Chapter 3 Utilities Indifference curves
Budget line Good 2 150 100 50 Good 1 0 50 100 150 Points on the budget line indicate all the bundles of goods that the consumer can afford.
Indifference Curve Indifference curve: locus of bundles that provide the consumer with the same level of satisfaction
Indifference curves Good 2 (x 2) b w 20 140 a 100 Good 1 (x 1) 10 0 Points on the same indifference curve represent bundles yielding the same amount of utility.
Indifference Curves • Indifference map • Set of indifference curves for a consumer • Every bundle • On an indifference curve • Indifference curves • Farther from origin ->Higher utility
The Shape of Indifference Curves • Indifference curves • Cannot slope upward • Nonsatiation assumption • Cannot cross each other • Transitivity and nonsatiation assumptions • Farther from the origin – higher utility • Nonsatiation assumption • Cannot cross each other • Are bowed in toward the origin • Convexity assumption • Farther from the origin - higher utility
Indifference curves cannot slope upward Good 2 (x 2) x y E B a D C Good 1 (x 1) 0 If an indifference curve ran from a to x, then bundle x would be no better than bundle a despite containing more of both goods. This upward slope of the indifference curve would be a violation of the nonsatiation assumption.
Indifference curves cannot cross each other Good 2 (x 2) b c I2 a I1 Good 1 (x 1) 0 If indifference curves I1 and I2 crossed at a, then by transitivity of preferences bundle b would be no better than bundle c despite containing more of both goods. This crossing of indifference curves would be a violation of the nonsatiation assumption
Farther from the origin -> higher utility Good 2 (x 2) w Bundle w must be preferred to bundle a because it contains more of both goods a Good 1 (x 1) 0
Bowed-in (b) (a) Good 2 (x 2) Good 2 (x 2) c c b b a a 0 0 Good 1 (x 1) Good 1 (x 1) (a) Bowed-out indifference curves violate convexity of preferences. Bundle c is a weighted average of bundles a and b, but yields lower utility level because it is on an indifference curve that is closer to the origin. (b) Bowed-in indifference curves satisfy the convexity of preferences. Bundle c, a weighted average of bundles a and b, yields a higher utility level
The Marginal Rate of Substitution • Marginal rate of substitution (MRS) • Particular point on indifference map • One consumer • Ratio of exchanging goods • Same utility • MRS = - ∆x2 / ∆x1
The Marginal Rate of Substitution • Diminishing marginal rate of substitution • From convexity • Move along the indifference curve • Same utility level • MRS decreases
Convex preferences and the MRS Good 2 (x 2) 100 +∆x2 -∆x1 60 c d a b -∆x1 +∆x2 10 9 I1 Good 1 (x 1) 0 10 20 100 110 As the consumer is given bundles containing more and more of good 2, she values an individual unit of good 2 less and less
Indifference Curves and Tastes • Flat indifference curves • Goods that yield no utility • Straight-line indifference curves • Goods that are perfect substitutes • MRS - constant along an indifference curve • In a two-good world • Indifference curve - straight line
(b) (a) Good 2 (x 2) Good 2 (x 2) 10 9 5 a 4 -∆x1 -∆x1 +∆x2 0 3 8 11 Good 1 (x 1) 0 Good 1 (x 1) +∆x2 (a) Flat indifference curves. The good measured on the horizontal axis is yielding no utility for the consumer. (b) Straight-line indifference curves: perfect substitutes. The same amount of good 2 is always needed to compensate the consumer for the loss of one unit of good 1.
Indifference Curves and Tastes • Right-angle indifference curves • Goods that are perfect complements • Must be consumed in a fixed ratio to produce utility • In a two-good world • Right angle indifference curves • Bowed-out indifference curves • Nonconvex preferences
(d) (c) Good 2 (x 2) Good 2 (x 2) +∆x2 +∆x2 -∆x1 +∆x2 -∆x1 11 I1 -∆x1 10 c b a b a 0 5 6 Good 1 (x 1) 0 Good 1 (x 1) (c) Right-angle indifference curves: perfect complements. Adding any amount of only one good to bundle a yields no additional utility. (d) Bowed-out indifference curves: nonconvex preferences and the MRS. As the consumer is given bundles containing more and more of good 2, he values an individual unit of good 2 more and more.
Perfect substitutes Pepsi 0 Coke Mary’s marginal rate of substitution is constant at any bundle of Pepsi and Coke.
Optimal Consumption Bundle • Optimal consumption bundle • Maximize consumer’s utility • Within the economically feasible set • Best bundle • According to consumer’s preferences • Characteristics of optimal bundles • Indifference curve tangent to budget line • Slope of indifference curve = MRS = -∆x2/∆x1 • Slope of budget line = price ratio = p1/p2 • MRS = p1/p2
The optimal consumption bundle Good 2 (x 2) B +1 -3 -4 +1 k e m n x z F 0 Good 1 (x 1) B’ At the optimal point e, the indifference curve is tangent to the boundary BB’ of the economically feasible consumption set.