Consumer theory 2.pdf

# Utility maximization and consumer choice This topic explores how consumers make choices to maximise their satisfaction given their budget constraints [5](#page=5). ### 1.1 Core concepts in consumer theory * **Utility:** Represents the satisfaction a consumer derives from consuming goods and services [6](#page=6). * **Marginal Utility (MU):** The additional satisfaction gained from consuming one more unit of a good [6](#page=6). * **Law of Diminishing Marginal Utility:** As a consumer consumes more of a good, the marginal utility they receive from each additional unit decreases [6](#page=6). ### 1.2 The equimarginal principle Consumers maximise their utility by allocating their spending such that the marginal utility per unit of currency spent is equal across all goods. This means the last dollar spent on any good yields the same additional satisfaction [11](#page=11) [6](#page=6) [7](#page=7). The mathematical representation of this principle is: $$ \frac{MU_x}{P_x} = \frac{MU_y}{P_y} $$ where $MU_x$ and $MU_y$ are the marginal utilities of goods X and Y, respectively, and $P_x$ and $P_y$ are their respective prices [11](#page=11) [8](#page=8). ### 1.3 Consumer preferences and indifference curves * **Assumptions about preferences:** Consumers' preferences are assumed to be complete, transitive, non-satiated, and convex [6](#page=6). * **Indifference curve:** A curve that illustrates all the different combinations of two goods that provide a consumer with the same level of utility or satisfaction [6](#page=6). * **Properties of indifference curves:** * Downward-sloping: To maintain the same utility, consuming more of one good requires consuming less of another [6](#page=6). * Never crossing: Different indifference curves represent different levels of utility, so they cannot intersect [6](#page=6). * Higher curves represent higher utility: Bundles on higher indifference curves provide greater satisfaction [6](#page=6). * Convex to the origin: Reflects the law of diminishing marginal utility and the decreasing marginal rate of substitution [6](#page=6). * **Marginal Rate of Substitution (MRS):** The rate at which a consumer is willing to exchange one good for another while maintaining the same level of utility. It is determined by the relative marginal utilities of the two goods. Mathematically, $MRS = \frac{MU_x}{MU_y}$ [11](#page=11) [7](#page=7) [8](#page=8). ### 1.4 The budget constraint The budget constraint represents the limit of a consumer's purchasing power, given their income and the prices of goods. The slope of the budget constraint is determined by the ratio of the prices of the two goods, representing the rate at which the market allows for trade [7](#page=7) [9](#page=9). ### 1.5 Achieving utility maximisation A consumer aims to achieve the highest possible level of utility while staying within their budget constraint. This occurs at the point where the consumer's indifference curve is tangent to their budget constraint [10](#page=10) [11](#page=11) [7](#page=7) [8](#page=8) [9](#page=9). At the optimal consumption bundle: * The bundle must lie on the budget line, meaning it is affordable [9](#page=9). * The bundle must represent the consumer's most preferred combination of goods [9](#page=9). * The consumer's willingness to trade (MRS) equals the market's rate of trade (price ratio) [11](#page=11) [8](#page=8). $$ MRS = \frac{MU_x}{MU_y} = \frac{P_x}{P_y} $$ > **Tip:** The utility-maximising bundle is the highest indifference curve that the consumer can reach given their budget constraint. Any bundle above the budget line is unaffordable, and any affordable bundle below the budget line yields less utility than a bundle on the budget line [10](#page=10). ### 1.6 Illustration of utility maximisation Consider a consumer choosing between food and clothing [10](#page=10). * Bundle E is desirable (high utility) but unaffordable as it lies above the budget line [10](#page=10). * Bundle D is affordable but yields lower utility than other affordable options [10](#page=10). * Bundles A, B, and C are affordable as they lie on the budget line [10](#page=10). * Bundle A is chosen because it lies on the highest possible indifference curve ($U_3$) that is still tangent to the budget constraint, thus providing the highest utility among affordable options [10](#page=10). ### 1.7 Implications of utility maximisation Even if consumers have different preferences between two goods, they will all achieve utility maximisation where their MRS equals the market price ratio ($ \frac{P_x}{P_y} $). This is because they all face the same market prices. This means that while their optimal bundles may differ in composition (e.g., one consumer may prefer more clothing than food compared to another), the underlying economic principle of equating marginal utility per dollar spent holds true for all [12](#page=12). > **Example:** Adam and Ben have equal incomes and face the same prices for food and clothing. Adam prefers food more than clothing, while Ben prefers clothing more than food. As a result, Adam's optimal bundle will contain more food than Ben's, and Ben's optimal bundle will contain more clothing than Adam's. However, both will have