Wolfram Demonstrations Project: "Constrained Optimization: Cobb-Douglas Utility and Interior Solutions Using a Lagrangian"

To use this simulation, you must download and install the Mathematica Viewer. Although this software is free, it is a sizable download. This activity is therefore optional.

This simulation presents an excellent illustration of how indifference curves and budget lines interact to produce optimal consumption (highest combined utility) of goods, given a set budget.

Our first task is to avoid letting the title of the simulation and the mathematical expressions fluster us! Once you have downloaded the software to your desktop, open the simulation and read the instructions.

A Lagrangian is an optimal point. Consumer optimization is all about finding the optimal point on the highest indifference curve (the red curve convex to the origin). An indifference curve shows different quantities of much of each of two goods (X1 and X2) a person can consume and end up with equivalent utility. You will work with an example of a college student who is trying to optimize his utility derived from eating pizza and drinking beer in the following quiz.

A vital constraint to consider in optimizing utility is budget. The slope of the budget line is controlled by the relative prices of both products. In the simulation, change the sliders that control the relative price of each good (X1 and X2) and note how the budget line changes. The cheaper either product is the more of that product can be bought, but that does not necessarily mean utility will respond proportionately. The income slider moves the budget line inward or outward; with more income we can buy proportionately more of both products. Notice how the budget line changes shape or moves up or down different indifference curves are tangent to the budget line. The indifference curve doesn't move, but we rather encounter different curves to which the budget line becomes tangent.

The "preference strength" for X1determines the shape of the indifference curve and consequently where it becomes tangent to the budget line. Experiment with varying preferences and recall that preference is another way of stating tastes. As preferences change, higher or lower indifferences curves intersect the budget line.

Start your exploration by setting all sliders to the middle of their respective bars. Then, try the following:

  • Examine the consequence of an overall price decline in both products. (Hint: Move both X sliders in turn, then at the same time.)
  • Examine the consequences of an overall price increase in both products in turn and at the same time.
  • Reset to the midpoints and explore income increases and decreases. How do these compare with price increases?
  • Reset to the midpoints and explore how changing preferences affect quantities purchased.
  • How could you represent luxuries and necessities with this demonstration?

Record your observations in your notes and consider sharing your conclusions on the discussion forum.