; p2 = 1; p1 is wage, write it as w;
w is opportunity cost of leisure. Labor supply problem: < w, 1, (R, C), u > Slope of the budget constraint = w; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choosing labor supply Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Endowment is (time/leisure, money/other consumption) = (R, C); Additional constraint: R ≤ 16; Labor supply ≡ L = 16 − R; p2 = 1; p1 is wage, write it as w;
w is opportunity cost of leisure. Labor supply problem: < w, 1, (R, C), u > Slope of the budget constraint = w; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choosing labor supply Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Endowment is (time/leisure, money/other consumption) = (R, C); Additional constraint: R ≤ 16; Labor supply ≡ L = 16 − R; p2 = 1; p1 is wage, write it as w;
w is opportunity cost of leisure. Labor supply problem: < w, 1, (R, C), u > Slope of the budget constraint = w; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Choosing labor supply Choice problem with endowment: < p1 , p2 , (ω1 , ω2 ), u >; Endowment is (time/leisure, money/other consumption) = (R, C); Additional constraint: R ≤ 16; Labor supply ≡ L = 16 − R; p2 = 1; p1 is wage, write it as w;
w is opportunity cost of leisure. Labor supply problem: < w, 1, (R, C), u > Slope of the budget constraint = w; Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Backwards bending labor supply curve Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Effect of wage increase on labor supply Suppose wage goes up; Total effect in labor supply is ....
Substitution effect: negative – less leisure as it becomes more expensive; Regular income effect: positive – more leisure as there’s more money to spend (if normal good); Endowment income effect: negative – less leisure as the ﬁxed available time decreases the gain in wage. Total effect is ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Effect of wage increase on labor supply Suppose wage goes up; Total effect in labor supply is ....
Substitution effect: negative – less leisure as it becomes more expensive; Regular income effect: positive – more leisure as there’s more money to spend (if normal good); Endowment income effect: negative – less leisure as the ﬁxed available time decreases the gain in wage. Total effect is ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Effect of wage increase on labor supply Suppose wage goes up; Total effect in labor supply is ....
Substitution effect: negative – less leisure as it becomes more expensive; Regular income effect: positive – more leisure as there’s more money to spend (if normal good); Endowment income effect: negative – less leisure as the ﬁxed available time decreases the gain in wage. Total effect is ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Effect of wage increase on labor supply Suppose wage goes up; Total effect in labor supply is ....
Substitution effect: negative – less leisure as it becomes more expensive; Regular income effect: positive – more leisure as there’s more money to spend (if normal good); Endowment income effect: negative – less leisure as the ﬁxed available time decreases the gain in wage. Total effect is ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Effect of wage increase on labor supply Suppose wage goes up; Total effect in labor supply is ....
Substitution effect: negative – less leisure as it becomes more expensive; Regular income effect: positive – more leisure as there’s more money to spend (if normal good); Endowment income effect: negative – less leisure as the ﬁxed available time decreases the gain in wage. Total effect is ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Choice Problem with Endowment Slutsky Equation with Endowment Application: choice of labor supply Effect of wage increase on labor supply Suppose wage goes up; Total effect in labor supply is ....
Substitution effect: negative – less leisure as it becomes more expensive; Regular income effect: positive – more leisure as there’s more money to spend (if normal good); Endowment income effect: negative – less leisure as the ﬁxed available time decreases the gain in wage. Total effect is ambiguous. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Goal of the welfare analysis Want to evaluate the welfare impact of economic situations;
policy, natural disaster, etc.. Measurement of welfare: utility. Initial choice: (¯1 , ¯2 ); xx
∗∗ Current choice: (x1 , x2 ); Want to measure the difference between the two ICs. (graph) Utility function/preferences are not observable and hence we need to estimate. Want a measure of “monetary equivalence” for the difference in utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Goal of the welfare analysis Want to evaluate the welfare impact of economic situations;
policy, natural disaster, etc.. Measurement of welfare: utility. Initial choice: (¯1 , ¯2 ); xx
∗∗ Current choice: (x1 , x2 ); Want to measure the difference between the two ICs. (graph) Utility function/preferences are not observable and hence we need to estimate. Want a measure of “monetary equivalence” for the difference in utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Goal of the welfare analysis Want to evaluate the welfare impact of economic situations;
policy, natural disaster, etc.. Measurement of welfare: utility. Initial choice: (¯1 , ¯2 ); xx
∗∗ Current choice: (x1 , x2 ); Want to measure the difference between the two ICs. (graph) Utility function/preferences are not observable and hence we need to estimate. Want a measure of “monetary equivalence” for the difference in utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Goal of the welfare analysis Want to evaluate the welfare impact of economic situations;
policy, natural disaster, etc.. Measurement of welfare: utility. Initial choice: (¯1 , ¯2 ); xx
∗∗ Current choice: (x1 , x2 ); Want to measure the difference between the two ICs. (graph) Utility function/preferences are not observable and hence we need to estimate. Want a measure of “monetary equivalence” for the difference in utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Goal of the welfare analysis Want to evaluate the welfare impact of economic situations;
policy, natural disaster, etc.. Measurement of welfare: utility. Initial choice: (¯1 , ¯2 ); xx
∗∗ Current choice: (x1 , x2 ); Want to measure the difference between the two ICs. (graph) Utility function/preferences are not observable and hence we need to estimate. Want a measure of “monetary equivalence” for the difference in utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Goal of the welfare analysis Want to evaluate the welfare impact of economic situations;
policy, natural disaster, etc.. Measurement of welfare: utility. Initial choice: (¯1 , ¯2 ); xx
∗∗ Current choice: (x1 , x2 ); Want to measure the difference between the two ICs. (graph) Utility function/preferences are not observable and hence we need to estimate. Want a measure of “monetary equivalence” for the difference in utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Goal of the welfare analysis Want to evaluate the welfare impact of economic situations;
policy, natural disaster, etc.. Measurement of welfare: utility. Initial choice: (¯1 , ¯2 ); xx
∗∗ Current choice: (x1 , x2 ); Want to measure the difference between the two ICs. (graph) Utility function/preferences are not observable and hence we need to estimate. Want a measure of “monetary equivalence” for the difference in utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Solutions Assuming we know the utility function.
Find out monetary equivalents of the difference in utilities; Without knowledge of the utility function.
Use observed consumptions to “estimate” the change in consumer’s welfare/utility. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Measuring difference in ICs
We know the utility function or ICs. Goal: measure the distance between two ICs.
Maximal utility level given one’s budget; The difference between the budgets provides a concrete measurement. Solution: measure the difference in parallel budget lines that are tangent to the ICs;
Tangent: constrained optimization; Parallel: use money to “buy out” the price change, thus keeping price ﬁxed. NOTE: the distance depends on the price ratio used.
Initial price: equivalent variation New price: compensating variation
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Two measurements of utility change Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Equivalent variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mEV :
How much money would the consumer need to obtain the new utility level under the old price? EV = mEV − m.
EV is the amount of money the consumer is willing to pay/receive to avoid the price change before it happens. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Equivalent variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mEV :
How much money would the consumer need to obtain the new utility level under the old price? EV = mEV − m.
EV is the amount of money the consumer is willing to pay/receive to avoid the price change before it happens. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Equivalent variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mEV :
How much money would the consumer need to obtain the new utility level under the old price? EV = mEV − m.
EV is the amount of money the consumer is willing to pay/receive to avoid the price change before it happens. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Equivalent variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mEV :
How much money would the consumer need to obtain the new utility level under the old price? EV = mEV − m.
EV is the amount of money the consumer is willing to pay/receive to avoid the price change before it happens. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Equivalent variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mEV :
How much money would the consumer need to obtain the new utility level under the old price? EV = mEV − m.
