Problem 4 - 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. \(x=9-3 p+q, y=10-2 p-5 q\)

Step-by-Step Solution

Answer

Find the partial derivatives: \( \frac{\text{d}x}{\text{d}p} = -3 \), \( \frac{\text{d}x}{\text{d}q} = 1 \), \( \frac{\text{d}y}{\text{d}p} = -2 \), \( \frac{\text{d}y}{\text{d}q} = -5 \). The relationship is neither complements nor substitutes.

Step 1 - Determine Partial Derivative of x with Respect to p

To find the partial derivative of demand for commodity x with respect to price p, differentiate the equation for x: \[ x = 9 - 3 p + q \] Differentiating with respect to p gives: \[ \frac{\text{d}x}{\text{d}p} = -3 \]

Step 2 - Determine Partial Derivative of x with Respect to q

Now, differentiate the equation for x with respect to q: \[ x = 9 - 3 p + q \] Differentiating with respect to q gives: \[ \frac{\text{d}x}{\text{d}q} = 1 \]

Step 3 - Determine Partial Derivative of y with Respect to p

Next, differentiate the equation for y with respect to p: \[ y = 10 - 2 p - 5 q \] Differentiating with respect to p gives: \[ \frac{\text{d}y}{\text{d}p} = -2 \]

Step 4 - Determine Partial Derivative of y with Respect to q

Finally, differentiate the equation for y with respect to q: \[ y = 10 - 2 p - 5 q \] Differentiating with respect to q gives: \[ \frac{\text{d}y}{\text{d}q} = -5 \]

Step 5 - Determine the Relationship Between Commodities

The cross partial derivatives, \( \frac{\text{d}x}{\text{d}q} = 1 \) and \( \frac{\text{d}y}{\text{d}p} = -2 \), are non-zero, indicating that changes in one commodity's price affect the other's demand. Because the partial derivative of x with respect to q is positive and the partial derivative of y with respect to p is negative, the commodities are neither straightforward complements nor substitutes. Picturing both commodities’ sensitivities is complex in this case.

Step 6 - Sketch the Demand Surfaces

Sketch the surfaces for x and y equations with 3D graphs, with axes p, q, and x or y respectively. For x: Plot \[ x = 9 - 3p + q \]. For y: Plot \[ y = 10 - 2p - 5q \]. These 3D plots should illustrate how each commodity's demand changes with respect to prices p and q.

Key Concepts

Marginal Demand

We begin with understanding the concept of marginal demand. Marginal demand refers to how the quantity demanded of a commodity changes when there is a small change in one of its influencing factors, like its price or the price of a related commodity. In our exercise, we calculate the marginal demand using partial derivatives.

Partial Derivatives:

• \(\frac{\text{d}x}{\text{d}p} = -3\), meaning a small increase in price \(p\) leads to a decrease in demand for \(x\) by 3 units.
• \(\frac{\text{d}x}{\text{d}q} = 1\), indicating an increase in demand for \(x\) by 1 unit for a small increase in \(q\).
• \(\frac{\text{d}y}{\text{d}p} = -2\) tells us that a small increase in \(p\) decreases the demand for \(y\) by 2 units.
• \(\frac{\text{d}y}{\text{d}q} = -5\) shows that a small increase in \(q\) results in a decrease in demand for \(y\) by 5 units.

These marginal demands help us understand the immediate impact of changing prices on the demand for each commodity. They are essential for making pricing decisions and understanding market sensitivity.

Related Commodities

Understanding how commodities relate to each other is crucial. Commodities are termed as complements or substitutes based on how the demand for one reacts to changes in the price of the other.

Complements:

• If increasing the price of one commodity leads to a decrease in demand for the other, they are complements.
• For example, if the rise in price of coffee leads to a drop in the demand for sugar, they're complements.

Substitutes:

• If increasing the price of one commodity leads to an increase in demand for the other, they are substitutes.
• For instance, if a hike in the price of tea causes a rise in demand for coffee, they're substitutes.

In this exercise, since \(\frac{\text{d}x}{\text{d}q}\) is positive and \(\frac{\text{d}y}{\text{d}p}\) is negative, the commodities are neither straightforward substitutes nor complements. Their relationship is more nuanced, likely affected by other underlying factors not captured in the partial derivatives alone.

Demand Surfaces

Demand surfaces help us visualize how demand for commodities \(x\) and \(y\) change with respect to prices \(p\) and \(q\). These surfaces can be plotted as 3D graphs.

For commodity \(x\): The equation \(x = 9 - 3p + q\) can be expressed as a plane in a 3D space where \(x\) is the vertical axis and \(p\) and \(q\) are the horizontal axes. Varying \(p\) and \(q\) will show how \(x\) changes.

For commodity \(y\): Similarly, the equation \(y = 10 - 2p - 5q\) forms another plane. Here, the demand for \(y\) will change differently compared to \(x\), reflecting its unique dependence on \(p\) and \(q\).

To sketch these demand surfaces:

• Create a grid of points in the \(p-q\) space.
• Calculate \(x\) and \(y\) values for each pair of \(p\) and \(q\).
• Plot these values in 3D to visualize the demand surfaces.