Utility Analysis: Cardinal and Ordinal Approaches

Introduction to Consumer Behaviour

The theory of consumer behaviour is a cornerstone of microeconomics that seeks to understand how an individual consumer makes decisions. The central issue is the problem of choice: a consumer has a limited income and must decide how to spend it on various goods and services to achieve the highest possible level of satisfaction.

A consumer's decision-making process is influenced by two primary factors:

1. Preferences: These are the subjective likes and dislikes of a consumer. They determine how much satisfaction a consumer gets from different combinations of goods, also known as consumption bundles. For simplicity, we often analyze a two-good world (e.g., bananas and mangoes). A bundle is represented as $(x_1, x_2)$, where $x_1$ is the quantity of good 1 and $x_2$ is the quantity of good 2.
2. Budget Constraint: This represents what the consumer can afford. It is determined by the consumer's fixed income and the prevailing market prices of the goods.

To analyze these choices, economists use the concept of utility.

What is Utility?

Utility is the want-satisfying capacity of a commodity. It is a conceptual measure of the satisfaction or happiness a consumer derives from consuming a good or service. The greater the need or desire for a commodity, the higher the utility derived from it.

Utility is a highly subjective concept. This means:

• It varies from person to person. A person who enjoys reading will derive high utility from a book, while another may not.
• It changes with place and time. The utility of a woollen coat is very high in Kashmir during winter but negligible in Chennai during summer.

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Cardinal Utility Analysis

Pioneered by neoclassical economists, this approach assumes that utility is a cardinal concept, meaning it can be measured and expressed in objective, numerical terms. The standard unit of measurement is referred to as a 'util'. For example, one could say, "This apple gives me 10 utils of satisfaction." This approach allows for direct comparison of utility from different goods.

Measures of Utility

• Total Utility (TU): This is the sum total of satisfaction derived from consuming a given quantity of a commodity. It is dependent on the quantity consumed. For example, $TU_4$ is the total utility from consuming 4 units of a good.
• Marginal Utility (MU): This is the additional utility derived from the consumption of one more (or one extra) unit of a commodity. It is the change in total utility resulting from a one-unit change in consumption.

The formula for the marginal utility of the $n^{th}$ unit is:

$MU_n = TU_n - TU_{n-1}$

Where $TU_n$ is the total utility from $n$ units and $TU_{n-1}$ is the total utility from $n-1$ units.

Conversely, Total Utility is the sum of all marginal utilities from each unit consumed:

$TU_n = MU_1 + MU_2 + \dots + MU_n = \sum_{i=1}^{n} MU_i$

Law of Diminishing Marginal Utility

This is a fundamental law of cardinal utility analysis. The Law of Diminishing Marginal Utility states that as a consumer consumes more and more units of a commodity, the marginal utility derived from each successive unit goes on diminishing, assuming the consumption of all other commodities remains constant.

The table below illustrates this law. As more units are consumed, the TU increases, but at a diminishing rate (because MU is falling). TU is maximized when MU becomes zero. After this point, consuming more units leads to negative MU (disutility), causing TU to fall.

Units Consumed	Total Utility (TU)	Marginal Utility (MU)
1	12	12
2	18	6 (18-12)
3	22	4 (22-18)
4	24	2 (24-22)
5	24	0 (24-24)
6	22	-2 (22-24)

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Derivation of Demand Curve in the Case of a Single Commodity

The Law of Demand states that, other things being equal, there is a negative relationship between the price of a commodity and the quantity demanded. Cardinal Utility Analysis, specifically the Law of Diminishing Marginal Utility (DMU), provides a powerful explanation for why the demand curve slopes downwards.

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The Condition for Consumer Equilibrium (Single Commodity Case)

A rational consumer will decide to purchase a unit of a commodity only if the satisfaction (utility) gained from it is at least equal to the satisfaction of the money they have to sacrifice in the form of its price. To make this comparison, we must express both the utility of the commodity and its price in the same units. We do this by expressing the marginal utility of the good in monetary terms.

The key condition for a consumer's equilibrium is:

Marginal Utility of the Good (in monetary terms) = Price of the Good

This can be expressed using the following formula:

$\frac{MU_x}{MU_M} = P_x$

Where,

• $MU_x$ = Marginal Utility of commodity X (in utils).
• $P_x$ = Price of commodity X (in monetary units, e.g., Rupees).
• $MU_M$ = Marginal Utility of Money. This is the utility derived from one unit of currency (e.g., one rupee) and is assumed to be constant for a rational consumer.

The term $\frac{MU_x}{MU_M}$ represents the value of the marginal utility of good X in terms of money. It is the maximum price a consumer is willing to pay for that unit of the commodity.

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Derivation of the Demand Curve

The demand curve can be derived directly from the Law of Diminishing Marginal Utility using the equilibrium condition. The logic is as follows:

1. A consumer will continue to purchase a good as long as the marginal utility they derive from it (in monetary terms) is greater than or equal to the market price ($\frac{MU_x}{MU_M} \ge P_x$).
2. According to the Law of Diminishing Marginal Utility, as the consumer buys more and more units of commodity X, the marginal utility ($MU_x$) from each successive unit keeps falling.
3. Since $MU_M$ is assumed to be constant, the value of the fraction $\frac{MU_x}{MU_M}$ must also fall as more units are consumed.
4. This means the consumer is willing to pay a lower price for each additional unit of the commodity because it provides less additional satisfaction.
5. Therefore, for the consumer to be persuaded to buy more units, the market price ($P_x$) must fall to match the declining value of $\frac{MU_x}{MU_M}$.

This establishes an inverse relationship between the price of the commodity and the quantity demanded. When the price is high, the consumer buys fewer units. As the price falls, the consumer is willing to buy more units. This relationship, when plotted on a graph, gives us a downward-sloping demand curve.

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Tabular and Graphical Illustration

Let's assume a consumer's marginal utility of money ($MU_M$) is constant at 4 utils per rupee ($MU_M = 4$). The marginal utility ($MU_x$) derived from consuming units of commodity X is given in the table below.

