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# Lecture 4 - Lecture 4 Utility Maximization I For whosoever...

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Lecture 4 Utility Maximization I For whosoever will save his life shall lose it: and whosoever will lose his life for my sake shall find it. Matthew 16:25 The Economics Outline 1. Budget Constraint 2. Utility Maximization (graph) 3. Corner Solutions 4. Marshallian Demand The Mathematics Outline 1. Constrained Maximization 2. Lagrangian Multiplier If you faced no constraints and if the nonsatiation axiom was not violated at any positive consumption values, then it would not be possible to maximize utility as you could always consume one more unit and increase utility. However, we all face constraints and in consumer theory we quantify the main constraint as an income constraint or budget constraint -- we are constrained in what we consume by how much we can afford. The Budget Constraint I = Income P x = Price of good x P y = Price of good y x = Quantity of good x y = Quantity of good y Budget Constraint: I y P x P y x + We can graph the budget constraint by first solving for y as follows: x P P P I P x P I y y x y y x = Dr. Steven Waters Econ 380 Page 1 of 7

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L4: Utility Maximization I The budget constraint then looks like this: Maximizing Utility with an Income Constraint We can demonstrate maximizing utility with a budget constraint graphically as follows: In this figure, the budget constraint (as identified by the straight line) shows the consumption bundles that the individual can purchase. By looking at the tangency point between the indifference curve and the budget constraint, point A, we find the maximum utility that the individual can attain given the budget constraint. Dr. Steven Waters Econ 380 Page 2 of 7
L4: Utility Maximization I From the above discussion of the budget constraint and from our previous discussion about the MRS, we know: 1. Slope of the budget constraint = y x P P 2. Slope of the indifference curve = y x MU MU

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