econ 201 may 7, 2009 indifference curves budget lines and demand curves 1

Econ 201May 7, 2009

Indifference CurvesBudget Lines

andDemand Curves

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Where Are We Going?

• Indifference (iso-utility) curves can be used with budget lines to provide a more rigorous derivation of consumer demand behavior

• Provide a richer insight into consumer behavior than simple demand curves

• Used to provide a mathematical foundation for analysis/modeling

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What Are Indifference Curves

• An indifference curve in microeconomic theory is a graph showing different bundles of goods, each measured as to quantity, between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another.

• One can equivalently refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer.

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What Do They Look Like?

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Perfect Substitutes

• Goods X and Y are perfect substitutes. The gray line perpendicular

to all curves indicates the curves are mutually parallel.

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Perfect Complements

• Three indifference curves where Goods X and Y are perfect substitutes. The

gray line perpendicular to all curves indicates the curves are mutually parallel.

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Properties of Indifference Curves

• Indifference curves are typically represented to be:• 1. defined only in the positive (+, +) quadrant of commodity-bundle quantities. • 2. negatively sloped. That is, as quantity consumed of one good (X) increases, total

satisfaction would increase if not offset by a decrease in the quantity consumed of the other good (Y). Equivalently, satiation, such that more of either good (or both) is equally preferred to no increase, is excluded. (If utility U = f(x, y), U, in the third dimension, does not have a local maximum for any x and y values.)

• 3. complete, such that all points on an indifference curve are ranked equally preferred and ranked either more or less preferred than every other point not on the curve. So, with (2), no two curves can intersect (otherwise non-satiation would be violated).

• 4. transitive with respect to points on distinct indifference curves. That is, if each point on I2 is (strictly) preferred to each point on I1, and each point on I3 is preferred to each point on I2, each point on I3 is preferred to each point on I1. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.

• 5. (strictly) convex (sagging from below). With (2), convex preferences implies a bulge toward the origin of the indifference curve. As a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep satisfaction unchanged, the substitution effect.

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Simpler Version of the Properties

• Indifference curves are negatively sloped– Graphically illustrates the trade off (in quantities)

required between the amounts of two goods while maintaining the same “utility” (or level of satisfaction)

– Tradeoff Marginal Rate of Substitution• Indifference curves are convex to the origin

– Diminishing marginal utility• As you greater/larger amounts of 1 good (e.g., X), it takes

increasingly greater amounts of Y to keep on the same U curve

• Indifference curves don’t cross (would violate rationality)

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Key Assumptions

• Rationality (called an ordering relationship in a more general mathematical context): Completeness + transitivity. For given preference rankings, the consumer can choose the best bundle(s) consistently among a, b, and c from lowest on up.

• Continuity: This means that you can choose to consume any amount of the good. For example, I could drink 11 mL of soda, or 12 mL, or 132 mL. I am not confined to drinking 2 liters or nothing. See also continuous function in mathematics.

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Why Do They Have That Shape?

• Implications of transitivity and rationality– U(I3) > U(I2) > U(I1) – Concavity gives curves their shape

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Budget Line/ConstraintTwo goods

Consider a world of two goods, called and , which can be purchased in quantities denominated by and , respectively. Let the price of be and the price of be

. Finally, let the income of the consumer be denoted by .

When the consumer purchases quantities and , his total spending is

The budget constraint states that total spending cannot exceed his revenue:

The graphical representation of the budget constraint is the budget line which represents the maximum quantity of the consumer can purchase for any given quantity of

The maximum quantity of that can be purchased (i.e., if ) is . The

maximum quantity of that can be purchased (i.e., if ) is .

When the consumer spends all his income we have

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Graphical Version of the Budget Constraint

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Utility Maximization

• Assume consumer is going to Max U, subject to a budget contraint– Get out to the highest U curve in the opportunity set (or budget)– Which is point C given this budget constraint (blue)

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Deriving the Demand Curve

• Decreasing the price of good Y; while keeping the price of good X constant– BC remains at same pt on x-axis (I/p(x))– BC shifts out on the x-axis; as price

decreases you can buy more Y• Max(Y) = I/p(Y)• P1 > P2 > P3 => Y1 < Y2 < Y3

– Yields greater utility

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Deriving the Demand Curve

• A higher price of A means that less of A can be purchased– budget line moves to the left, intersecting the vertical axis at a lower point.

– Point c is no longer possible and the consumer must move to a new position,

which, assuming utility maximization, will be point b.

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Deriving the Demand Curve

More than 1 price change (3)

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Ordinal vs. Cardinal Utility

Ordinal Utility

Interpersonal comparisons in utility can not be made as scales differ for different individuals

Cardinal

Would imply interpersonal comparisons could be made as scales are absolute, e.g., Centrigade/Farenheit

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