Units of Commodity X	Marginal Utility ($MU_x$ in utils)	Marginal Utility in Money Terms ($\frac{MU_x}{MU_M}$ in $\text{₹} \ $)	Price ($P_x$) a Consumer is Willing to Pay (in $\text{₹} \ $)
1	20	20 / 4 = 5	5
2	16	16 / 4 = 4	4
3	12	12 / 4 = 3	3
4	8	8 / 4 = 2	2
5	4	4 / 4 = 1	1

From the table above, we can derive the individual's demand schedule:

Price ($P_x$ in $\text{₹} \ $)	Quantity Demanded ($Q_x$)
5	1
4	2
3	3
2	4
1	5

When we plot this demand schedule on a graph, we get a downward-sloping demand curve.

Thus, the Law of Diminishing Marginal Utility explains the fundamental reason behind the negative slope of the demand curve and the Law of Demand itself.

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Ordinal Utility Analysis

Many economists, notably J.R. Hicks, criticized the cardinal approach, arguing that utility is a psychological phenomenon that cannot be measured numerically. The Ordinal Utility Analysis was developed as an alternative. It is based on a more realistic assumption: a consumer cannot quantify utility but can rank different consumption bundles in order of preference. A consumer can state that they prefer Bundle A to Bundle B, or Bundle B to Bundle A, or are indifferent between them, but they cannot say by how much they prefer one over the other.

Indifference Curve (IC)

An indifference curve is a locus of points representing various bundles of two goods that provide the same level of satisfaction to the consumer. Because every bundle on the curve gives the consumer equal utility, the consumer is said to be "indifferent" among them.

Marginal Rate of Substitution (MRS)

The slope of an indifference curve at any point is known as the Marginal Rate of Substitution (MRS). It measures the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility.

For two goods, X (on the horizontal axis) and Y (on the vertical axis), the MRS is defined as:

$MRS_{xy} = \left| \frac{\Delta Y}{\Delta X} \right|$

It represents the amount of Good Y that the consumer is willing to give up to obtain one more unit of Good X.

Law of Diminishing Marginal Rate of Substitution

This law is the cornerstone of the ordinal approach. It states that as a consumer increases the consumption of Good X, the amount of Good Y they are willing to sacrifice for an additional unit of Good X decreases. This is because as the consumer gets more of Good X, the marginal utility of X falls. Simultaneously, as they have less of Good Y, its marginal utility rises. Consequently, they are willing to give up less and less of the increasingly valuable Good Y. This law is the reason why a standard indifference curve is convex to the origin.

Combination	Quantity of Bananas (Good X)	Quantity of Mangoes (Good Y)	MRS ($\left| \frac{\Delta Y}{\Delta X} \right|$)
A	1	15	-
B	2	12	3 Mangoes : 1 Banana
C	3	10	2 Mangoes : 1 Banana
D	4	9	1 Mango : 1 Banana

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Monotonic Preferences

In ordinal utility analysis, we make a simple and rational assumption about a consumer's preferences called monotonic preferences. This is the formal name for the "more is better" assumption.

A consumer's preferences are said to be monotonic if and only if, between any two consumption bundles, the consumer prefers the bundle that has more of at least one good and no less of the other good.

This implies two things:

• If bundle A $(x_1, x_2)$ and bundle B $(y_1, y_2)$ are such that $x_1 > y_1$ and $x_2 = y_2$, then the consumer will prefer bundle A to bundle B.
• If bundle A has more of both goods, i.e., $x_1 > y_1$ and $x_2 > y_2$, then bundle A is definitely preferred to bundle B.

This assumption guarantees that the marginal utility of any good is always positive (i.e., the consumer is never satiated) and is a key reason behind the fundamental shape and properties of indifference curves.

Answer:

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Shape and Properties of an Indifference Curve

The assumption of monotonic preferences, along with the Law of Diminishing Marginal Rate of Substitution, gives indifference curves their characteristic shape and properties.

1. Indifference Curves are Downward Sloping

An indifference curve must slope downwards from left to right. This is a direct consequence of monotonic preferences. If the curve were upward sloping, a point further up and to the right would represent more of both goods. According to monotonic preferences, this point must be preferred and therefore cannot lie on the same indifference curve. To keep the consumer's satisfaction level constant, an increase in the quantity of one good ($\Delta x_1 > 0$) must be compensated by a decrease in the quantity of the other good ($\Delta x_2 < 0$).

2. Indifference Curves are Convex to the Origin

This is the most common shape of an indifference curve and is caused by the Law of Diminishing Marginal Rate of Substitution (MRS). As a consumer moves down the curve (acquiring more of the good on the X-axis), they are willing to give up less and less of the good on the Y-axis. The MRS continuously falls, which gives the curve its bowed-in, convex shape.

Exception: Indifference Curve for Perfect Substitutes

In the rare case of perfect substitutes—two goods which provide the exact same level of utility (e.g., a five-rupee coin and a five-rupee note)—the MRS is constant. The consumer is always willing to substitute them at the same rate (e.g., 1:1). In this case, the indifference curve is not convex but is a straight line sloping downwards.

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Indifference Map

An indifference map is a collection of indifference curves that represents a consumer's preferences over all possible consumption bundles. It is a complete map of the consumer's tastes. Each curve in the map represents a different level of utility.

Key properties of an indifference map are:

1. Higher Curves, Higher Utility: A higher indifference curve (one further from the origin) represents a higher level of satisfaction than a lower one. This is because any bundle on a higher curve contains more of at least one good, which is preferred due to monotonic preferences. The consumer's objective is to reach the highest possible indifference curve.
2. Curves Never Intersect: Two indifference curves can never cross each other. If they did, it would imply a logical contradiction. The point of intersection would mean that a single bundle provides two different levels of satisfaction, which is impossible. It would also violate the assumption of transitivity of preferences.

