Lecture 4 _Jan 28_ - Economics 100A Lecture #4: Thursday,...

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Economics 100A Lecture #4: Thursday, Jan. 28 1. Utility functions, again 2. Consumption under uncertainty
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Utility function n Assigns a number to each basket so as to “represent” a consumer’s preferences: If A > B, then U(X A ,Y A ) > U(X B ,Y B ) n In particular, utility takes same value along an indifference curve: For all (X,Y) ˜ (X 0 ,Y 0 ), U(X,Y) = U(X 0 ,Y 0 ) = u 0
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Utility of two goods Slope = MU X
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Properties of utility functions n Invariant to an increasing transformation: V(X,Y) = f(U(X,Y)) ¤ Transformed function also represents the consumer’s preferences ¤ Example: f(u) = log(u) is strictly increasing n Marginal utility of any good (holding all others fixed) is positive ¤ MU X = ? U/ ? X = ?U(X,Y)/?X > 0 ¤ Similarly for MU Y
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Properties (cont’d) n Diminishing Marginal Rate of Substitution ¤ A consumer is willing to give up less of a good to get one more unit of a second good, the more of the second good she has to begin with (idea: prefer variety or a “mixture” of goods) ¤ MRS XY = - ? Y / ? X| ? U=0 decreases with increases in X (along a given indifference curve) ¤ Indifference curve is “convex to origin” ¤ Does not require diminishing marginal utility of each good, i.e., ? (MU X )/ ? X = ? 2 U(X,Y)/?X 2 < 0
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Diminishing MRS 8 2 Food (# pounds) 4 8 Clothing (# shirts) A B C 4 ? ? ? 2 MRS XY = -?Y/?X| ? U = -(-4)/2 = 2 MRS XY = -?Y/?X| ? U = -(-2)/4 = 1/2
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Perfect substitutes n U(X,Y) = X + Y L MRS XY = MU X /MU Y = 1/1 ¤ Note: constant MRS indicates perfect substitutes Pepsi Coke 1 2 3 4 1 2 3 4 ˜
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Perfect complements n U(X,Y) = minimum {X, 2Y} ¤ MRS XY = 0 if X > 2Y ¤ MRS XY = +8 if X < 2Y ¤ Not defined if X = 2Y n ¤ X = sugar in teaspoons ¤ Y = coffee in 8oz cups ¤ Strongly prefer 2 tsp sugar per 1 cup of coffee X (sugar) Y (coffee) 2 1 2 4 Y=X/2 -?Y/?X = -(-1)/8 = 0 -?Y/?X = -(-1)/0 = +8
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Cobb-Douglas utility function n U(X,Y) = AX a Y ß where A, a, ß > 0 ¤ Indifference curve: AX a Y ß = u 0 L Y = (u 0 A -1 X -a ) 1/ß n Marginal utilities:
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This note was uploaded on 03/18/2010 for the course ECON 100A taught by Professor Woroch during the Spring '08 term at University of California, Berkeley.

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Lecture 4 _Jan 28_ - Economics 100A Lecture #4: Thursday,...

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