Transcript Player 2
Doctoral Program and Advanced Degree in Sustainable Energy Systems Doctoral Program in Mechanical Engineering Ecological Economics Week 2 Tiago Domingos Assistant Professor Environment and Energy Section Department of Mechanical Engineering Preferences • Notation – Strictly preferred: ( x1 , x2 ) ( y1 , y2 ) – Indifferent: ( x1 , x2 ) ~ ( y1 , y2 ) – Weakly preferred: ( x1 , x2 )( y1 , y2 ) • Assumptions about preferences – Complete: we assume that any two bundles can be compared. That is, given any x-bundle and any y-bundle, we assume that x1, x2 y1, y2 , or y1 , y2 x1 , x2 , or both, in which case the consumer is indifferent between the two bundles. Or in mathematics, a binary relation R over a set X is total if it holds for all a and b in X that a is related to b or b is related to a (or both). a, b X , aRb bRa Violation: extreme situations involving life or death choices where ranking the alternatives might be difficult or even impossible Preferences • Assumptions about preferences – Reflexive: we assume that any bundle is at least as good as itself: x1 , x2 x1 , x2 . A reflexive relation R on set X is one where for all a in X, a is R-related to itself. a X , aRa Violation: Endowment effect The value of a good increases when it becomes part of a persons endowment. Value of a good A = ai Purchase of good A Value of good A = af > ai Loss aversion Preferences • The marginal benefit for the first glass of water is higher than the same amount of diamonds… but the price of diamonds is higher… – Use value: in our daily live, the first glass of water is more important than the diamonds – Exchange value: the price of diamonds is much higher than the price of water (the same amont) because of scarcity – Surplus : difference between use value and exchange value Preferences • Assumptions about preferences – Transitive: if x1 , x2 y1 , y2 and y1 , y2 z1 , z2 , then we assume that x1 , x2 z1 , z2 . In other words, if the consumer thinks that bundle X is at least as good as Y and that Y is at least as good as Z, then the consumer thinks that X is at least as good as Z Violation: Rugby Players A player is chosen over other if two of his characteristics are better Weight Velocity Discipline Player 1 150 Low Average Player 2 120 High Low Player 3 100 Average High Player 1 preferred to Player 2, Player 2 preferred to Player 3 but Player 3 preferred to Player 1 Preferences Strictly preferred Indifferent Weakly preferred Complete Reflexive Transitive Preferences • Indifference versus incomplete – Consider that I am indifferent to basket players, if they have the same height. If I consider only those kind of players, then the relation is always complete. If I consider all the players, including those who have different height, then the relation is not complete because I cannot chose between two players with different height. – Being indifferent is not being incomplete. The fact that the bundle oranges is indifferent to the bundle bananas, does not mean that I cannot chose. Preferences • Assumptions about preferences – Monotonic: we assume that more is better, that is, that we are talking about goods, not bads. More precisely, if x1 , x2 is a bundle of goods and y1 , y2 is a bundle of goods with at least as much of both goods and more of one, then y1 , y2 x1 , x2 – Convex: we assume that averages are preferred to extremes. That is, if we take two bundles of goods x1 , x2 and y1 , y2 on the same indifference curve and take a weighted average of the two bundles such as tx1 1 t x2 , ty1 1 t y2 , 0 t 1 then the average bundle will be at least as good as or strictly preferred to each of the two extreme bundles. (Example: olives and icecream) Preferences • Indiference curves – Graph the set of bundles that are indifferent to some bundle. y1 , y2 – Indifference curves are like contour lines on a map. – Note that indifference curves describing two distinct levels of preferences cannot cross Preferences • Perfect Substitutes Preferences • Perfect Complements Preferences • Bads Preferences • Neutral Preferences • Satiation Preferences • Marginal rate of substitution x2 x1 MRS Utility • Interpreting – Summarizes preferences – A utility function assigns a number to each bundle of goods so that more preferred bundles get higher numbers – that is, u(x1, x2) > u(y1, y2) if and only if (x1, x2) ≻ (y1, y2) – only the ordering of bundles counts, so this is a theory of ordinal utility • Ordinal vs Cardinal utility – The size of the utility difference between two bundles of goods is suposed to have some sort of significance. Bundle U1 U2 U3 A 3 17 -1 B 2 10 -2 C 1 0.002 -3 Utility • Marginal utility (may be called “oportunity cost”) MU 1 U U ; MU 2 x1 x2 MRS MU 1 MU 2 See Varian, Chapter 4 – APPENDIX Utility • Cobb-Douglas preferences u x1 , x2 x1c x2d and x1 , x2 , c, d 0 ln u x1 , x2 c ln x1 d ln x2 See Varian, Chapter 4 – EXAMPLE: Cobb-Douglas Preferences Choice • Geometric solution Choice • Analytical solution – Problem formulation: max U x1 , x2 s.t. p1 x1 p2 x2 m and x1 , x2 0 – It can be solved using Lagrange multipliers: x1 , x2 , U x1 , x2 p1 x1 p2 x2 m x 0 1 MU 1 p1 0 MU 1 MU 2 p2 0 MU 2 p2 0 p1 x2 p x p x m p x p x m 2 2 1 1 1 1 2 2 0 Gives the convertion between money and utility Choice • Analytical solution – It must be noticed that x1 0 x2 0 p1 0 p2 0 – If the Lagrangian solution does not respect it, we have a corner solution given by: x1 0 x2 m p2 x2 0 x1 m p1 Choice • Example: Cobb-Douglas Demand Functions ux1, x2 x1c x2d – Monotonic tranformation: ln u x1, x2 c ln x1 d ln x2 – The problem: max c ln x1 d ln x2 x1 , x2 such that p1 x1 p2 x2 m – The solution: c x cx2 p1 1 cd dx p 1 2 p x p x m x d 2 2 1 1 2 c d m p1 m p2 Demand Effect on demand of a good Change in income Quantity demanded increases with income– normal goods. For some normal goods, the quantity demanded increases more than proportionally with income – luxury goods. For other normal goods, the quantity demanded increases less than proportionally with income – necessary goods. Quantity demanded decreases with an increase in income – inferior good (example: low quality food) Change in own price The quantity demanded for good 1 increases when its price decreases – ordinary good The quantity demanded for good 1 decreases with its price – Giffen good Change in the price of the other good The demand for good 1 increases when the price of good 2 increases – good 1 is a substitute for good 2 The demand for good 1 decreases when the price of good 2 increases – good 1 is a complement to good 2 Assignments • Exercise 4 from Preferences • Exercise 2 from Choice • and the following exercise (2nd Season Exam of 2004-05). Joana likes chocolate cake and icecream, but after eating 10 slices of cake, she is full of it, and eating more chocolate cake leaves her less satisfied. Joana always prefers to eat more ice-cream than less. Joana’s parents allow her to leave aside everything she dislikes eating. Sketch indifference curves that are consistent with this description.