380 L8-Expenditure or Cost Minimization

380 L8-Expenditure or Cost Minimization - Expenditure or...

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Expenditure or Cost Minimization Background and Motivation Question: If the price of gasoline rises from $2.50 to $3.50 per gallon, how much will that affect John’s standard or cost of living? (Or personalize it to you) This is a common statement but many things are unclear. “Standard of living” likely means something to do with utility but what exactly is “living” and what are costs. The most common way economists approach such issues is for “living” to mean “remaining on the original utility function”. If U=100 was attained when the price of gas was $2.50, and 50 gallons of gasoline were consumed, one answer might be that costs would go up $50 or $1 times 50 gallons. This is a “back of the envelope calculation” and ignores that John might decide to drive less, drive more efficiently, or get a more fuel efficient car. There is a conceptually clear way to approach this problem: deriving compensated demands and the expenditure function. The Expenditure Problem It is easy to show that a person maximizing utility subject to a budget constraint is a cost or expenditure minimizer. Indeed, it is sometimes called the dual to utility maximization. The problem is stated as (1) , (, ,) [ : ( ,) ] xy x yx y Ep pU M in px py U Uxy =+ - Note that total expenditure is px py + . Expenditure looks just like a budget constraint except it isn’t constrained. In utility maximization, it is constrained by income but here we are choosing x and y so as to reach a given indifference curve defined by U . Consider what this looks like graphically. If one solves Ep xp y and solving for y in terms of prices and E gives: (2) x yy pp =-
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which is just like the budget constraint in utility maximization except that E replaces I. Graphically, Note that at x c and y c , we have the familiar condition that the slope of the expenditure function x y p p is equal to the slope of the utility function which is –MRS. Thus, the expenditure minimizing demands are characterized by x y p MRS p just as in utility maximization subject to the budget constraint. The difference here is rather than being on a given budget line, one must here remain on a given indifference curve. y y* x* x U E(p x ,p y ,I) utility is fixed, the budget line is not Figure 1-Minimizing Expenditure 0
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Lagrangian Solution Now let’s go through more formally. The problem in (1) can be written as a Lagrangian: (3) ,, (( , ) ) xy x y Max L p x p y U U x y  Hopefully, you are starting to get the idea about how to form Lagrangians. Note that we are thinking of U as fixed (at least for now). It is some number like 100. First order conditions for a calculus solution to the problem in (3) are: (4) 0 x LU p x x   (5) 0 y p yy (6) (, ) 0 L UUxy .
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This note was uploaded on 04/07/2011 for the course ECON 380 taught by Professor Showalter,m during the Winter '08 term at BYU.

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380 L8-Expenditure or Cost Minimization - Expenditure or...

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