EV is the amount of money the consumer is willing to pay/receive to avoid the price change before it happens. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Equivalent variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mEV :
How much money would the consumer need to obtain the new utility level under the old price? EV = mEV − m.
EV is the amount of money the consumer is willing to pay/receive to avoid the price change before it happens. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Compensating variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mCV :
How much money would the consumer need to keep the old utility level under the new price? EV = mCV − m.
CV is the amount of money the consumer has to be compensated to offset the impact of the price change after the change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Compensating variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mCV :
How much money would the consumer need to keep the old utility level under the new price? EV = mCV − m.
CV is the amount of money the consumer has to be compensated to offset the impact of the price change after the change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Compensating variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mCV :
How much money would the consumer need to keep the old utility level under the new price? EV = mCV − m.
CV is the amount of money the consumer has to be compensated to offset the impact of the price change after the change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Compensating variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mCV :
How much money would the consumer need to keep the old utility level under the new price? EV = mCV − m.
CV is the amount of money the consumer has to be compensated to offset the impact of the price change after the change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Compensating variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mCV :
How much money would the consumer need to keep the old utility level under the new price? EV = mCV − m.
CV is the amount of money the consumer has to be compensated to offset the impact of the price change after the change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Compensating variation Graph. Example: u(x1 , x2 ) = x1 x2 , p1 = 2, p2 = 1, p1 = 1, m = 100 Compute mCV :
How much money would the consumer need to keep the old utility level under the new price? EV = mCV − m.
CV is the amount of money the consumer has to be compensated to offset the impact of the price change after the change. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Comparing EV and CV The distance between two curves varies depending on the slope of the lines we use. In general, CV = EV . Exception: CV = EV when preferences are quasilinear. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Comparing EV and CV The distance between two curves varies depending on the slope of the lines we use. In general, CV = EV . Exception: CV = EV when preferences are quasilinear. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Comparing EV and CV The distance between two curves varies depending on the slope of the lines we use. In general, CV = EV . Exception: CV = EV when preferences are quasilinear. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Comparing EV and CV The distance between two curves varies depending on the slope of the lines we use. In general, CV = EV . Exception: CV = EV when preferences are quasilinear. (graph) Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus A proxy for utility: total willingness to pay
In real life, we don’t observe preferences; Use consumption data to estimate demand function; What we have (roughly): the demand function x1 (p1 ; p2 , m). Inverse demand function: p1 (x1 ; p2 , m) = p2 MRS;
∗ ∗ For any x1 , p1 (x1 ) measures the consumer’s marginal willingness to pay for one more unit of x1 given his current consumption; ∗ Total willingness to pay for x1 units of good 1 = sum of all ∗ marginal willingness to pay when he consumes less than x1 ; Calculus: area below the inverse demand function;
∗ x1 p1 (x1 ; p2 , m)dx1 .
0 ∗ Monetary approximation for the utility of consuming x1 units of good 1.
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus How good is the approximation?
The issue: in general, MRS for good 1 depends on the consumer’s consumption of good 2.
∗ Consequence: total willingness to pay for x1 = utility of ∗. consuming x1 Exception: quasilinear preferences u(x1 , x2 ) = v(x1 ) + x2 ; Demand function is derived by solving: v (x1 ) = p1 ; x Fundamental theorem of calculus: v(x) = 0 p1 dx1 ; Total willingness to pay measures utility perfectly. The approximation is good enough for many purposes. We’ll hide good 2 in the background from now on: demand function x(p)
Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Consumer’s Surplus Consumer’s surplus when consuming x∗ = net willingness to pay = total willingness to pay  payment =
x∗ 0 p(x)dx − p(x∗ )x∗ . CS measures the consumer’s net gain from trade. Change in consumer’s surplus (∆CS) measures the change in consumer’s welfare.
