g30d, week 2socscistaff.bham.ac.uk/jensen/week2(consumertheory-cont)-slides.… · martin k. jensen...

OutlineUtility Representation theorem (from Week 1)

The Feasible SetThe Consumption DecisionExistence and Uniqueness

The MRS

G30D, Week 2

Martin K. Jensen (U. B’ham)

October 2012

Martin K. Jensen (U. B’ham) G30D, Week 2

The MRS

1 Utility Representation theorem (from Week 1)

2 The Feasible Set

3 The Consumption Decision

4 Existence and Uniqueness

5 The MRS

The MRS

Theorem

(Utility Representation Theorem) Let a preference relation �on X = Rn

+ satisfy assumptions 1-5 of week 1. Then there exists autility representation u : X → R which is a continuous, stronglymonotone, and strictly quasi-concave.

You can find this result in appendix 2, chapter 2 of GR in a slightlydifferent language. You should read through this appendix, but youwill not be required to know the proof.

The MRS

Imagine going to the supermarket. There are n goods, sayn = 2000. Each will have a price, say p1 = 1, p500 = 5 and soon.

The vector of all prices, p = (p1, . . . , pn) ∈ Rn++ is called the

price vector.If you wish to buy a specific vector of goods x = (x1, . . . , xn)you will have to pay exactly:

p1x1 + p2x2 + . . .+ pnxn =n∑

pixi (1)

If you have income M > 0, you can buy any the vector ofgoods x precisely if it is in the feasible set (a.k.a. thebudget set):

{x ∈ X :n∑

pixi ≤ M} (2)

The MRS

Imagine going to the supermarket. There are n goods, sayn = 2000. Each will have a price, say p1 = 1, p500 = 5 and soon.The vector of all prices, p = (p1, . . . , pn) ∈ Rn

++ is called theprice vector.

If you wish to buy a specific vector of goods x = (x1, . . . , xn)you will have to pay exactly:

p1x1 + p2x2 + . . .+ pnxn =n∑

pixi (1)

{x ∈ X :n∑

pixi ≤ M} (2)

The MRS

++ is called theprice vector.If you wish to buy a specific vector of goods x = (x1, . . . , xn)you will have to pay exactly:

p1x1 + p2x2 + . . .+ pnxn =n∑

pixi (1)

{x ∈ X :n∑

pixi ≤ M} (2)

The MRS

++ is called theprice vector.If you wish to buy a specific vector of goods x = (x1, . . . , xn)you will have to pay exactly:

p1x1 + p2x2 + . . .+ pnxn =n∑

pixi (1)

{x ∈ X :n∑

pixi ≤ M} (2)

The MRS

The objective of the consumer in a competitive economy is tochoose the consumption vector within her feasible set whichyields the highest level of satisfaction. The highest level ofsatisfaction translates into “on the highest indifference curve”in an indifference diagram.

Assume that the consumer has utility function u : X → R.

The consumer’s decision problem is then:

max u(x1, . . . , xn)

{ ∑i pixi ≤ M

xi ≥ 0 for i = 1, . . . , n(3)

The MRS

max u(x1, . . . , xn)

{ ∑i pixi ≤ M

xi ≥ 0 for i = 1, . . . , n(3)

The MRS

max u(x1, . . . , xn)

{ ∑i pixi ≤ M

xi ≥ 0 for i = 1, . . . , n(3)

The MRS

Theorem

If M > 0, pi > 0 for all i = 1, . . . , n, and u is a continuous,strongly monotone, and strictly quasi-concave utility function thenthe consumer’s decision problem has exactly one solution for everygiven price vector p and income level M.

The solution to the consumer’s decision problem given pricesp and income W is denoted byx(p,M) = (x1(p,M), . . . , xn(p,M)). This is the demandfunction.

The MRS

Theorem

If M > 0, pi > 0 for all i = 1, . . . , n, and u is a continuous,strongly monotone, and strictly quasi-concave utility function thenthe consumer’s decision problem has exactly one solution for everygiven price vector p and income level M.

