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Review of Previous Lecture
Main goal: Derive consumer demand (what and how much consumers choose to consume).
What do consumers consume? What they like. Bundle is … Preferences tell us …
• “nice” if satisfy 3 assumptions: (1) (2) (3)
Indifference curves depict ... Describe indifference curves: marginal rate of substitution.
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Review of Previous Lecture
Units of Food
Units of Clothing
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Properties of indifference curves
1. Direction of improvement is “north, east and north-east”
2. Each basket lies on a single indifference curve
3. Indifference curves have negative slope
4. Indifference curves do not cross
5. Indifference curves are not “thick”
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The Utility Function
Definition: A utility function attaches a number to each bundle.
We say that a utility function u(x) represents preferences if
bundle A is preferred to bundle B if and only if u(A)>u(B).
Theorem. Preferences are rational (complete and transitive) if and only if thereexists a utility function that represents the preference.
Another way to express preferences is through utility functions.
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Types of Ranking
Ordinal ranking: At the exam, Betty did best, Joe did second best, Harry didthird best, and so on.Cardinal ranking: Betty got 80, Joe got 75, Harry got 74 and so on.
A ranking is ordinal when only the order is important. A ranking is cardinalwhen the order and the absolute performance are both important.
The ordinal ranking is the important ranking in consumer theory.
Any transformation of a utility function that preserves the original ranking ofbundles is an equally good representation of preferences.
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The Utility Function: Example
Example:
Utility functions:
1. U(x,y) = xy2
2. U(x,y) = (xy) 0.5
3. U(x,y) = x2+y2
Bundles (x,y):
(7,3), (5,5), (4,5), (3,7), and (1,12)
x y U(x,y)= xy2 U(x,y) = (xy) 0.5 U(x,y)= x2 +y2
7 3 63 4,58 58
5 5 125 5,00 50
4 5 100 4,47 41
3 7 147 4,58 58
1 12 144 3,46 145
Q: What is each consumer’s favourite bundle? What is each consumer’s least favouritebundle?
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The Utility Function: Example
Example:
Utility functions:
1. U(x,y) = xy2
2. U(x,y) = (xy) 0.5
3. U(x,y) = x2+y2
Bundles (x,y):
(7,3), (5,5), (4,5), (3,7), and (1,12)
x y U(x,y)= xy2 U(x,y) = (xy) 0.5 U(x,y)= x2 +y2
7 3 63 4,58 58
5 5 125 5,00 50
4 5 100 4,47 41
3 7 147 4,58 58
1 12 144 3,46 145
Q: What is each consumer’s favourite bundle? What is each consumer’s least favouritebundle?
Based on the information, are these three preferences monotone?
Which of these preferences have ICs that are bowed towards the origin?
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The Utility Function: Example
Example:
1. U(x,y) = xy2
2. U(x,y) = (xy) 0.5
3. U(x,y) = x2+y2
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1 3 5 7 9 11 13 15 17
y
x
Indifference curves:
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Marginal Utility
Definition:The marginal utility of good x, MUx, is the additional utility that the consumer getsfrom consuming a little more of x when the consumption of all the other goods inthe consumer’s basket remains constant.
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Marginal Utility
Example:
1. U(x,y) = xy2
2. U(x,y) = (xy) 0.5
3. U(x,y)= x2+y2
Is marginal utility diminishing?
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Marginal Rate of Substitution
Along the indifference curve, the utility does not change. Technically:
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Marginal Rate of Substitution
Example:
1. U(x,y) = xy2
2. U(x,y) = (xy) 0.5
3. U(x,y)= x2+y2
Is MRS diminishing?
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Special Functional Forms
Cobb-Douglas: U = axy
IC1
IC2
y
Key property: The MRS diminishes, indifference curves are convex.
x
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Special Functional Forms
Perfect substitutes: U = ax + by
Key property: The consumer is willing to substitute a/b units of y for 1 more unit of x everywhere on the indifference curve. Slope of IC does not depend on x or y!
y
IC1 IC2 IC3
x
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Special Functional Forms
Perfect complements: U = min{x,y}
y
IC1
IC2
Key property: The consumer can only enjoy one good when it is consumed with the other.
MRS not defined!
x
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Special Functional Forms
Key property:MRS increases: the more x you have, the more y you are willing to trade for it. Examples?
IC1
IC2
y
x
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Special Functional Forms
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ICs have same slope on any vertical line
IC2
IC1
Quasi-linear preferences: U = v(x) + ay
Key property:MRS depends only on the quantity of x
x
y
