Consumption- Preferences Representatio Economists make three assumptions about preferences. These are 1. Completeness: Either r y or y > r for all r and y. In other words, all the different bundles can be compared 2. Reflexivity: r>r for all r. In other words, all the bundles are at least as good as themselves. 3. Transitivity: If z>y and y >2 then r> z for all r, y and z. In other words. if a consumer prefers bundle r to bundle y and bundle y to bundle z then the consumer also prefers bundle r to bundle z. At a first glance these do not seem unreasonable. Nevertheless they are assumptions and not facts. Theorem: Any preferences that satisfy the above three assumptions can be represented by a utility functio The utility function is u(). It takes each bundle of goods and a number to it. The theorem states that there is a utility function u()such that r >y if and only if u(r)>u(y). That is, a consumer prefers bundle r to bundle y if and only if they get bigger" utility"from bundle r. The ordering induced by a consumer's preferences be very simply represented by numbers. The numbers u() themselves might have no meaning it is the relative size that matters. Utility functions allow the construction of indiffe Consumption-Preferenees Indifference Curves 1 As the theorem says,“ utility numbers”can to each bundle of goods. This can be graphically represented. Suppose all the bundles with a utility number greater than 5 are found and a line is drawn to separate them from the bundles with a utility number less than 5. This line is called an indifference cu = In the picture above u(r)>5 and u(y)< 5 therefore r>y. Also z> z but more information is required to rank y and z. More indifference curves need to be drawn. Clearly they can be drawn for any utility number With the current assumptions indifference curves can be of various shapes. However, they cannot cross. Why not?Consumption — Preferences 3 Representation • Economists make three assumptions about preferences. These are: 1. Completeness: Either x º y or y º x for all x and y. In other words, all the different bundles can be compared. 2. Reflexivity: x º x for all x. In other words, all the bundles are at least as good as themselves. 3. Transitivity: If x º y and y º z then x º z for all x, y and z. In other words, if a consumer prefers bundle x to bundle y and bundle y to bundle z then the consumer also prefers bundle x to bundle z. • At a first glance these do not seem unreasonable. Nevertheless they are assumptions and not facts. • Theorem: Any preferences that satisfy the above three assumptions can be represented by a utility function. • The utility function is written as u(·). It takes each bundle of goods and assigns a number to it. The theorem states that there is a utility function u(·) such that x º y if and only if u(x) ≥ u(y). That is, a consumer prefers bundle x to bundle y if and only if they get bigger “utility” from bundle x. The ordering induced by a consumer’s preferences can be very simply represented by numbers. • The numbers u(·) themselves might have no meaning — it is the relative size that matters. Utility functions allow the construction of indifference curves. Consumption — Preferences 4 Indifference Curves 1 • As the theorem says, “utility numbers” can be assigned to each bundle of goods. This can be graphically represented. Suppose all the bundles with a utility number greater than 5 are found and a line is drawn to separate them from the bundles with a utility number less than 5. This line is called an indifference curve. ................................................................................................................................................................................................................................................................................ 0 x2 x1 ·x · y · z u = 5 u > 5 u < 5 . .......................................................................................................................... • In the picture above u(x) > 5 and u(y) < 5 therefore x º y. Also x º z but more information is required to rank y and z. More indifference curves need to be drawn. Clearly they can be drawn for any utility number. • With the current assumptions indifference curves can be of various shapes. However, they cannot cross. Why not?