Economics 50: Intermediate Microeconomics Microeconomics Summer 2010 Stanford University Michael Bailey Lecture Lecture 1: Supply Supply, Demand, Demand, and Elastici Elasticities ties1
Overview A market is a collection of agents who want to trade. Market demand and supply is the summation of
the individual demand and supply. A very useful metric is how much much quantit quantity y changes changes with respect to price. price. The slope of the demand demand
curve is one way to measure this, but the slope depends on the units of measurement. The own price elasticity, the percentage change in quantity demanded per percentage change in price,
is a unit-less measure of the response and is denoted by "Q;P = "Q;P < 1 )
own-price elastic
"Q;P = 1 )
own-price unit elastic
0 < " Q;P < 1 )
own-price inelastic
"Q;P > 0 )
Gi¤en Good (upward sloping demand)
% %Q Q % %P P :
Every linear demand curve has an own-price inelastic and elastic region. Elasticit Elasticity y can change depending depending on the starting starting point before the price change. The Arc Elasticit Elasticity y is
used more often in practice because it gives the same value no matter which points are used as the starting or ending points: ARC"Q;P =
Q1 Q0 Q1 +Q0 P 1 P 0 P 1 +P 0
for starting points (P 0 ; Q0 ) and ending points (P 1 ; Q1 ):
The own-price elasticity of a good is determined by the availability of substitutes and complements,
the portion of one’s budget the good occupies, and time horizon considered (short run vs. long run). If we make an assumption about the shape of the demand curve, we can identify the parameters of the
demand function if we know the elasticity at a given point on the curve. Change in expenditure for a good per change in price is a function of the own-price elasticity of the
good. 1 These
lecture notes draw heavily from the notes of Luke Stein and Manuj Garg, the previous instructors of Econ 50 during the summer (and presumably presumably they based based their their notes notes heavily heavily on previous previous instructor instructors, s, so on and so forth), as well well as the class class notes of Ran Abrimitzky and Mark Tendall, the academic year professors of Econ 50.
1
The income elasticity of demand is the percentage change in quantity demanded per percentage change
in income "Q;I =
% %Q Q % %I I :The
cross-pri cross-price ce elasticit elasticity y of demand demand is the percen p ercentage tage change in quantit quantity y
demanded demanded per percentage percentage change in another another good’s price "Q;P y = "Q;I > 0 )
Normal Good
"Q;I < 0 )
Inferior Good
"Q;I > 1 )
Luxury Good
"Q;P y > 0 )
Good X and Y are Substitutes
"Q;P y < 0 )
Good X and Y are Complements
% %Q Q % %P P y :
Markets A market is a collecti collection on of agents agents who want want to trade. trade. Note that the de…nition de…nition of someone who "wants" "wants" to trade could could be interpre interpreted ted in a very very general way. way. An agent might might only be willing willing to trade for the item at prices that are never seen in the market, and thus never engages in trade in the market, and yet still might be considered considered as part of the market. In practice, practice, it is usually those who are actively actively trading trading in the market market that are considered part of the market and those who could potentially be a part of the market if the prices are low, or high, enough are considered part of a potential market. The reservation price is the maximum price the buyer is willing to pay to acquire the good, and the minimum price the seller is willing to accept to sell the good. The individual demand is a function that expresses the quantity of the good demanded by the individual at each each price. price. The market demand is a function that expresses the quantity of the good demanded by all the individuals in the market at each price. Example 1 Market for Ipods
We organize the market by dividing up the agents into "buyers" and "sellers". For simplicity we will say that the buyers each want one, and only one, Ipod and the sellers each own one, and only one, Ipod. We sort the buyers by those with the highest reservation price to those with the lowest, and the same with the sellers. Figure 1 shows the amount of Ipods demanded by John, the buyer with the highest reservation price, at each price. price. Notice Notice that John demands demands 1 Ipo Ipod when the price price is below below 100$ and 0 otherwise otherwise.. Figure Figure 2 shows the amount of Ipods demanded by Mary at each price.
2
Figure 1: John’s Demand for an Ipod
Figure 2: Mary’s Demand for an Ipod
3
These are individual demand functions. If John and Mary were the only buyers in the market, the amount of Ipods demanded by the buyers for each price would be represented by Figure 3. Figure 3 is the market demand function if John and Mary are the only buyers in the market.
Figure 3: Market Demand for John and Mary
We can do the same thing for the sellers and construct individual supply functions and the market supply function. Incorporating a few more buyers and sellers and the market demand and supply functions begin to look linear, see Figure 4. Notice the market curves jump when a buyer or seller enters the market. If there are many buyers and sellers, as a market at the national level would have, the discrete bumps in the graph would be smoothed out and we would get the typical "Demand and Supply" graph in Figure 5.
