User talk:Jackftwist/PED Notes

Definition
PED is a measure of the sensitivity (or responsiveness) of the quantity of a good or service demanded to changes in its price. The formula for the coefficient of price elasticity of demand for a given product is:


 * $$E_d = \frac{\%\ \mbox{change in quantity demanded}}{\%\ \mbox{change in price}} = \frac{\Delta Q_d/Q_d}{\Delta P/P} = \frac{\Delta Q_d}{\Delta P}\times\frac{P}{Q_d} $$ ,

where Qd and P are the quantity demanded and its corresponding price at a point on the demand curve, ∆Qd is the change in quantity demanded between that point and a second point on the curve, and ∆P is the corresponding change in price.

This measure of elasticity is sometimes referred to as the own-price elasticity of demand for a good, i.e., the elasticity of demand with respect to the good's own price, in order to distinguish it from the elasticity of demand for that good with respect to the change in the price of some other good, i.e., a complementary or substitute good. The latter type of elasticity measure is called a cross-price elasticity of demand.

Properties of Price Elasticity of Demand
As the difference between the two prices or quantities increases, the accuracy of the PED given by the formula above decreases for a combination of two reasons. First, the PED for a good is not necessarily constant; as explained below, PED can vary at different points along the demand curve, due to its percentage nature. Second, percentage changes are not symmetric; instead, the percentage change between any two values depends on which one is chosen as the starting value and which as the ending value. For example, if quantity demanded increases from 10 units to 15 units, the percentage change is 50%, i.e., (15 − 10) ÷ 10 (converted to a percentage). But if quantity demanded decreases from 15 units to 10 units, the percentage change is −33.3%, i.e., (15 − 10) ÷ 15.
 * 1) PED will usually be negative. The above formula usually yields a negative value, due to the inverse nature of the relationship between price and quantity demanded, as described by the "law of demand". For example, if the price increases by 5% and quantity demanded decreases by 5%, then the elasticity at the initial price and quantity = −5%/5% = −1.  Because the PED is negative for the vast majority of goods and services, economists often refer to price elasticity of demand as a positive value (i.e., in absolute value terms).  In a few extremely rare exceptions, the calculated value of PED is not negative:
 * 2) As discussed below, PED = 0 in the special case where the demand curve for a good is perfectly inelastic (i.e., vertical).
 * 3) Two small classes of goods, Veblen and Giffen goods, have a calculated PED value (not just absolute value) greater than 0.
 * 4) Elasticity is a pure or dimensionless number. In the formula for computing the PED, the units of measurement for both price and quantity demanded cancel out.  As a result, PED is completely independent of units of measurement. For any point on the demand curve, PED will be the same whether price is denominated in dollars, pounds, euros, etc., and whether quantity demanded is measured in ounces, kilograms, barrels, and so on.  Changing the units of measurement does not affect the value of PED.  In contrast, slope does depend on the units used to measure both price and quantity, so changing the units of measurements may also change the slope.
 * 5) Elasticity is not the same thing as the slope of the demand curve. The slope of the demand curve is ∆P/∆Qd, with price on the vertical axis and quantity demanded on the horizontal.  But as the last equation in the definition above shows, PED equals the reciprocal of the slope (i.e., ∆Qd/∆P) multiplied by the ratio of the price to the quantity demanded.

Two alternative elasticity measures avoid or minimise these shortcomings of the basic elasticity formula: point-price elasticity and arc elasticity.

Point-price elasticity
One way to avoid the accuracy problem described above is to minimise the difference between the starting and ending prices and quantities. This is the approach taken in the definition of point-price elasticity, which uses differential calculus to calculate the elasticity for an infinitesimal change in price and quantity at any given point on the demand curve. If the equation for the demand function, $$Q_d = f(P)$$, is known, then the point-price elasticity of demand at any point $$(Q_d, P)$$ on the demand curve is:
 * $$E_d = \frac{P}{Q_d}\times\frac{dQ_d}{dP}$$

In other words, it is equal to the absolute value of the first derivative of quantity demanded with respect to price $$(dQ_d/dP)$$ multiplied by the point's price $$(P)$$ divided by its quantity demanded ($$Q_d)$$.