allocated their spending such that the last dollar spent on food gives them the same additional utility as the last dollar spent on clothing [12](#page=12). --- # Comparative statics of consumer choice This section examines how consumer choices change in response to variations in income and prices, illustrating the derivation of demand curves and explaining the effects of changes in income on normal and inferior goods. ### 2.1 Linking consumer choice and demand The consumer choice framework allows us to understand the relationship between consumer decisions and individual demand curves. This connection helps explain shifts in prices due to changing tastes, the benefits consumers derive from products, how consumption patterns evolve with wealth, the impact of one good's price on another's demand, and how consumers react to price changes. The focus here will be on variations in prices and income [14](#page=14). ### 2.2 Effect of income changes Changes in income, while holding relative prices constant, shift the budget constraint. An increase in income causes an outward, parallel shift of the budget constraint, whereas a decrease causes an inward, parallel shift. Whether higher income leads to increased consumption depends on the nature of the goods [15](#page=15). #### 2.2.1 Normal goods For normal goods, an increase in income is associated with rising consumption [15](#page=15). > **Example:** If a consumer's income increases, and both premier league tickets and movies are normal goods, the consumer will likely consume more of both. However, premier league tickets might be considered more of a luxury good, leading to a proportionally higher increase in demand compared to movies [16](#page=16). Figure 3 illustrates an income increase for normal goods: an outward shift of the budget constraint from $BC1$ to $BC2$ results in a new optimal consumption bundle at a higher utility level, consuming more of both goods [16](#page=16). #### 2.2.2 Inferior goods For inferior goods, an increase in income is associated with falling consumption [15](#page=15). > **Example:** If a consumer's income increases, and mac and cheese is an inferior good while sirloin steak is a normal good, the consumer will increase their consumption of sirloin steak and decrease their consumption of mac and cheese [17](#page=17). Figure 4 illustrates an income increase with inferior goods: an outward shift of the budget constraint from $BC1$ to $BC2$ leads to a new optimal bundle where consumption of the normal good (sirloin steak) increases, and consumption of the inferior good (mac and cheese) decreases [17](#page=17). ### 2.3 Effect of price changes Changes in the price of a good, while holding income constant, cause the budget constraint to pivot. An increase in the price of a good pivots the budget line inward, reducing the affordable quantity of that good. This pivot occurs around the intercept of the other good, which remains unchanged. Conversely, a decrease in the price of a good pivots the budget line outward, increasing the affordable quantity. These price changes alter the consumer's purchasing power and real income in terms of the goods [18](#page=18). #### 2.3.1 Simultaneous price changes If both prices change in the same direction and by the same amount, the slope of the budget constraint remains the same, but real income in terms of both goods changes. This situation is equivalent to a change in income. For instance, if both prices triple, it's akin to income falling to one-third of its initial level; if both prices halve, it's like income doubling. The resulting impact on consumer choice mirrors that of income changes. The crucial factor is whether relative prices change or not [19](#page=19). ### 2.4 Deriving demand curves from price changes By systematically altering the price of one good while keeping other factors constant, we can trace changes in the budget constraint, determine optimal consumption bundles, map the change in quantity demanded for each price variation, and ultimately derive the demand curve. This highlights a direct link between utility maximization in consumer choice and the resulting individual demand curve [20](#page=20). > **Example:** Consider a scenario where the price of food decreases while the price of clothing remains constant. As the price of food falls, the budget constraint pivots outwards. At each lower price of food, the consumer achieves a higher level of utility by adjusting their consumption bundle. This leads to an increase in the quantity of food demanded at each successively lower price, graphically forming a downward-sloping demand curve for food [21](#page=21). Figure 5 and Figure 6 illustrate how changes in the price of food, leading to outward pivots of the budget constraint ($BC1$ to $BC2$ to $BC3$) and corresponding increases in utility (U1 to U2 to U3), result in increased consumption of food (from $QA_f$ to $QB_f$ to $QC_f$) and consequently derive the downward-sloping demand curve for food [20](#page=20) [21](#page=21). ### 2.5 Key insights from demand curve derivation Several key insights emerge from linking consumer choice to demand curves: * The level of utility achieved by the consumer changes as we move along the