When observing an indifference map, the vertical space between adjacent curves may seem to increase as one moves horizontally to the right. This perceived widening is a common optical illusion. The actual vertical distance between the curves does not necessarily change in this way, a fact that can be confirmed by direct measurement on the graph.

Optical Illusion of Widening Vertical Distance Between Indifference Curves

A person is described as having convex preferences if their indifference curves are convex, meaning they bow inwards towards the origin. This shape, which becomes progressively flatter when moving to the right, reflects a diminishing marginal rate of substitution (MRS). The underlying reason is that when an individual has an abundance of one good compared to another, they are more willing to trade a larger amount of the abundant good for a single unit of the scarcer good. As a result, the MRS—the absolute value of the indifference curve's slope—declines as one moves along the curve. This preference structure also means that an individual will prefer a 'mixture' or a weighted average of two bundles over either of the original bundles, if both provide the same utility.

Convex Preferences

While the convex shape of indifference curves is a common feature reflecting a preference for smoothing, individuals' specific time preferences can differ, even with the same endowment. This variation in impatience is shown by the different slopes of their indifference curves at a given point. A steeper indifference curve indicates greater impatience, as the individual requires more future compensation for a present sacrifice. [9] Conversely, a shallower curve suggests greater patience.

Comparing Individual Time Preferences via Indifference Curve Slopes

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.

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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.

Indifference Curve

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

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)

The principle of diminishing marginal rate of substitution (MRS) posits that as an individual has more of one good relative to another, their willingness to trade away a unit of the scarcer good for an additional unit of the abundant good decreases. This phenomenon is a direct result of diminishing marginal utility; as the quantity of a good increases, the extra satisfaction (marginal utility) gained from one more unit falls. Consequently, the individual values the abundant good less at the margin, leading to a lower MRS and flatter indifference curves. This characteristic is what gives indifference curves their typical convex shape.

Diminishing Marginal Rate of Substitution

The 'more is better' principle implies that utility rises with an increase in desirable goods like free time ($t$) and consumption ($c$). This is mathematically expressed by requiring the partial derivatives of the utility function for both goods to be positive. Therefore, for the utility function to be increasing, the marginal utility of free time ($\frac{\partial U}{\partial t}$) and the marginal utility of consumption ($\frac{\partial U}{\partial c}$) must be greater than zero. When these conditions are met, it is said that utility 'depends positively' on both free time and consumption.

Condition for an Increasing Utility Function

A convex decreasing function is characterized by a slope that, while always negative, increases (becomes less negative) as the input variable `x` increases. This causes the function's curve to become progressively flatter. This characteristic corresponds to a positive second derivative ($f''(x) > 0$). In economics, this is the standard mathematical representation for the shape of most indifference curves.

Convex Decreasing Function

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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

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

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

A vertical line on a work-leisure graph identifies all bundles with the same amount of free time but different levels of consumption. This graphical tool is used for two main analyses. First, it demonstrates the 'more is better' principle, as higher points on the line are preferred due to greater consumption. Second, it shows how the Marginal Rate of Substitution (MRS) changes with consumption levels. As one moves up this vertical line, the indifference curves intersecting it become progressively steeper, indicating that the MRS increases with higher levels of consumption for a fixed amount of free time.

Comparing Bundles and MRS Along a Vertical Line

There is a close relationship between the marginal utilities of different goods and the Marginal Rate of Substitution (MRS) between them. The MRS, which reflects a consumer's willingness to trade one good for another, is determined by the ratio of the goods' marginal utilities.

Marginal Utility and the Marginal Rate of Substitution

A fundamental principle that applies to any utility function is that the Marginal Rate of Substitution (MRS) is equivalent to the ratio of the marginal utilities of the two goods in question. For two goods, this relationship is expressed by a formula where the MRS equals the marginal utility of the good on the horizontal axis divided by the marginal utility of the good on the vertical axis. This principle holds true universally across different types of utility functions.

