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# lect2 - Utility Maximization Intermediate Microeconomics...

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Utility Maximization Intermediate Microeconomics Amy Brown University of California, Los Angeles August 6, 2008 A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 1 / 32

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Outline of Lecture 1 Review of Budget Sets 2 Graphical Constrained Optimization 3 Mathematical Solutions 4 Break (5 minutes) 5 Marshallian Demand Functions and Indirect Utility Function 6 Application to Subsidies and Taxation This should cover chapter 4 and a part of chapter 5 in Nicholson. A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 2 / 32
Review of Budget Sets Consumers have limited resources of time and income. The collection of all consumption bundles that are within reach is called the budget set . Budget set depends on the consumer°s income and the prices of goods. Suppose M is the consumer°s income or budget, while p x is the price of good x and p y the price of good y . Then the feasible consumption set is M ° p x x + p y y A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 3 / 32

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Budget Set and the E/ect of Income Increases Graphically, M ° p x x + p y y A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 4 / 32
E/ect of Price Change on Budget Constraint A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 5 / 32

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Utility Maximization Maximize utility subject to the budget constraint: max x , y U ( x , y ) subject to M ° p x x + p y y using the choice variables x and y . Monotonicity implies that the budget constraint is always binding. A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 6 / 32
Graphical Solution to Utility Maximization Marginal rate of substitution equals price ratio: Consumer is willing to trade o/ the two goods at the same ratio as the market. A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 7 / 32

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Substitution Method Solve budget constraint for x or y y = M p y ± p x p y x Substitute into U ( x , y ) and solve a single variable problem max x U ° x , M p y ± p x p y x ± Take ±rst-order conditions using product and chain rules U x ± p x p y U y = 0 MRS = U x / U y = p x p y A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 8 / 32
Lagrange Method De±ne λ , the Lagrange multiplier. Form Lagrangian function: L ( x , y ; λ ) = U ( x , y ) + λ ( M ± p x x ± p y y ) Take ±rst order conditions with respect to x , y and λ and set them to zero: 1 L x : U x ± p x λ = 0 2 L y : U y ± p y λ = 0 3 L ∂λ : M ± p x x ± p y y = 0 Solve for the optimum. Again, notice that MRS = U x / U y = p x p y A. Brown (UCLA) Econ 11 Lecture 2 08/06/08 9 / 32

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Second Order Conditions First order conditions need not always characterize an optimum.
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lect2 - Utility Maximization Intermediate Microeconomics...

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