Demand and elasticity Based on sales data over the past...

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function $D(p)=40-2 p,$ where $D(p)$ is the number of DVDs that can be sold in one day at a price of $p$ dollars. a. According to the model, how many DVDs can be sold in a day at a price of $\$ 10 ?$ b. According to the model, what is the maximum price that can be charged (above which no DVDs can be sold)? c. Find the elasticity function for this demand function. d. For what prices is the demand elastic? Inelastic? e. If the price of DVDs is raised from $\$ 10.00$ to $\$ 10.25,$ what is the exact percentage decrease in demand (using the demand function)? I. If the price of DVDs is raised from $\$ 10.00$ to $\$ 10.25,$ what is the approximate percentage decrease in demand (using the elasticity function)?

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function $D(p)=40-2 p,$ where $D(p)$ is the number of DVDs that can be sold in one day at a price of $p$ dollars.
a. According to the model, how many DVDs can be sold in a day at a price of $\$ 10 ?$
b. According to the model, what is the maximum price that can be charged (above which no DVDs can be sold)?
c. Find the elasticity function for this demand function.
d. For what prices is the demand elastic? Inelastic?
e. If the price of DVDs is raised from $\$ 10.00$ to $\$ 10.25,$ what is the exact percentage decrease in demand (using the demand function)?
I. If the price of DVDs is raised from $\$ 10.00$ to $\$ 10.25,$ what is the approximate percentage decrease in demand (using the elasticity function)?

Key Concepts

Elastic and Inelastic Demand

Elastic demand occurs when the percentage change in quantity demanded is greater than the percentage change in price (elasticity greater than one in absolute value), whereas inelastic demand occurs when the percentage change in quantity demanded is less than the percentage change in price (elasticity less than one in absolute value). Understanding whether demand is elastic or inelastic helps businesses anticipate how changes in price will affect their total revenue.

Demand Function

A demand function represents the relationship between the price of a good and the quantity demanded. In economic analysis, it is used to model consumer behavior, showing how many units of a product will be purchased at different prices. This relationship helps in predicting sales and in making pricing decisions.

Price Elasticity of Demand

Price elasticity of demand measures how sensitive the quantity demanded is to a change in the product's price. It is defined as the percentage change in quantity demanded divided by the percentage change in price. This concept is crucial for understanding consumer responsiveness and for making informed pricing strategies.

Critical Price (Zero-Demand Price)

The critical price, often referred to as the choke price or zero-demand price, is the price at which the quantity demanded falls to zero. In linear demand models, this threshold indicates the maximum price consumers are willing to pay before they stop purchasing the product entirely. It is an important concept for determining pricing limits.

Exact vs. Approximate Percentage Change (Using Elasticity)

Exact percentage change in demand is calculated using the actual demand function values before and after a price change. Alternatively, the elasticity function can be used to approximate the percentage change in demand, especially for small changes in price. This approximation relies on the linear responsiveness indicated by the derivative of the demand function, allowing for easier analysis in dynamic pricing decisions.

Transcript

00:06 So we're given the demand function as a function of price p to be 40 minus 2p.

00:13 And if we want to know how many dvds we can sell if we had a price of $10, then that just means to input 10 for our price p, evaluate the function so we can see that we would sell 20 dvds at a price of $10.

00:40 The next part, we want to know what's the maximum price we can charge for the dvd.

00:48 In order to find the maximum price, we want to set our demand function equal to zero and solve for p.

00:55 So 40 minus 2p equals 0, add 2p to both sides, divide by 2.

01:06 We can see that our max price is $20 per dvd.

01:17 Next, we want to find the elasticity of demand as a function of p.

01:24 So let's recall the following, and that is that elasticity of price p is equivalent to the derivative of our demand function with respect to p times the price p over the demand function d.

01:46 In this case, since our demand function is 40 minus 2p, we know that the derivative is negative 2.

01:59 Since we want the elasticity function in terms of a price p, we know to leave p in our function as p, divided by our demand function 40 minus 2p.

02:17 So next we want to simplify this.

02:19 So we obtain negative 2p across the numerator divided by 40 minus 2p in the denominator.

02:28 We can recognize that the numerator and denominator have a common factor of negative 2.

02:35 So we're going to factor out of negative 2 from the numerator.

02:39 We're going to also factor out a negative 2 from the denominator.

02:44 40 divided by negative 2 is negative 20.

02:47 Negative 2p divided by negative 2 is positive p.

02:54 Next we will reduce the negative 2s to obtain a positive 1.

03:00 And as we rewrite the function, we can recall that we can reorder the order of addition and subtraction.

03:10 So instead of negative 20 plus p, i'll write that as p minus 20.

03:16 So p minus 20 is our elast.

03:19 Elasticity function in terms of p.

03:35 So next, let's recall from part c, we know the elasticity function.

03:41 And from part b, we know the maximum price of a dvd.

03:47 So next for part d, we can find when demand is elastic.

03:54 We know that demand is elastic when our elasticity is is less than negative 1.

04:07 So we want to solve for p such that our elasticity function is less than negative 1.

04:14 So we'll start by substituting p divided by p minus 20 for e.

04:25 And next we want to solve one that is less than negative 1.

04:30 To solve this inequality, we want to multiply both sides by p minus 20.

04:35 However, we want to recall the idea that, since our maximum price is $20, we know that p minus 20 for any price that's greater than zero, but again, less than our maximum price of 20, we want to recognize that p minus 20, excuse me, that's a typo, p minus 20 is a negative value, meaning that it's less than zero.

05:01 So if we multiply both sides of this inequality by p minus 20, we need to understand that we, we're really multiplying by negative expression.

05:12 By multiplying by negative expression, we've got to change the direction of our inequality.

05:19 So now we have p is greater than negative 1 times the quantity p minus 20, which we will go ahead and distribute.

05:30 Negative 1 through to give us negative p plus 20.

05:37 Solving for p, we're going to add p to both sides of our inequality.

05:41 So we get 2p is greater than 20.

05:44 Divide both sides by 2, which is a positive number, so it does not affect our inequality direction.

05:50 20 divided by 2 is 10.

05:54 So for elasticity of demand, when our demand is elastic, we need a price of $10.

06:03 Next, we want to find when demand is inelastic.

06:12 Demand is inelastic when our elasticity is greater than negative one.

06:23 So again, substituting our elasticity function into our inequality...

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