September 15, 2008 Elasticity of supply The price elasticity of supply measures how quantity offered of a good responds to a change in the good s price. It is defined the same as price elasticity of demand, only for points on a supply curve: ɛ s = (Q 2 Q 1 )/[(Q 1 + Q 2 )/2] (P 2 P 1 )/[(P 1 + P 2 )/2]. It shows how flexible sellers are to a change in price. Example: Land has an inelastic supply, but the amount of land used for farming has a more elastic (still not very elastic) supply, as more forests can be cut down for farming, etc. So the supply curve for land is almost vertical. An important determinant of elasticity of supply of a good is the time frame considered. Over a long period of time supply tends to be more elastic. Over time, firms can change their production technologies to be able to supply more a good if its price has increased, or supply less of it if its price has decreased. Another good that has low elasticity of supply, at least in the short run is hotel rooms. The higher elasticity of supply and elasticity of demand are, the less of a price increase a leftward shift in supply will cause. The lower both elasticity of demand and elasticity of supply are, the bigger a change in price due to a leftward shift in supply will be. Thus when considering whether to impose a tax, which would shift the supply curve, the government considers both the elasticity of supply and the elasticity of demand for a good. A tax on hotel rooms was imposed in NYC in the early 90 s. Since (shortrun) supply elasticity for hotel rooms is low but demand elasticity is high, the prices charged actually fell. 1
A gasoline tax was raised for New York State in the mid-90 s. The (shortrun) demand for gasoline has a low elasticity but the supply for gasoline to New York State has a high elasticity. This is because sellers can sell in so many other places. The price went up for consumers, but sellers kept the same price. Examples of supply curves and their elasticities. Perfectly inelastic supply: Elasticity equals zero. P Q=100 Q 2
Less inelastic, but still inelastic: A 22% increase in price ((P 2 P 1 )/[(P 1 + P 2 )/2] = 0.22)leads to a 10% increase in quantity supplied ((Q 2 Q 1 )/[(Q 1 + Q 2 )/2] = 0.095). P 5 4 100 110 Q 3
Unit elastic supply: A 22% increase in price leads to a 22% increase in quantity supplied. (P 2 P 1 )/[(P 1 + P 2 )/2] = 0.22 and (Q 2 Q 1 )/[(Q 1 + Q 2 )/2] = 0.22. P 5 4 100 125 Q 4
Elastic supply: A 22% increase in price leads to a 67% increase in quantity supplied. P 5 4 100 200 Q 5
Infinitely elastic supply: Any amount of the good is supplied at a particular price P. None of it is supplied at any price below P, and an infinite amount of the good is supplied at a price above P. P 4 Q 6
Any linear supply curve with equation Q = a + bp, with a > 0 and b > 0 is inelastic everywhere. proof. ɛ s = [(Q 2 Q 1 )/(P 2 P 1 )] (Q 1 + Q 2 )/(P 1 + P 2 ) = b (Q 1 + Q 2 )/(P 1 + P 2 ) = b (a + bp 1 + a + bp 2 )/(P 1 + P 2 ) = 1 (2a/b + P 1 + P 2 )/(P 1 + P 2 ) = P 1 + P 2 2a/b + P 1 + P 2 < 1. Note that the larger a is and the smaller b is, the more inelastic the supply curve is. Any linear supply curve with equation Q = bp with b > 0 is unit elastic everywhere. proof. ɛ s = [(Q 2 Q 1 )/(P 2 P 1 )] (Q 1 + Q 2 )/(P 1 + P 2 ) = (bp 2 bp 1 )/(P 2 P 1 ) (bp 1 + bp 2 )/(P 1 + P 2 ) = 1. Any linear supply curve with equation Q = a + bp with a < 0 and b > 0 is elastic between any two points P 1, P 2. proof. ɛ s = [(Q 2 Q 1 )/(P 2 P 1 )] (Q 1 + Q 2 )/(P 1 + P 2 ) = b (Q 1 + Q 2 )/(P 1 + P 2 ) = b (a + bp 1 + a + bp 2 )/(P 1 + P 2 ) = 1 (2a/b + P 1 + P 2 )/(P 1 + P 2 ) = 7
since 2a/b < 0. P 1 + P 2 2a/b + P 1 + P 2 > 1. An example of an partially inelastic supply curve that has P > 0 when Q = 0. It is more realistic to assume that suppliers will not sell anything at any price below some positive price. That corresponds to a supply curve with P > 0 when Q = 0. Any supply curve of the form Q = (P a) 1/n is inelastic for P > P for some P. For instance the supply curve Q = P 1 (so P = 1 + Q 2 ) is inelastic for P 1, P 2 > 2. 8
Another example of a supply curve that is inelastic after a certain point would be a supply curve that has a kink. The slope of the supply curve changes at some price level. P inelastic elastic elastic inelastic Q 9
