In this problem, p is in dollars and q is the number of units.

Suppose that the demand for a product is given by 
(p + 8)

q + 4
 = 1980.
(a) Find the elasticity when 
p = $82.
 (Round your answer to two decimal places.)


(b) Tell what type of elasticity this is.
Demand is elastic.
Demand is inelastic.    
Demand is unitary.

(c) How would a price increase affect revenue?
An increase in price increases revenue.
An increase in price decreases revenue.    
Revenue is unaffected by price.

The demand equation provided is:

\[
(p + 8)q + 4 = 1980
\]

We are tasked with calculating the elasticity of demand at $ p = 82 $, classifying the type of elasticity, and determining how a price increase would affect revenue.

### (a) **Find the Elasticity When $ p = 82 $**

1. First, we need to solve for $ q $ in terms of $ p $.
   
\[
(p + 8)q + 4 = 1980 \quad \text{or} \quad (p + 8)q = 1976
\]

\[
q = \frac{1976}{p + 8}
\]

This is the demand function, $ q(p) $. Next, the elasticity of demand, $ E(p) $, is given by the formula:

\[
E(p) = \left| \frac{p}{q} \cdot \frac{dq}{dp} \right|
\]

First, differentiate $ q(p) $ with respect to $ p $:

\[
q(p) = \frac{1976}{p + 8}
\]

Using the quotient rule for differentiation, we get:

\[
\frac{dq}{dp} = -\frac{1976}{(p + 8)^2}
\]

Now plug into the elasticity formula:

\[
E(p) = \left| \frac{p}{q} \cdot \frac{dq}{dp} \right|
\]

Substituting $ q = \frac{1976}{p + 8} $ and $ \frac{dq}{dp} = -\frac{1976}{(p + 8)^2} $:

\[
E(p) = \left| \frac{p}{\frac{1976}{p + 8}} \cdot \left(-\frac{1976}{(p + 8)^2}\right) \right|
\]

Simplifying:

\[
E(p) = \left| \frac{p \cdot (p + 8)^2}{1976} \cdot \left(-\frac{1976}{(p + 8)^2}\right) \right|
\]

\[
E(p) = \left| \frac{-p}{p + 8} \right| = \frac{p}{p + 8}
\]

Now, substitute $ p = 82 $:

\[
E(82) = \frac{82}{82 + 8} = \frac{82}{90} \approx 0.91
\]

Thus, the elasticity at $ p = 82 $ is approximately $ E(82) = 0.91 $.

### (b) **Classifying the Type of Elasticity**

The demand is classified based on the elasticity value $ E(p) $:

- If $ E(p) > 1 $, demand is **elastic**.
- If $ E(p) = 1 $, demand is **unitary**.
- If $ E(p) < 1 $, demand is **inelastic**.

Since $ E(82) = 0.91 $, which is less than 1, the demand is **inelastic**.

### (c) **Effect of Price Increase on Revenue**

For inelastic demand (when $ E(p) < 1 $), an increase in price **increases revenue** because the percentage decrease in quantity demanded is smaller than the percentage increase in price.

Thus, **an increase in price increases revenue** in this case.

---

Would you like more details on any part of the calculation?

---

### Follow-up Questions:
1. What happens to elasticity as $ p $ increases?
2. How does elasticity affect total revenue in general?
3. What is the significance of unitary elasticity in pricing decisions?
4. Can demand become elastic at higher prices?
5. How would a change in the demand equation affect elasticity?

**Tip:** Elasticity helps businesses understand how sensitive customers are to price changes, influencing pricing strategies.

In this problem, p is in dollars and q is the number of units. Suppose that the demand for a product is given by (p + 8)q + 4 = 1980. (a) Find the elasticity when p = $82. (b) Tell what type of elasticity this is. (c) How would a price increase affect revenue?

Learn how to calculate the elasticity of demand for a product given the demand equation (p + 8)q + 4 = 1980. The solution covers how to find elasticity at p = 82, classify the type of elasticity, and explain how a price increase affects revenue. This is a college-level economics problem suitable for students studying elasticity and revenue implications.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation