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November 17th, 2009, 08:14 AM   #1
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How to find the utility maximizing bundle of goods?

Hi All!

I am happy to find this forum as I often have problems with mathematics in my economics studies. This is my first question.

My microeconomics textbook is not really clear about how to find the utility maximizing bundle of goods for a consumer. The budget constraint is usually a straight downward sloping line given by the formula p1x1+p2x2=m where p1 is the price of good 1 and x1 is the quantity of good 1. m is total income. The slope of the budget constraint line is -p1/p2. The budget constraint determines possible bundles of goods.

The second part is indifference curves which determines which bundles are desirable as compared to other bundles. We can assume well behaved indifference curves which means it is downward sloping and strictly convex. A simple formula giving these properties is x1x2=U. It could also be a Cobb-Douglas function or some other function. The slope of this function (MRS for Marginal Rate of Substitution) is different at different places of the curve. It is U (utility) that I want to maximize with the constraint mentioned above.

Together the budget constraint and the indifference curves give us all necessary information in determining an utility function and an optimal bundle of goods. Intuitively it is easy to see that the optimal bundle is where the highest possible indifference curve is tangent to the budget constraint, that is when they are tangent. In mathematical terms: -p1/p2=MRS

I perfectly understand the concepts of budget constraint and indifference curves but I need help with mathematically finding the correct indifference curve that is tangent to the budget constraint. Once I have found the right value of U and the coordinates where -p1/p2=MRS I have found the utility maximizing bundle of goods. How do I do this? I hope there's a universal solution for all well behaved indifference curves or at least some methodology that can be applied to several different kinds of functions.

Thank you for your time!
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