EconGraphs Logo BETA
Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
If you catch a typo or error, or just have a suggestion, please submit a note here. Thanks!

Optimal Choices Characterized by a Tangency Condition

In some cases, a consumer’s optimal bundle will be characterized by the tangency condition \(MRS = {p_1 \over p_2}\) Let’s think about what this means intiuitively, mathematically, and visually.

It can sometimes be helpful to plot the tangency condition equation; in this case the optimum occurs at the intersection of the tangency condition curve and the budget line:

Notice which of the factors affect which of the conditions:

When does the tangency condition find the optimal bundle?

The optimal bundle is not always characterized by a tangency condition.

When the budget constraint is a simple straight line, we can establish some conditions that guarantee an optimum at a tangency condition along the budget line:

Putting these all together means that the solution lies along the budget line; that the MRS is greater than the price ratio at the the vertical intercept, smoothly decreases along the budget line, and is less than the price ratio at the horizontal intercept. Therefore, by a continuity argument, there must be a single point at which the MRS equals the price ratio. Perhaps more intuitively and succinctly: monotonicity pulls the consumer up and to the right, convexity pulls them toward an interior solution, and conditions on the endpoints of the budget line ensure that the solution lies in the first quadrant.

When one or more of the above conditions fail, the solution may not be characterized by a tangency condition:

The Lagrange Method of Finding an Optimal Choice

In situations where the optimum is characterized by a tangency condition, the method of Lagrange multipliers from multivariate calculus may be used to find the optimal choice. This is a large enough topic that it deserves its own page.

Next: Corner Solutions
Copyright (c) Christopher Makler /