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 � on X = R n + 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 p 1 = 1, p 500 = 5 and so on. 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 p 1 = 1, p 500 = 5 and so on. The vector of all prices, p = ( p 1 , . . . , p n ) ∈ R n ++ 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 p 1 = 1, p 500 = 5 and so on. The vector of all prices, p = ( p 1 , . . . , p n ) ∈ R n ++ is called the price vector . If you wish to buy a specific vector of goods x = ( x 1 , . . . , x n ) you will have to pay exactly: n � p 1 x 1 + p 2 x 2 + . . . + p n x n = (1) p i x i i =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 p 1 = 1, p 500 = 5 and so on. The vector of all prices, p = ( p 1 , . . . , p n ) ∈ R n ++ is called the price vector . If you wish to buy a specific vector of goods x = ( x 1 , . . . , x n ) you will have to pay exactly: n � p 1 x 1 + p 2 x 2 + . . . + p n x n = (1) p i x i i =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 ): n � { x ∈ X : p i x i ≤ M } (2) i =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 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 ( x 1 , . . . , x n ) � � i p i x i ≤ M (3) s.t. x i ≥ 0 for i = 1 , . . . , 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 Theorem If M > 0 , p i > 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 , p i > 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 ) = ( x 1 ( p , M ) , . . . , x n ( 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 ( x 1 , x 2 ) = x α 1 x β (4) 2 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 ). 1 This 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 ( x 1 , x 2 ) = x α 1 x β (4) 2 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 : 1 � x α u ( x 1 , x 2 , . . . , x n ) = ( i ) (5) α i where α > 0. 1 This 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 MRS 21 : MRS 21 = u 1 ( x 1 , x 2 ) (6) u 2 ( x 1 , x 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 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 MRS 21 : MRS 21 = u 1 ( x 1 , x 2 ) (6) u 2 ( x 1 , x 2 ) Here u 1 ( x 1 , x 2 ) is the partial derivative of u with respect to x 1 (and likewise u 2 ( x 1 , x 2 ) is the partial derivative w.r.t. x 2 ). 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 D x 1 u ( x 1 , x 2 ) or u ′ 1 ( x 1 , x 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 In the case of a Cobb-Douglas utility function, x β 1 x β − 1 u 1 ( x 1 , x 2 ) = α x α − 1 2 and u 2 ( x 1 , x 2 ) = β x α , and so: 1 2 x β 21 = α x α − 1 = α x 2 MRS CD 1 2 (7) 1 x β − 1 x 1 β x α β 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 In the case of a Cobb-Douglas utility function, x β 1 x β − 1 u 1 ( x 1 , x 2 ) = α x α − 1 2 and u 2 ( x 1 , x 2 ) = β x α , and so: 1 2 x β 21 = α x α − 1 = α x 2 MRS CD 1 2 (7) 1 x β − 1 x 1 β x α β 2 The marginal rate of substitutes is equal to minus the slope of the indifference curve at the point ( x 1 , x 2 ). Martin K. Jensen (U. B’ham) G30D, Week 2
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