Question

Graphing complements is complicated because a complementary relationship between goods (under Hicks' definition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more) goods. In his review of complementarity, Samuelson provides a way of illustrating the concept with a two-dimensional indifference curve diagram (see the Suggested Readings). To examine this construction, assume there are three goods that a consumer might choose. The quantities of these are denoted by \(x_{1}, x_{2},\) and \(x_{3} .\) Now proceed as follows. a. Draw an indifference curve for \(x_{2}\) and \(x_{3},\) holding the quantity of \(x_{1}\) constant at \(x_{1}^{0} .\) This indifference curve will have the customary convex shape. b. Now draw a second (higher) indifference curve for \(x_{2}, x_{3},\) holding \(x_{1}\) constant at \(x_{1}^{0}-h .\) For this new indifference curve, show the amount of extra \(x_{2}\) that would compensate this person for the loss of \(x_{1} ;\) call this amount \(j .\) Similarly, show that amount of extra \(x_{3}\) that would compensate for the loss of \(x_{1}\) and call this amount \(k\) c. Suppose now that an individual is given both amounts \(j\) and \(k\), thereby permitting him or her to move to an even higher \(x_{2}, x_{3}\) indifference curve. Show this move on your graph, and draw this new indifference curve. d. Samuelson now suggests the following definitions: If the new indifference curve corresponds to the indifference curve when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are independent. If the new indifference curve provides more utility than when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are complements. If the new indifference curve provides less utility than when \(x_{1}=x_{1}^{0}-2 h,\) goods 2 and 3 are substitutes. Show that these graphical definitions are symmetric. e. Discuss how these graphical definitions correspond to Hicks' more mathematical definitions given in the text. f. Looking at your final graph, do you think that this approach fully explains the types of relationships that might exist between \(x_{2}\) and \(x_{3} ?\)

Short answer

Question: Analyze the relationships among three goods (\(x_1, x_2, x_3\)) using a two-dimensional indifference curve diagram, and determine if the graphical definitions of independent, complementary, and substitutive goods correspond to Hicks' mathematical definitions. Discuss the sufficiency of this approach for explaining relationships between \(x_2\) and \(x_3\). Answer: The graphical definitions of independent, complementary, and substitutive goods derived from the indifference curve analysis show that if the new indifference curve corresponds to the indifference curve when \(x_1 = x_1^0 - 2h\), then goods 2 and 3 are independent; if the new indifference curve provides more utility than when \(x_1 = x_1^0 - 2h\), they are complements; and if the new indifference curve provides less utility than when \(x_1 = x_1^0 - 2h\), they are substitutes. These graphical definitions are symmetric and correspond to the mathematical definitions provided by Hicks. However, the approach may have limitations in fully explaining the possible relationships that might exist between \(x_2\) and \(x_3\). Additional investigation, such as examining consumer behavior, preferences, and constraints, may provide a more complete understanding.

Step-by-step solution

(a) Drawing indifference curve for \(x_2\) and \(x_3\) holding \(x_1\) constant

Draw an indifference curve for \(x_2\) and \(x_3\) in a two-dimensional plane, holding the quantity of \(x_1\) constant at \(x_1^0\). This curve will have a convex shape typical of indifference curves.

(b) Drawing a higher indifference curve while holding \(x_1\) constant at \(x_1^0 - h\)

Draw another (higher) indifference curve for \(x_2\) and \(x_3\), this time holding \(x_1\) constant at \(x_1^0 - h\). Show the amount of extra \(x_2\) that would compensate the person for the loss of \(x_1\) (call this amount \(j\)) and the amount of extra \(x_{3}\) that would compensate for the loss of \(x_1\) (call this amount \(k\)).

(c) Showing the move to a higher \(x_2, x_3\) indifference curve after receiving amounts \(j\) and \(k\)

Now assume that the individual is given both amounts \(j\) and \(k\). This allows them to move to an even higher \(x_2, x_3\) indifference curve. Show this move on the graph and draw the new indifference curve.

(d) Defining goods as independent, complements, or substitutes according to Samuelson's definitions

Using the new indifference curve and following Samuelson's definitions: - If the new indifference curve corresponds to the indifference curve when \(x_1 = x_1^0 - 2h\), then goods 2 and 3 are independent. - If the new indifference curve provides more utility than when \(x_1 = x_1^0 - 2h\), then goods 2 and 3 are complements. - If the new indifference curve provides less utility than when \(x_1 = x_1^0 - 2h\), then goods 2 and 3 are substitutes. Show that these graphical definitions are symmetric.

(e) Discussing correspondence between graphical definitions and Hicks' mathematical definitions

Discuss how the graphical definitions of independent, complementary, and substitutive goods derived from the indifference curve analysis correspond to the more mathematical definitions provided by Hicks in the text.

(f) Evaluating the sufficiency of this approach for explaining relationships between \(x_2\) and \(x_3\)

Looking at the final graph, consider whether this approach fully and accurately explains the possible relationships that might exist between \(x_2\) and \(x_3\). Discuss any limitations or potential improvements.

Key concepts

Complementary Goods

Complementary goods are products or services that tend to be used together, meaning the consumption of one enhances the consumption of the other. For instance, smartphones and data plans, or coffee and sugar, are typical examples of complementary goods. In microeconomic theory, when the price of one complementary good rises, the demand for the other tends to decrease, since consumers would derive less satisfaction from the use alone.

Indifference curve analysis helps to visualize this relationship. If an individual has less of one good, they would require more of its complement to maintain the same level of utility. This is depicted on an indifference curve by pinpointing the additional amount of one good needed to compensate for a reduction in another, preserving utility. If the new indifference curve after the adjustment lies above the original, this reflects the complementary nature of the goods in contributing to utility.

Substitute Goods

Substitute goods, in contrast, are goods that can replace each other in consumption, serving a similar purpose for the consumer. Examples include butter and margarine, or tea and coffee. In the context of indifference curves, substitute goods can be identified when a reduction in the quantity of one good can be offset by an equivalent increase in the other, with no loss in overall utility.

Illustrated graphically, if providing a consumer with more of one good does not compensate for a loss of another good, leading to a lower utility level on the new indifference curve, it indicates that the goods are substitutes. As such, the consumer doesn't derive a combined utility from their simultaneous consumption, which differentiates substitutes from complements in indifference curve analysis.

Microeconomic Theory

Microeconomic theory is the branch of economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources. It encompasses various concepts, including demand and supply, elasticity, consumer behavior, and market structures. Indifference curve analysis is a fundamental concept within microeconomic theory that demonstrates consumer preferences.

Through indifference curves, we can analyze how consumers would substitute between goods as prices change. This model assumes rationality, meaning consumers aim to maximize utility. Therefore, understanding how individuals react to changes in availability or the prices of goods (be it complementary or substitutive) is key for businesses, policymakers, and economists to predict market outcomes and consumer behavior. The graphical approach, combined with Hicks' mathematical definitions, forms a comprehensive basis for analyzing consumer choice, though real-world complexities sometimes necessitate more nuanced models.