Slide 1Econ 410: Micro TheoryMathematical Utility MaximizationMonday, September 10th, 2007Slide 2Visualizing Utility FunctionsSlide 3Recall from last time… Utility functions and Marginal Utility Applying Partial Derivatives to Utility Recall that marginal utility is the increase in utility associated with an increase in good X. As such, it can be represented mathematically as the partial derivative of the utility function with respect to good X. Example: U(X,Y) = 10X2Y MUX= MUY= XYXYXU20),(210),(XYYXUSlide 4Utility Maximization Assumptions The Consumer’s Maximization Problem Max U(X,Y) subject to PXX + PYY = I Specific vs. General Functional Forms Assuming a specific utility function allows us to find a consumer’s demand function for a good We can still find out plenty of properties about utility using general functional forms Constrained OptimizationSlide 5Utility Maximization The Consumer’s Maximization Problem Max U(X,Y) subject to PXX + PYY = I Constrained OptimizationSlide 6Method of Lagrange Multipliers The method of Lagrange multipliers is a technique to maximize (or minimize) a function subject to one or more constraints Steps State the Problem Differentiate the Lagrangian Solve the resulting equationsSlide 7Stating the Problem Original Problem: Max U(X,Y) subject to PXX + PYY = I Rewrite the budget constraint PXX + PYY – I = 0 Lagrangian: L = U(X,Y) - (PXX + PYY – I) RationaleSlide 8Differentiating To maximize the Lagrangian, we differentiate it with respect to X, Y, and and set the derivatives equal to zero. Mathematically 0),( XXPYXMUXL0),( YYPYXMUYL0YPXPILYXVisuallySlide 9Solving the Equations Results of Differentiation0),( XXPYXMUXL Rewriting these equations, we find three conditions that hold at the optimum: MUX= PX MUY= PY PXX + PYY = I0),( YYPYXMUYL0YPXPILYXSlide 10The first two equations from optimization show us this result mathematically: MUX= PXMUY= PY Solving: andThe Equal Margin Principle Recall from last class: Only when a consumer has equalized the marginal utility per dollar of expenditure across all goods will he or she have maximized utilityXXyyPMUPMUyXyxMUMUPPSlide 11What is Anyway? is the Lagrange multiplier, and represents the extra utility that is able to be obtained when the problem’s constraint is relaxed Since we our currently using a budget constraint in the utility maximization problem, represents the marginal utility of income for the consumer For the skeptical (or mathematically inclined)… The meaning of can be shown formally by totally differentiating the utility function with respect to I.Slide 12For next time… Solve the practice utility maximization problem given out in class Read Section 4.1 of your textbook Wednesday’s Agenda We’ll go over the practice problem and use other examples of utility functions to apply these concepts We’ll discuss the notion of “duality” in consumer theory, and begin our discussion of Chapter
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