Example: x(p) = 10 − p, p = 1, p = 2. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Consumer’s Surplus Consumer’s surplus when consuming x∗ = net willingness to pay = total willingness to pay  payment =
x∗ 0 p(x)dx − p(x∗ )x∗ . CS measures the consumer’s net gain from trade. Change in consumer’s surplus (∆CS) measures the change in consumer’s welfare.
Example: x(p) = 10 − p, p = 1, p = 2. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Consumer’s Surplus Consumer’s surplus when consuming x∗ = net willingness to pay = total willingness to pay  payment =
x∗ 0 p(x)dx − p(x∗ )x∗ . CS measures the consumer’s net gain from trade. Change in consumer’s surplus (∆CS) measures the change in consumer’s welfare.
Example: x(p) = 10 − p, p = 1, p = 2. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Consumer’s Surplus Consumer’s surplus when consuming x∗ = net willingness to pay = total willingness to pay  payment =
x∗ 0 p(x)dx − p(x∗ )x∗ . CS measures the consumer’s net gain from trade. Change in consumer’s surplus (∆CS) measures the change in consumer’s welfare.
Example: x(p) = 10 − p, p = 1, p = 2. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus EV, CV and CS Quasilinear preferences: EV = CV = ∆CS In general, EV = CV = ∆CS. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus EV, CV and CS Quasilinear preferences: EV = CV = ∆CS In general, EV = CV = ∆CS. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Supply curve and inverse supply curve Producer’s theory is the mirror image of consumer’s theory. Supply curve: the relationship between price and quantity supplied;
Typically upward sloping; Supply function: xs (ps ); Inverse supply function: ps (xs ).
Price measures marginal willingness to receive. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Supply curve and inverse supply curve Producer’s theory is the mirror image of consumer’s theory. Supply curve: the relationship between price and quantity supplied;
Typically upward sloping; Supply function: xs (ps ); Inverse supply function: ps (xs ).
Price measures marginal willingness to receive. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Supply curve and inverse supply curve Producer’s theory is the mirror image of consumer’s theory. Supply curve: the relationship between price and quantity supplied;
Typically upward sloping; Supply function: xs (ps ); Inverse supply function: ps (xs ).
Price measures marginal willingness to receive. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Supply curve and inverse supply curve Producer’s theory is the mirror image of consumer’s theory. Supply curve: the relationship between price and quantity supplied;
Typically upward sloping; Supply function: xs (ps ); Inverse supply function: ps (xs ).
Price measures marginal willingness to receive. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Supply curve and inverse supply curve Producer’s theory is the mirror image of consumer’s theory. Supply curve: the relationship between price and quantity supplied;
Typically upward sloping; Supply function: xs (ps ); Inverse supply function: ps (xs ).
Price measures marginal willingness to receive. Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Producer’s surplus
Total willingness to receive for supplying x∗ = area below the inverse supply curve =
x∗ ps (xs )dxs .
0 Producer’s surplus when supplying x∗ = total payment  total willingness to receive = p (x )x −
0 s ∗ ∗ x∗ ps (xs )ds . Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Producer’s surplus
Total willingness to receive for supplying x∗ = area below the inverse supply curve =
x∗ ps (xs )dxs .
0 Producer’s surplus when supplying x∗ = total payment  total willingness to receive = p (x )x −
0 s ∗ ∗ x∗ ps (xs )ds . Jing Li Intermediate Microeconomics Slutsky Equation Quasilinear Preferences Buying and Selling Consumer’s Surplus Compensating and Equivalent Variations Consumer’s Surplus Producer’s Surplus Producer’s surplus
Total willingness to receive for supplying x∗ = area below the inverse supply curve =
x∗ ps (xs )dxs .
0 Producer’s surplus when supplying x∗ = total payment  total willingness to receive = p (x )x −
0 s ∗ ∗ x∗ ps (xs )ds . Jing Li Intermediate Microeconomics ...
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This note was uploaded on 09/27/2009 for the course ECON 101 taught by Professor Dannicatambay during the Spring '08 term at UPenn.
 Spring '08
 DANNICATAMBAY
 Microeconomics

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