demand curve [22](#page=22). * At every point on the demand curve, the consumer is maximizing their utility [22](#page=22). * A lower price for a product is associated with a higher level of total utility for the consumer [22](#page=22). * Conversely, with lower prices, the marginal utility derived from the good decreases [22](#page=22). * This implies that the marginal rate of substitution (MRS) must be decreasing, which is consistent with the relative price going down, as at utility-maximizing points, the MRS equals the relative prices [22](#page=22). ### 2.6 Understanding seemingly counterintuitive demand changes It is possible for the demand for one good to consistently increase (e.g., food as its relative price decreases) while the demand for another good decreases and then increases slightly (e.g., clothing as its relative price increases). This apparent paradox can be explained within the framework of consumer choice and income and substitution effects when relative prices change [23](#page=23). --- # Consumer responses to price changes and decomposition of effects This section explores how consumers alter their purchasing decisions when prices change, breaking down these reactions into substitution and income effects using different analytical methods. ### 3.1 Consumer responses to price changes When the price of a good changes relative to another, two fundamental shifts occur in the consumer's decision-making environment. Firstly, one good becomes relatively more expensive, while the other becomes relatively cheaper, altering the trade-off between them. Secondly, the consumer's overall purchasing power, or real income, is affected [24](#page=24). #### 3.1.1 The substitution effect The substitution effect refers to the change in a consumer's consumption choices that arises solely from a change in the relative prices of goods. This effect is always negative; if the price of one good increases relative to another, the consumer will reduce their consumption of the more expensive good and increase consumption of the relatively cheaper one [24](#page=24). #### 3.1.2 The income effect The income effect represents the change in a consumer's consumption choices resulting from a change in their real income, with relative prices held constant. This effect can be either negative or positive, depending on whether the good in question is a normal good or an inferior good [25](#page=25). #### 3.1.3 Total effect of a price change The total effect of a price change on a consumer's consumption is the sum of the substitution effect and the income effect. It represents the observed net change in consumption of a good following a price alteration [25](#page=25). The relationship can be formally expressed as: $$ \text{Total Effect} = \text{Substitution Effect} + \text{Income Effect} $$ > **Tip:** Remember that "income" in the context of price changes often refers to changes in purchasing power rather than changes in actual earnings [32](#page=32). #### 3.1.4 Example: Clothing demand and food price decrease Consider a scenario where the price of food decreases. This causes the budget constraint to pivot outwards, increasing the set of affordable bundles [26](#page=26). * **From initial point A to a new point B:** The price of food falls. * **Substitution Effect:** Food becomes cheaper, leading consumers to substitute away from clothing and towards food. This effect is negative for clothing consumption [26](#page=26). * **Income Effect (for normal goods):** The decrease in food price increases the consumer's real income. For normal goods, this leads to an increase in demand for both food and clothing [26](#page=26). * If the substitution effect for clothing dominates, its consumption will fall (e.g., $Q_{clothing}^c < Q_{clothing}^a$) [26](#page=26). * **From point B to point C:** As food becomes even cheaper, the income effect strengthens. * If the income effect for clothing now dominates the substitution effect, its consumption will rise (e.g., $Q_{clothing}^c > Q_{clothing}^b$) [26](#page=26). * **Conclusion:** The demand for clothing may initially fall due to substitution effects but then rise as a stronger income effect takes hold [26](#page=26). ### 3.2 Decomposing price effects: Hicks vs. Slutsky Economists use two primary methods to disentangle the substitution and income effects of a price change: the Hicks decomposition and the Slutsky decomposition. Both methods aim to isolate the substitution effect, but they differ in how they conceptualize the income effect [28](#page=28). #### 3.2.1 Hicks decomposition: Holding utility constant The Hicks decomposition aims to isolate the substitution effect by keeping the consumer's utility level constant (#page=28, 26). This is achieved by conceptually shifting the new budget line (after the price change) inwards until it is tangent to the *original* indifference curve. This requires a hypothetical reduction in the consumer's income to neutralize the change in purchasing power, allowing us to observe how consumption would change based purely on the altered relative prices while maintaining the same level of satisfaction. The movement along the original indifference curve represents the pure substitution effect [28](#page=28) [29](#page=29). > **Tip:** The Hicksian method asks: "How would the consumer adjust consumption if they had to maintain their current level of satisfaction but faced the new relative prices?" [28](#page=28). ##### 3.2.1.1 Hicks decomposition for normal goods Starting at an initial optimum (E1, bundle B1) on budget line XY and indifference curve I1. A price fall for good Z pivots the budget line to XZ, leading to a new optimum E2 (bundle B2) on a higher indifference curve I2. The total effect is B2 - B1. To find the substitution effect (SE), the new budget line XZ is shifted inwards to PQ, parallel to XZ, until it is tangent to the original indifference curve I1 at point E3 (bundle B3). This movement from B1 to B3 represents the SE. The income effect (IE) is the remaining part of the total effect, from B3 to B2. For a normal good, both SE and IE contribute positively to the demand for good Z when its price falls [33](#page=33) [34](#page=34). $$ \text{Total Effect} = (\text{B2} - \text{B1}) $$ $$ \text{Substitution Effect} = (\text{B3} - \text{B1}) $$ $$ \text{Income Effect} = (\text{B2} - \text{B3}) $$ ##### 3.2.1.2 Hicks decomposition for inferior goods For an inferior good, the process for isolating the substitution effect is identical to that for normal goods, by returning the consumer to the same utility level before the price change. However, because good Z is now an inferior good, the income effect is negative. While the substitution effect still increases demand for good Z (as it becomes relatively cheaper), the negative income effect decreases demand due to increased purchasing power. The overall effect on demand for good Z remains positive (as the price fell), but it is smaller than for a normal good [37](#page=37). ##### 3.2.1.3 Hicks decomposition for Giffen goods Giffen goods represent a special case where the total price effect is negative, meaning demand falls when the price falls. In Hicksian decomposition, the substitution effect is positive (as the good becomes relatively cheaper). However, for a Giffen good, the income effect is strongly negative and large enough to outweigh the positive substitution effect. Consequently, the overall price effect is negative, leading to a decrease in demand when the price falls [39](#page=39). #### 3.2.2 Slutsky decomposition: Holding purchasing power constant The Slutsky decomposition isolates the substitution effect by keeping the consumer's *purchasing power* constant, specifically in terms of their ability to afford the original consumption bundle (#page=28, 26). This is achieved by shifting the new budget line so that it passes through the original consumption bundle. This hypothetical scenario allows the consumer to still afford their initial purchase at the new relative prices. Any change in consumption from this point is attributed to the substitution effect alone [28](#page=28) [30](#page=30). > **Tip:** The Slutsky method asks: "How would the consumer adjust consumption if they faced the new relative prices but had just enough extra income to purchase their original bundle?" [28](#page=28). ##### 3.2.2.1 Slutsky decomposition for normal goods Starting at the initial optimum E1 (bundle B1) on budget line XY and indifference curve I1. A price fall for good Z pivots the budget line to XZ, leading to a new optimum E2 (bundle B2) on I2. The total effect is B2 - B1. To find the substitution effect (SE), the new budget line XZ is shifted inwards to PQ such that it passes through the original bundle B1 (#page=35, 32). This ensures the original bundle is still affordable at the new relative prices. The new utility-maximizing bundle is found at E3 (bundle B3) on a higher indifference curve I3. The movement from B1 to B3 represents the SE. The income effect (IE) is the remaining change, from B3 to B2. For a normal good, both SE and IE increase demand when the price falls [35](#page=35) [36](#page=36). $$ \text{Substitution Effect} = (\text{B3} - \text{B1}) $$ $$ \text{Income Effect} = (\text{B2} - \text{B3}) $$ $$ \text{Total Effect} = (\text{B2} - \text{B1}) $$ ##### 3.2.2.2 Slutsky decomposition for inferior goods Similar to Hicks' method, the process for isolating the substitution effect under Slutsky's approach involves returning the consumer to a state where they can afford their original bundle at the new relative prices. As good Z is an inferior good, the income effect works in the opposite direction to the substitution effect, reducing demand as purchasing power increases. While the substitution effect increases demand for good Z (due to its lower relative price), the negative income effect decreases it [38](#page=38). ##### 3.2.2.3 Slutsky decomposition for Giffen goods Under Slutsky's method, the Giffen good scenario is also observable. The substitution effect (SE) from a price fall is positive. However, the income effect (IE) is strongly negative and outweighs the substitution effect, such that $IE > SE$. This results in a negative total effect (TE), meaning demand falls when the price of a Giffen good falls [40](#page=40). > **Tip:** It is a valuable exercise to consider how these decompositions change if the price of a good *increases* or if we analyze the good on the y-axis