Problem 63
Question
Determine whether the demand functions describe complementary or substitute product relationships. Using the notation of Example 4, let \(x_{1}\) and \(x_{2}\) be the demands for products \(p_{1}\) and \(p_{2}\), respectively. $$ x_{1}=150-2 p_{1}-\frac{5}{2} p_{2}, \quad x_{2}=350-\frac{3}{2} p_{1}-3 p_{2} $$
Step-by-Step Solution
Verified Answer
The given demand functions describe complementary product relationships.
1Step 1: Understand the Demand Functions
Firstly, look at the given demand functions. The notation \(x_{1}\) and \(x_{2}\) are the demands for products \(p_{1}\) and \(p_{2}\) respectively. These functions give the demands \(x_{1}\) and \(x_{2}\) in terms of \(p_{1}\) and \(p_{2}\). The terms -2\(p_{1}\) and -\(\frac{5}{2}\) \(p_{2}\) in the equation of \(x_{1}\) and -\(\frac{3}{2}\) \(p_{1}\) and -3 \(p_{2}\) in the equation of \(x_{2}\) reflect how the demand of each product is affected by the prices of the other product.
2Step 2: Identify Substitutes or Complements
From the definition of substitutes and complements, you thus need to check the sign of the coefficient of \(p_{2}\) in the demand function of \(x_{1}\) and similarly, check the sign of the coefficient of \(p_{1}\) in the demand function of \(x_{2}\). If they are positive, they are substitutes. If they are negative, then they are complements.
3Step 3: Check Coefficients
The coefficient of \(p_{2}\) in the function for \(x_{1}\) is -\(\frac{5}{2}\) (negative) and the coefficient of \(p_{1}\) in the function for \(x_{2}\) is -\(\frac{3}{2}\) (negative). These coefficients are negative which means that when the price of the product increases, the demand decreases. This indicates that the products are complementary, not substitutes.
Key Concepts
Demand FunctionsEconomics in AlgebraPrice-Demand Relationship
Demand Functions
Understanding demand functions is crucial for analyzing market dynamics. In economics, a demand function, such as \(x_1 = 150 - 2p_1 - \frac{5}{2}p_2\), represents the relationship between the quantity demanded of a good (in this case \(x_1\)) and the factors affecting it, typically including the good's own price \(p_1\) and the prices of other goods \(p_2\).
Demand functions allow us to predict how consumers will react to changes in prices. If a product's price increases and the demand decreases, as suggested by a negative coefficient in its demand function, we can infer certain behavioral patterns among consumers. In our textbook exercise, the negative coefficients indicate that as the price for one product increases, the demand for that product decreases, which is a typical response in a market.
Algebra is indispensable when working with demand functions, as it allows for manipulation and solving of equations to derive meaningful insights, such as identifying whether two goods are substitutes or complements. The step-by-step solution to the exercise provides a clear example of applying algebra to determine the nature of the relationship between two products.
Demand functions allow us to predict how consumers will react to changes in prices. If a product's price increases and the demand decreases, as suggested by a negative coefficient in its demand function, we can infer certain behavioral patterns among consumers. In our textbook exercise, the negative coefficients indicate that as the price for one product increases, the demand for that product decreases, which is a typical response in a market.
Algebra is indispensable when working with demand functions, as it allows for manipulation and solving of equations to derive meaningful insights, such as identifying whether two goods are substitutes or complements. The step-by-step solution to the exercise provides a clear example of applying algebra to determine the nature of the relationship between two products.
Economics in Algebra
Algebraic expressions in economics, like demand functions, are not just abstract mathematical constructs; they represent real-world economic behaviors and relationships. By using algebra we can simplify complex scenarios into manageable equations.
This approach enables us to visualize the price-demand relationship and the influence of complementary or substitute goods on demand. In the given problem, algebra helps us to identify the nature of goods by analyzing the signs of the coefficients that multiply the prices of other goods. Negative coefficients suggest that the goods under consideration behave as complements.
By mastering algebraic techniques such as solving equations, graphing functions, and interpreting coefficients, students can gain deeper insights into economic concepts and make well-informed predictions about how changes in prices can affect demand.
This approach enables us to visualize the price-demand relationship and the influence of complementary or substitute goods on demand. In the given problem, algebra helps us to identify the nature of goods by analyzing the signs of the coefficients that multiply the prices of other goods. Negative coefficients suggest that the goods under consideration behave as complements.
By mastering algebraic techniques such as solving equations, graphing functions, and interpreting coefficients, students can gain deeper insights into economic concepts and make well-informed predictions about how changes in prices can affect demand.
Price-Demand Relationship
The price-demand relationship is fundamental in economics and is usually depicted through the demand curve, which graphically shows how the quantity demanded of a good or service varies with its price.
Generally, there is an inverse relationship: as the price of a product increases, the quantity demanded tends to decrease, all else being equal. This relationship can be expressed algebraically, and in the context of our exercise, is embodied within the demand functions. The negative signs in front of the prices in the demand equations (\(x_1\) and \(x_2\)) directly reflect this inverse relationship.
It's essential to note how understanding the price-demand relationship can inform strategies around pricing, marketing, and product development. In the case of complementary goods, for instance, a rise in the price of one good can lead to a decrease in demand for both, illustrating an interdependent market behavior that must be carefully considered by businesses and economists alike.
Generally, there is an inverse relationship: as the price of a product increases, the quantity demanded tends to decrease, all else being equal. This relationship can be expressed algebraically, and in the context of our exercise, is embodied within the demand functions. The negative signs in front of the prices in the demand equations (\(x_1\) and \(x_2\)) directly reflect this inverse relationship.
It's essential to note how understanding the price-demand relationship can inform strategies around pricing, marketing, and product development. In the case of complementary goods, for instance, a rise in the price of one good can lead to a decrease in demand for both, illustrating an interdependent market behavior that must be carefully considered by businesses and economists alike.
Other exercises in this chapter
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Determine whether the demand functions describe complementary or substitute product relationships. Using the notation of Example 4, let \(x_{1}\) and \(x_{2}\)
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