Suppose that basket B (x = 5, y = 10) is on Matt's linear budget line, Matt's MRSX,Y (B) = 1/4, PX =$2 and PY = $2. Matt's preferences are represented by smooth curves.

Question

Suppose that basket B (x = 5, y = 10) is on Matt's linear budget line, Matt's MRSX,Y (B) = 1/4, PX =$2 and PY = $2.  Matt's preferences are represented by smooth curves.

BlackTom AI · Accepted Answer

We begin by restating the scenario and the options clearly so the reasoning is grounded in the setup.
Question restated: Matt is on a linear budget line with B at (x = 5, y = 10). The marginal rate of substitution of x for y at B is MRSxy(B) = 1/4. Prices are Px = $2 and Py = $2, and Matt’s preferences are represented by smooth indifference curves.
Option analyzed: 'B cannot be Matt's optimal choice. His optimal choice will be to the South East of B.'
First, note the standard condition for an interior optimum on a linear budget line with smooth preferences: the consumer’s MRSxy should equal the price ratio Px/Py at the optimum. Here, Px/Py = 2/2 = 1. The given MRSxy(B) = 1/4 is not equal to 1, so B is not the tangency point where the indifference curve is just touching the budget line.
Second, compare MRSxy(B) to the price ratio to infer which direction along the budget line would increase utility if we move from B. Since MRSxy = 1/4 is less than Px/Py = 1, the consumer is willing to trade only 0.25 units of y for 1 extra unit of x, which is a weaker preference for x relative to the market’s price for x. In practical terms, to reach a higher indifference curve given the budget constraint, Matt would want relatively more y and relatively less x, i.e., move toward the northwest along the budget line, not toward the southeast.
Third, consider what moving southeast implies. Southeast of B on the budget line means increasing x and decreasing y, which would involve giving up more y per unit of x than the consumer is actually willing to trade (since MRS is only 0.25 while the market requires a trade rate of 1). This mismatch means such a move would not improve utility as efficiently as a move toward the northwest where y is increased and x is decreased, bringing the MRS closer to the price ratio.
Fourth, what would a move toward the Northwest imply about the optimal point? It would be the direction in which you substitute y for x along the budget, increasing y relative to x until the MRSxy aligns with Px/Py (or until you hit a corner if no interior solution exists). In this case, since the budget line is linear and preferences are smooth, the interior optimum would occur where MRSxy = 1, which will be at some point northwest of B if feasible by your budget.
In summary, the claim that the optimal choice lies to the Southeast of B is inconsistent with the fact that MRSxy(B) = 1/4 while Px/Py = 1. The more accurate directional intuition is that the optimal point lies to the Northwest of B (more y, less x), assuming feasibility on the budget line. Therefore, the provided option misidentifies the direction and mischaracterizes the optimization condition.

SIPAIA6400_001-002_2025_3 - Microeconomic Analysis for International and Public Affairs Problem Set 4, Individual Portion

Suppose that basket B (x = 5, y = 10) is on Matt's linear budget line, Matt's MRSX,Y (B) = 1/4, PX =$2 and PY = $2. Matt's preferences are represented by smooth curves.

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