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Properties of Indifference Curves

1. Indifference curves slope downwards from left to right: This is a direct consequence of the assumption of monotonic preferences. Monotonic preferences mean that a consumer always prefers a bundle that contains more of at least one good and no less of the other. Therefore, to keep the utility level constant (i.e., to stay on the same IC), if the quantity of one good is increased, the quantity of the other good must be decreased.
2. A higher indifference curve represents a higher level of utility: Due to monotonic preferences, any bundle on a higher IC is superior to any bundle on a lower IC because it represents more of at least one good. A diagram showing a family of indifference curves for a consumer is called an indifference map. The consumer's goal is to reach the highest possible indifference curve.
3. Two indifference curves never intersect each other: If two indifference curves were to intersect, it would violate the principle of transitivity and monotonicity of preferences. For example, if IC1 and IC2 intersect at point A, it would mean A is equivalent to B (on IC1) and A is equivalent to C (on IC2). This would wrongly imply that B is equivalent to C, even though one of the bundles clearly contains more of both goods and lies on a higher indifference curve.

Special Case: Indifference Curve for Perfect Substitutes

If two goods are perfect substitutes (e.g., a ₹5 coin and a ₹5 note), the consumer is willing to exchange them at a constant rate. In this case, the MRS does not diminish; it remains constant. Therefore, the indifference curve for perfect substitutes is a straight line sloping downwards.

The Consumer's Budget: Budget Set and Budget Line

While preferences and indifference curves explain what a consumer wants to consume, they do not account for the reality of limited income and the prices of goods. The consumer's budget represents what the consumer can actually afford. This constraint is fundamental to the problem of choice.

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Budget Set and Budget Line

Let's consider a consumer with a fixed income, say $M$, to be spent on two goods: Good 1 (e.g., bananas) and Good 2 (e.g., mangoes). The market prices for these goods are $p_1$ and $p_2$ per unit, respectively. The consumer can only purchase combinations, or bundles, of these goods that do not exceed their total income.

The Budget Constraint

The expenditure on Good 1 is the price per unit times the quantity purchased ($p_1x_1$). Similarly, the expenditure on Good 2 is $p_2x_2$. The consumer's budget constraint dictates that the total expenditure on both goods cannot be more than their income $M$. This is expressed by the inequality:

$p_1x_1 + p_2x_2 \le M$

The Budget Set

The budget set is the collection of all consumption bundles $(x_1, x_2)$ that the consumer can afford to buy with their income at the given market prices. Graphically, it is the entire triangular area bounded by the two axes and the budget line. Any point within this area or on its boundary is an affordable bundle.

The Budget Line

The budget line is a crucial part of the budget set. It represents all the bundles that cost exactly equal to the consumer's entire income. A rational consumer with monotonic preferences will always choose a bundle on the budget line, as any point below it would mean some income is left unspent, and a more preferred bundle could be afforded. The equation of the budget line is:

$p_1x_1 + p_2x_2 = M$

Answer:

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Price Ratio and the Slope of the Budget Line

To understand the properties of the budget line, we can rearrange its equation into the slope-intercept form ($y = c + mx$):

$p_2x_2 = M - p_1x_1$

$x_2 = \frac{M}{p_2} - \frac{p_1}{p_2}x_1$

From this, we can identify key components:

• Vertical Intercept ($\frac{M}{p_2}$): This is the maximum amount of Good 2 the consumer can buy if they spend their entire income on Good 2 ($x_1=0$).
• Horizontal Intercept ($\frac{M}{p_1}$): This is the maximum amount of Good 1 the consumer can buy if they spend their entire income on Good 1 ($x_2=0$).
• Slope ($-\frac{p_1}{p_2}$): The slope of the budget line is the ratio of the prices of the two goods, known as the price ratio. It is negative because the line is downward sloping. The slope represents the rate at which the market allows the consumer to substitute one good for another. It is the opportunity cost of consuming one more unit of Good 1 in terms of Good 2. For every additional unit of Good 1 the consumer wants to buy, they must give up $\frac{p_1}{p_2}$ units of Good 2.

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Derivation of the Slope of the Budget Line

The slope of a line measures its steepness and represents the rate of change between the vertical and horizontal variables. For the budget line, the slope tells us the rate at which the market allows a consumer to substitute one good for another. It is the opportunity cost of consuming one more unit of the good on the horizontal axis in terms of the good on the vertical axis.

We can derive this slope, which is equal to the negative of the price ratio ($-\frac{p_1}{p_2}$), using two common methods.

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Method 1: Using Two Points on the Budget Line (The Delta Method)

This method formally calculates the slope as the change in the vertical variable ($\Delta x_2$) divided by the change in the horizontal variable ($\Delta x_1$).

1. Start with the Premise: Any bundle $(x_1, x_2)$ that lies on the budget line must satisfy the equation $p_1x_1 + p_2x_2 = M$.
2. Consider Two Points: Let's take any two points on the budget line.
   • Point A: $(x_1, x_2)$
   • Point B: $(x_1 + \Delta x_1, x_2 + \Delta x_2)$, where $\Delta$ (delta) represents a small change.
3. Write the Equation for Each Point: Since both points are on the budget line, they must both satisfy the equation.

   $p_1x_1 + p_2x_2 = M \quad \dots(\text{Equation 1})$

   $p_1(x_1 + \Delta x_1) + p_2(x_2 + \Delta x_2) = M \quad \dots(\text{Equation 2})$

4. Subtract Equation 1 from Equation 2: To find the relationship between the changes ($\Delta x_1$ and $\Delta x_2$), we subtract the first equation from the second.

   $(p_1x_1 + p_1\Delta x_1 + p_2x_2 + p_2\Delta x_2) - (p_1x_1 + p_2x_2) = M - M$

   The terms $p_1x_1$ and $p_2x_2$ cancel out, leaving:

   $p_1\Delta x_1 + p_2\Delta x_2 = 0$

5. Solve for the Slope ($\frac{\Delta x_2}{\Delta x_1}$): Now, we rearrange the equation to isolate the slope.

   $p_2\Delta x_2 = -p_1\Delta x_1$

   $\frac{\Delta x_2}{\Delta x_1} = -\frac{p_1}{p_2}$

This derivation shows that the slope of the budget line is equal to the negative of the ratio of the prices.

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Method 2: Rearranging the Budget Line Equation

This method uses the standard slope-intercept form of a linear equation, $y = mx + c$, where 'm' is the slope and 'c' is the y-intercept.

