Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a survey it was determined that the demand equation for VCRs is given by $$x=f(p, q)=10,000-10 p-e^{0.5 q}$$ The demand equation for blank VCR tapes is given by $$y=g(p, q)=50,000-4000 q-10 p$$ where \(p\) and \(q\) denote the unit prices, respectively, and \(x\) and \(y\) denote the number of VCRs and the number of blank VCR tapes demanded each week. Determine whether these two products are substitute, complementary, or neither.

Step-by-Step Solution

Answer

The cross-price elasticity of demand for VCRs and blank VCR tapes is given by: $$E_x_y = \frac{40000 + 5e^{0.5q}}{(10,000-10 p-e^{0.5 q}) × (50,000-4000 q-10 p)}$$ Since the term \(5e^{0.5q}\) is always positive (and therefore E_x_y is always positive), we can conclude that VCRs and blank VCR tapes are substitute products. The demand for one product increases as the price of the other product increases.

Step 1: Compute the partial derivatives of the demand functions

For both demand equations, calculate the four partial derivatives with respect to p and q. We will need them in the cross-price elasticity formula. \(\frac{\partial x}{\partial p} = -10\) \(\frac{\partial x}{\partial q} = -\frac{1}{2}e^{0.5q}\) \(\frac{\partial y}{\partial p} = -10\) \(\frac{\partial y}{\partial q} = -4000\)

Step 2: Calculate the cross-price elasticity of demand

Now we can plug these partial derivatives into the formula for the cross-price elasticity of demand: $$E_x_y = \frac{(-10) × (-4000) - (-\frac{1}{2}e^{0.5q}) × (-10)}{(10,000-10 p-e^{0.5 q}) × (50,000-4000 q-10 p)}$$ Simplifying the expression yields: $$E_x_y = \frac{40000 + 5e^{0.5q}}{(10,000-10 p-e^{0.5 q}) × (50,000-4000 q-10 p)}$$

Step 3: Analyze the cross-price elasticity

From the cross-price elasticity formula, we can see that the sign of E_x_y depends on the term \(5e^{0.5q}\), which is always positive (since q is non-negative). Thus E_x_y is always positive regardless of the values of p and q. Since E_x_y is positive, we can conclude that VCRs and blank VCR tapes are substitute products. The demand for one product increases as the price of the other product increases.

Key Concepts

Partial derivatives

Partial derivatives are essential tools in calculus, used to measure how a function changes as one variable changes while others are held constant. In the context of economics, and particularly in demand equations, partial derivatives can help us understand how the quantity demanded of a product changes as its price or the price of another product changes.
For instance, in the given demand equations, we have demanded functions for two products: VCRs and VCR tapes. To analyze how sensitive the quantity demanded is to changes in prices, we compute partial derivatives with respect to each price. The partial derivative of VCRs with respect to its own price \(p\) is \(-10\), indicating that for a unit increase in the price of VCRs, the demand decreases by 10 units, all else being equal.
Understanding these derivatives gives us a mathematical lens to see how prices influence demand, setting the stage to further explore relationships between different goods, such as substitutes and complements.

Demand equation

A demand equation is a mathematical function representing the relationship between the quantity demanded of a good and various factors affecting this demand, primarily prices. It typically takes the form of a formula where demand quantity is expressed as a function of price and other relevant variables. This helps us predict consumer behavior under different price scenarios.
In the example, we have two demand equations: one for VCRs and one for blank VCR tapes, expressed in terms of their unit prices \(p\) and \(q\). These demand equations not only encapsulate the direct effect of changes in the own price of a good but also facilitate the analysis of cross-price effects.
This kind of analysis is crucial for understanding market dynamics, informing pricing strategies, and gauging potential impacts on demand from pricing decisions.

Substitute goods

Substitute goods are products that can replace each other in consumption. When the price of one good increases, the demand for its substitute generally rises, as consumers switch to the relatively cheaper alternative. This is quantified using the cross-price elasticity of demand, a key indicator that helps in determining the nature of goods.
In the example, VCRs and blank VCR tapes are identified as substitutes because the cross-price elasticity \(E_x_y\) is positive. This indicates that an increase in the price of VCRs leads to an increase in demand for VCR tapes, and vice versa.
Understanding substitutability is significant for businesses as it impacts marketing strategies, pricing, and competitive positioning.

Complementary goods

Complementary goods are products that are typically consumed together, thus one good's demand is directly related to the price of its complement. When the price of one complementary good rises, leading to a decrease in its demand, the demand for the other good also tends to decrease because they are consumed jointly.
Although VCRs and VCR tapes might seem like complementary goods, given they were often used together, the analysis through cross-price elasticity shows they function as substitutes in this scenario. This means their demand patterns align more with substitutes, and price changes won't necessarily decrease the use of both together.
Recognizing goods as complements can influence company supply chains, pricing decisions, and promotional activities, as changes in one product's pricing can impact the entire bundle's sales performance.