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Difficulty: HardIndifference Curves and Their Properties

A consumer is indifferent among five bundles of Commodity XX and Commodity YY as presented in the schedule below:

CombinationCommodity XX (units)Commodity YY (units)
P116
Q211
R37
S44
T52

Based on this schedule, what is the Marginal Rate of Substitution of XX for YY (MRSxyMRS_{xy}) when the consumer moves from combination R to combination S, and what fundamental geometric property of the indifference curve does the overall trend of MRSxyMRS_{xy} demonstrate?

  1. 3 units of YY for 1 unit of XX; diminishing MRSxyMRS_{xy} resulting in convexity to the originAnswer
  2. B
    4 units of YY for 1 unit of XX; constant MRSxyMRS_{xy} resulting in a linear downward slope
  3. C
    3 units of YY for 1 unit of XX; total utility falling to zero at the point of maximum consumption
  4. D
    7 units of YY for 1 unit of XX; increasing MRSxyMRS_{xy} resulting in concavity to the origin

Answer

The Marginal Rate of Substitution of XX for YY from combination R to S is 3 units of YY per unit of XX, and the overall pattern demonstrates a diminishing MRSxyMRS_{xy}, which explains why indifference curves are convex to the origin.
The change in good YY given up when moving from bundle R (3X, 7Y) to bundle S (4X, 4Y) is 7−4=37 - 4 = 3 units of YY for 1 unit of XX. Furthermore, evaluating the whole schedule shows that the MRSxyMRS_{xy} steadily declines (5→4→3→25 \rightarrow 4 \rightarrow 3 \rightarrow 2), which directly explains the fundamental property of standard indifference curves being convex to the origin.

Step-by-Step Solution

1
Calculate the Marginal Rate of Substitution (MRSxyMRS_{xy}) when moving from combination R to combination S.
MRSxy=−ΔYΔX=7−44−3=31=3MRS_{xy} = \frac{-\Delta Y}{\Delta X} = \frac{7 - 4}{4 - 3} = \frac{3}{1} = 3 units of YY per unit of XX.
MRS measure the quantity of good YY a consumer is willing to give up to gain one additional unit of good XX while maintaining the same level of utility.
2
Analyze the trend of MRSxyMRS_{xy} across all successive consumption bundles in the schedule.
From P to Q: MRS=5MRS = 5; Q to R: MRS=4MRS = 4; R to S: MRS=3MRS = 3; S to T: MRS=2MRS = 2.
As consumption of XX increases, the consumer values additional units of XX relatively less in terms of YY.
3
Relate the diminishing trend of MRSxyMRS_{xy} to the corresponding property of the indifference curve.
The diminishing MRSxyMRS_{xy} gives rise to an indifference curve that is convex to the origin.
Because the slope of the curve (−dYdX=MRSxy-\frac{dY}{dX} = MRS_{xy}) decreases in magnitude as XX increases, the curve flattens out toward the right, forming a convex shape.

Key Concept

Indifference Curve Convexity and Diminishing Marginal Rate of Substitution (MRS)
Estimated Time:2m 0s
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Topics

Indifference Curves and Their Properties

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