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Preferences

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Rationality in Economics u Behavioral Postulate: A decisionmaker always chooses its most preferred alternative from its set of available alternatives. u So to model choice we must model decisionmakers preferences.

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Indifference Curves u Take a bundle x. The set of all bundles equally preferred to x is the indifference curve containing x. u Indifference curve: locus of consumption bundles for which the consumer is indifferent. u Indifference curves cannot cross.

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Indifference Curves x2x2x2x2 x1x1x1x1 x x x x x x

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Indifference Curves x2x2 x1x1 x All bundles in I 1 are strictly preferred to all in I 2. y z All bundles in I 2 are strictly preferred to all in I 3. I1I1 I2I2 I3I3

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Slopes of Indifference Curves u When more of a commodity is always preferred, the commodity is a good. u If every commodity is a good then indifference curves are negatively sloped.

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Slopes of Indifference Curves Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve.

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Slopes of Indifference Curves u If less of a commodity is always preferred then the commodity is a bad.

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Slopes of Indifference Curves Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve.

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Slopes of Indifference Curves Worse Better Bad 2 Good 1

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Slopes of Indifference Curves Bad 2 Worse Better Bad 1 Two bads a negatively sloped indifference curve.

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Extreme Cases of Indifference Curves: Perfect Substitutes u 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.

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Extreme Cases of Indifference Curves: Perfect Substitutes x2x2x2x2 x1x1x1x1 8 8 15 15 Slopes are constant at - 1. I2I2 I1I1 Bundles in I 2 all have a total of 15 units and are strictly preferred to all bundles in I 1, which have a total of only 8 units in them.

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Extreme Cases of Indifference Curves: Perfect Complements u 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.

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Extreme Cases of Indifference Curves: Perfect Complements x2x2x2x2 x1x1x1x1 I1I1 45 o 5 9 59 Each of (5,5), (5,9) and (9,5) contains 5 pairs so each is equally preferred.

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Extreme Cases of Indifference Curves: Perfect Complements x2x2x2x2 x1x1x1x1 I2I2 I1I1 45 o 5 9 59 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.

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Preferences Exhibiting Satiation u A bundle strictly preferred to any other is a satiation point or a bliss point. u What do indifference curves look like for preferences exhibiting satiation?

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Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Satiation (bliss) point

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Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Better Better Better Satiation (bliss) point

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Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Better Better Better Satiation (bliss) point

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Indifference Curves for Discrete Commodities u A commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or cheese. u A commodity is discrete if it comes in unit lumps of 1, 2, 3, … and so on; e.g. aircraft, ships and refrigerators.

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Indifference Curves for Discrete Commodities u Suppose commodity 2 is an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference curves look like?

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Indifference Curves With a Discrete Good Gas- oline Aircraft 01234 Indifference curves are collections of discrete points.

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Well-Behaved Preferences u A preference relation is well- behaved if it is –monotonic and convex. u Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good).

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Well-Behaved Preferences u 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.

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Well-Behaved Preferences -- Convexity x2x2x2x2 y2y2y2y2 x 2 +y 2 2 x1x1x1x1 y1y1y1y1 x 1 +y 1 2 x y z = x+y 2 is strictly preferred to both x and y.

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Well-Behaved Preferences -- Convexity x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 x y z =(tx 1 +(1-t)y 1, tx 2 +(1-t)y 2 ) is preferred to x and y for all 0 < t < 1.

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Non-Convex Preferences x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y.

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More Non-Convex Preferences x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y.

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Slopes of Indifference Curves u The slope of an indifference curve is its marginal rate-of-substitution (MRS). u How can a MRS be calculated?

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Marginal Rate of Substitution x2x2x2x2 x1x1x1x1 x MRS at x is the slope of the indifference curve at x

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Marginal Rate of Substitution x2x2x2x2 x1x1x1x1 MRS at x is = dx 2 /dx 1 at x x 2 x 1 x

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Marginal Rate of Substitution x2x2x2x2 x1x1 dx 2 dx 1 dx 2 = MRS dx 1 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. x

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MRS & Ind. Curve Properties Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve MRS < 0.

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MRS & Ind. Curve Properties Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve MRS > 0.

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MRS & Ind. Curve Properties Good 2 Good 1 MRS = - 5 MRS = - 0.5 MRS always increases with x 1 (becomes less negative) if and only if preferences are strictly convex.

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MRS & Ind. Curve Properties u 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. u 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.

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