Some markets might have higher elasticity of supply at low prices and lower elasticity of supply at high prices. This is because some firms have limited production capacity. At low levels of production, a small percentage increase in price can lead to a large percentage increase in production. There may be unused equipment or plants that can be put to use. At high levels of production, it may cost substantially more to increase production further in response to a price increase. That is why a small percentage price increase at high levels of production may only lead to a small percentage increase in production. Applications Agronomists discover a new hybrid of wheat that is more productive than previously existing varieties. This means that more of the new hybrid can be produced with the same value of inputs. Will wheat farmers be better off or worse off? The supply curve for all farmers who adopt the new hybrid will shift to the right. So the total supply curve will shift to the right, because it now costs less to farmers to produce a given amount of wheat. The demand curve for wheat will stay the same there is no reason for it to shift. This will lead to a decrease in the equilibrium price for wheat and an increase in the equilibrium amount of wheat. What will happen to farmers total revenue? The elasticity of demand between the two equilibrium points determines whether total revenue rises or falls. The demand for wheat is likely to be inelastic, as wheat is a basic ingredient in food. This means that the total revenue for farmers will fall when the price decreases. Why will farmers switch to the new hybrid even though the change makes them worse off? Suppose no farmers switch to the new hybrid. Then consider an individual farmer. The farmer could switch to the new hybrid and produce more at a lower 10
cost without the total supply curve being much affected. Thus the equilibrium price will hardly decrease, and the quantity sold by that farmer will increase significantly from the farmer s viewpoint, so that farmer s revenue will increase. The same is true for any individual farmer. By switching to the new hybrid, any farmer can increase their production and amount sold significantly as a fraction of their total production while only slightly decreasing the price. So every farmer will switch to the new hybrid. However the combined effects of this switch increase supply and decrease price significantly. Programs exist that pay farmers to leave some of their land fallow. Goal is to help farmers by reducing the amount of farm products supplied to raise prices. No individual farmer would choose to leave land unused unless paid to do so. These policies help farmers at the expense of consumers. Does drug interdiction increase or decrease drug-related crime? To discourage use of illegal drugs, US government spends billions dollars a year to reduce inflow of drugs into the country. When the government increases the number of federal agents working for the war on drugs, it makes the cost of selling drugs higher. Therefore the supply curve for drugs shifts to the left. The equilibrium quantity of drugs falls and the equilibrium price of drugs rises. What happens to drug-related crime? Since those drug users who still buy drugs with the higher price now need more money, they may engage in more crime to get that money. So the amount of drug-related crime increases. Another policy would be to try to shift the demand curve for drugs to the left through drug education. This would lead to both a decrease in amount consumed and a decrease in price. Thus there would be less drug-related crime and fewer drugs used. Proponents of drug interdiction argue that while demand is inelastic in the short run, demand may be more elastic in the long run. A higher price would discourage new users and over time lead to fewer drug addicts. Thus there would be less drug-related crime in the long-run even though there is more in 11
the short run. But there is a problem drug dealers offer lower prices than the market price to beginning users to let them become addicted, then raise the price for them. So it is not sure that a higher equilibrium price really discourages first-time users. 12