Evaluating Optimization Methods for Atypical Preferences

Calculating Optimal Consumption

A consumer is choosing an optimal bundle of two goods, given a fixed budget. Their preferences are represented by standard, smooth, convex-to-the-origin indifference curves. Under which condition will the solution found using the tangency method (where the Marginal Rate of Substitution equals the price ratio) be guaranteed to be the same as the solution found using the substitution method (where the budget constraint is substituted into the utility function)?

For a consumer choosing between two goods that are perfect substitutes, the optimal consumption bundle can always be found by identifying the point where the indifference curve is tangent to the budget constraint.

A consumer seeks to maximize their satisfaction from two goods, given a limited budget. Match each analytical method for finding the optimal consumption bundle with its corresponding description.

Economic Intuition of the Tangency Condition

A consumer wants to maximize their utility from two goods, X and Y, subject to a budget constraint. Arrange the following steps in the correct order to find the optimal consumption bundle using the substitution method.

Choosing the Right Optimization Tool

A consumer has a utility function U(X, Y) = X * Y and faces a budget constraint PxX + PyY = M, where Px is the price of good X, Py is the price of good Y, and M is income. To solve for the optimal consumption bundle by embedding the budget constraint into the utility function, one first expresses Y in terms of X from the budget constraint and substitutes this into the utility function. The resulting utility function, expressed solely in terms of X, is U(X) = ____.

A consumer's preferences for goods A and B are such that their Marginal Rate of Substitution (MRS) is always 3 (meaning they are always willing to trade 3 units of B for 1 unit of A). The price of good A is $10 and the price of good B is $5. If one attempts to find the optimal consumption bundle by applying the condition that the MRS must equal the price ratio, what will be the result?