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These notes essentially correspond to chapter 3 of the text.
1 Elasticities
An important concept in economics is elasticity. Recall that when we are mea- suring elasticity we want to see how responsive a quantity measure (demanded or supplied) is to a change in a dollar measure (a price or income). In par- ticular, elasticities measure the percentage change in the quantity measure to a percentage change in a dollar measure. Some of the more common elasticities and their uses are discussed below.^1
1.1 Own-price elasticity of demand
The own-price elasticity of demand is commonly called the price elasticity of demand (P ED). When we calculate this elasticity we are measuring the per- centage change in quantity demanded (%QD ) to a given percentage change in the own price of the good (%Pown). Mathematically, we have:
P ED =
%QD
%Pown The basic formula to Önd the percentage change in anything is to take the new amount, subtract o§ the old amount, and then divide the di§erence by the old amount. So we can write:^2
%QD =
QnewD QoldD QoldD
and %Pown =
P (^) ownnew P (^) ownold P (^) ownold So our elasticity can then be written as:
P ED =
QnewD QoldD QoldD P (^) ownnew P (^) ownold P (^) ownold Rearranging some terms,
P ED =
QnewD QoldD P (^) ownnew P (^) ownold
P (^) ownold QoldD Letting QnewD QoldD = QD and P (^) ownnew P (^) ownold = Pown (note that these are NOT percentage changes but simply the net change in quantity and price), we have: (^1) For a quick refresher on PED, you can see my principles of micro notes. The link to them is: http://www.belkcollege.uncc.edu/azillant/prmicroch7out.pdf (^2) In your principles of micro class you may have calculated the arc price elasticity of demand where you had %QD = (^) QnewQnew +^ QnewQold 2
. This formula was used so that you would get the
same measure of elasticity regardless of whether you started from the old quantity or the new quantity.
P ED =
QD
Pown
P (^) ownold QoldD Now, take a generic simple demand function of the form (why you would want to do this will become clear in a few steps):
QD = a bPown We know that:
QnewD = a bP (^) ownnew QoldD = a bP (^) ownold Now, subtract the bottom equation from the top equation to get:
QnewD QoldD = a bP (^) ownnew a + bP (^) ownold Simplify, recalling that QnewD QoldD = QD and P (^) ownnew P (^) ownold = Pown:
QD = bP (^) ownnew + bP (^) ownold
QD = b
P (^) ownnew P (^) ownold
QD = b (Pown)
QD
Pown
= b
So we now have a very simple result for our P ED. If we have a demand function (NOTE that this is a demand function and NOT an inverse demand function), then:
P ED = b
P (^) ownold QoldD A verbal way of stating this is that PED is the ratio of a price to its cor- responding quantity (as given by the demand function) times the coe¢ cient on Pown (which is b in the example). For those of you with a calculus background, recall that for inÖntesimal (very small) changes in Pown, the relationship (^) PQownD
becomes the derivative, (^) dPdQownD , or if the demand function is a more complex ver- sion with income and prices of substitutes and complements, we get the partial derivative, (^) @P@QownD. Hopefully, this makes some sense as (^) PQownD is simply a rate of change, which is exactly what a derivative is. Now that the gory mathematical details have been laid out, What does it all mean? We say that the demand for a good is elastic when the j%QD j > j%Pownj or when (^) j%j%PQownD^ jj > 1. We say that the demand for a good is
inelastic when j%QD j < j%Pownj or when (^) j%j%PQownD^ jj < 1. In the rare case
1.2.1 Normal and Inferior goods
The sign of the IE determines if the good is a normal good or an inferior good. A normal good is a good with a positive IE. We call it normal because if income increases then the quantity demanded of the good also increases (recall this rule from the principles class). An inferior good is a good with a negative IE. We call them inferior because as consumers earn more income they shift away from these goods, which is contrary to what we usually believe happens when consumers earn more income. Examples of such inferior goods are used clothing and Ramen noodles.