4
Figure 4: Market Demand and Supply with Some Market Participants
Figure 5: Market Demand and Supply with Many Market Participants
5
Individual demand could be a function of many things including tastes, weather, time of the day, etc. In this class, we only consider a simpli…ed demand that is a function of market prices (P x ; P y ; : : : Pz ), and the individuals income. X i (P x ; P y ; : : : Pz ; I i ) = Individual demand for good X of agent i P x
= Price of good x
I i
= Income of Individual i
Example 2 Linear Demand
X i (P x ; P y ; : : : Pz ; I i ) = a bx P x by P y ::: bz P z + I i
The market demand is the summation of the individual quantity demand functions of the individuals in the market and is assumed to be a function of market prices and the incomes of the agents in the market. For goods X and Y we would denote market demand by Qx and Qy and individual demand by X i and Y i respectively.
Qx (P x ; P y ; : : : Pz ; I i ; I j ; : : : ; I n ) = Market demand for good X n
=
X X (P ; P ; : : : P ; I ) i
x
y
i
z
i=1
Example 3 Linear Demand: Suppose there are 10 agents in the market with linear demand X i (P x ; P y ; : : : Pz ; I i ) =
a bx P x by P y ::: bz P z + I i ; what is the market demand curve? 10
i
j
n
Qx (P x ; P y ; : : : Pz ; I ; I ; : : : ; I ) =
X X (P ; P ; : : : P ; I ) i
x
y
z
i
i=1 10
=
X a b P b P ::: b P + I x x
y
y
z
z
i
i=1
10
=
X I 10(a b P b P ::: b P ) + x x
y y
z
i
z
i=1
For each demand curve, there is an associated inverse demand curve, which relates the price required to generate the quantity demanded . To get the inverse demand from the demand function, just solve for the price as a function of quantity demanded. P x (Qi ; P y ; : : : Pz ; I i ) = Inverse demand for good X of agent i 6
Example 4 Linear Demand
X i (P x ; P y ; : : : Pz ; I i ) = X i = a bP x + I i bP x P x
= a + I i X i =
a 1 i 1 i + I X b b b
= Inverse Demand
We typically use, and plot, the inverse demand curve. It is useful to remember that if price is on the y-axis, the graph is of inverse demand (price as a function of quantity). If quantity is on the y axis, we are working with the demand function. Figure 6: Demand Function Q = a bP
7
Figure 7: Inverse Demand P =
a b
1b Q
The Law of Demand is that demand functions will be downward sloping. This requires that quantity demanded decreases as price rises. A good where quantity demanded rises with the price is called a Gi¤en Good and examples are rare. In theory, it could be that the income e¤ect outweighs the substitution e¤ect
(more of this in later lectures) and so a price increase of an inferior good would cause demand to rise. For example, suppose that you need to eat 2 potatoes to live. You would de…nitely like to have some meat with your potatoes, but meat is expensive. Suppose your income is $10, potatoes are $2 each, and a serving of meat is $6. You would then consume 2 potatoes and one serving of meat. If the price of potatoes rises to $3, you would then consume 3 potatoes as you can no longer a¤ord meat. In this example, potatoes are a Gi¤en Good. Notice that in our model of individual demand, X i and P x are explained by the model and a;b; and I i are not explained by the model but given in the problem. Variables that are explained by the model are endogenous , and variables not explained or "outside" the model are exogenous .
Equilibrium A market is in equilibrium when quantity demanded equals quantity supplied. Recall from Economics 1A that price and quantity changes are movements along the demand (supply) curve and are called changes in quantity demanded (quantity supplied), but if the relationship between quantity demanded (or supplied) changes for all prices, this is a shift in the demand (supply) curve and is called a change in demand (supply). Demand shifters include changes in tastes, income, market characteristics, among many others. Supply
8
shifters include changes in costs, working conditions, government regulations, among many others. Notice that the shape of the demand and supply curves plays a big part in where the equilibrium is, and more importantly how quantity changes in response to price changes. In Figure 8, an increase in demand leads to both a price and quantity increase, whereas in Figure 9, it only leads to a price increase. To Figure 8: An increase in income has shifted demand from Demand0 to Demand1 , leading to a new equilibrium E 1 : Both price and quantity demanded increase.
understand market behavior, we must know how changes in price will a¤ect changes in quantity. One way to measure this is by taking the change in quantity demanded divided by the change in price,
Q P
( =
change, Q = (Q1 Q0 ) where Q0 is the initial quantity demanded and Q1 is the new quantity demanded). This is equal to the slope of the demand curve (Figure 6), or the inverse of the slope of the inverse demand curve (Figure 7). One problem with this measure is that it is sensitive to the units used. Quantity demanded has quantity units like gallons, or Ipods. Price has units of dollars or Euros. Thus Euro, or Ipods per cent.
9
Q P
would have units like gallons per
Figure 9: An increase in income has shifted demand from Demand0 to Demand1 , leading to a new equilibrium E 1 : Because Supply is …xed, only price increases.
