Question

The demand function for a product is given by \(p=20-0.02 x, \quad 0

Short answer

The price elasticity of demand when \(x=560\) is -1.4. The total revenue is maximized at \(x=500\) with the price \(p=10\). At this point, the price elasticity of demand does have unit elasticity with the value -1.

Step-by-step solution

Compute the price elasticity of demand

We know that price \(p=20-0.02x\) and quantity \(x=560\). The derivative \(p'(x) = -0.02\). Hence the price elasticity of demand when \(x=560\) will be \(\varepsilon = (20-0.02*560)*560/(-0.02*560) = -1.4\)

Find the values of \(x\) and \(p\) that maximize the total revenue

The total revenue \(R\) is expressed as the product of the quantity \(x\) and the price for that quantity \(p(x) = (20-0.02x)x = 20x - 0.02x^2\). To maximize total revenue, take the derivative \(R'(x) = 20 - 0.04x\), set it to zero and solve for \(x\). \nSo, \(20 - 0.04x = 0\) implies \(x= 20/0.04 = 500\). Substituting \(x=500\) into \(p=20-0.02*500\) gives \(p=10\).

Proving unit elasticity

Let us check the value of the price elasticity at \(x=500\), which we computed in step 2. Unit elasticity occurs when the value of the elasticity equals to -1 (or 1 in absolute terms). Compute \(\varepsilon = p*x'/p'*x = (20-0.02*500)*500/(-0.02*500) = -1\). Hence, the price elasticity of demand has unit elasticity at \(x=500\) and \(p=10\).

Key concepts

Demand Function

The demand function describes the relationship between the price of a good and the quantity demanded by consumers. In our exercise, the demand function is given by \(p = 20 - 0.02x\), where \(p\) is the price and \(x\) is the quantity. This function tells us that as the quantity \(x\) increases, the price \(p\) will decrease, reflecting a typical downward-sloping demand curve.
This demand function is linear, which means that the relationship between \(p\) and \(x\) is a straight line. The constant \(-0.02\) is the slope of the line, indicating how much the price changes when the quantity changes by one unit.

• The negative sign indicates an inverse relationship between price and quantity.
• The intercept \(20\) is the price at which the quantity demanded would be zero (hypothetically, when \(x = 0\)).
• The domain of our interest here is \(0 < x < 1000\).

Total Revenue Maximization

Total revenue (TR) is calculated as the product of price \(p\) and quantity \(x\), given by the function \(R(x) = x \cdot p(x) = x (20 - 0.02x) = 20x - 0.02x^2\). This quadratic function represents a parabola opening downwards, indicating there is a maximum point.
To find this maximum, we use the first derivative of the revenue function, \(R'(x)\), which gives us \(R'(x) = 20 - 0.04x\).

• Setting \(R'(x) = 0\) gives the quantity \(x\) that maximizes the revenue, which is \(x = 500\).
• Placing \(x = 500\) into the demand function, we solve for the price \(p = 10\).

The maximum revenue occurs at these values of \(x\) and \(p\), elucidating the idea of optimizing business profits by balancing production levels and pricing.

Derivative in Economics

In economics, derivatives help measure rates of change. For instance, in our exercise, the derivative of the demand function \(p(x)\) reveals how the price changes with respect to changes in quantity demanded.
Here, since \(p'(x) = -0.02\), it indicates that for each additional unit of the product, the price decreases by \(0.02\). This derivative is crucial as it provides a precise measurement of responsiveness within the demand function.
Further, the derivative is used to find the maximum total revenue by setting the derivative \(R'(x) = 20 - 0.04x\) to zero, providing us a critical point. Utilizing derivatives in this manner aids decision-making by pinpointing optimal production and pricing strategies.

Unit Elasticity

Unit elasticity refers to a situation where the percentage change in quantity demanded is equal to the percentage change in price, hence the absolute value of elasticity is 1. In the context of our exercise, we aim to verify this property when \(x = 500\).
The elasticity formula we use is \(\varepsilon = \frac{p \cdot x'}{p' \cdot x}\). Substituting the known values, \((20-0.02 \cdot 500) \cdot 500 / (-0.02 \cdot 500)\), results in an elasticity of -1.

• This implies that at \(x=500\) and \(p=10\), a 1% increase in price results in a 1% decrease in quantity demanded, or vice versa.
• Unit elasticity highlights a balance where changing price does not affect total revenue.

Understanding unit elasticity is important for grasping the broader implications of pricing and consumer response, allowing businesses to adjust to market conditions effectively.