Elasticity Business and Economics Applications

In summary: At the unitary elastic point, marginal revenue is zero, and so revenue is at its maximum.In summary, we determined the elasticity of demand for a product with a demand function of P=20-0.02x, and found that demand is elastic for prices between $0 and $500, unit elastic at a price of $500, and inelastic for prices between $500 and $1000. We also determined that revenue is maximized at a price of $500, where demand is unit elastic, and decreases as price increases or decreases.
  • #1
monster123
1
0
Elasticity

The Demand function for a product is modeled by

P=20-0.02x, less than or equal to x less than or equal to 1000

Where p is the price per unit in dollars and x is the number of units.

A. Determine when the demand is elastic, inelastic, and of unit elasticity.

B. Use the result of part (a) to describe the behavior of the revenue function.I started the problem using n=p/x/dp/dx and plugged in the numbers into the formula and did the derivative after computing I end up with an answer of -999 with absolute value of 999 is this correct? if not what could be my mistake?
 
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  • #2
According to Wikipedia, the "Point-price elasticity of demand" is given by:

\(\displaystyle E_d=\frac{P}{Q_d}\times\d{Q_d}{P}\)

Here, we have:

\(\displaystyle P=\frac{1000-x}{50}\implies x=1000-50P\)

\(\displaystyle Q_d=x\)

And so we find:

\(\displaystyle E_d=\frac{\dfrac{1000-x}{50}}{x}\times(-50)=1-\frac{1000}{x}\)

A. We have the following:

Demand is elastic: $E_d<-1$

\(\displaystyle 1-\frac{1000}{x}<-1\)

\(\displaystyle 2-\frac{1000}{x}<0\)

\(\displaystyle \frac{x-500}{x}<0\)

Our critical values are:

\(\displaystyle x\in\{0,500\}\)

We then find the inequality is true on the interval:

\(\displaystyle (0,500)\)

Next, we have:

Demand is relatively inelastic:

\(\displaystyle -1<E_d<0\)

What do you get when solving this inequality?
 
  • #3
To follow up, first I want to point out that it is said the price elasticity of demand for a good is perfectly elastic when:

\(\displaystyle E_d=-\infty\)

And we see that:

\(\displaystyle \lim_{x\to0^{+}}E_d=-\infty\)

Okay, back to the case of relative inelasticity:

\(\displaystyle -1<1-\frac{1000}{x}<0\)

\(\displaystyle -2<-\frac{1000}{x}<-1\)

\(\displaystyle 2>\frac{1000}{x}>1\)

\(\displaystyle \frac{1}{2}<\frac{x}{1000}<1\)

\(\displaystyle 500<x<1000\)

So, the elasticity of demand is relatively inelastic on:

$(500,1000)$

Perfectly inelastic:

\(\displaystyle E_d=0\)

\(\displaystyle 1-\frac{1000}{x}=0\)

\(\displaystyle x=1000\)

Unit elasticity:

\(\displaystyle E_d=-1\)

\(\displaystyle 1-\frac{1000}{x}=-1\)

\(\displaystyle x=500\)

Let's summarize our findings in the following table:

Elasticity type
Quantity demanded
Perfectly elastic
$x=0$​
Relatively elastic
$0<x<500$​
Unitary elastic
$x=500$​
Relatively inelastic
$500<x<1000$​
Perfectly inelastic
$x=1000$​

Now, to see the effect on revenue $R$, we have:

\(\displaystyle R'=P\left(1+\frac{1}{E_d}\right)\)

\(\displaystyle R'(x)=\frac{1000-x}{50}\left(1+\frac{x}{x-1000}\right)=\frac{1000-x}{50}-\frac{x}{50}=\frac{500-x}{25}\)

On a graph with both a demand curve and a marginal revenue curve, demand will be elastic at all quantities where marginal revenue is positive. Demand is unit elastic at the quantity where marginal revenue is zero. Demand is inelastic at every quantity where marginal revenue is negative.

We have:

\(\displaystyle R'(x)>0\) for $0\le x<500$

\(\displaystyle R'(x)=0\) for $x=500$

\(\displaystyle R'(x)<0\) for $500<x\le1000$

This agrees with our table. When demand is elastic, revenue moves with the price, that is, as price increases, so does revenue, and as price decreases so does revenue. When demand is inelastic, revenue moves against price, that is, as price increases revenue decreases, and as price decreases, revenue increases.
 

FAQ: Elasticity Business and Economics Applications

1. What is elasticity in business and economics?

Elasticity in business and economics refers to the measure of the responsiveness of a product or service to a change in price. It is used to determine how a change in price affects the quantity demanded or supplied of a particular good or service.

2. Why is elasticity important in business and economics?

Elasticity is important in business and economics because it helps firms make decisions regarding pricing strategies, production levels, and revenue. It also helps in understanding consumer behavior and market trends.

3. What are the different types of elasticity?

There are four main types of elasticity: price elasticity of demand, price elasticity of supply, income elasticity of demand, and cross-price elasticity of demand. Each type measures the responsiveness of quantity demanded or supplied to a specific factor such as price, income, or the price of related goods.

4. How is elasticity calculated?

Elasticity can be calculated using the following formula: Elasticity = (% change in quantity / % change in price). This gives a numerical value that indicates the degree of responsiveness of quantity to a change in price. A value greater than 1 indicates elasticity, a value less than 1 indicates inelasticity, and a value equal to 1 indicates unit elasticity.

5. What are some real-world applications of elasticity in business and economics?

Elasticity is used in various business and economic scenarios, such as setting prices for goods and services, determining the effect of taxes on consumer behavior, predicting the impact of changes in income on demand for certain goods, and analyzing the impact of substitute or complementary products on consumer choices.

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