Classifying normal goods ñnecessities and luxuries If a good is a nor- mal good we can classify it as either a luxury good or a necessity. A necessity is a good with an IE between 0 and 1. Consider electricity. If you receive a moderate increase in your income you are unlikely to change your electric- ity purchases given this change in income. Perhaps you will consume a slight amount more, perhaps not. Since there is such a small change in quantity demanded when your income rises, we classify goods such as electricity as ne- cessities. However, with even small to moderate increases in income we may see large increases in quantity demanded of other items. Suppose you receive a $15 per week raise and this causes you to increase the number of times that you eat out per week to increase from 1 to 2. You have seen a 100% increase in the amount of times you eat out per week even though you only had a minor increase in income. This would be considered a luxury good.
1.3 Cross-price elasticity of demand
Cross-price elasticity of demand measures how responsive quantity demanded is to a change in the price of a di§erent good
%P B^
. Technically, it is the percentage change in quantity demanded in response to a percentage change in the price of another good. Mathematically,
X price =
%QAD
%P B
Using the same steps as above, we can Önd that:
X price =
QAD
P B^
P B;old QA;oldD Specify the demand function as:
QAD = a bPown + cY + dP B We can then Önd that the formula for X price elasticity is:
X price = d
P B;old QA;oldD
Verbally, the cross-price elasticity of a good is equal to the ratio of the price of good B to the quantity demanded of good A times the coe¢ cient on the price of good B in the demand function. It is important to note that d can be positive or negative, and that the sign has important implications for the two goods as discussed below.
1.3.1 Substitutes and Complements
The sign of the cross-price elasticity measure is important in determining which goods are substitutes and which goods are complements. If the cross-price elasticity between two goods is positive, then the two goods are substitutes. The logic is that if the quantity demanded of one good increases when the price of another good increases, consumers must be substituting away from the good with the now relatively higher price by replacing purchases of the second good with more purchases of the Örst good. If the cross-price elasticity between two goods is negative, then the two goods are complements. The logic is that if the quantity demanded of one good decrease when the price of another good increases, consumers are responding to the price increase of the second good by cutting their purchases of the Örst good. This implies that the goods are consumed together, which means that they are complements.
1.4 Price elasticity of supply
Price elasticity of supply (P ES) measures how responsive quantity supplied (QS ) is to a change in the own-price of a good (Pown). Technically, it is the percentage change in quantity supplied in response to a percentage change in the own-price of the good. Mathematically,
P ES =
%QS
%Pown We Önd that PES is:
P ES =
QS
Pown
P (^) ownold QoldS Using a SUPPLY function such as:
QS = z + vPown We can Önd that the term (^) PQownS = v. So our formula for the PES is:
P ES = v
P (^) ownold QoldS Verbally, the price elasticity of supply for a good is equal to the ratio of the price to the quantity supplied times the coe¢ cient on the price of the good
2.1 Example
I will use the processed pork market, and we will suppose that the government wants to place a per-unit tax (use as the symbol for the tax) on the producers of processed pork. We know that the supply and demand conditions before the tax are:
QD = 286 20 Pown QS = 88 + 40Pown The initial equilibrium price and quantity in this market are $3.30 and 220 units (see chapter 2 notes on how to get this result). When the government places this tax on the suppliers, nothing happens to the demand curve, so we still have QD = 286 20 Pown. However, we know that for every unit sold the producer will have to give back to the government. The easiest way to incorporate into the supply function is to rewrite it as the inverse supply function:
Pown = 2 :2 + : 025 QS Now, all we need to do is subtract the tax, , from Pown because the producer will receive only Pown for each unit. The after-tax inverse supply function is then:
Pown = 2 :2 + : 025 QS Rewriting this as the supply function, we get:
QS = 88 + 40 (Pown ) Or,
QS = 88 + 40Pown 40 Now, if we know the amount of the tax, all we need to do is plug in the amount of the tax and then solve for the equilibrium price and quantity as we did in chapter 2. Suppose the tax is $1: 05 per-unit sold. Then our supply function becomes: QS = 88 + 40Pown 42. Subtracting 42 from 88 will give us supply and demand functions of:
QD = 286 20 Pown QS = 46 + 40Pown Now all we need to do is solve for the equilibrium price and quantity. Set 286 20 Pown = 46 + 40Pown and solve for Pown. You should get Pown = $4. Now plug $4 in for the price to Önd that QS = QD = 206. You should note that the buyers in this market pay $4, but that the sellers only keep $2.95 because they must give $1.05 to the government.^3 (^3) To see this done with a graph, go to my principles of micro chapter 4 lecture. It is not the same example, but the graph works in the same manner. The link is http://www.belkcollege.uncc.edu/azillant/prmicroch4out.pdf.
