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### Transcript of 1. Preferences and Utility

Preferences and Utility(Chapters 3, 4)

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1. 1a.

Preferences A single consumption good

Consider a world with a single consumption good. A corresponding consumption bundle would list the amount of that single good, (x1). If the good is really a good (rather than a bad or a neutral), we know how any individual would feel about two different bundles: the bundle with more of the good would be strictly preferred. We could also define weak preference and indifference. All the notions of preference would depend on concepts of equality and inequality of real numbers. weak preference: , the usual greater than or equal to strict preference: >, the usual strictly greater than indifference: =, the usual equal to

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1b. Two consumption goodsConsider a world with two consumption goods. A corresponding consumption bundle would least the amount of each good, (x1, x2). If both goods are really goods (rather than bads), then if one bundle had more of each of the goods, it would be strictly preferred by any individual. However, if one bundle had more of the first good but less of the second good than the second bundle, then different individuals might rank the two bundles differently. As an analog to the equality and inequality concepts for numbers, we introduce preference concepts for bundles of two (or more) goods.

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weak preference, at least as good as or weakly preferred tostrict preference, preferred to is derived from weak preference: (x1, x2) is strictly preferred to (y1, y2) if it is weakly preferred to (y1, y2), and (y1, y2) is not weakly preferred to it. indifference is derived from weak preference: (x1, x2) is indifferent to (y1, y2) if it is weakly preferred to (y1, y2), and (y1, y2) is weakly preferred to it. Maintained assumption: Unless stated otherwise, we will assume that each of the two goods may be consumed in any nonnegative amount (i.e., quantities need not to be integers).

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2. Economic Assumptions about Preferences* Compare the following assumptions to what you know about inequalities and numbers. 2a. Rationality assumption: weak preferences are complete and transitive complete means any two bundles can be ranked transitive means the ranking has no cycles (i.e., it is not possible to find three bundles such that the first is strictly better than the second, which is strictly better than the third, which is strictly better than the first) The Rationality Assumption implies strict preference and indifference are also transitive. However they are not complete (why?).

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2b.

Continuity assumption:

if we consider a sequence of bundles, each of which is as good as (x1, x2), the limit of the sequence must also be as good as (x1, x2). Similarly, if the bundles in the sequence are no better than (x1, x2) then the limit is no better than (x1, x2). 2c. Monotonicity or nonsatiation assumption

version 1: if (x1, x2) has more of each good than (y1, y2), then it is strictly preferred version 2: if (x1, x2) has more of one good and no less of other good than (y1, y2), then it is strictly preferred.

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3. Indifference CurvesAn indifference curve is a set of bundles all of which are indifferent to each other. An indifference curve is a level curve in the sense that all bundles on the curve have the same level of preference. There is a different indifference curve for each different level of preference. With the assumptions from section 2, distinct indifference curves cannot intersect and movement to northeast leads to preferred indifference curves

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4. Utility Functions Utility functions allow us to examine consumer behavior using calculus rather than set theory. A utility function is a function that assigns numbers to consumption bundles. A utility function u represents preferences u(x1, x2)u(y1, y2) if and only if (x1, x2) is weakly preferred to (y1, y2). If the rationality and continuity assumptions hold, then there exists a utility function representing preferences. If there is a utility function representing preferences, it is not unique. The number assigned matters only in an ordinal sense. This means all that matters is whether ones bundle is higher than that of another bundle. The actual numbers assigned can be positive or negative, large or small. With a utility function, indifference curves are just level curves of the utility function.

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5. Marginal rate of substitution MRS at the point (x1, x2) is (the absolute value of) the slope of the line tangent to the indifference curve at the point (x1, x2).

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If there is a differentiable utility function, then MRS is the ratio of marginal utilities

MRS=U1 (x1, x2)/U2 (x1, x2) where U1 (x1, x2) is the partial derivative of the utility function with respect to the first good and U2 (x1, x2) is the partial derivative of the utility function with respect to the second good.

