Problem 3 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 9 , demand equations for two related commodities are given. In each exercise, determine the four partial marginal demands. Determine if the commodities are complementary, substitutes, or neither. In Exercises 1 through 6 , draw sketches of the two demand surfaces. \(x=-3 p+5 q+15, y=2 p-4 q+10\)

Step-by-Step Solution

Answer

The partial marginal demands are: \( \frac{\partial x}{\partial p} = -3 \), \( \frac{\partial x}{\partial q} = 5 \), \( \frac{\partial y}{\partial p} = 2 \), \( \frac{\partial y}{\partial q} = -4 \). The commodities are substitutes.

Step 1 - Find \(\frac{\partial x}{\partial p}\)

Differentiate the equation for commodity x with respect to p. Given: \[ x = -3p + 5q + 15 \]\[ \frac{\partial x}{\partial p} = -3 \]

Step 2 - Find \(\frac{\partial x}{\partial q}\)

Differentiate the equation for commodity x with respect to q.\[ x = -3p + 5q + 15 \]\[ \frac{\partial x}{\partial q} = 5 \]

Step 3 - Find \(\frac{\partial y}{\partial p}\)

Differentiate the equation for commodity y with respect to p. Given: \[ y = 2p - 4q + 10 \]\[ \frac{\partial y}{\partial p} = 2 \]

Step 4 - Find \(\frac{\partial y}{\partial q}\)

Differentiate the equation for commodity y with respect to q.\[ y = 2p - 4q + 10 \]\[ \frac{\partial y}{\partial q} = -4 \]

Step 5 - Determine the relationship between commodities

Look at the signs of the cross partial derivatives (\( \frac{\partial x}{\partial q} \) and \( \frac{\partial y}{\partial p} \)). Here, \( \frac{\partial x}{\partial q} = 5 \) and \( \frac{\partial y}{\partial p} = 2 \). Both derivatives are positive, indicating that the commodities are substitutes.

Step 6 - Sketch the demand surfaces (Optional)

Sketch how x and y vary with respect to p and q. This helps visualize the demand surfaces. Remember, these are 3D surfaces where one axis is p, one is q, and one is either x or y.

Key Concepts

marginal demand

The concept of marginal demand refers to how the demand for a product changes when the price or other factors related to another product change. It is a crucial concept in economics because it shows the sensitivity of demand for one product to changes in price or other factors.
The partial derivatives found in the exercise, such as \(\frac{\partial x}{\partial p}\) and \(\frac{\partial x}{\partial q}\), represent the marginal demand. For instance, \(\frac{\partial x}{\partial p} = -3\) means that if the price of commodity p increases by 1 unit, the demand for commodity x will decrease by 3 units. Similarly, \(\frac{\partial x}{\partial q} = 5\) indicates that if the price of commodity q increases by 1 unit, the demand for commodity x will increase by 5 units.
Marginal demand helps businesses and policymakers make informed decisions. Understanding how demand reacts to various factors allows for better pricing strategies and resource allocation.

complementary goods

Complementary goods are products whose demands are linked together. When the price or availability of one of these goods changes, it directly affects the demand for the other. A classic example is a pair of shoes and shoe polish.
If increasing the price of one good leads to a decrease in the demand for both goods, they are called complementary. In mathematical terms, if the partial derivatives of demand functions such as \(\frac{\partial x}{\partial q}\) and \(\frac{\partial y}{\partial p}\) are negative, then the goods are complementary. This is not the case in our exercise, where both corrections are positive (5 and 2, respectively), showing that the goods are not complementary.

substitute goods

Substitute goods are those that can replace each other in consumption. When the price of one good increases, the demand for its substitute typically rises because consumers switch to the cheaper alternative. Think of coffee and tea as an example.
In the exercise provided, since both \(\frac{\partial x}{\partial q}\) and \(\frac{\partial y}{\partial p}\) are positive, it suggests that an increase in the price of one commodity leads to an increase in the demand for the other. This positive relationship between the cross partial derivatives indicates that x and y are substitute goods. This concept is essential for understanding consumer behavior and how products compete in the market.
Understanding substitutes helps businesses identify competitors and assess market dynamics. This strategic insight is crucial for marketing and pricing strategies.