1. Start with the Budget Line Equation:

   $p_1x_1 + p_2x_2 = M$

2. Solve for the variable on the vertical axis ($x_2$): To get the equation into the desired form, we isolate $x_2$.

   $p_2x_2 = M - p_1x_1$

   $x_2 = \frac{M}{p_2} - \frac{p_1}{p_2}x_1$

3. Compare to the Slope-Intercept Form ($y = mx + c$):
   • The 'y' variable is $x_2$.
   • The 'x' variable is $x_1$.
   • The y-intercept, 'c', is $\frac{M}{p_2}$.
   • The slope, 'm', is $-\frac{p_1}{p_2}$.

This method quickly confirms that the slope of the budget line is indeed $-\frac{p_1}{p_2}$.

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Economic Interpretation of the Slope

The mathematical result has a crucial economic meaning. The slope, $-\frac{p_1}{p_2}$, represents the opportunity cost of consuming Good 1 as determined by the market.

• The negative sign indicates the trade-off. To get more of one good, you must give up some of the other.
• The magnitude $\frac{p_1}{p_2}$ is the price ratio. It tells you exactly how much of Good 2 you must sacrifice to afford one additional unit of Good 1.

Answer:

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Changes in the Budget Set

The set of affordable bundles is not fixed. It changes whenever the consumer's income or the price of either good changes.

1. Change in Income

• Increase in Income: If income ($M$) increases while prices remain constant, the consumer's purchasing power rises. Both the horizontal and vertical intercepts increase. Since the price ratio ($-\frac{p_1}{p_2}$) does not change, the slope remains the same. This results in a parallel outward shift of the budget line.
• Decrease in Income: If income decreases, purchasing power falls. The budget line makes a parallel inward shift. The set of affordable bundles shrinks.

2. Change in Price

• Change in Price of Good 1 ($p_1$): Suppose $p_1$ falls, while $M$ and $p_2$ are unchanged. The vertical intercept ($\frac{M}{p_2}$) remains fixed. However, the horizontal intercept ($\frac{M}{p_1}$) moves to the right as the consumer can now buy more of Good 1. The slope ($-\frac{p_1}{p_2}$) becomes less steep (flatter). This causes the budget line to pivot outwards around the vertical intercept. An increase in $p_1$ would cause the line to become steeper and pivot inwards.

A change in the price of Good 2 would cause a similar pivot around the horizontal intercept.

Optimal Choice of the Consumer: The Equilibrium Point

After understanding the consumer's preferences (indifference map) and constraints (budget line), we can now determine their optimal choice. The consumer's problem is to choose the consumption bundle from their budget set that provides the maximum possible satisfaction.

In economics, we assume the consumer is a rational individual. A rational consumer clearly knows their preferences and, in any given situation, always tries to achieve the best possible outcome for themselves. In graphical terms, the consumer's goal is to reach the highest attainable indifference curve given their budget constraint.

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Equality of the Marginal Rate of Substitution and the Price Ratio

The condition for the consumer's optimum, where the budget line is tangent to an indifference curve, is mathematically expressed as the equality between the Marginal Rate of Substitution (MRS) and the price ratio. This equality is not just a mathematical coincidence; it represents the precise point where a rational consumer finds no further incentive to reallocate their budget, thus maximizing their utility.

Let's break down the two components of this crucial equilibrium condition:

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The Two Sides of the Equilibrium Equation

At the point of equilibrium, the following condition must hold:

$MRS_{xy} = \frac{P_x}{P_y}$

1. The Marginal Rate of Substitution ($MRS_{xy}$)

• What it is: The slope of the indifference curve.
• What it represents: The consumer's subjective or internal valuation of the goods. It is the rate at which the consumer is willing to substitute Good Y for one additional unit of Good X while keeping their satisfaction level constant. It is determined purely by the consumer's personal preferences.
• In terms of Marginal Utility: $MRS_{xy} = \frac{MU_x}{MU_y}$

2. The Price Ratio ($\frac{P_x}{P_y}$)

• What it is: The slope of the budget line.
• What it represents: The market's objective or external valuation of the goods. It is the rate at which the consumer is able to substitute Good Y for Good X in the market. It is determined by market prices and is the same for all consumers.

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Why Equilibrium Occurs Only When the Two Rates are Equal

A rational consumer will continuously adjust their consumption as long as their personal valuation of a good differs from the market's valuation. Equilibrium is only achieved when these two valuations align perfectly. To understand why, let's consider the two cases of disequilibrium.

Assume the price ratio is $\frac{P_x}{P_y} = 2$. This means the market requires the consumer to give up 2 units of Y to get 1 unit of X.

Case 1: Disequilibrium where $MRS_{xy} > \frac{P_x}{P_y}$

Suppose the consumer is at a point where their $MRS_{xy} = 3$.

• Interpretation: The consumer is willing to give up 3 units of Y for 1 unit of X. However, the market only requires them to give up 2 units of Y for 1 unit of X.
• Rational Action: From the consumer's perspective, Good X is a "bargain" in terms of Good Y. They value it more highly than the market does. Therefore, a rational consumer will increase their satisfaction by consuming more of Good X and less of Good Y.
• Movement towards Equilibrium: As the consumer consumes more X and less Y, they move down along their budget line. Due to the Law of Diminishing MRS, their MRS will begin to fall. This process continues until their MRS falls from 3 to 2, at which point $MRS_{xy} = \frac{P_x}{P_y}$, and they have reached the equilibrium point of tangency.

Case 2: Disequilibrium where $MRS_{xy} < \frac{P_x}{P_y}$

Suppose the consumer is at a point where their $MRS_{xy} = 1$.

• Interpretation: The consumer is only willing to give up 1 unit of Y for 1 unit of X. However, the market requires them to give up 2 units of Y for 1 unit of X.
• Rational Action: From the consumer's perspective, Good X is "overpriced" in terms of Good Y. They value it less than the market does. Therefore, a rational consumer will increase their satisfaction by consuming less of Good X and more of Good Y.
• Movement towards Equilibrium: As the consumer consumes less X and more Y, they move up along their budget line. Due to the Law of Diminishing MRS (in reverse), their MRS will begin to rise. This process continues until their MRS rises from 1 to 2, at which point $MRS_{xy} = \frac{P_x}{P_y}$, and they have reached equilibrium.

