MHB Elasticity Business and Economics Applications

[monster123]

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?

 

[MarkFL]

According to Wikipedia, the "Point-price elasticity of demand" is given by:

$$E_d=\frac{P}{Q_d}\times\d{Q_d}{P}$$

Here, we have:

$$P=\frac{1000-x}{50}\implies x=1000-50P$$

$$Q_d=x$$

And so we find:

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

A. We have the following:

Demand is elastic: $E_d<-1$

$$1-\frac{1000}{x}<-1$$

$$2-\frac{1000}{x}<0$$

$$\frac{x-500}{x}<0$$

Our critical values are:

$$x\in\{0,500\}$$

We then find the inequality is true on the interval:

$$(0,500)$$

Next, we have:

Demand is relatively inelastic:

$$-1<E_d<0$$

What do you get when solving this inequality?

 

[MarkFL]

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:

$$E_d=-\infty$$

And we see that:

$$\lim_{x\to0^{+}}E_d=-\infty$$

Okay, back to the case of relative inelasticity:

$$-1<1-\frac{1000}{x}<0$$

$$-2<-\frac{1000}{x}<-1$$

$$2>\frac{1000}{x}>1$$

$$\frac{1}{2}<\frac{x}{1000}<1$$

$$500<x<1000$$

So, the elasticity of demand is relatively inelastic on:

$(500,1000)$

Perfectly inelastic:

$$E_d=0$$

$$1-\frac{1000}{x}=0$$

$$x=1000$$

Unit elasticity:

$$E_d=-1$$

$$1-\frac{1000}{x}=-1$$

$$x=500$$

Let's summarize our findings in the following table:

[table="width: 400, class: grid, align: left"]
[tr]
[td]

Elasticity type​

[/td]
[td]

Quantity demanded​

[/td]
[/tr]
[tr]
[td]Perfectly elastic[/td]
[td]

$x=0$​

[/td]
[/tr]
[tr]
[td]Relatively elastic[/td]
[td]

$0<x<500$​

[/td]
[/tr]
[tr]
[td]Unitary elastic[/td]
[td]

$x=500$​

[/td]
[/tr]
[tr]
[td]Relatively inelastic[/td]
[td]

$500<x<1000$​

[/td]
[/tr]
[tr]
[td]Perfectly inelastic[/td]
[td]

$x=1000$​

[/td]
[/tr]
[/table]​

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

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

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

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:

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

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

$$R'(x)<0$$ 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.