Elasticity The price along the West Coast of the United...

Elasticity The price along the West Coast of the United States for Japanese spruce logs (in dollars per cubic meter) based on the demand (in thousands of cubic meters per day) has been approximated by $$p=0.604 q^{2}-20.16 q+263.067.$$ Source: Kamiak Econometrics. (a) Find the elasticity when the demand is 11 thousand cubic meters a day. (Hint: To find d $/ d p$ when $p$ is expressed in terms of $q,$ you may use the fact that $$\frac{d q}{d p}=\frac{1}{d p / d q}.$$ Review the explanation of the derivative of the natural logarithm to see why this is true.) (b) Is the demand for spruce logs elastic or inelastic? (c) What happens to the elasticity as $q$ approaches 16.6887$?$ Discuss the limitations of this model for the price as a function of the demand.

Elasticity The price along the West Coast of the United States for Japanese spruce logs (in dollars per cubic meter) based on the demand (in thousands of cubic meters per day) has been approximated by
$$p=0.604 q^{2}-20.16 q+263.067.$$
Source: Kamiak Econometrics.
(a) Find the elasticity when the demand is 11 thousand cubic meters a day. (Hint: To find d $/ d p$ when $p$ is expressed in terms of $q,$ you may use the fact that
$$\frac{d q}{d p}=\frac{1}{d p / d q}.$$
Review the explanation of the derivative of the natural logarithm to see why this is true.)
(b) Is the demand for spruce logs elastic or inelastic?
(c) What happens to the elasticity as $q$ approaches 16.6887$?$ Discuss the limitations of this model for the price as a function of the demand.

Key Concepts

Price Elasticity of Demand

This concept measures how sensitive the quantity demanded of a good is to a change in its price. It is defined as the percentage change in quantity demanded divided by the percentage change in price. This measure is central in economics to determine how much the quantity demanded will respond to a change in price, and it distinguishes between elastic demand (high responsiveness) and inelastic demand (low responsiveness).

Inverse Function Derivative

When a function is given in a form where the dependent variable is expressed in terms of the independent variable, sometimes it is necessary to compute the derivative of the inverse function. This is achieved using the relation that the derivative of the inverse function is the reciprocal of the derivative of the original function; that is, if p is expressed in terms of q, then dq/dp is 1/(dp/dq). This technique is commonly used in elasticity calculations where prices are given as a function of quantity.

Elastic versus Inelastic Demand

This key concept differentiates between the cases where the demand is highly sensitive to price changes (elastic demand) and where it is not (inelastic demand). If the absolute value of the price elasticity is greater than 1, the demand is considered elastic, meaning that a small change in price results in a large change in quantity demanded. Conversely, if the absolute value is less than 1, the demand is inelastic, indicating that price changes have a relatively small effect on quantity demanded.

Limitations of Mathematical Models

Mathematical models, such as the price function in this scenario, provide simplified representations of real-world scenarios. They may have limitations, especially near points where the function behavior changes dramatically (for example, when derivatives become very large or small), which may lead to unrealistic predictions. It is important to recognize that models may break down or lose predictive accuracy near critical values or under extreme conditions, indicating the need for careful interpretation and the consideration of other real-world factors.

Transcript

00:01 Okay, so we have p equal 2 .604 q squared minus 20 .16q plus 263 .067.

00:08 You'll notice this is in terms of a p equals equation on a q equals, but i'm going to switch to the book, which you should be able to see on the screen, to the part a, which is to find the elasticity when the demand is 11, we're given this hint right here.

00:24 So if we go back to the video, we have part a.

00:32 Which wants us to find e, which is equal to negative p over q times dq dp.

00:41 But when the hint says, so we're given this hint, which i just showed, was that dq, dp, which we don't have, because we don't have a function of q, is just equal to one over the derivative of p with respect to q.

00:58 So we do have an equation for two, for p, so we can find this.

01:03 So this right here with the star next to it, that's what we need to find first.

01:08 So we're going to go over here, that star, which is dp, dq, is equal to, we're going to take this derivative.

01:17 This derivative is 1 .208q minus 20 .16.

01:27 So now we have that e is equal to negative p, which is .604q squared minus 20 .16q plus 263 .067.

01:45 That's p, so negative p divided by q times 1 .208q minus 20 .16.

01:57 16 and all i'm doing here and i'll go e is equal to negative p over q times one over dp dq all i'm doing is writing that as negative p over q times dp dq i'm just not writing it as two separate fractions okay so now that we have that we also know that we have q is equal to 11.

02:32 So we know we have q equal to 11 and we want to know what our e is.

02:37 So when q is equal to our 11, we have e which is equal to negative.

02:45 And then if you keep your parentheses there, i'm going to let you guys do most of this part by yourself since it is just a plug -in -child equation essentially.

02:54 You're going to have negative 0 .6 .604 q squared and q is 11 so let's go ahead and plug in our 11.

03:16 11 squared minus 20 .16 times 11 plus 263 .067 all of that divided by q which is 11 times 1 .208 times 11 minus 20 .16.

03:42 Okay, so now that you have this, you've plugged in your 11 for q, you should be able to solve this.

03:49 You get an e that's equal to 1 .51.

03:53 This is the answer to part a.

03:57 So a is that e is equal to 1 .51.

04:03 Again, i'm going to pull up the question so we can look at them together.

04:07 A was fine, the elasticity, we did that.

04:09 B is to decide if the demand is elastic or inelastic...