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Intermediate microeconomics: Lecture 3

Utility and indiﬀerence curve Budget constraint Consumer’s problem

Perfect complement Perfect substitute

Intermediate microeconomics: Lecture 3

March 14, 2014

Preference over bundles

Intermediate microeconomics: Lecture 3

Utility and indiﬀerence curve Budget constraint Consumer’s problem

Perfect complement

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Two goods 1 and 2. Let (x, y ) be quantity of good 1 and good 2. Now, we can deﬁne preference over (x, y )

Perfect substitute

Rational preference

Intermediate microeconomics: Lecture 3

Utility and indiﬀerence curve Budget constraint Consumer’s problem

Perfect complement Perfect substitute

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Let x (α ) = α x + (1 − α )x ′ and y (α ) = α y + (1 − α )y ′ . ¯ ¯ Preference is convex if (¯(α ), y (α )) x ¯ any α ∈ (0, 1) (x, y ), (x ′ , y ′ ) for

Consumer’s problem

Perfect complement Perfect substitute

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Or U(¯ (α ), y (α )) ≥ U(x, y ), U(x ′ , y ′ ) for any α ∈ (0, 1). x ¯

Substitution

Intermediate microeconomics: Lecture 3

Utility and indiﬀerence curve Budget constraint Consumer’s problem

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Suppose x is decreased to x − ∆ where ∆ denotes a “small change.” How much extra y do you need to get the same utility? Intuitively, how much extra y do you need to be compensated to achieve the same happiness?

Perfect complement Perfect substitute

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Marginal rate of substitution

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Intermediate microeconomics: Lecture 3

By choosing ∆ which is close to 0, we can formulate the idea of substitution as marginal rate of substitution First, by “total-diﬀerentiation,” we get dU(x, y ) = U1 (x, y )dx + U2 (x, y )dy = 0

Utility and indiﬀerence curve Budget constraint Consumer’s problem

Perfect complement Perfect substitute

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Then, if the change is dx, the compensation dy to keep the level of U(x, y ) is dy U1 (x, y ) =− . dx U2 (x, y )

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MRS is deﬁned as − dy = dx

U1 (x ,y ) U2 (x ,y )

MRS

Intermediate microeconomics: Lecture 3

Perfect complement Perfect substitute

Figure:

Cobb-Douglas

Intermediate microeconomics: Lecture 3

Utility and indiﬀerence curve Budget constraint

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Cobb-Douglas function U(x, y ) = x y

α (1−α )

Consumer’s problem

Perfect complement Perfect substitute

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The MRS is MRS(x, y ) =

α x α −1 y (1−α ) α y = (1 − α )x α y −α 1−α x

Perfect substitute

Intermediate microeconomics: Lecture 3

Perfect complement Perfect substitute

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Consider iPhone and “Galaxy” U(x, y ) = α x + (1 − α )y .

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The MRS is MRS(x, y ) =

α 1−α

Perfect substitute

Intermediate microeconomics: Lecture 3

Perfect complement Perfect substitute

Figure:

Perfect Complement

Intermediate microeconomics: Lecture 3

Utility and indiﬀerence curve Budget constraint

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Sometimes a loss of one good cannot be compensated by any gain of the other good. Without a phone, a phone case is useless. U(x, y ) = min{x, y }

Consumer’s problem

Perfect complement Perfect substitute

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If y = x, then there is no y that can compensate x − ∆. On the other hand, if y < x, as long as y ≤ x − ∆ there is no need to compensate.

Perfect complement

Intermediate microeconomics: Lecture 3

Perfect complement Perfect substitute

Figure:

Limited budget

Intermediate microeconomics: Lecture 3

Utility and indiﬀerence curve Budget constraint

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