The solution to the consumer’s decision problem given pricesp and income W is denoted byx(p,M) = (x1(p,M), . . . , xn(p,M)). This is the demandfunction.

The MRS

Let n = 2 (two goods), and consider the Cobb-Douglasutility function:

u(x1, x2) = xα1 xβ2 (4)

where α, β > 0 (if α+ β < 1 then u is strictly concave and soalso strictly quasi-concave;1 but u is strictly quasi-concaveregardless of the sum of α and β).

Let n be any natural number, and consider the ConstantElaticity of Substitution (CES) utility function:

u(x1, x2, . . . , xn) = (∑i

xαi )1α (5)

where α > 0.

1This is an important result which you should remember: Any functionwhich is strictly concave is also strictly quasi-concave.

The MRS

Let n = 2 (two goods), and consider the Cobb-Douglasutility function:

u(x1, x2) = xα1 xβ2 (4)

where α, β > 0 (if α+ β < 1 then u is strictly concave and soalso strictly quasi-concave;1 but u is strictly quasi-concaveregardless of the sum of α and β).

Let n be any natural number, and consider the ConstantElaticity of Substitution (CES) utility function:

u(x1, x2, . . . , xn) = (∑i

xαi )1α (5)

where α > 0.

1This is an important result which you should remember: Any functionwhich is strictly concave is also strictly quasi-concave.

The MRS

Both the Cobb-Douglas and CES utility functions aredifferentiable. This makes it possible to derive the marginalrate of substitution (MRS) between two goods. Take n = 2(two goods), and consider the MRS between good 1 and 2,denoted MRS21:

MRS21 =u1(x1, x2)

u2(x1, x2)(6)

Here u1(x1, x2) is the partial derivative of u with respect to x1(and likewise u2(x1, x2) is the partial derivative w.r.t. x2). Asyou can see on p.26 of GR, this is notation of the book.However, you will often find that people prefer one ofDx1u(x1, x2) or u′1(x1, x2).

The MRS

Both the Cobb-Douglas and CES utility functions aredifferentiable. This makes it possible to derive the marginalrate of substitution (MRS) between two goods. Take n = 2(two goods), and consider the MRS between good 1 and 2,denoted MRS21:

MRS21 =u1(x1, x2)

u2(x1, x2)(6)

Here u1(x1, x2) is the partial derivative of u with respect to x1(and likewise u2(x1, x2) is the partial derivative w.r.t. x2). Asyou can see on p.26 of GR, this is notation of the book.However, you will often find that people prefer one ofDx1u(x1, x2) or u′1(x1, x2).

The MRS

In the case of a Cobb-Douglas utility function,u1(x1, x2) = αxα−11 xβ2 and u2(x1, x2) = βxα1 x

β−12 , and so:

MRSCD21 =

αxα−11 xβ2

βxα1 xβ−12

The marginal rate of substitutes is equal to minus the slope ofthe indifference curve at the point (x1, x2).

It measures how much extra of good 2 would be required inreturn for a unit of the 1’st good if the consumer is to remainequally happy.

The MRS decreases when x1 is increased and x2 is keptconstant. Intuitively, the consumer gets “more and more fedup” with the first good and so wants to exchange less and lessof the second good in return for it.

The MRS

β−12 , and so:

MRSCD21 =

αxα−11 xβ2

βxα1 xβ−12

The MRS

β−12 , and so:

MRSCD21 =

αxα−11 xβ2

βxα1 xβ−12

The MRS

β−12 , and so:

MRSCD21 =

αxα−11 xβ2

βxα1 xβ−12

The MRS

The fact that the MRS decreases along indifference curves isreferred to as the principle of the diminishing marginalrate of substitution.

Mathematically, it holds because u is strictly quasi-concave(why?)

The MRS

The fact that the MRS decreases along indifference curves isreferred to as the principle of the diminishing marginalrate of substitution.

Mathematically, it holds because u is strictly quasi-concave(why?)

g30d, week 2socscistaff.bham.ac.uk/jensen/week2(consumertheory-cont)-slides.… · martin k. jensen...

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