Marginal Rate of Substitution as the Ratio of Marginal Utilities

The steepness of an indifference curve at a particular point directly reflects an individual's valuation of one good relative to another. A steeper curve corresponds to a higher Marginal Rate of Substitution (MRS), which is the absolute value of the slope. This high MRS indicates that the individual places a greater relative value on the good measured on the horizontal axis. For example, for a choice between free time and grain, a steeper indifference curve shows that a person values an additional unit of free time more highly compared to an additional unit of grain.

Steeper Indifference Curve and Higher Marginal Rate of Substitution

In the context of solving a constrained choice problem with calculus, the Marginal Rate of Substitution (MRS) can be represented by the derivative of the utility component associated with a specific good. For instance, if utility from free time $t$ is given by a function $v(t)$, the MRS can be defined as $MRS = v'(t)$. This formulation is used alongside the derivative for the MRT to establish the first-order condition for optimization, $MRS = MRT$.

MRS as a Derivative of a Utility Component Function

Angela's optimal choice, which maximizes her utility, is at point A, corresponding to 16 hours of free time and 46 bushels of grain. At this point, the two critical trade-offs she faces are perfectly balanced. The trade-off she is subjectively willing to make between grain and free time, her Marginal Rate of Substitution (MRS), becomes equal to the objective trade-off she is constrained to make by her production technology, the Marginal Rate of Transformation (MRT). This condition (MRS = MRT) signifies the point of tangency between her feasible frontier and the highest possible indifference curve, as depicted in Figure 5.7.

Angela's Optimal Choice (Point A) where MRS = MRT

A horizontal line on a graph of two goods shows bundles with a constant amount of the good on the y-axis but increasing amounts of the good on the x-axis (e.g., free time). As one moves rightward along this line, the indifference curves become flatter, signifying a decrease in the Marginal Rate of Substitution (MRS). The underlying reason is that as the good on the horizontal axis becomes more abundant, its marginal utility diminishes relative to the scarcer good on the vertical axis. Consequently, the individual becomes less willing to sacrifice the scarce good for an additional unit of the abundant one.

Abundance, Marginal Utility, and MRS Along a Horizontal Line

Quasi-linear preferences represent a specific preference structure that serves as a valuable analytical simplification, despite being a restrictive assumption. Their primary utility in economic modeling is that they permit the measurement of benefits and costs (utility) in monetary terms. This structure is defined by an individual's desire for one good being independent of the quantity of another good they possess. Graphically, this results in indifference curves that are vertical translations of each other, meaning their slopes are identical for any given amount of the good on the horizontal axis. Consequently, the Marginal Rate of Substitution (MRS) is determined solely by the quantity of the good on the horizontal axis.

Quasi-linear Preferences

The marginal rate of substitution (MRS) at any given point on an indifference curve reflects the relative scarcity of the two goods. When a good is scarce relative to another, such as having low present consumption compared to high future consumption, an individual places a higher value on an additional unit of it. This higher valuation means they are willing to trade a larger quantity of the abundant good for the scarce one, resulting in a high MRS and a steep indifference curve at that point. As the individual moves along the curve and the initially scarce good becomes more plentiful, the MRS diminishes and the curve flattens.

Relationship between Relative Scarcity and the Marginal Rate of Substitution

In the Browneville model, the shape of citizens' indifference curves is a direct consequence of the diminishing marginal utility (MU) of wages. Because the Marginal Rate of Substitution (MRS) between environmental quality and wages is equal to the marginal utility of wages, the fact that MU diminishes as wages rise means the MRS must also diminish. This diminishing MRS is represented graphically by the indifference curves becoming progressively flatter as wages increase, resulting in their characteristic convex shape.

Diminishing Marginal Utility of Wages and the Shape of Indifference Curves in the Browneville Model

For a Cobb-Douglas utility function, $u(t,c) = t^\alpha c^\beta$, the assumption that the parameters $\alpha$ and $\beta$ are positive is what ensures the 'more is better' principle is satisfied. Given that consumption ($c$) and free time ($t$) are positive, positive values for $\alpha$ and $\beta$ guarantee that the marginal utilities for both goods, $\frac{\partial u}{\partial t}$ and $\frac{\partial u}{\partial c}$, will also be positive. This means that utility will always increase with an increase in either free time or consumption.

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