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Unformatted text preview: Dr. Steven Waters Econ 380 Page 1 of 10 Lecture 3 Preferences & Utility II There are two ways to get enough: one is to continue to accumulate more and more. The other is to desire less.  G.K. Chesterton The Economics Outline 1. MRS=Marginal Rate of Substitution 2. Some Types of Utility Functions 2.1. Perfect Substitutes 2.2. Perfect Complements 2.3. CobbDouglas 2.4. CES The Mathematics Outline 1. Total Differential 2. Implicit Function Theorem A Few Notational Issues Sometimes I use subscripts to denote partial derivatives and sometimes I use subscripts just to denote which thing I’m talking about. For example, when I write: 1 or 1 U U x the subscripts here mean that you need to take the partial derivative of the utility function. However, in the notation 1 1 or or or 1 1 P P MU MU x x the subscripts just mean that I’m talking about the good 1 x . If you ever have a question as to what the notation means, make sure you ask. Some Mathematical Tools Last time we began talking about indifference curves. We will find it helpful to calculate the slope of the indifference curve. To do that, we will need a few mathematical tools at our disposal. L3: Preferences & Utility II Dr. Steven Waters Econ 380 Page 2 of 10 Total Differential A total differential shows how a function changes as there are small changes around all of the independent variables. Univariate Case Function: ) ( x f y = Total Differential: dx f dx x f dy x = ∂ ∂ = Multivariate Case Function: ) ,..., , , ( 3 2 1 k x x x x f y = Total Differential: n n dx f dx f dx f dy + + + = L 2 2 1 1 Implicit Functions and Their Derivatives Given ) ,..., , , , ( 3 2 1 = k x x x x y F , the total differential of F is: 2 2 1 1 = + + + + n n y dx F dx F dx F dy F L In order to calculate the derivative 1 dx dy , holding the other variables constant, one need merely manipulate the total differential to show: y F F dx dy 1 1 = Example Given: 768 2 3 5 2 2 = + y y x , you cannot solve this equation explicitly for y as a function of x . However, there is an implicit relation between y and x and we can still take the derivative of y with respect to x . Putting the equation into the form dictated by the implicit function theorem, 768 2 3 5 ) , ( 2 2 = + = y y x y x F , we can then calculate the following: 1 3 5 2 6 10 = = = y x y x F F dx dy y x L3: Preferences & Utility II Dr. Steven Waters Econ 380 Page 3 of 10 Total Differentials and the Indifference Curve Suppose U = U(x,y) Taking a total differential we get dy U dx U dy y U dx x U dU y x + = ∂ ∂ + ∂ ∂ = Along an indifference curve we have dU=0 , so y x U U dx dy = (this is the slope of the indifference curve) The Implicit Function Theorem and the Indifference Curve Suppose U = U(x,y) We can write this as an implicit function as follows: y y x x U F U F y x U U y x F = = = ) , ( ) , ( The implicit function theorem tells us: y x y x y x U U U U F F dx dy = = = (once again, the slope of the indifference curve) Manipulating the total differential and using the implicit function theorem got us the...
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This note was uploaded on 11/30/2011 for the course STAT 380 taught by Professor Stevens during the Spring '11 term at Brigham Young University, Hawaii.
 Spring '11
 Stevens

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