ch5_lec - Varian Chapter 5: Utility Maximization John Rust,...

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Varian Chapter 5: Utility Maximization John Rust, Juan Diaz, Sung Jin Cho University of Maryland <http://gemini.econ.umd.edu/jrust/econ306/ch5_lec.pdf> February 5, 2003 1
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What Do Consumers Do? They maximize their utility subject to their budget constraint. That is, they choose the consumption bundle inside their budget constraint that gives them the highest possible utility. In this lecture we will show how this is done in 2 different ways, 1. Visually, via graphs 2. Mathematically, using calculus and Lagrange multipliers 2
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Recall the defnition oF the budget set Defnition: The budget set B is the set of all feasible consumption bundles that cost less than or equal to the consumer’s income y , i.e. B = ± x | p 0 x y (1) Recall that the notation p 0 x denotes inner product p 0 x = n i = 1 p i x i (2) So, p 0 x is the total cost of the consumption bundle x . Thus, the budget set B is simply the set of all consumption bundles whose total cost is less than or equal to the consumer’s total income y . 3
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02.01 4
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Deriving the slope of the budget constraint, - p 1 / p 2 Notice that in the two good case, n = 2 , the budget constraint is p 1 x 1 + p 2 x 2 y (3) However the equation when the budget constraint is binding is p 1 x 1 + p 2 x 2 = y (4) Solving this equation for x 2 as a function of x 1 we get x 2 = - p 1 p 2 x 1 + 1 p 2 y (5) But this is just a line with slope - p 1 / p 2 . 5
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0 2 4 6 8 10 0 2 4 6 8 10 0 5 10 15 20 25 30 35 x 1 Maximize u(x 1 ,x 2 )=4 x 1 .3 x 2 .7 +20 subject to 2x 1 +4x 2 20 x 2 6
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Graphical explanation of utility maximization The consumer must fnd the consumption bundle ( x 1 , x 2 ) with the highest utility inside their budget set B = { ( x 1 , x 2 ) | p 1 x 1 + p 2 x 2 y } . IF the consumer likes more goods better than Fewer goods (i.e. the consumer’s utility Function u ( x 1 , x 2 ) is monotonically increasing in both x 1 and x 2 , then it is easy to see that the consumer will spend all oF her income y , i.e. the consumer’s budget constraint p 1 x 1 + p 2 x 2 y will be binding at the optimal consumption bundle ( x 1 , x 2 ) : p 1 x 1 + p 2 x 2 = y (6) In addition, iF the optimal consumption oF each good is positive, i.e. x 1 > 0 and x 2 > 0 , then at the optimal bundle ( x 1 , x 2 ) , the consumer’s highest indiFFerence curve will be tangent to the budget line. 7
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Graphical explanation of utility maximization, (continued) The tangency condition (slope of indifference curve = slope of budget line) can be written mathematically as - u x 1 ( x 1 , x 2 ) u x 2 ( x 1 , x 2 ) = - p 1 p 2 (7) The quantity on the left hand side of the above equation, i.e. the slope of the consumer’s indifference curve, is called the marginal rate of substitution (MRS). The marginal rate of substitution represents the amount of good
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ch5_lec - Varian Chapter 5: Utility Maximization John Rust,...

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