2.2 Some additional calculations
Now that we have found the new equilibrium price and quantity, we can see how much revenue the government receives, how much society loses from the tax, and who actually bears the burden of the tax ñthe consumer or the producer.
2.2.1 Government revenue
This is a fairly quick calculation to make with a per-unit tax. The government receives the amount of the tax, , for every unit sold in the after-tax market. So all we need is the new equilibrium quantity, which we found to be 206, and the tax rate, which was $1.05 per-unit. Multiply the two of them together to Önd how much revenue the government generates. In this case, it is $216.30.
2.2.2 Deadweight loss
If a market is allowed to operate unimpeded then all of the gains from trade should be captured by the buyers and sellers in the market. The gains from trade can be deÖned as the total amount of consumer and producer surplus that a market can generate. When a market has some restriction imposed on it, such as a price control or a tax, or if the market is not competitive, as in the case of a monopoly, the market is no longer unimpeded. This generates a deadweight loss to society. Deadweight loss is best deÖned as gains from trade that fail to be captured by the market due to some restriction in the market. Look at the picture below, with a linear demand curve and a linear supply curve. The triangular area shaded in black as the deadweight loss due to the tax. When the market was untaxed, consumers and producers shared these gains. Now, however, since the tax is placed on the market, no one receives these potential gains from trade because no units beyond Q ñ after-tax are traded. Thus, this is societyís loss.
An alternative method of Önding the burden of the tax that the consumers must bear is to use the following formula:
consumer burden =
P ES
P ES P ED
You should note that P ES is the price elasticity of supply at the before-tax equilibrium price, which was 0 : 6. Also, P ED is the price elasticity of demand at the before-tax equilibrium price, which was 0 : 3. Plugging these numbers in we get that the consumers must bear a burden of (^) : 6 :(^6 :3) = 23 , which is exactly what we found up above. We can also use this formula to see what happens to the consumers burden as demand becomes more elastic, holding supply constant. Suppose the price elasticity of demand is now ( 0 :9). The consumers burden is : 6 : 6 ( :9) =^
2 5.^ So, as demand becomes more elastic the seller is not able to pass as much of the burden along to the consumer. Alternatively, if supply were to become more elastic, say to 0 : 9 , then we would have the consumers burden as : 9 : 9 ( :3) =^
3 4.^ Thus, if the supply is more elastic the producer can pass more of the burden on to the consumer.
2.2.4 Government revenue and elasticity
The government would prefer to tax goods with both inelastic demand and inelastic supply. The more inelastic the curves, the less the equilibrium quantity traded will fall after the tax is imposed. As an extreme example, suppose that either the supply curve or the demand curve is PERFECTLY inelastic at a quantity level of 220. Suppose the equilibrium price is $4. If the government places a $1 tax on the item, the quantity traded will not fall at all since one curve is perfectly inelastic. While all the burden will fall on whoever has the perfectly inelastic curve, the government will gain a great deal of revenue as the quantity will not fall. To convince yourself of this fact, draw two markets on the same graph, one with a supply and demand curve that are very inelastic and another with a supply and demand curve that are very elastic. Make sure the initial equilibrium price and quantity in both markets is the same. Then place a tax of the same amount on the sellers in each market. You should see the quantity traded in the market with the elastic curves fall by a greater amount than the quantity traded in the market with the inelastic curves. Thus, the government will gain a greater amount of revenue in the market with the inelastic curves.