In conclusion, the equality of the MRS and the price ratio is the logical culmination of a rational consumer's optimization process. It is the only point where the consumer's personal trade-off rate is in sync with the market's trade-off rate, leaving no room for any further beneficial reallocation of their income.

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Locating the Consumer's Equilibrium

To find the optimal bundle, we can superimpose the indifference map onto the budget set. The consumer will choose the bundle that lies on the highest indifference curve that they can afford.

• Points below the budget line are not optimal: For any point inside the budget line (e.g., Point A), there is always some point on the budget line (e.g., Point E) which has more of at least one good. Due to monotonic preferences, this point on the line would be preferred. Thus, a rational consumer will always spend their entire income.
• Points on higher indifference curves are unaffordable: Bundles on indifference curves like IC3 are more desirable, but they lie entirely outside the consumer's budget set and are therefore unattainable.
• Points where the budget line crosses an indifference curve are not optimal: At points like B and D, where the budget line intersects IC1, the consumer can still move to a higher indifference curve (IC2) by reallocating their spending along the budget line.

The optimal choice is therefore the point where the budget line just touches, or is tangent to, the highest possible indifference curve (Point E on IC2). At this point of tangency, the consumer's satisfaction is maximized for the given budget. This point is known as the consumer's equilibrium or consumer's optimum.

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Conditions for Consumer's Equilibrium

At the point of tangency (equilibrium), two crucial conditions are satisfied:

1. The Slope Condition: The slope of the indifference curve must be equal to the slope of the budget line.
2. The Convexity Condition: The indifference curve must be convex to the origin at the point of tangency. This ensures that it is a point of maximum, not minimum, satisfaction.

The slope condition is the primary mathematical condition for equilibrium. We know that:

• The slope of the indifference curve is the Marginal Rate of Substitution ($MRS_{xy}$).
• The slope of the budget line is the Price Ratio ($\frac{P_x}{P_y}$).

Therefore, at equilibrium, the condition is:

$MRS_{xy} = \frac{P_x}{P_y}$

This can also be expressed in terms of marginal utilities as:

$\frac{MU_x}{MU_y} = \frac{P_x}{P_y}$

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The Economic Intuition Behind the Equilibrium Condition

The equilibrium condition $MRS_{xy} = \frac{P_x}{P_y}$ has a powerful economic meaning. It signifies a point where the consumer's subjective valuation of the goods perfectly aligns with the market's objective valuation.

• $MRS_{xy}$ is the rate at which the consumer is willing to trade Good Y for one more unit of Good X. It is their personal, internal trade-off rate.
• $\frac{P_x}{P_y}$ is the rate at which the market allows the consumer to trade Good Y for one more unit of Good X. It is the external, market-determined trade-off rate.

Let's consider why any other point is not an equilibrium:

• Case 1: $MRS > \frac{P_x}{P_y}$ (e.g., at Point B)

  Suppose $MRS=3$ and the price ratio is $2$. This means the consumer is willing to give up 3 units of Y for 1 unit of X, but the market only requires them to give up 2 units of Y. This is a "good deal" for the consumer. A rational consumer will increase their consumption of Good X by sacrificing Good Y. As they do so, due to the Law of Diminishing MRS, their MRS will fall until it becomes equal to the price ratio at the equilibrium point E.

• Case 2: $MRS < \frac{P_x}{P_y}$ (e.g., at Point D)

  Suppose $MRS=1$ and the price ratio is $2$. This means the consumer is only willing to give up 1 unit of Y for 1 unit of X, but the market requires them to give up 2 units of Y. This is a "bad deal". The consumer values Good Y more than the market does at this point. They will increase their satisfaction by consuming less of Good X and more of Good Y. This will cause their MRS to rise until it equals the price ratio at point E.

Thus, the consumer is in a state of rest or equilibrium only when their personal rate of substitution is exactly equal to the market rate of substitution. At this point, there is no incentive to reallocate their income.

Demand and the Law of Demand

What is Demand?

In economics, demand is a multifaceted concept. It is not simply the desire or need for a commodity. To constitute effective demand, a desire must be backed by both the willingness to buy and the ability (purchasing power) to buy. Formally, demand refers to the various quantities of a commodity that a consumer is willing and able to purchase at various possible prices during a specific period of time.

The quantity of a good that a consumer chooses to buy, their optimal choice, is influenced by a set of key variables. The relationship between the quantity demanded and these determinants is known as the demand function. It can be expressed as:

$Q_x = f(P_x, P_r, M, T, E)$

Where:

• $Q_x$ = Quantity demanded of good X.
• $P_x$ = The price of the good X itself.
• $P_r$ = The prices of related goods (which can be substitutes or complements).
• $M$ = The consumer's income.
• $T$ = The consumer's tastes, preferences, and habits.
• $E$ = The consumer's expectations about future prices or income.

The demand function establishes a relationship between the quantity a consumer demands and the factors that influence this decision.

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The Demand Curve and the Law of Demand

To analyze the effect of price on demand, we invoke the crucial assumption of ceteris paribus, a Latin phrase meaning "all other things being held constant." By assuming that income ($M$), prices of related goods ($P_r$), tastes ($T$), and expectations ($E$) are unchanged, we can isolate the relationship between the quantity demanded of a good and its own price ($P_x$). This simplified relationship is what we typically refer to as the demand function, $Q_x = f(P_x)$, and its graphical representation is the demand curve.

The Law of Demand

The Law of Demand is a fundamental principle of economics that describes the inverse relationship between price and quantity demanded. It states that, ceteris paribus, as the price of a commodity increases, the quantity demanded for it falls, and as the price decreases, the quantity demanded for it rises.

This inverse relationship is why the demand curve is typically downward sloping from left to right.

