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## Preferences

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**Rationality in Economics**• Behavioral Postulate:A decisionmaker always chooses its most preferred alternative from its set of available alternatives. • So to model choice we must model decisionmakers’ preferences.**Indifference Curves**• Take a bundle x’. The set of all bundles equally preferred to x’ is the indifference curve containing x’. • Indifference curve: locus of consumption bundles for which the consumer is indifferent. • Indifference curves cannot cross.**Indifference Curves**x2 x’ ~ x” ~ x”’ x’ x” x”’ x1**Indifference Curves**I1 All bundles in I1 are strictly preferred to all in I2. x2 x z I2 All bundles in I2 are strictly preferred to all in I3. y I3 x1**Slopes of Indifference Curves**• When more of a commodity is always preferred, the commodity is a good. • If every commodity is a good then indifference curves are negatively sloped.**Slopes of Indifference Curves**Good 2 Two goodsa negatively sloped indifference curve. Better Worse Good 1**Slopes of Indifference Curves**• If less of a commodity is always preferred then the commodity is a bad.**Slopes of Indifference Curves**Good 2 One good and onebad a positively sloped indifference curve. Better Worse Bad 1**Slopes of Indifference Curves**Bad 2 Worse Better Good 1**Slopes of Indifference Curves**Bad 2 Two badsa negatively sloped indifference curve. Worse Better Bad 1**Extreme Cases of Indifference Curves: Perfect Substitutes**• If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes and only the total amount (when the substitution rate is 1) of the two commodities in bundles determines their preference rank-order. The consumer is willing to substitute one good for the other at a constant rate.**Extreme Cases of Indifference Curves: Perfect Substitutes**x2 Slopes are constant at - 1. 15 • I2 Bundles in I2 all have a totalof 15 units and are strictly preferred to all bundles in I1, which have a total of only 8 units in them. 8 I1 x1 8 15**Extreme Cases of Indifference Curves: Perfect Complements**• If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements and only the number of pairs of units of the two commodities determines the preference rank-order of bundles.**Extreme Cases of Indifference Curves: Perfect Complements**x2 Each of (5,5), (5,9) and (9,5) contains5 pairs so each is equally preferred. 45o 9 5 I1 x1 5 9**Extreme Cases of Indifference Curves: Perfect Complements**x2 Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is less preferred than the bundle (9,9)which contains 9 pairs. 45o 9 I2 5 I1 x1 5 9**Preferences Exhibiting Satiation**• A bundle strictly preferred to any other is a satiation point or a bliss point. • What do indifference curves look like for preferences exhibiting satiation?**Indifference Curves Exhibiting Satiation**x2 Satiation(bliss)point x1**Indifference Curves Exhibiting Satiation**x2 Better Better Satiation(bliss)point Better x1**Indifference Curves Exhibiting Satiation**x2 Better Better Satiation(bliss)point Better x1**Indifference Curves for Discrete Commodities**• A commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or cheese. • A commodity is discrete if it comes in unit lumps of 1, 2, 3, … and so on; e.g. aircraft, ships and refrigerators.**Indifference Curves for Discrete Commodities**• Suppose commodity 2 is an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference “curves” look like?**Indifference Curves With a Discrete Good**Gas-oline Indifference “curves”are collections ofdiscrete points. Aircraft 0 1 2 3 4**Well-Behaved Preferences**• A preference relation is “well-behaved” if it is • monotonic and convex. • Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good).**Well-Behaved Preferences**• Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the 50-50 mixture of the bundles x and y is at least as preferred as x or y.**Well-Behaved Preferences -- Convexity**x x2 x+y is strictly preferred to both x and y. x2+y2 z = 2 2 y y2 x1+y1 x1 y1 2**Well-Behaved Preferences -- Convexity**x x2 z =(tx1+(1-t)y1, tx2+(1-t)y2) is preferred to x and y for all 0 < t < 1. y y2 x1 y1**Non-Convex Preferences**x2 Better The mixture zis less preferred than x or y. z y2 x1 y1**More Non-Convex Preferences**x2 Better The mixture zis less preferred than x or y. z y2 x1 y1**Slopes of Indifference Curves**• The slope of an indifference curve is its marginal rate-of-substitution (MRS). • How can a MRS be calculated?**Marginal Rate of Substitution**x2 MRS at x’ is the slope of theindifference curve at x’ x’ x1**Marginal Rate of Substitution**x2 MRS at x’ is = dx2/dx1 at x’ x’ Dx2 Dx1 x1**Marginal Rate of Substitution**dx2 = MRS ´ dx1 so, at x’, MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. x2 x’ dx2 dx1 x1**MRS & Ind. Curve Properties**Good 2 Two goodsa negatively sloped indifference curve Better MRS < 0. Worse Good 1**MRS & Ind. Curve Properties**Good 2 One good and onebad a positively sloped indifference curve Better MRS > 0. Worse Bad 1**MRS & Ind. Curve Properties**Good 2 MRS = - 5 MRS always increases with x1 (becomes less negative) if and only if preferences are strictly convex. MRS = - 0.5 Good 1**MRS & Ind. Curve Properties**• Interpretation: as the amount consumed of one good decreases more and more, an increasingly higher amount of the other is required in order to maintain the level of satisfaction. • Or: as the amount consumed of one good increases more and more, a smaller reduction in the consumption of the other is required in order to maintain the level of satisfaction.