Question

When a wholesaler sold a product at \(\$ 40\) per unit, sales were 300 units per week. After a price increase of \(\$ 5\), however, the average number of units sold dropped to 275 per week. Assuming that the demand function is linear, what price per unit will yield a maximum total revenue?

Short answer

$70 per unit will yield the maximum total revenue.

Step-by-step solution

Define the Variables

Let's denote the price increase from original $40 by \( x \), then the new price will be \( 40 + x \) and the corresponding quantity sold will be \( 300 - 5x \). The coefficient (-5) is based on the rate at which quantity sold decreases with a price increase, per the problem statement.

Define the Revenue Function

The total revenue (R) is given by price per unit multiplied by quantity sold. So, \( R = (40 + x)(300 - 5x) \).

Simplify the Revenue Function

Expand the terms in the revenue function to simplify it: \( R = 12000 + 300x - 5x^2 \).

Find the Maximum Revenue

For a quadratic function \( ax^2 + bx + c \), the value of \( x \) that maximizes the function is given by \( -b/2a \). For our revenue function, \( a = -5 \) and \( b = 300 \), so the optimum \( x \) value (price increase) that maximizes revenue is \( x = -300/(2*(-5)) = 30 \).

Calculate the Maximum Price

Substitute \( x = 30 \) back into the equation for price to find the price per unit that will yield maximum total revenue: New Price \( = 40 + x = 40 + 30 = 70 \) dollars. Hence, a price of 70 dollars per unit will yield maximum total revenue.

Key concepts

Demand Function

A demand function is a mathematical representation of the relationship between the price of a product and the quantity demanded by consumers. It typically reflects how the quantity demanded decreases as the price increases, and vice versa. In our wholesaler's scenario, the demand function is assumed to be linear based on observed changes in demand with variations in price. The initial data points give us two coordinates: at a price of \(40, quantity is 300 units; and at \)45, quantity drops to 275 units. From here, the linear relationship is identified as a decrease of 5 units for every $1 increase in price, leading to a demand function in the form of quantity sold as a function of price, which can be expressed as Q = 300 - 5(P - 40), where Q is the quantity and P is the price per unit. Understanding and defining the demand function correctly is pivotal since it directly impacts revenue optimization strategies.

Revenue Function

The revenue function, symbolized as R, calculates the total income generated from selling a certain number of products at a specific price. It is obtained by multiplying the selling price per unit by the number of units sold ( R = Price × Quantity ). In our exercise, with the demand function defined, we construct the revenue function to determine the maximum possible revenue. By introducing a variable x to denote the increase in price, and understanding that an increase in price can reduce the quantity demanded, we can define a mathematical equation for revenue as a function of the variable x (R = (40 + x)(300 - 5x)) . Simplifying this into a quadratic expression allows us to use properties of quadratic functions to find the maximum revenue, thus underscoring the importance of a properly constructed revenue function.

Quadratic Functions

Quadratic functions take the form f(x) = ax^2 + bx + c and are graphically represented by a parabola. These functions are essential in economics for modeling phenomena such as revenue calculations. The parabola opens upward or downward depending on the sign of the coefficient a . In our case, the coefficient is negative, indicating that the parabola opens downward, which is characteristic when finding a maximum value. To find the vertex of the parabola, which represents either the maximum or minimum value based on the function, we use the formula x = -b/(2a) . For revenue maximization, locating this vertex is crucial as it provides the price increase that leads to the highest possible total revenue.

Price Elasticity of Demand

Price elasticity of demand measures how sensitive the quantity demanded of a good or service is to changes in its price. It is defined as the percentage change in quantity demanded divided by the percentage change in price. Elasticity can be categorized as elastic (greater than 1), inelastic (less than 1), or unitary (equal to 1). Understanding elasticity helps businesses anticipate how a change in price could affect demand and subsequently, revenue. In the context of our problem, the decrease in quantity sold for every increase in price points towards a negative elasticity of demand. This implies that the wholesaler must find a delicate balance between price and demand to avoid diminishing returns, which is precisely what the maximum total revenue calculation aims to achieve.