Reasons for the Downward Slope of the Demand Curve

1. Law of Diminishing Marginal Utility (DMU): As a consumer consumes more units of a good, the marginal utility (additional satisfaction) from each successive unit decreases. Therefore, the consumer will only be willing to buy an additional unit if its price is lower. The price a consumer is willing to pay is directly related to the marginal utility they expect to receive.
2. Substitution Effect: When the price of a good falls, it becomes relatively cheaper compared to its substitutes. A rational consumer will substitute the now cheaper good for other, relatively more expensive goods. For example, if the price of coffee falls while the price of tea remains constant, consumers will buy more coffee and less tea. This leads to an increase in the quantity demanded of the cheaper good.
3. Income Effect: A change in the price of a good affects the consumer's real income or purchasing power. When the price of a good falls, the consumer's real income increases, meaning they can afford to buy more of the good with the same money income. For most goods (normal goods), this increase in purchasing power leads to a higher quantity demanded.
4. New Consumers: When the price of a commodity falls, some consumers who were previously unable to afford it can now purchase it. This entry of new buyers into the market increases the total quantity demanded.

For most goods, the substitution and income effects work in the same direction to reinforce the Law of Demand.

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Deriving a Demand Curve from Indifference Curves and Budget Constraints

The downward slope of the demand curve can be formally derived using the principles of ordinal utility analysis. The process involves observing how a consumer's optimal choice changes as the price of one good changes, holding other factors constant. The path traced by these optimal points is called the Price Consumption Curve (PCC).

Let's consider a consumer in equilibrium, consuming two goods, X and Y.

1. Initial Equilibrium: The consumer is initially at equilibrium at point E1, where the budget line BL1 is tangent to the indifference curve IC1. The price of good X is P1, and the consumer purchases Q1 units of it.
2. Price Change: Suppose the price of good X falls from P1 to P2, while income and the price of good Y remain unchanged. This fall in price causes the budget line to pivot outwards to a new position, BL2. It becomes flatter, indicating that the consumer can now buy more of good X.
3. New Equilibrium: The consumer can now reach a higher indifference curve. The new equilibrium is at point E2, where the new budget line BL2 is tangent to a higher indifference curve, IC2. At this new equilibrium, the consumer buys Q2 units of good X.
4. Further Price Change: If the price of X falls further to P3, the budget line pivots out again to BL3, and a new equilibrium is reached at E3 on IC3, with quantity Q3 purchased.

By plotting the price-quantity combinations from each equilibrium point—(P1, Q1), (P2, Q2), and (P3, Q3)—on a separate graph, we can trace out the individual's demand curve. Since a lower price leads to a higher quantity purchased ($P1 > P2 > P3$ implies $Q1 < Q2 < Q3$), the resulting demand curve is downward sloping.

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Normal, Inferior, and Giffen Goods

The relationship between a consumer's income and their demand for a good is a key determinant of the good's nature.

• Normal Goods: These are goods for which there is a positive relationship between income and demand. As a consumer's income rises, they demand more of these goods. The income effect is positive. Examples include high-quality food, branded clothing, and vacations. The demand curve for a normal good shifts rightward with an increase in income.
• Inferior Goods: These are goods for which there is a negative relationship between income and demand. As income rises, consumers buy less of these goods, typically because they switch to better-quality substitutes. The income effect is negative. Examples include coarse cereals (like bajra), travel by public bus instead of a taxi, or second-hand goods. The demand curve for an inferior good shifts leftward with an increase in income.
• Giffen Goods: A Giffen good is a rare and extreme type of inferior good. It is a good where the negative income effect is stronger than the positive substitution effect. When the price of a Giffen good falls, the consumer's real income increases so significantly that they drastically reduce their consumption of this good to buy more of a superior good. This results in the quantity demanded falling as the price falls, leading to an upward-sloping demand curve. This is a theoretical exception to the Law of Demand.

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Substitutes and Complements

The demand for a good is also influenced by the prices of related goods.

• Substitutes: These are goods that can be used in place of each other to satisfy the same want (e.g., tea and coffee, Coke and Pepsi). There is a positive relationship between the demand for a good and the price of its substitute. If the price of coffee rises, consumers will demand more tea. This is reflected as a rightward shift in the demand curve for tea. The cross-price elasticity for substitutes is positive.
• Complements: These are goods that are consumed together to satisfy a particular want (e.g., cars and petrol, pen and ink). There is a negative relationship between the demand for a good and the price of its complement. If the price of petrol rises significantly, the demand for cars is likely to decrease. This is reflected as a leftward shift in the demand curve for cars. The cross-price elasticity for complements is negative.

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Movement Along the Demand Curve vs. Shift in the Demand Curve

It is crucial to distinguish between a "change in quantity demanded" and a "change in demand."

Movement Along the Demand Curve (Change in Quantity Demanded)

This occurs when the quantity of a good purchased changes solely due to a change in its own price, while all other factors remain constant (ceteris paribus).

• A fall in price causes a downward movement along the curve, known as an expansion or extension of demand.
• A rise in price causes an upward movement along the curve, known as a contraction of demand.

Shift in the Demand Curve (Change in Demand)

This occurs when the entire demand curve moves to a new position because of a change in any factor other than the good's own price. This indicates that at any given price, consumers are willing to buy more or less than before.

• A rightward shift is called an increase in demand.
• A leftward shift is called a decrease in demand.

Factors Causing a Shift in the Demand Curve:

1. Change in Income: An increase in income shifts the demand curve rightward for normal goods and leftward for inferior goods.
2. Change in Prices of Related Goods: A rise in the price of a substitute shifts the demand curve rightward. A rise in the price of a complement shifts it leftward.
3. Change in Tastes and Preferences: A favourable change (e.g., a good becomes fashionable) shifts the curve rightward. An unfavourable change (e.g., a health scare related to a product) shifts it leftward.
4. Change in Expectations: If consumers expect the price of a good to rise in the future, they may buy more of it today, shifting the current demand curve rightward.

Market Demand and Price Elasticity of Demand

Market Demand

While individual demand explains the behaviour of a single consumer, market demand represents the aggregate behaviour of all consumers in the market for a particular good. It is defined as the total quantity of a good that all consumers taken together are willing and able to buy at each possible price during a given period.

The market demand curve is derived by the horizontal summation of the individual demand curves of all consumers in the market. This means that at each price, we add up the quantities demanded by every individual to find the total market quantity demanded.