instead of the x-axis [40](#page=40). ### 3.3 Types of goods Understanding different classifications of goods is essential for analyzing price and income effects [31](#page=31). #### 3.3.1 Normal Goods For normal goods, an increase in income leads to an increase in demand for the good. These goods exhibit a positive income effect [31](#page=31). #### 3.3.2 Inferior Goods When income increases, the demand for inferior goods decreases. These goods have a negative income effect, which is typically smaller than the substitution effect [31](#page=31). #### 3.3.3 Giffen Goods Giffen goods are a rare type of inferior good where the negative income effect is so strong that it completely dominates the substitution effect. This leads to an upward-sloping demand curve, where demand for the good rises as its price rises (#page=31, 35) [31](#page=31) [39](#page=39). ### 3.4 Interpretation of "income" in price changes In the context of analyzing consumer responses to price changes, the term "income" often refers to a change in purchasing power rather than an alteration in actual earnings or wages. For instance, a decrease in the price of a good, like food, enhances a consumer's ability to purchase goods and services, even if their nominal income remains unchanged. This conceptualization of income allows for the analysis of income effects by focusing on changes in "real" consumption possibilities [32](#page=32). --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Utility | Utility represents the satisfaction a consumer derives from consuming goods and services. It is a measure of happiness or contentment associated with consumption. | | Marginal Utility (MU) | Marginal utility is the additional satisfaction gained from consuming one more unit of a good or service. It quantifies the incremental benefit of consuming an extra unit. | | Law of Diminishing Marginal Utility | This law states that as a person consumes more units of a good, the additional satisfaction (marginal utility) gained from each subsequent unit tends to decrease. | | Equimarginal Principle | This principle suggests that consumers maximize their utility by allocating their spending such that the marginal utility per unit of currency spent is equal across all goods consumed. Mathematically, it is expressed as $\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$. | | Substitution Effect | The substitution effect refers to the change in consumption of a good resulting from a change in its relative price, with the consumer's utility level held constant. Consumers tend to substitute towards relatively cheaper goods. | | Income Effect | The income effect describes the change in consumption of a good that results from a change in a consumer's purchasing power (real income) due to a price change, with relative prices held constant. This effect can be positive for normal goods and negative for inferior goods. | | Indifference Curve | An indifference curve represents all the combinations of two goods that provide a consumer with the same level of satisfaction or utility. Combinations on higher indifference curves are preferred over those on lower ones. | | Budget Constraint | The budget constraint illustrates all the possible combinations of goods that a consumer can afford given their income and the prices of the goods. It defines the boundary of affordable consumption bundles. | | Marginal Rate of Substitution (MRS) | The marginal rate of substitution is the rate at which a consumer is willing to trade one good for another while maintaining the same level of utility. It represents the slope of the indifference curve. | | Comparative Statics | Comparative statics is an analytical method used to determine how equilibrium quantities or prices change when one of the underlying parameters of a model is changed. In consumer theory, it examines how choices change with income or price variations. | | Normal Goods | Normal goods are those for which demand increases as consumer income rises, and decreases as income falls, assuming prices remain constant. They have a positive income elasticity of demand. | | Inferior Goods | Inferior goods are those for which demand decreases as consumer income rises, and increases as income falls, assuming prices remain constant. They have a negative income elasticity of demand. | | Giffen Goods | Giffen goods are a rare type of inferior good where the income effect is so strong and negative that it outweighs the substitution effect. As a result, the demand for a Giffen good increases when its price rises, leading to an upward-sloping demand curve. | | Hicks Decomposition | The Hicks decomposition is a method to separate the total effect of a price change into substitution and income effects. It holds the consumer's utility level constant by constructing a hypothetical budget line tangent to the original indifference curve at the new relative prices. | | Slutsky Decomposition | The Slutsky decomposition is another method to separate the total effect of a price change into substitution and income effects. It holds the consumer's purchasing power constant by constructing a hypothetical budget line that passes through the original consumption bundle at the new relative prices. |