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Chapter 2: Problem 14

Demand for tomato plants. Sunshine Gardens determines the following demand function during early summer for tomato plants: $$ q=D(x)=\frac{2 x+300}{10 x+11} $$ where \(q\) is the number of plants sold per day when the price is \(x\) dollars per plant. a) Find the elasticity. b) Find the elasticity when \(x=3\) c) At \(\$ 3\) per plant, will a small increase in price cause total revenue to increase or decrease?

Short Answer

Expert verified
a) Elasticity formula derived from demand. b) Elasticity at \(x=3\) is 0.712. c) Total revenue increases with a price increase.

Step by step solution

01

Define Elasticity Formula

Elasticity, denoted by \(E(x)\), measures the sensitivity of demand to changes in price. It is defined as \[ E(x) = \left| \frac{x}{q} \cdot \frac{dq}{dx} \right| \], where \(q = D(x)\) is given and \(dq/dx\) is the derivative of the demand function with respect to \(x\).
02

Find the Derivative of Demand Function

The demand function is \( q = \frac{2x + 300}{10x + 11} \). Using the quotient rule for differentiation: \( h(x) = u(x)/v(x) \), where \(u(x) = 2x + 300\) and \(v(x) = 10x + 11\), \[ \frac{dq}{dx} = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} = \frac{(10x+11) \cdot 2 - (2x+300) \cdot 10}{(10x+11)^2} \].
03

Simplify the Derivative

Compute the derivative: \( u'(x) = 2 \) and \( v'(x) = 10 \). Substitute these into the derivative formula: \[ \frac{dq}{dx} = \frac{(20x + 22) - (20x + 3000)}{(10x + 11)^2} = \frac{22 - 3000}{(10x + 11)^2} = \frac{-2978}{(10x + 11)^2} \].
04

Substitute into Elasticity Formula

Substitute \( q = \frac{2x + 300}{10x + 11} \), \( \frac{dq}{dx} = \frac{-2978}{(10x + 11)^2} \) into the elasticity formula: \[ E(x) = \left| \frac{x}{\frac{2x + 300}{10x + 11}} \cdot \frac{-2978}{(10x + 11)^2} \right| = \left| \frac{-2978x}{2x + 300} \cdot \frac{10x + 11}{(10x + 11)^2} \right| \].
05

Simplify the Elasticity Expression

The elasticity expression becomes: \[ E(x) = \left| \frac{-2978x}{2x + 300} \cdot \frac{1}{10x + 11} \right| = \left| \frac{-2978x}{(2x + 300)(10x + 11)} \right| \].
06

Calculate Elasticity at Specific Price

Now find \( E(3) \): \[ E(3) = \left| \frac{-2978 \times 3}{(2 \times 3 + 300)(10 \times 3 + 11)} \right| \]. Calculate: \( 2 \times 3 + 300 = 306 \) and \(10 \times 3 + 11 = 41 \), so \[ E(3) = \left| \frac{-8934}{306 \times 41} \right| = \left| \frac{-8934}{12546} \right| = 0.712 \].
07

Interpret Elasticity at x = 3

Since \( E(3) = 0.712 < 1 \), the demand is inelastic at \( x = 3 \). This means that a small increase in price will lead to a less than proportionate decrease in quantity demanded, causing total revenue to increase.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Price Sensitivity
Understanding price sensitivity is crucial when examining how changes in price affect the demand for goods or services. Price sensitivity is sometimes defined as elasticity of demand. In simpler terms, it measures how much the quantity demanded of a product changes when its price changes. If a product is said to have high price sensitivity, even a small change in price can cause a significant change in the quantity demanded.
For instance, consider luxury items; they often experience greater drops in sales with price increases, indicating high price sensitivity. Conversely, essential goods, like bread or milk, have low price sensitivity because people need them regardless of price changes.
  • High price sensitivity: Large change in demand for small price changes.
  • Low price sensitivity: Small or no change in demand despite price changes.
This concept helps businesses determine pricing strategies to optimize revenue.
Derivative of Demand
The derivative of the demand function plays a key role in deriving the elasticity of demand. By understanding the demand function and how it changes, one can find the derivative, which tells us the rate at which demand changes as price changes. In mathematical terms, the derivative of a demand function, denoted as \( \frac{dq}{dx} \), gives you the slope of the demand curve at any given point.
For example, the demand function for tomato plants is given as \( q = \frac{2x + 300}{10x + 11} \). To find its derivative, we use the quotient rule for differentiation, which can be complex but essentially breaks down the function into parts to find the rate of change.
  • Derivative \( \frac{dq}{dx} \) informs the elasticity calculation.
  • It helps understand how responsive demand is to price changes.
Knowing \( \frac{dq}{dx} \) is essential for accurately calculating and interpreting elasticity.
Inelastic Demand
Inelastic demand refers to a situation where the quantity demanded changes by a smaller percentage than the price change. This is indicated when the elasticity of demand is less than 1. In practical terms, this means that consumers are not very sensitive to price changes for that product. For the demand for tomato plants discussed in the exercise, an elasticity value of 0.712 indicates inelastic demand.
Products with inelastic demand typically can have their prices raised without significantly affecting sales volume. Examples often include essential or necessary goods, such as prescription drugs or basic food items. With tomato plants, at a price point of \(3 dollars\), a small increase in price would likely lead to a minor reduction in quantity demanded, causing total revenue to rise.
  • \(E(x) < 1\): Demand is inelastic.
  • Small price changes result in smaller changes to demand.
This concept is vital for understanding how price adjustments can be strategically used to maximize revenue without losing a large customer base.

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