Answer:

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Price Elasticity of Demand ($e_D$)

The Law of Demand tells us the direction of change in quantity demanded when price changes, but it doesn't tell us by how much. The Price Elasticity of Demand ($e_D$) is a quantitative measure that shows the degree of responsiveness or sensitivity of the quantity demanded of a good to a change in its own price. It is defined as the percentage change in quantity demanded divided by the percentage change in price.

Formula and Derivation

The definitional formula is:

$e_D = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}} = \frac{\% \Delta Q_d}{\% \Delta P}$

We can expand this to a more practical formula:

Percentage change in quantity demanded $(\% \Delta Q) = \frac{\text{Change in Quantity}}{\text{Original Quantity}} \times 100 = \frac{\Delta Q}{Q} \times 100$

Percentage change in price $(\% \Delta P) = \frac{\text{Change in Price}}{\text{Original Price}} \times 100 = \frac{\Delta P}{P} \times 100$

Substituting these back into the elasticity formula:

$e_D = \frac{(\Delta Q / Q) \times 100}{(\Delta P / P) \times 100} = \frac{\Delta Q}{Q} \times \frac{P}{\Delta P}$

Rearranging the terms gives us the standard point elasticity formula:

$e_D = \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}$

Since price and quantity demanded are inversely related (due to the Law of Demand), the term $\frac{\Delta Q}{\Delta P}$ is always negative. Consequently, $e_D$ is always a negative number. For ease of comparison, economists usually refer to its absolute value, $|e_D|$.

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Degrees of Price Elasticity of Demand

Based on the value of $|e_D|$, demand can be classified into five categories:

1. Perfectly Inelastic Demand ($|e_D| = 0$): The quantity demanded does not change at all, regardless of the price. The demand curve is a vertical line. This is rare but can apply to life-saving drugs for which there are no substitutes.
2. Inelastic Demand ($|e_D| < 1$): The percentage change in quantity demanded is less than the percentage change in price. Demand is not very responsive. This is typical for necessities like salt, basic food items, and fuel.
3. Unitary Elastic Demand ($|e_D| = 1$): The percentage change in quantity demanded is exactly equal to the percentage change in price. Total expenditure on the good remains constant when its price changes.
4. Elastic Demand ($|e_D| > 1$): The percentage change in quantity demanded is greater than the percentage change in price. Demand is highly responsive to price changes. This is typical for luxuries like foreign holidays, expensive cars, and designer goods.
5. Perfectly Elastic Demand ($|e_D| = \infty$): Consumers are willing to buy an infinite quantity at a specific price, but the quantity demanded drops to zero if the price increases even slightly. The demand curve is a horizontal line. This occurs in perfectly competitive markets.

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Elasticity along a Linear Demand Curve

A common misconception is that the elasticity is the same at all points on a straight-line (linear) demand curve because its slope is constant. This is incorrect. The elasticity of demand varies at every point along a linear demand curve.

Consider a linear demand curve with the equation: $q = a - bp$.

Here, 'a' is the quantity intercept, and '-b' is the slope ($\frac{\Delta q}{\Delta p} = -b$).

Substituting this into the elasticity formula:

$e_D = \frac{\Delta q}{\Delta p} \times \frac{p}{q} = (-b) \times \frac{p}{a-bp} = -\frac{bp}{a-bp}$

This formula shows that the elasticity depends on the price 'p'. As 'p' changes, the value of $e_D$ also changes.

• At the vertical axis (Y-axis): Quantity demanded is zero ($q=0$) and price is at its maximum ($p = a/b$). Here, elasticity is infinite ($|e_D| = \infty$).
• At the horizontal axis (X-axis): Price is zero ($p=0$). Here, elasticity is zero ($|e_D| = 0$).
• At the midpoint of the demand curve: Here, price is $p = a/2b$. Elasticity is unitary ($|e_D| = 1$).
• Above the midpoint: Demand is elastic ($|e_D| > 1$).
• Below the midpoint: Demand is inelastic ($|e_D| < 1$).

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Geometric Measure of Elasticity along a Linear Demand Curve

The elasticity at any point on a linear demand curve can be easily measured using a geometric method, often called the point elasticity method.

The formula is:

$|e_D|$ at a point = $\frac{\text{Length of the Lower Segment of the Demand Curve}}{\text{Length of the Upper Segment of the Demand Curve}}$

Derivation

Consider the linear demand curve AB in the figure. We want to find the elasticity at point D, where price is $p_0$ and quantity is $q_0$.

The elasticity formula is $e_D = \frac{\Delta q}{\Delta p} \times \frac{p_0}{q_0}$.

1. The term $\frac{\Delta q}{\Delta p}$ is the reciprocal of the slope of the demand curve AB. For a small change around point D, this is represented by the ratio $\frac{CD}{CE}$. The slope itself is $\frac{CE}{CD}$, so $\frac{\Delta q}{\Delta p} = \frac{CD}{CE}$.
2. At point D, $p_0 = OD = CE$ and $q_0 = Oq_0 = ED$. (Mistake in NCERT source; $p_0$ should be $Op_0$, $q_0$ should be $Oq_0$. Correcting based on standard convention and the provided diagram logic). Let's use the triangle ratios from the source text for consistency. The source text uses $p_0/q_0$ as $Op_0/Oq_0$.
3. $e_D = \frac{CD}{CE} \times \frac{Op_0}{Oq_0}$.
4. Now consider the triangles $\triangle ECD$ and $\triangle p_0BD$. They are similar triangles. Therefore, the ratio of their sides is equal: $\frac{CD}{CE} = \frac{p_0D}{p_0B}$.
5. Also note that $p_0D = Oq_0$. So, $e_D = \frac{Oq_0}{p_0B} \times \frac{Op_0}{Oq_0} = \frac{Op_0}{p_0B}$.
6. Now, consider the large triangle $\triangle BOA$ and the small triangle $\triangle Bp_0D$. They are also similar. Therefore, the ratio of their corresponding sides is equal: $\frac{Op_0}{p_0B} = \frac{DA}{DB}$.
7. Combining these results, we get: $|e_D| = \frac{DA}{DB}$. This is the ratio of the lower segment (DA) to the upper segment (DB).

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Constant Elasticity Demand Curves

While elasticity varies along a linear demand curve, some non-linear curves exhibit constant elasticity at every point.

• Perfectly Inelastic Demand Curve (Panel a): This is a vertical line, indicating that the quantity demanded remains fixed at $q^*$ regardless of the price. Since $\Delta q$ is always zero, the elasticity is zero ($|e_D|=0$).
• Perfectly Elastic Demand Curve (Panel b): This is a horizontal line, indicating that consumers will buy any amount at price $p^*$, but nothing above it. A tiny change in price causes an infinite change in quantity, so the elasticity is infinite ($|e_D|=\infty$).
• Unitary Elastic Demand Curve (Panel c): This curve takes the shape of a rectangular hyperbola. It has the unique property that the percentage change in quantity is always equal to the percentage change in price. Therefore, the elasticity is one ($|e_D|=1$) at every point on the curve.

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Factors Determining Price Elasticity of Demand for a Good

The elasticity of demand for a good is influenced by several factors:

1. Nature of the Commodity:
   • Necessities: Goods essential for life (e.g., salt, food grains, medicines) have inelastic demand because consumers will buy them even if their prices rise.
   • Luxuries: Goods that are not essential and are associated with comfort and prestige (e.g., sports cars, foreign vacations) have elastic demand. Their purchase can be postponed if prices rise.
2. Availability of Close Substitutes: If a good has many close substitutes (e.g., different brands of soft drinks), its demand will be elastic. A small price increase will cause consumers to switch to the substitutes. If a good has no close substitutes (e.g., gasoline), demand will be inelastic.
3. Proportion of Income Spent: Goods on which a consumer spends a very small fraction of their income (e.g., a matchbox, a newspaper) tend to have inelastic demand. A price change does not significantly impact the consumer's budget. Goods that command a large part of the budget (e.g., a car, housing rent) have more elastic demand.
4. Time Period: Demand tends to be more elastic in the long run than in the short run. In the long run, consumers have more time to adjust their consumption habits and find substitutes.
5. Number of Uses: A commodity with multiple uses (e.g., electricity, milk) will have a more elastic demand. If its price rises, it will be restricted to its most important uses, causing a significant fall in quantity demanded.

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Elasticity and Expenditure

Total Expenditure (or Total Revenue from the seller's perspective) is calculated as Price times Quantity ($TE = P \times Q$). The relationship between a price change and the resulting change in total expenditure is entirely dependent on the price elasticity of demand. This is known as the Total Outlay Method.

The logic is as follows:

• If demand is elastic ($|e_D| > 1$): A price increase will lead to a proportionally larger decrease in quantity demanded. The fall in quantity outweighs the rise in price, causing total expenditure to decrease. Conversely, a price decrease will cause total expenditure to increase. Price and TE move in opposite directions.
• If demand is inelastic ($|e_D| < 1$): A price increase will lead to a proportionally smaller decrease in quantity demanded. The rise in price outweighs the fall in quantity, causing total expenditure to increase. Conversely, a price decrease will cause total expenditure to decrease. Price and TE move in the same direction.
• If demand is unitary elastic ($|e_D| = 1$): A price increase will lead to a proportionally equal decrease in quantity demanded. The two effects exactly offset each other, and total expenditure remains unchanged.

Elasticity	If Price Increases	If Price Decreases
Elastic ($|e_D| > 1$)	Total Expenditure Decreases	Total Expenditure Increases
Inelastic ($|e_D| < 1$)	Total Expenditure Increases	Total Expenditure Decreases
Unitary Elastic ($|e_D| = 1$)	Total Expenditure Remains Constant	Total Expenditure Remains Constant

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Rectangular Hyperbola

A rectangular hyperbola is a specific mathematical curve defined by the equation $xy = c$, where 'c' is a constant. In the context of economics, if the variables are price (p) and quantity (q), the equation becomes $pq = c$.

This equation means that the product of price and quantity is always constant. Since Total Expenditure (TE) is defined as $p \times q$, a demand curve in the shape of a rectangular hyperbola implies that the total expenditure on the good remains constant, no matter what the price is.

As we established in the total outlay method, when total expenditure remains constant following a price change, the demand is unitary elastic. Therefore, a demand curve shaped like a rectangular hyperbola has a price elasticity of demand equal to 1 at every point on the curve.

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Relationship between Elasticity and Change in Expenditure on a Good (Formal Derivation)

We can formally prove the relationship between elasticity and expenditure.

Let initial expenditure be $E_1 = pq$.

Let the price change to $p + \Delta p$ and quantity change to $q + \Delta q$.

The new expenditure is $E_2 = (p + \Delta p)(q + \Delta q)$.

The change in expenditure, $\Delta E = E_2 - E_1 = (p + \Delta p)(q + \Delta q) - pq$

$\Delta E = pq + p\Delta q + q\Delta p + \Delta p \Delta q - pq$

$\Delta E = p\Delta q + q\Delta p + \Delta p \Delta q$

For very small changes in price and quantity, the term $\Delta p \Delta q$ is negligible and can be ignored. So, the approximate change in expenditure is:

$\Delta E \approx p\Delta q + q\Delta p$

Factoring out $\Delta p$:

$\Delta E = \Delta p (p\frac{\Delta q}{\Delta p} + q)$

Now, factor out $q$ from the term in the parentheses:

$\Delta E = \Delta p \left[q \left(\frac{p}{q}\frac{\Delta q}{\Delta p} + 1\right)\right]$

We know that the price elasticity of demand is $e_D = \frac{\Delta q}{\Delta p} \frac{p}{q}$. Substituting this in:

$\Delta E = \Delta p [q(1 + e_D)]$

This equation shows:

• If demand is elastic ($e_D < -1$, so $|e_D|>1$), then $(1+e_D)$ is negative. Thus, $\Delta E$ and $\Delta p$ will have opposite signs. (If price rises, expenditure falls).
• If demand is inelastic ($ -1 < e_D < 0$, so $|e_D| < 1$), then $(1+e_D)$ is positive. Thus, $\Delta E$ and $\Delta p$ will have the same sign. (If price rises, expenditure rises).
• If demand is unitary elastic ($e_D = -1$, so $|e_D|=1$), then $(1+e_D)$ is zero. Thus, $\Delta E = 0$. (Expenditure does not change).

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