Problem 48 - Calculus Single Variable

Question

The demand curve for a given commodity is the set of all points \((p, q)\) in the \(p q\) -plane where \(q\) is the number of units of the product that can be sold at price \(p .\) The elasticity of demand for the product at price \(p\) is defined to $$ \text { be } E(p)=-q^{\prime}(p) \cdot p / q(p) \text { . } $$ a. Suppose that a demand curve for a commodity is given by $$ p+q+2 p^{2} q+3 p q^{3}=1000 $$ when \(p\) is measured in dollars and the quantity \(q\) of items sold is measured by the \(1000 .\) For example, the point \((p, q)=(6,3.454)\) is on the curve. That means that 3454 items are sold at $$ 6 .\( What is the slope of the demand curve at the point (6,3.454)\)?\( b. What is elasticity of demand for the product of part at \)p=\$ 6 ?$

Step-by-Step Solution

Answer

The slope is \(-0.301\) and the elasticity is approximately 0.523 at \( p = 6 \).

Step 1: Understand the Demand Curve Equation

The demand curve is given by \( p + q + 2p^2q + 3pq^3 = 1000 \). This represents the relationship between price \( p \) and quantity \( q \). We need to find the slope of this curve at a specific point.

Step 2: Determine the Point of Interest

The point provided is \((p, q) = (6, 3.454)\), meaning at price \(p = 6\), the quantity \(q = 3.454\). This corresponds to selling 3454 units since \(q\) is measured in thousands.

Step 3: Implicit Differentiation to Find Slope

Differentiate the demand curve equation implicitly with respect to \( p \) to get \( 1 + \frac{dq}{dp} + 4pq + 2p^2\frac{dq}{dp} + 3q^3 + 9pq^2\frac{dq}{dp} = 0 \). Solve for \( \frac{dq}{dp} \) to find the slope at the given point.

Step 4: Substitute Point Values

Plug in \( p = 6 \) and \( q = 3.454 \) into the differentiated equation. Simplify to solve for \( \frac{dq}{dp} \), which gives the slope.

Step 5: Simplify to Find the Slope

After substitution and simplification: \( 1 + \frac{dq}{dp} + 82.896 + \frac{dq}{dp}(272.86) = 0 \). Solving the equation gives \( \frac{dq}{dp} = -0.301 \).

Step 6: Elasticity Formula Substitution

Apply the elasticity formula \( E(p) = -q'(p) \cdot \frac{p}{q(p)} \) using \( p = 6 \) and \( \frac{dq}{dp} = -0.301 \), and \( q(p) = 3.454 \).

Step 7: Calculate Elasticity at \( p = 6 \)

Substitute \( q'(p) = -0.301 \), \( p = 6 \), and \( q(p) = 3.454 \) into the elasticity formula: \[ E(6) = 0.301 \times \frac{6}{3.454} = 0.523 \].

Step 8: Final Step: Interpretation

The elasticity at \( p = 6 \) is approximately 0.523, which indicates that the demand changes by 0.523% for a 1% change in price.

Key Concepts

Demand Curve

The concept of a demand curve is fundamental in economics as it represents the relationship between the price level and the quantity of a commodity that consumers are willing and able to purchase. Typically, the demand curve is downward sloping, illustrating that as prices fall, the quantity demanded generally increases, and vice versa.
A demand curve can be visualized in the coordinate plane with price (\(p\)) on the vertical axis and quantity (\(q\)) on the horizontal axis. The set of all points \((p, q)\) in this plane represents different combinations of price and quantity for which the product can be sold. In the provided exercise, the demand curve is defined by the equation \( p+q+2p^2q+3pq^3=1000 \). This complex equation shows an intricate relationship between \(p\) and \(q\) due to the involvement of higher-degree terms.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of an equation that defines a relationship between two variables without explicitly solving for one variable in terms of the other. In the context of the demand curve problem provided, we need the derivative of the implicit function \( p+q+2p^2q+3pq^3=1000 \) to determine how quantity \(q\) changes with respect to price \(p\). Rather than express \(q\) explicitly in terms of \(p\), we differentiate each term of the equation with respect to \(p\).
During this process, we treat \(q\) as a function of \(p\) and use the chain rule to differentiate terms involving \(q\), introducing \(\frac{dq}{dp}\). This results in the differentiated equation: \(1 + \frac{dq}{dp} + 4pq + 2p^2\frac{dq}{dp} + 3q^3 + 9pq^2\frac{dq}{dp} = 0\). Solving this equation for \(\frac{dq}{dp}\) provides the slope of the demand curve at a specific point.

Slope Calculation

The slope of the demand curve at a specific point indicates how much the quantity \(q\) will change for a small change in price \(p\). Given the differentiated demand curve equation, finding the slope at \((p, q) = (6, 3.454)\) involves substituting these values into the equation \(1 + \frac{dq}{dp} + 82.896 + \frac{dq}{dp}(272.86) = 0\). By solving this equation for \(\frac{dq}{dp}\), we calculate the slope to be \(-0.301\).
This numerical value tells us that at the point where the price is 6 dollars, a small increase in price will result in a decrease in quantity demanded, which is consistent with the typical behavior of most demand curves. Understanding the slope helps in assessing how sensitive the quantity demanded is to price changes.

Price and Quantity Relationship

The price and quantity relationship in demand curves reflects the general economic principle of demand elasticity — the responsiveness of the quantity demanded to changes in price. In our exercise, after finding the slope of the demand curve, the elasticity of demand is calculated using the formula \( E(p) = -q'(p) \cdot \frac{p}{q(p)} \).
Substituting the values from the exercise, where \(q'(p) = -0.301\), \(p = 6\), and \(q(p) = 3.454\), we find \(E(6) = 0.301 \times \frac{6}{3.454} = 0.523\). This figure indicates the elasticity of demand at \(p = 6\), suggesting that for a 1% increase in the price, the quantity demanded would decrease by approximately 0.523%.
This is a measure of inelastic demand, as the absolute value of elasticity is less than one, indicating that consumers are not highly responsive to changes in price. Understanding such relationships can help businesses set optimal pricing strategies.