G30D, Week 2 Martin K. Jensen (U. Bham) October 2012 Martin K. - - PowerPoint PPT Presentation

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

G30D, Week 2

Martin K. Jensen (U. B’ham) October 2012

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

1 Utility Representation theorem (from Week 1) 2 The Feasible Set 3 The Consumption Decision 4 Existence and Uniqueness 5 The MRS

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

Theorem (Utility Representation Theorem) Let a preference relation

• n X = Rn

+ satisfy assumptions 1-5 of week 1. Then there exists a

utility representation u : X → R which is a continuous, strongly monotone, and strictly quasi-concave. You can find this result in appendix 2, chapter 2 of GR in a slightly different language. You should read through this appendix, but you will not be required to know the proof.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

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

• n.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

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

• n.

The vector of all prices, p = (p1, . . . , pn) ∈ Rn

++ is called the

price vector.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

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

• n.

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

• i=1

pixi (1)

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

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

• n.

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

• i=1

pixi (1) If you have income M > 0, you can buy any the vector of goods x precisely if it is in the feasible set (a.k.a. the budget set): {x ∈ X :

n

• i=1

pixi ≤ M} (2)

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

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

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

The objective of the consumer in a competitive economy is to choose the consumption vector within her feasible set which yields the highest level of satisfaction. The highest level of satisfaction translates into “on the highest indifference curve” in an indifference diagram. Assume that the consumer has utility function u : X → R.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

The objective of the consumer in a competitive economy is to choose the consumption vector within her feasible set which yields the highest level of satisfaction. The highest level of satisfaction 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) s.t.

i pixi ≤ M

xi ≥ 0 for i = 1, . . . , n (3)

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness 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 then the consumer’s decision problem has exactly one solution for every given price vector p and income level M.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness 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 then the consumer’s decision problem has exactly one solution for every given price vector p and income level M. The solution to the consumer’s decision problem given prices p and income W is denoted by x(p, M) = (x1(p, M), . . . , xn(p, M)). This is the demand function.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

Let n = 2 (two goods), and consider the Cobb-Douglas utility function: u(x1, x2) = xα

1 xβ 2

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

1This is an important result which you should remember: Any function

which is strictly concave is also strictly quasi-concave.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

Let n = 2 (two goods), and consider the Cobb-Douglas utility function: u(x1, x2) = xα

1 xβ 2

(4) where α, β > 0 (if α + β < 1 then u is strictly concave and so also strictly quasi-concave;1 but u is strictly quasi-concave regardless of the sum of α and β). Let n be any natural number, and consider the Constant Elaticity of Substitution (CES) utility function: u(x1, x2, . . . , xn) = (

• i

xα

i )

1 α

(5) where α > 0.

1This is an important result which you should remember: Any function

which is strictly concave is also strictly quasi-concave.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

Both the Cobb-Douglas and CES utility functions are

• differentiable. This makes it possible to derive the marginal

rate 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)

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

Both the Cobb-Douglas and CES utility functions are

• differentiable. This makes it possible to derive the marginal

rate 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). As you can see on p.26 of GR, this is notation of the book. However, you will often find that people prefer one of Dx1u(x1, x2) or u′

1(x1, x2).

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

In the case of a Cobb-Douglas utility function, u1(x1, x2) = αxα−1

1

xβ

2 and u2(x1, x2) = βxα 1 xβ−1 2

, and so: MRSCD

21 = αxα−1 1

xβ

2

βxα

1 xβ−1 2

= α β x2 x1 (7)

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

In the case of a Cobb-Douglas utility function, u1(x1, x2) = αxα−1

1

xβ

2 and u2(x1, x2) = βxα 1 xβ−1 2

, and so: MRSCD

21 = αxα−1 1

xβ

2

βxα

1 xβ−1 2

= α β x2 x1 (7) The marginal rate of substitutes is equal to minus the slope of the indifference curve at the point (x1, x2).

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

In the case of a Cobb-Douglas utility function, u1(x1, x2) = αxα−1

1

xβ

2 and u2(x1, x2) = βxα 1 xβ−1 2

, and so: MRSCD

21 = αxα−1 1

xβ

2

βxα

1 xβ−1 2

= α β x2 x1 (7) The marginal rate of substitutes is equal to minus the slope of the indifference curve at the point (x1, x2). It measures how much extra of good 2 would be required in return for a unit of the 1’st good if the consumer is to remain equally happy.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

In the case of a Cobb-Douglas utility function, u1(x1, x2) = αxα−1

1

xβ

2 and u2(x1, x2) = βxα 1 xβ−1 2

, and so: MRSCD

21 = αxα−1 1

xβ

2

βxα

1 xβ−1 2

= α β x2 x1 (7) The marginal rate of substitutes is equal to minus the slope of the indifference curve at the point (x1, x2). It measures how much extra of good 2 would be required in return for a unit of the 1’st good if the consumer is to remain equally happy. The MRS decreases when x1 is increased and x2 is kept

• constant. Intuitively, the consumer gets “more and more fed

up” with the first good and so wants to exchange less and less

• f the second good in return for it.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

The fact that the MRS decreases along indifference curves is referred to as the principle of the diminishing marginal rate of substitution.

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

Outline Utility Representation theorem (from Week 1) The Feasible Set The Consumption Decision Existence and Uniqueness The MRS

The fact that the MRS decreases along indifference curves is referred to as the principle of the diminishing marginal rate of substitution. Mathematically, it holds because u is strictly quasi-concave (why?)

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