Introduction of microeconomic, Lecture notes of Microeconomics

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Microeconomics BEEM

Pauline Vorjohann

University of Exeter

Lecture 7: Consumer behavior (part 2)

Differentiability

Differentiability of ≽ ensures that the partial derivatives of the representing utility function u are well defined such that we can use them to calculate the MRS:

MRS(x 1 , x 2 ) =

∂u/∂x 1 ∂u/∂x 2

Cobb Douglas

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 x 2.

Cobb Douglas

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 x 2.

≽ is monotone:

Cobb Douglas

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 x 2.

≽ is monotone:x∗^ is on the budget line, i.e. p 1 x 1 ∗ + p 2 x 2 ∗ = w. ≽ is differentiable (indifference curves are smooth):

Cobb Douglas

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 x 2.

≽ is monotone:x∗^ is on the budget line, i.e. p 1 x 1 ∗ + p 2 x 2 ∗ = w. ≽ is differentiable (indifference curves are smooth):MRS(x) = x 2 /x 1.

Cobb Douglas

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 x 2.

≽ is monotone:x∗^ is on the budget line, i.e. p 1 x 1 ∗ + p 2 x 2 ∗ = w. ≽ is differentiable (indifference curves are smooth):MRS(x) = x 2 /x 1. ≽ is also (strictly) convex: (a), (b), or (c) applies. Any bundle with x 1 , x 2 > 0 is strictly preferred to any bundle with x 1 = 0 or x 2 = 0: (a) applies, i.e. MRS(x∗) = x 2 ∗ /x 1 ∗ = p 1 /p 2.

Cobb Douglas

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 x 2.

≽ is monotone:x∗^ is on the budget line, i.e. p 1 x 1 ∗ + p 2 x 2 ∗ = w. ≽ is differentiable (indifference curves are smooth):MRS(x) = x 2 /x 1. ≽ is also (strictly) convex: (a), (b), or (c) applies. Any bundle with x 1 , x 2 > 0 is strictly preferred to any bundle with x 1 = 0 or x 2 = 0: (a) applies, i.e. MRS(x∗) = x 2 ∗ /x 1 ∗ = p 1 /p 2.

⇒ Unique solution to the consumer’s problem:

x∗(p 1 , p 2 , w ) = (w /(2p 1 ), w /(2p 2 ))

Cobb Douglas

Perfect substitutes

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 + x 2.

Perfect substitutes

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 + x 2.

≽ is differentiable:MRS(x) = 1.

Perfect substitutes

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 + x 2.

≽ is differentiable:MRS(x) = 1. ≽ is also (strongly) monotone and convex: (a), (b), or (c) applies.

Perfect substitutes

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 + x 2.

≽ is differentiable:MRS(x) = 1. ≽ is also (strongly) monotone and convex: (a), (b), or (c) applies. For p 1 < p 2 : x 2 ∗ = 0 and MRS(x∗) = 1 > p 1 /p 2 ((c) applies). For p 1 > p 2 : x 1 ∗ = 0 and MRS(x∗) = 1 < p 1 /p 2 ((b) applies).

Perfect substitutes

Suppose ≽ is represented by u(x 1 , x 2 ) = x 1 + x 2.

≽ is differentiable:MRS(x) = 1. ≽ is also (strongly) monotone and convex: (a), (b), or (c) applies. For p 1 < p 2 : x 2 ∗ = 0 and MRS(x∗) = 1 > p 1 /p 2 ((c) applies). For p 1 > p 2 : x 1 ∗ = 0 and MRS(x∗) = 1 < p 1 /p 2 ((b) applies). For p 1 = p 2 : Either x 1 ∗ , x 2 ∗ > 0, or x 1 ∗ = 0, or x 2 ∗ = 0 and MRS(x∗) = 1 = p 1 /p 2 ((a), (b), or (c) applies).