A systematic method for constructing an indifference curve involves a sequential questioning process to identify bundles of goods that yield the same utility. Starting with an initial bundle, like point A (15 hours of free time, €540), the individual is asked how much of one good (consumption) they would trade for an additional unit of the other (free time). For instance, if Karim is willing to sacrifice €94 for one extra hour of free time, it establishes that he is indifferent between point A and point E (16 hours, €446). This process is repeated for subsequent points (E to F, F to G, etc.) to map out a series of equally satisfying combinations. These points are then plotted and connected to form the curve.

Activity: Constructing an Indifference Curve

The characteristic shape of an indifference curve is generally expected to be both downward-sloping and convex to the origin. The downward slope is a result of the 'more is better' assumption. The convexity, which causes the curve to become flatter as one moves to the right, is a direct result of diminishing marginal rate of substitution (MRS). This principle stems from the plausible assumption that the more of one good an individual possesses, the more willing they are to trade it for the other good. Consequently, the MRS (the absolute value of the slope) decreases as the quantity of the good on the horizontal axis increases. If an indifference curve is expressed as an equation with one good as a function of the other, it forms a convex function.

Shape of an Indifference Curve

An indifference map is a graphical tool that visualizes a person's preferences by displaying a set of indifference curves. While the map shows which combinations of goods yield higher or lower utility by their position on different curves, it provides an incomplete picture of overall preferences. This is because it may not be possible to determine the preferred bundle between two points that lie in the space between the drawn indifference curves.

The Indifference Map

Indifference curves are graphical representations based on a set of standard assumptions regarding an individual's preferences between two goods. These assumptions are what determine the characteristic shape and properties of the curves.

Assumptions about Preferences in Indifference Curve Analysis

Calculus provides multiple methods for analyzing indifference curves. One approach is to rearrange the indifference curve equation to express one good as an explicit function of the other, and then differentiate to find the slope. An alternative and often more general technique is implicit differentiation. This method is applied directly to the utility function and considers how one good (e.g., consumption) must change in response to a small change in the other (e.g., free time) to keep the utility level constant. For a more rigorous mathematical treatment of these techniques—including the use of derivatives to determine properties like quasi-linearity, calculate the MRS, and apply implicit differentiation—readers can consult Sections 14.1, 15.1, 17.1, and 17.3 of 'Mathematics for Economists: An Introductory Textbook' by Malcolm Pemberton and Nicholas Rau (4th ed., 2015 or 5th ed., 2023).

Calculus-Based Methods for Analyzing Indifference Curves

A core principle for standard indifference curves is that individuals prefer having more of a good to less, an assumption known as non-satiation. Consequently, any bundle of goods located above and to the right of an indifference curve is strictly preferred to any bundle on the curve itself. This preference exists because the new bundle contains a greater quantity of both goods, resulting in a higher level of utility.

The 'More is Better' Principle for Indifference Curves

A key similarity between the preferences of Angela (for grain and free time) and those of Karim in Unit 3 is that both are modeled as being convex. For both individuals, this property is represented by indifference curves that slope downward and become progressively flatter as one moves to the right, signifying a diminishing marginal rate of substitution.

Convexity of Angela's and Karim's Preferences

An indifference curve is a graphical representation connecting all combinations ('bundles') of goods that provide an individual with an equal level of utility or satisfaction. Since all points on the curve yield the same satisfaction, the individual is indifferent between them. A key property of indifference maps is that higher curves (those further from the origin) represent greater levels of utility. For instance, in Angela's model, bundles on indifference curves like IC4 or IC5 are preferred to any bundle on a lower curve like IC3.

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An individual is considered indifferent between various combinations, or 'bundles', of goods—like consumption and free time—if each bundle provides them with the exact same level of utility or satisfaction. Consequently, the individual has no preference for one bundle over another. Graphically, all such bundles are located on the same indifference curve.

Indifference Between Consumption-Leisure Bundles

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

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Indifference Curve

The Marginal Rate of Substitution (MRS) quantifies an individual's willingness to trade one good for another while maintaining the same level of utility. While the slope of an indifference curve is mathematically a negative number, the MRS is defined as a positive quantity representing the magnitude of this trade-off. Specifically, the MRS is the absolute value of the slope of the indifference curve. It is common to hear the phrase ‘the MRS is equal to the slope,’ but this is a simplification; what is strictly meant is that the MRS is equal to the absolute value of the slope. For two goods, y (on the vertical axis) and z (on the horizontal axis), the slope is written as $-\frac{dy}{dz}$, making the MRS a positive value.

Marginal Rate of Substitution (MRS)

In his work-leisure model, Karim is indifferent between the bundle at Point A (15 hours of free time, €520 consumption) and the bundle at Point E (16 hours of free time, €430 consumption). This is because both points lie on the same indifference curve, providing him with an equal level of utility. This specific relationship illustrates a trade-off where Karim is willing to sacrifice €90 in consumption for one additional hour of free time.

Indifference between Karim's Bundles A and E

In Karim's work-leisure model, he is indifferent between the bundle at Point H (19 hours of free time, €260 consumption) and the bundle at Point D (20 hours of free time, €240 consumption). This indifference is shown graphically by the fact that both points lie on the same indifference curve. This illustrates a trade-off where Karim is willing to exchange €20 of consumption for one additional hour of free time.

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