Problem 19 - Applied Calculus

Question

The quantity of a product demanded by consumers is a function of its price. The quantity of one product demanded may also depend on the price of other products. For example, the demand for tea is affected by the price of coffee; the demand for cars is affected by the price of gas. The quantities demanded, \(q_{1}\) and \(q_{2}\), of two products depend on their prices, \(p_{1}\) and \(p_{2}\), as follows: $$ \begin{array}{l} q_{1}=150-2 p_{1}-p_{2} \\ q_{2}=200-p_{1}-3 p_{2} \end{array} $$ (a) What does the fact that the coefficients of \(p_{1}\) and \(p_{2}\) are negative tell you? Give an example of two products that might be related this way. (b) If one manufacturer sells both products, how should the prices be set to generate the maximum possible revenue? What is that maximum possible revenue?

Step-by-Step Solution

Answer

The products are substitutes. Set prices to \(p_1=40\) and \(p_2=30\) for maximum revenue of 3400.

Step 1: Understanding the Effect of Price on Demand

The given equations show the demand for two products as functions of their prices:\[q_1 = 150 - 2p_1 - p_2\]\[q_2 = 200 - p_1 - 3p_2\]The negative coefficients of \(p_1\) and \(p_2\) in both equations indicate that as the price of a product increases, the demand for that product decreases. This is consistent with the law of demand. For example, tea and coffee can be substitutes. If the price of tea \(p_1\) increases, its demand \(q_1\) might decrease, and if the price of coffee \(p_2\) increases, the demand for tea could similarly decrease due to less disposable income available to spend on tea.

Step 2: Formulating the Revenue Function

Revenue for each product is calculated by multiplying the price by the quantity sold. So, the total revenue \(R\) from both products is:\[ R = p_1 \cdot q_1 + p_2 \cdot q_2 \]Substituting the expressions for \(q_1\) and \(q_2\), we have:\[R = p_1(150 - 2p_1 - p_2) + p_2(200 - p_1 - 3p_2)\]

Step 3: Simplifying the Revenue Function

Expand and simplify the revenue function:\[R = 150p_1 - 2p_1^2 - p_1p_2 + 200p_2 - p_2p_1 - 3p_2^2\]This simplifies to:\[R = -2p_1^2 - 3p_2^2 - 2p_1p_2 + 150p_1 + 200p_2\]

Step 4: Maximizing Revenue Using Calculus

To find the prices that maximize revenue, we need to calculate the partial derivatives of \(R\) with respect to \(p_1\) and \(p_2\), set them to zero, and solve for \(p_1\) and \(p_2\).\(\frac{\partial R}{\partial p_1} = -4p_1 - p_2 + 150 = 0\)\(\frac{\partial R}{\partial p_2} = -6p_2 - p_1 + 200 = 0\)Solving these two equations simultaneously gives us the values of \(p_1\) and \(p_2\).

Step 5: Solving the Equations to Find Optimal Prices

Solving the system of equations, we first express \(p_1\) in terms of \(p_2\) from the first:\[p_1 = \frac{150 - p_2}{4}\]Substitute in the second equation:\[6p_2 + \frac{150 - p_2}{4} = 200\]Solve this equation to find \(p_2\), and then substitute back to find \(p_1\). This yields \(p_1 = 40\) and \(p_2 = 30\).

Step 6: Calculating the Maximum Revenue

Substituting the found prices \(p_1 = 40\) and \(p_2 = 30\) back into the demand equations, we find the quantities:\[q_1 = 150 - 2(40) - 30 = 40\]\[q_2 = 200 - 40 - 3(30) = 60\]The maximum revenue is:\[R = 40 \cdot 40 + 30 \cdot 60 = 1600 + 1800 = 3400\]

Key Concepts

Price Elasticity

Price elasticity of demand is a crucial concept in economics that illustrates how sensitive the quantity demanded of a good is to a change in its price. It's a measure of responsiveness, and it's expressed as the ratio of the percentage change in quantity demanded to the percentage change in price. If the demand for a product significantly changes with a small change in price, the demand is said to be elastic. Conversely, if the demand barely changes, it's considered inelastic. For example, luxury goods often have elastic demand, while necessities like bread may have inelastic demand.

In our original exercise, the negative coefficients in the demand functions \(q_1 = 150 - 2p_1 - p_2\) and \(q_2 = 200 - p_1 - 3p_2\) indicate the price elasticity of demand. It shows that for both products, an increase in price decreases the quantity demanded, which is consistent with the law of demand. These equations reveal the inverse relationship between price and quantity, a key feature of elastic demand.

Substitute Goods

Substitute goods are products that can replace each other in consumer usage. The increase in the price of one leads to an increase in the demand for the other. For instance, if tea and coffee are substitute goods, when the price of tea increases, consumers might buy more coffee instead. This concept is crucial when analyzing multi-product demand functions.

In the exercise, the demand functions \(q_1 = 150 - 2p_1 - p_2\) and \(q_2 = 200 - p_1 - 3p_2\) reflect the presence of substitute goods. Here, an increase in the price of one product affects the demand for the other product. Therefore, the coefficients in front of \(p_2\) in \(q_1\) and \(p_1\) in \(q_2\) illustrate the cross-price elasticity, where the price of one good impacts the demand for another. This interdependence highlights the concept of substitute goods.

Revenue Maximization

Revenue maximization involves setting the price levels of products in such a way that the company's total revenue is at its highest possible point. To achieve this, businesses must understand the relationship between prices, demand, and revenue.

In the given exercise, the revenue function, \[R = -2p_1^2 - 3p_2^2 - 2p_1p_2 + 150p_1 + 200p_2\], illustrates the total revenue derived from the two products. To find the optimal prices for revenue maximization, it's essential to differentiate the revenue function with respect to prices \(p_1\) and \(p_2\), set these derivatives to zero, and then solve the resulting equations for the optimal price points. This leads to calculating \(p_1 = 40\) and \(p_2 = 30\), which, after substituting back into the demand functions, yield a maximum revenue of 3400.

This process highlights the importance of calculus in finding peak revenue conditions and demonstrates how a strategic pricing approach can significantly impact business profitability.