Problem 51 - Calculus Single Variable

Question

By the demand curve for a given commodity, we mean 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 \(p .\) Use the differential approximation to estimate the demand \(q(p)\) for a commodity at a given price \(p\). 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 \(1000 .\) The point \(\left(p_{0}, q_{0}\right)=\) (6.75,3.248) is on the curve. That means that 3248 items are sold at \(\$ 6.75 .\) What is the slope of the demand curve at the point (6.75,3.248)\(?\) Approximately how many units will be sold if the price is increased to \$6.80? Decreased to \$6.60?

Step-by-Step Solution

Answer

The slope at (6.75, 3.248) is -0.253. At $6.80, approx. 3235 units sold; at $6.60, approx. 3286 units sold.

Step 1: Understanding the Problem

We are asked to find how the demand changes when the price changes slightly, using a given demand curve equation. We're also given a particular point on the curve to start from.

Step 2: Implicit Differentiation of the Demand Curve

The demand curve is given by the equation \(p + q + 2p^2 q + 3pq^3 = 1000\). We use implicit differentiation to find \( \frac{dq}{dp} \). Differentiate both sides with respect to \(p\): \( 1 + \frac{dq}{dp} + (4pq + 2p^2 \frac{dq}{dp}) + (3q^3 + 9p q^2 \frac{dq}{dp}) = 0 \).

Step 3: Simplifying the Implicit Derivative Expression

Rearrange the terms to isolate \( \frac{dq}{dp} \): \( \frac{dq}{dp} (1 + 2p^2 + 9pq^2) = - (1 + 4pq + 3q^3) \). Therefore, \( \frac{dq}{dp} = - \frac{1 + 4pq + 3q^3}{1 + 2p^2 + 9pq^2} \).

Step 4: Calculating the Slope at the Given Point

Plug \(p_0 = 6.75\) and \(q_0 = 3.248\) into our derivative \( \frac{dq}{dp} \): \( \frac{dq}{dp} \approx - \frac{1 + 4(6.75)(3.248) + 3(3.248)^3}{1 + 2(6.75)^2 + 9(6.75)(3.248)^2} = -0.253 \). This is the slope of the demand curve at the point \((6.75, 3.248)\).

Step 5: Estimating Demand for a Price Increase

For a price increase to \$6.80, the price increases by \( \Delta p = 0.05\). We use \( \Delta q \approx \frac{dq}{dp} \cdot \Delta p \). Thus, \( \Delta q \approx -0.253 \times 0.05 = -0.01265 \), so \( q(p) \approx 3.248 - 0.01265 = 3.23535 \). Multiply by 1000: \( 3235.35 \) units.

Step 6: Estimating Demand for a Price Decrease

For a price decrease to \$6.60, \( \Delta p = -0.15\). Using \( \Delta q \approx \frac{dq}{dp} \cdot \Delta p \), \( \Delta q \approx -0.253 \times (-0.15) = 0.03795 \). Thus, \( q(p) \approx 3.248 + 0.03795 = 3.28595 \). Multiply by 1000: \( 3285.95 \) units.

Key Concepts

Implicit Differentiation

Implicit differentiation is a method used when dealing with equations where it’s difficult or impossible to solve for one variable in terms of another. For example, with the given demand curve equation, both price \( p \) and quantity \( q \) are mixed within the expression, making direct differentiation with respect to \( p \) challenging.
By differentiating implicitly, we take the derivative of both sides with respect to \( p \) and treat \( q \) as a function of \( p \). This means when we differentiate terms involving \( q \), we multiply by \( \frac{dq}{dp} \).
This step allows us to eventually solve for \( \frac{dq}{dp} \) in terms of \( p \) and \( q \), which represents the rate of change of quantity with respect to price along the demand curve.

Demand Curve

A demand curve represents the relationship between the price of a good or service and the quantity demanded. It is typically downward sloping, indicating that as the price decreases, more of the good is demanded.
In this problem, the demand curve is represented by the equation \( p + q + 2p^2q + 3pq^3 = 1000 \). Each point on this curve indicates a specific price and quantity where the market is in equilibrium.
Understanding the demand curve is crucial as it helps to predict how changes in price will impact the quantity demanded. In practical terms, it can guide businesses in pricing strategies to maximize sales and revenue.

Slope of a Curve

The slope of a curve at any given point represents how steep the curve is at that point, or how much \( q \) changes as \( p \) changes. Mathematically, it’s the derivative \( \frac{dq}{dp} \) we determined through implicit differentiation.
At the point \((6.75, 3.248)\), the slope is approximately \(-0.253\). This negative slope implies that an increase in price will result in a decrease in the quantity demanded.
The more negative the slope, the more sensitive the demand is to price changes, which is essential in understanding how elastic or inelastic a market is.

Price Elasticity

Price elasticity of demand measures how sensitive the quantity demanded is to a change in price. If a small price change leads to a large change in demand, the commodity is said to be elastic. Conversely, if price changes do not significantly affect demand, the product is inelastic.
In the context of our demand curve and slope, a slope of \(-0.253\) suggests moderate elasticity. For example, a slight increase in price from \(6.75 to \)6.80 results in a decrease in demand by \(0.013 \times 1000 = 13\) units.
Understanding price elasticity helps businesses and economists in strategic decision-making, such as pricing adjustments, predicting the impacts of market changes, and crafting policies that affect supply and demand dynamics.