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In this article, we will see how we can use the Lagrange Multiplier to find the optimal bundle. We will, specifically, see how it is done using a Cobb-Douglas function as an example.
Defining the Utility Function
The first question to ask is the following: What exactly is a utility function? We remember that we have defined a bundle of goods to be denoted as:
We can then denote the utility of this bundle, which is simply a function that measures preferences over a set of goods and services. In other words, the utility represents the satisfaction that the consumers receive for choosing a given bundle. This utility is given by a function, which we can denote this like so:
We can then give a specific example of a utility function — for example the Cobb-Douglas function. It is given by:
Using Lagrange to Find the Optimal Bundle
We have now defined the concept of utility and the utility function. We can now see how we can find the optimal bundle — i.e., the bundle where the consumer receives the greatest amount of satisfaction. We can do this by using the Lagrange Multiplier. We can take an example of a specific utility function to see how it is done. We can choose the following utility function:
Since we assume that we need to respect the budget line, then we have the following constraint:
