Problem 15 - Calculus Concepts

Question

DVDs The table shows the number of DVDs producers will supply at given prices. Supply Schedule for DVDs (Producers will not supply DVDs when the market price falls below \(\$ 5.00\) ) a. Find a model giving the quantity supplied as a function of the price per DVD. b. How many DVDs will producers supply when the market price is \(\$ 15.98 ?\) c. At what price will producers supply 2.3 million DVDs? d. Calculate the producer revenue and producer surplus when the market price is \(/\) 19.99 .$ $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Price } \\ \text { (dollars per DVD) } \end{array} & \begin{array}{c} \text { Quantity } \\ \text { (million DVDs) } \end{array} \\ \hline 5.00 & 1 \\ \hline 7.50 & 1.5 \\ \hline 10.00 & 2 \\ \hline 15.00 & 3 \\ \hline 20.00 & 4 \\ \hline 25.00 & 5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

a. Supply function: \(Q = 0.2P\); b. 3.196 million DVDs; c. Price: \(\$11.50\); d. Revenue: \(\$79.92\) million; Surplus: \(\$29.99\) million.

Step 1: Determine the Relationship Type

Observe the table of prices and quantities supplied to recognize a potential linear relationship. Each dollar increase appears to correspond to a consistent increase in the quantity supplied.

Step 2: Formulate the Linear Equation

Identify that the relationship is linear and can be written in the slope-intercept form: \(Q = mP + c\). Substitute the values from any two points, such as (5, 1) and (25, 5), into the formula to solve for the slope \(m\) and intercept \(c\).

Step 3: Calculate the Slope, m

Using the points (5, 1) and (25, 5), calculate the slope as follows: \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 1}{25 - 5} = \frac{4}{20} = 0.2\).

Step 4: Find the Y-Intercept, c

Now that we have \(m = 0.2\), substitute one of the points, such as (5, 1), into the equation \(1 = 0.2(5) + c\) to solve for \(c\). Simplifying gives \(1 = 1 + c\), so \(c = 0\).

Step 5: Write the Supply Function

The supply function is \(Q = 0.2P\), where \(Q\) is the quantity in millions of DVDs and \(P\) is the price per DVD in dollars.

Step 6: Calculate Quantity Supplied at \(\$15.98\)

Using the supply function \(Q = 0.2P\), substitute \(P = 15.98\) to find \(Q\):\[Q = 0.2 \times 15.98 = 3.196\text{ million DVDs}\].

Step 7: Calculate Price for 2.3 Million DVDs

For \(Q = 2.3\), use the supply function. Rearrange for \(P\), \(P = \frac{Q}{0.2}\), gives \(P = \frac{2.3}{0.2} = 11.5\). Thus, producers supply 2.3 million DVDs at \(\$11.50\).

Step 8: Calculate Producer Revenue at \(\$19.99\)

Determine \(Q\) at \(\$19.99\) using the supply function: \(Q = 0.2 \times 19.99 = 3.998 \text{ million DVDs}\). Revenue is \(P \times Q = 19.99 \times 3.998 \times 10^6 = 79.92 \times 10^6\) dollars.

Step 9: Calculate Producer Surplus at \(\$19.99\)

Producer surplus is the area above the supply curve and below the market price. Using the graph method, calculate the area of the triangle formed: Base \(= 3.998\)--height \(= 19.99 - 5 = 14.99\). Surplus \(= \frac{1}{2} \times 3.998 \times 14.99 \approx 29.99\) million.

Key Concepts

Linear Functions

In economics, linear functions are often used to model the relationship between variables. In the context of supply and demand, a linear function can beautifully model how the quantity of goods supplied correlates with their price.
This is because, in some cases, the quantity supplied increases in a direct mechanism as the price goes up. This consistent change forms a straight line when plotted on a graph.
For instance, consider the supply function for DVDs: \( Q = 0.2P \). Here, \( Q \) represents the quantity of DVDs supplied in millions and \( P \) is the price per DVD in dollars.
Here's how linear functions help:

• The slope \( m \) tells you the rate at which the quantity supplied changes with price. In our function, the slope is 0.2, meaning for every dollar increase in price, producers supply 0.2 million more DVDs.
• The intercept \( c \) is the quantity supplied when the price is zero. For DVDs, our intercept is 0, meaning producers only start supplying when the price is positive.

The use of linear functions allows for quick calculations and predictions by simply plugging values into the equation. This facilitates decision-making for producers regarding how much to supply at various price points.

Producer Revenue

Producer revenue is the total income that producers receive from selling their goods. It's calculated by multiplying the market price by the quantity supplied.
In our example, at a market price of \( \$19.99 \), the quantity supplied is \( \approx 3.998 \) million DVDs. Therefore, the producer revenue is calculated as follows:
\[Revenue = P \times Q = 19.99 \times 3.998 \times 10^6 = 79.92 \times 10^6 \text{ dollars}\]
This revenue is crucial for producers as it represents the inflow of cash from their business operations. It helps them cover production costs and potentially earn profits.
When understanding producer revenue:

• It provides insight into financial health and sustainability of production.
• Higher prices typically lead to higher revenue, assuming constant demand.
• Changes in supply due to price adjustments directly affect revenue.

For students, grasping this concept helps them understand the economic incentives for producers to adjust supply based on price changes.

Producer Surplus

Producer surplus is a measure of producer welfare and represents the difference between what producers actually receive for selling a good and the minimum amount they would be willing to accept for producing it.
This concept is visually represented as the area above the supply curve and below the market price on a supply-demand graph.
To compute producer surplus in our DVD example, we look at the triangle formed between the supply curve, the price line at \( \$19.99 \), and the quantity axis:
\[\text{Surplus} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 3.998 \times (19.99 - 5) = 29.99 \text{ million}\]
Understanding producer surplus allows students to appreciate:

• How market conditions can benefit producers beyond their basic production costs.
• The factors leading to producer happiness and how this influences their supply decisions.
• It's an indicator of the efficiency of resource allocation within a market.

The producer surplus concept helps frame discussions around market dynamics, benefits of trade, and the incentives for production under varying economic conditions.