Generally MRS depends on the point (x1, x2).

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ExampleLet x be the amount of the first good and y be the amount of the second good. Then consumption bundles are pairs of numbers, of the form (x, y). Suppose preferences satisfy assumptions 2a, 2b, and 2c, and that all indifference curves are straight lines with slope -1. Some of the indifference curves are shown in the figure below. We can think of a simple utility function that represents these preferences if we use the equations of the lines. The indifference curve through (1, 0) also goes through (0, 1) and has equation x + y = 1

. The indifference curve through (2, 0) also goes through (0, 2) and has equation x + y = 2. . The indifference curve through (3, 0) also goes through (0, 3) and has equation x + y = 3. Etc.11

Consider as a possible utility functionu(x, y) = x + y. Does this satisfy the requirements? Yes. Comparing two bundles of goods, it assigns a higher number to the bundle on the better indifference curve. This is not the only utility function that represents these preferences. As alternatives we could use u*(x, y) = x + y - 103 (note the fact that u* is sometimes negative is irrelevant) or u**(x, y) = (x + y)2.

Note at any point (x, y), the MRS is 1. (The indifference curves all have slope 1, and the ratio of marginal utilities is 1.)

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7.

Important special classes of preferences

We will consider several important classes of preferences. Each class is made up of many different preferences that have some important common feature. (We will also use these same classes when we discuss production later.) The previous example can be generalized to a special class of preferences, generalized perfect substitutes.

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7a.

The class of (generalized) perfect substitutes

For any member of this class of preferences, all indifference curves are parallel, straight lines. The preferences can be represented by a utility function of the form

u(x, y)=ax + bywhere a and b are positive numbers such that a/b is equal to the slope of the indifference curves. Note that the marginal rate of substitution is constant: MRS = a/b at every point (x, y). Examples: For an individual who thinks all colas are the same, if the goods are liter bottles of Coke and liter bottles of Pepsi, then the MRS is always 1. For most individuals, if the goods are \$5 bills and \$10 bills, then the MRS is always 1/2.14

7b. The class of (generalized) perfect complements It is easiest to start by considering some examples. For most purposes, a left shoe is worthless without the matching right shoe. One of each is required to make a useful object, a pair of shoes. A bicycle frame must be combined with two tires to make a useful object. For some martini drinkers, a reasonable martini requires exact proportions of gin and vermouth.In each example, the ratio of the amounts of the two goods was important. Two right shoes and one left shoe makes just one pair of shoes, and is no better than one right and one left shoe. The corresponding indifference curves are as in the figure below. The figure shows two indifference curves in addition to the axes (solid lines) and a dashed line. Each indifference curve is made up of a vertical and a horizontal segment, with a kink where they meet. The dashed line shows where the kinks would be for other, unshown, indifference curves. It is not an indifference curve.

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The preferences can be represented by a utility function of the formu(x, y) = minimum {x/a, y/b} where a and b are positive numbers and minimum means take the smaller of the two numbers (e.g., the smaller of the number of left shoes and the number of right shoes determines the number of usable pairs of shoes). The ratio a/b is determined by the slope of the dashed line in the figure since the point (x, y) = (a, b) lies on that line (x/a = a/a = 1 = b/b = y/b). Note that the marginal rate of substitution is undefined at the kink and 0 (on the horizontal segment) or (on the vertical segment) elsewhere.

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7c.

The classes of parallel preferences (or quasi-linear utility)

(There are two related classes, one corresponding to vertical shifts and the other to horizontal shifts.) For these preferences, all indifference curves look the same, and are formed by taking one indifference curve and shifting it (vertically for one class, horizontally for the other). An example of the vertical shift case is given in the figure. The preferences are said to be parallel. These preferences can be represented by (quasi-linear) utility functions of the form

u(x, y) = f(x) + y (for the vertical shift case)or u(x, y) = x + g(y) (for the horizontal shift case) The utility function is said to be quasi-linear because in each case one of the goods appears as a linear term while the other appears as the argument in an increasing function.

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