380 L5 Utility Maximization Subject to a Budget Constraint

# 380 L5 Utility Maximization Subject to a Budget Constraint...

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Utility Maximization Subject to a Budget Constraint The standard static theory of the consumer is that consumers solve this problem (1) ,, (, ) . . , , [ (, ) : ] x y xy x y Max U x y s t p x p y I or Max U x y p x p y I  This reads maximize by choosing x and y the utility function such that the budget constraint is satisfied (or subject to constraint). Points inside the budget constraint are called feasible. Note that if the problem in (1) were stated without the budget constraint, then consumers would choose x and y so that they become satiated in each good. What prevents me from becoming satiated on Porsches is the budget constraint. Why don’t I even own one or a BMW given that I “like cars”? Why do we make the choices we do? The problem in (1) is what is called a constrained maximization problem. It isn’t quite right as stated. More correctly: (2) , [( ,) : , 0 ] x y Max U x y p x p y I x y  That is, one can’t usually choose negative consumption. Unfortunately, adding the two additional constraints causes considerable technical difficulties for calculus and are sometimes avoided by saying “I assume an interior solution (on 2 R )” and avoid corner- point solutions. Practically, the problem itself dictates whether one needs to pay serious attention to this issue. For example, where I to study the demand behavior of practicing LDS people and wine was one of the goods, (2) would be important. It is sometimes called the “zeros” problem in demand. It is also clear that if U isn’t differentiable, then calculus can’t be used to find a solution to the problem. For this class, one can generally deal with both non-differentiability and corner solutions using graphical techniques mixed with calculus. First, the standard calculus solutions are discussed and then the discussion closes with an examination of these problems. A Graphical Interior Solution Because of non-satiation, the solution to (1) will occur on the budget constraint. To begin with consider the graphical interior solution to the problem in (2). The solution is depicted by the highest indifference curve touching the budget constraint. Such a solution is depicted at C in the graph below from the text:

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Point D yields higher utility than B or C but is not-feasible. B is feasible but does not correspond to the highest possible utility in the feasible set. Thus, of all of the feasible choices in the shaded portion, the optimal point is at C where the budget constraint and the indifference curves just touch or are tangent. Tangency implies that the slopes of the budget constraint and the indifference curve are equal. In earlier discussions, the slope of the budget constraint is x y p p while the slope of the indifference curve is MRS . This is stated in the following result.
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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 L5 Utility Maximization Subject to a Budget Constraint...

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