In terms of partial-differential calculus, point-price elasticity of demand can be defined as follows: Let $$x(p,w)$$ be the demand for goods $$x_1,x_2,\dots,x_L$$ as a function of parameters price and wealth, and let $$x_l(p,w)$$ be the demand for good $$l$$. The elasticity of demand for good $$x_l(p,w)$$ with respect to price $$p_k$$ is


 * $$E_{x_l,p_k} = \frac{\partial x_l(p,w)}{\partial p_k}\cdot\frac{p_k}{x_l(p,w)} = \frac{\partial \log x_l(p,w)}{\partial \log p_k}$$

Arc elasticity
Another solution to the problem of having a PED dependent on which of the two given points on a demand curve is chosen as the "original" point and which as the "new" one is to compute the percentage change in P and Q relative to the average of the two prices and the average of the two quantities, rather than just the change relative to one point or the other. Loosely speaking, this gives an "average" elasticity for the section of the actual demand curve—i.e., the arc of the curve—between the two points. As a result, this measure is known as the arc elasticity, in this case with respect to the price of the good. The arc elasticity is defined mathematically as:


 * $$E_d = \frac{\frac{P_1 + P_2}{2}}{\frac{Q_{d_1} + Q_{d_2}}{2}}\times\frac{\Delta Q_d}{\Delta P} = \frac{P_1 + P_2}{Q_{d_1} + Q_{d_2}}\times\frac{\Delta Q_d}{\Delta P}$$

This method for computing the price elasticity is also known as the "midpoints formula", because the average price and average quantity are the coordinates of the midpoint of the straight line between the two given points. The arc elasticity also has the added advantage of eliminating the symmetry problem associated with simple percentage changes, as discussed above. However, because this formula implicitly assumes the section of the demand curve between those points is linear, the greater the curvature of the actual demand curve is over that range, the worse this approximation of its elasticity will be.

Constant elasticity functions
A demand function of the form $$P=aQ^{-\beta}$$ where a and beta are constant, has constant price elasticity of $$1/\beta$$. In fact these type of functions are commonly used in empirical studies - and this is where all those numbers in the "Selected price elasticities" come from. Otherwise, the elasticity wouldn't be constant and a single number (w/o additional information) wouldn't be very informative. So the section contradicts somewhat the material in the point price elasticity section (which I tried to clarify a bit).radek (talk) 02:44, 6 March 2010 (UTC)
 * Yes, I do believe you've shown up my own lack of knowledge about PEDs in real life: anything you could add or amend would very much be appreciated. - Jarry1250 [Humorous? Discuss.] 12:19, 6 March 2010 (UTC)

--Jackftwist (talk) 20:06, 5 April 2010 (UTC)

Negative sign vs. absolute value
Thanks for clarifying those elasticity definitions in the PED article. The article is inconsistent about using the negative sign—as are most of the textbooks—so I'd been wanting to clean that section up for awhile, but have been preoccupied with other projects (limited resources vs. unlimited wants and all that). One of those projects is planning a rather substantial revision of the article in pursuit of FA status. Several members of the Econ Project have been discussing this, and we'd certainly welcome your help. (Most of the revision would involve some reorganization of what's there, but there'll be some additional material, too.) We'll put something about the proposed changes on the article's talk page eventually (before actually making any major revisions) to give anyone interested the opportunity to comment and contribute.

I noticed that you removed the absolute value signs, which weren't consistent with the notation in the table. I've been surveying over a dozen different introductory and intermediate texts to see how they treat a number of different points, including this one, so we can follow the consensus practice, assuming there is one. So far, I haven't found any texts that keep the negative sign consistently throughout their discussion. Most of them just say up front that PED is always negative (with minor exceptions), so we adopt the convention of treating it as positive because it's easier to talk about that way. The rest eventually explicitly switch from the negative sign to absolute value. All these approaches have their advantages and disadvantages, of course.

One of the things I'd like to do in any revision is to be completely consistent in treating the negative sign, because switching back and forth is bound to confuse many of those who read the article, who're likely new to the topic anyway. I'm leaning strongly toward what seems to be the majority approach, which basically how the intro to the PED is currently written: i.e., (as mentioned above) PED is usually negative, so we usually drop the negative sign by convention. Overall, I think this approach makes the rest of the article easier for the typical reader to understand. Personally, even after teaching this topic for years, I still find it a lot easier to think about the elasticity of a demand curve increasing as the coefficient increases (in absolute value). I find it less intuitive, and inevitably get something backwards eventually, when I have to think about a smaller negative value meaning demand is more elastic!

I'd welcome your thoughts on the subject. Thanks again for your edits. --Jackftwist (talk) 17:26, 4 June 2010 (UTC)

Example of info footnotes plus reference footnotes & bibliography
See Brander-Spencer Model wiki/Brander_Spencer_model  --Jackftwist (talk) 17:28, 3 July 2010 (UTC)