Question

A firm faces the following linear inverse demand for its product, \(P=60-2 Q\), Find the firm's total revenue function, \(T R(Q)\). Plot the total revenue function, (Hint: using \(P=60-2 Q\), plot a demand schedule for an arbitrary number of quantities - say, from 5 to 25 in fives.)

Short answer

The total revenue function is \(TR(Q) = 60Q - 2Q^2\). Total revenue is maximized at \(Q=15\).

Step-by-step solution

Understand the Demand Function

The firm faces a linear inverse demand function given by \(P=60-2Q\), where \(P\) is the price and \(Q\) is the quantity of the product. This function indicates that as the quantity increases, the price decreases by 2 units for each additional unit of \(Q\).

Express Price in Terms of Quantity

To find the total revenue, express price explicitly as a function of the quantity using the given demand function, where \(P=60-2Q\).

Derive the Total Revenue Function

Total revenue \(TR(Q)\) is calculated as \(TR(Q) = P \times Q\). Substitute the expression for \(P\) from the demand function: \[ TR(Q) = (60-2Q) \times Q = 60Q - 2Q^2 \]. The total revenue function is thus \(TR(Q) = 60Q - 2Q^2\).

Create a Demand Schedule

Generate a demand schedule by calculating the price \(P\) for each quantity \(Q = 5, 10, 15, 20, 25\) using the demand function \(P=60-2Q\): - For \(Q=5, P=60-2(5)=50\) - For \(Q=10, P=60-2(10)=40\)- For \(Q=15, P=60-2(15)=30\)- For \(Q=20, P=60-2(20)=20\)- For \(Q=25, P=60-2(25)=10\)

Calculate Total Revenue for Each Quantity

Use the total revenue function \(TR(Q) = 60Q - 2Q^2\) to find the total revenue for each \(Q\):- For \(Q=5, TR=60(5) - 2(5)^2 = 250\)- For \(Q=10, TR=60(10) - 2(10)^2 = 400\)- For \(Q=15, TR=60(15) - 2(15)^2 = 450\)- For \(Q=20, TR=60(20) - 2(20)^2 = 400\)- For \(Q=25, TR=60(25) - 2(25)^2 = 250\)

Plot the Total Revenue Function

Plot the values of \(TR(Q)\) for each \(Q\) from 5 to 25. The plot will help visualize how total revenue changes as quantity increases. The plot typically forms a parabola opening downwards, showing maximum total revenue at a certain quantity before it declines.

Key concepts

Inverse Demand

In economics, the inverse demand function is a way to express price as a function of quantity demanded. This differs from the typical demand function which relates the quantity demanded to the price.
In the context of our problem, the firm faces an inverse demand function given by \(P = 60 - 2Q\). This implies that as more of the product is offered for sale (i.e., as \(Q\) increases), the price \(P\) consumers are willing to pay decreases. Essentially, with every additional unit sold, the price drops by 2 units.
This understanding helps firms understand how much they can charge for a product given a certain level of supply, and adjust their production strategy accordingly.

Linear Demand

A linear demand function is one where the relationship between price and quantity demanded is a straight line. This type of demand is simple and predictable.
In our exercise, the equation \(P = 60 - 2Q\) represents a linear demand because if you plotted price () against quantity (), it would form a straight line with a negative slope. This slope, represented by the coefficient (which is -2 here), tells us the rate at which price decreases as quantity increases.
Linear demand makes it easier for firms to predict their potential revenue and make decisions about how much of a product to supply at different price points.

Demand Schedule

A demand schedule is a table that shows the relationship between the price of a good and the quantity demanded. This is useful for visualizing how changes in price affect quantity demanded.
To create a demand schedule based on our function \(P = 60 - 2Q\), we calculate price at specific quantities. For example:

• When \(Q=5\), \(P=50\)
• When \(Q=10\), \(P=40\)
• When \(Q=15\), \(P=30\)
• When \(Q=20\), \(P=20\)
• When \(Q=25\), \(P=10\)

These calculated values provide a clear picture of how prices decrease as quantity increases, aiding firms in making informed pricing strategies.

Revenue Maximization

Revenue maximization involves finding the quantity that yields the highest possible total revenue. This is a critical concept for firms aiming to optimize their earnings.
The total revenue function emerges from the product of price and quantity, expressed as \(TR(Q) = P \times Q\). For our scenario, using the inverse demand function we derived year revenue function: \(TR(Q) = 60Q - 2Q^2\).
Graphing this function can reveal insights about revenue patterns. Typically, a parabola is formed, where total revenue first increases, reaches a maximum point, and then decreases if further quantities are produced. Firms look for this peak to ensure their production levels align with maximum revenue, balancing out the adverse effects of lower prices at higher outputs.