preference relation - carnegie mellon university...properties: 1. monotonicity 2. convexity...
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Preferences
Preference Relation
The consumer strictly prefers bundle X to bundle Y:
The consumer is indifferent between X and Y:
),(),( 2121 yyxx f
),(~),( 2121 yyxx
Weak Preference
If
or
Then:
),(),( 2121 yyxx f
),(~),( 2121 yyxx
),(),( 2121 yyxx !
How are the relations related?
Q: What do these two relations imply?
),(),( 2121 yyxx !),(),( 2121 xxyy !
How are the relations related?
A: The consumer is indifferent between X and Y:
),(~),( 2121 yyxx
Assumption I: Complete Preferences
For any two bundles X and Y:X preferred to Y:
Y preferred to X:
Indifference:
),(),( 2121 yyxx !
),(),( 2121 xxyy !
),(~),( 2121 yyxx
Assumption II: Reflexive
Any bundle X is at least as good as itself:
),(),( 2121 xxxx !
Assumption III: Transitive
If:
And:
Then:
),(),( 2121 yyxx !
),(),( 2121 zzyy !
),(),( 2121 zzxx !
Indifference curves
Indifference curve
Weakly preferred set
2x
1x1x
2x
Q: Can indifference curves cross?
2x
1x
Z
X
Y
Perfect substitutes
2x
1x
Perfect complements
2x
1x
Satiation
2x
1x
2x
1x
Well-behaved preferences
Let’s impose some extra assumptions to rule out less interesting situations Well-behaved preferences satisfy two properties:
1. Monotonicity2. Convexity
Monotonicity
Consider two bundles:
where Y has at least as much of both goods and more of one.Then:
Monotonicity implies that indifference curves have negative slopes
),(),,( 2121 yyxx
),(),( 2121 xxyy f
Indifference curves have negative slopes
2x
1x
2x
1x
Convexity
Consider two bundles:
Convexity implies that, for ),(~),( 2121 yyxx
))1(,)1(( 2211 ytxtytxt −+−+
),( 21 xx!
10 ≤≤ t
Convex preferences
2x
1x1x1y2x
2y
Z
Non-convex preferences
2x
1x1x1y2x
2y
Z
Marginal rate of substitution
The MRS is the slope of the indifference curve at a point
MRS=derivative of indifference curve
2x
1x
),( 21 xx
1x
2x
Interpretation of MRS
The MRS measures the rate at which the consumer is willing to substitute one good for the other. If good 2 is measured in dollars, the MRS measures the consumer’s willingness to pay for an extra unit of good 1.
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