09-10-08 - Econ 302: Intermediate Microeconomic Theory...

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Econ 302: Intermediate Microeconomic Theory Andres Elberg University of Illinois at Urbana-Champaign September 10, 2008 Lecture 5: Choice September 10, 2008 1 / 45
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Review In our last lecture we introduced the concept of the utility function to summarize consumer preferences. A utility function is a rule which assigns a unique number to each bundle in the consumption set and this assignment is such that more preferred bundles are associated with larger numbers. If preferences are complete and transitive (as well as continuous) then there is always a utility function which can represent those preferences. Lecture 5: Choice September 10, 2008 2 / 45
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Review We can restate the de&nition of an indi/erence curve as the set of bundles that provide the same level of utility. A utility function allows us to label indi/erence curves With monotonic preferences, higher indi/erence curves are associated with higher utility A positive monotonic transformation of a utility function represents exactly the same preference ordering. Graphically, applying a positive monotonic transformation to a utility function amounts to relabeling all indi/erence curves. Lecture 5: Choice September 10, 2008 3 ± 45
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Review The general equation of an indi/erence curve is: u ( x 1 , x 2 ) = u where u is a constant level of utility Example Suppose the utility function of a consumer is u ( x 1 , x 2 ) = x 1 x 2 . An indi/erence curve of this consumer is described by the equation x 2 = u x 1 Lecture 5: Choice September 10, 2008 4 ± 45
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Review An important notion in consumer theory is the concept of marginal utility . It is the change in utility associated to a marginal increase in the consumption of one of the goods (while keeping the amount of the other good constant). For a change Δ x 1 in the amount consumed of good 1 the associated change in utility is Δ u ( x 1 , x 2 ) Δ x 1 = u ( x 1 + Δ x 1 , x 2 ) & u ( x 1 , x 2 ) Δ x 1 Lecture 5: Choice September 10, 2008 5 / 45
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Review Marginal utility equals this rate of change when Δ x 1 gets arbitrarily small (i.e. as Δ x 1 ! 0) MU 1 & lim Δ x 1 ! 0 u ( x 1 + Δ x 1 , x 2 ) ± u ( x 1 , x 2 ) Δ x 1 & u ( x 1 , x 2 ) x 1 Similarly, the marginal utility with respect to good 2 is de&ned as MU 2 & lim Δ x 2 ! 0 u ( x 1 , x 2 + Δ x 2 ) ± u ( x 1 , x 2 ) Δ x 2 & u ( x 1 , x 2 ) x 2 Lecture 5: Choice September 10, 2008 6 / 45
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Review We can obtain an expression for the slope of an indi/erence curve by totally di/erentiating the equation of an indi/erence curve, u ( x 1 , x 2 ) = u Since the level of utility u does not change along an indi/erence curve (i.e. du = 0), the total di/erential of this equation is u ( x 1 , x 2 ) x 1 dx 1 + u ( x 1 , x 2 ) x 2 dx 2 = 0 Lecture 5: Choice September 10, 2008 7 ± 45
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Review Rearranging, dx 2 dx 1 = & u ( x 1 , x 2 ) / x 1 u ( x 1 , x 2 ) / x 2 = & MU 1 MU 2 Thus, the Marginal Rate of Substitution equals the ratio of marginal utilities: MRS = & & & & dx 2 dx 1 & & & & = u ( x 1 , x 2 ) / x 1 u ( x 1 , x 2 ) / x 2 = MU 1 MU 2 Lecture 5: Choice September 10, 2008 8 / 45
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Review Notice that while marginal utility has no behavioral content (it&s magnitude has no meaning), the ratio of marginal utilities (i.e. the MRS) does.
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09-10-08 - Econ 302: Intermediate Microeconomic Theory...

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