Elasticity The own-price elasticity of demand is the percentage change in quantity divided by the percentage change in price and is denoted by "Q;P =
%Q %P
(or "X;P x if referring to individual demand for good X ): This is a
unit-less measure of the responsiveness of quantity demanded to price. Q1 Q0 Q0 P 1 P 0 P 0
%Q = %P
gallons gallons dollars dollars
Q1 Q0 Q0 P 1 P 0 P 0
=
Remark 5 For a downward sloping demand curve, the elasticity will be negative. It is convention in eco-
nomics to report the absolute value of the elasticity. In this class, we will not use this convention. Example 6 Linear Demand: If the inverse demand function is given by P x = 40 Q, what is the own-price
elasticity of demand as the price changes from $20 to 19$? From $34 to $35? Q1 Q0 Q0 P 1 P 0 P 0
"Q;P =
Q1 Q0 Q0 P 1 P 0 P 0
=
"Q;P =
21 20 20 19 20 20
5 6 6 35 34 34
=
=
10
1 20 1 20
1 6 1 34
= 1
=
=
17 3
Figure 10: Example 6
If we want to compute elasticity at a point, we need to consider an in…nitesimal changes in the price at that point: %Q "Q;P = lim = lim P 0 %P P 0 !
!
Q Q P P
= lim P
Q P @Q P = 0 P Q @P Q
!
Example 7 Linear Demand: Find the own-price elasticity as a function of the price for the demand function
X i = a bP x at any point.
@X i @P P x X i "X;P x
= b P x a bP x P x = b a bP x =
Notice that "X;P x (P x = 0) limP x
!
a b
=0
"X;P x (P x ) = 1
"X;P x (P x =
a 2b )
= 1
Since this was derived from a general linear curve, this is true for downward sloping demand curves, the midpoint has elasticity 1, and the elasticity converges to 0 and 1 at the endpoints, see Figure 11. The
11
terminology we use is that if the elasticity is: "Q;P < 1 )
own-price elastic
"Q;P = 1 )
own-price unit elastic
0 < " Q;P < 1 )
own-price inelastic
"Q;P > 0 )
Gi¤en Good (upward sloping demand)
Figure 11: All Linear Demand Curves have an Own-Price Elastic, Unit Elastic, and Inelastic Region.
Thus every linear demand curve has an inelastic region, and an elastic region, separated by the midpoint. (note that if "Q;P > 0; the demand is upward sloping). It is critical to understand the intuition behind this result. Why would a linear curve with an unchanging slope yield such di¤erent estimates for a percentage change in quantity demanded per percentage change in price depending on the starting point? When quantity is very small, a small increase could be a very large percentage increase. Consider the inverse demand curve P x = 40 Q: If Q = 0:001; P = 39:999 and a change in the price of $1 would be small in percentage terms (1=40 = 2:5%); but it would lead to a huge percentage change in quantity (1=:001 = 1000%) : That is why the elasticity is converging to - 1: Similarly if we considered the point P = 0:001; Q = 39:999; we would see huge percentage changes in the price for price changes and very small percentage changes in quantity, thus leading to an elasticity near 0. Remark 8 If every linear demand has an elastic and inelastic region, why do we call some linear demand
curves elastic and others inelastic? This is an abuse of terminology, there is no way you can say that a linear demand curve is elastic or inelastic because every linear demand curve has an elastic and inelastic portion
12
separated by the midpoint. What is usually meant is that over the regions of the lines being compared, one is relatively more own-price elastic than the other. Example 9 Constant Elasticity Demand Function: Let demand be given by X i = aP x b , what is the own
price elasticity as a function of the price?
@X i @P P x X i
= abP x
b 1
P x P xb+1 = a aP x b P 1 = abP x (b+1) x a =
b
"X;P x
= b
The elasticity is a constant, b; and so the percentage change in quantity demanded X i is proportional to the percentage change in the price at all prices. Remark 10 Remember that the formula for elasticity with the derivative is for in…nitesimal changes in the
price. If you take discrete changes in the price, you will get a di¤erent elasticity estimate. For example, for the constant elasticity function with a = 1; b = 1, consider a price change from 1 to
1 2;
quantity demanded
increases from 1 to 2, and "X;P x = 2; which is not equal to 1 from the above calculation. However, as you let the price change get smaller and smaller, the elasticity will converge to 1: Remark 11 X i = aP x
b
is the only constant elasticity demand function.
Figure 12: X = aP
b
Price
for a = 1 and b = 2; 1; 2
5
4
3
2
1
0 0
1
2
3
13
4
5
Quantity
Figure 13: X = aP
b
Price
for a = 2; 1; 2 and b = 1
5 4 3 2 1 0 1
2
3
4
-1
5
Quantity
-2 -3 -4 -5
Arc Elasticity Suppose we computed the elasticity for a price increase from $1 to $2. Would this be equal to the elasticity for a price decrease from $2 to $1? Earlier, we found in the case of P x = 40 Q that for a price increase from $34 to $35 the own-price elasticity is 17 3 . The own-price elasticity for a price decrease from $35 to $34 is: Q1 Q0 Q0 P 1 P 0 P 0
"Q;P =
6 5 5 34 35 35
=
=
1 5 1 35
=
35 17 = 7 6 = 5 3
The problem is that the base price and base quantity is changing. In practice, economists use a measure of the elasticity that does not depend upon the starting point used, this is called the Arc Elasticity and is equal to: ARC"Q;P =
Q AV ERAGE(Q) P AV ERAGE(P )
Q1 Q0 (Q1 +Q0 )=2 P 1 P 0 (P 1 +P 0 )=2
=
Example 12 Constant Elasticity Demand Function: Q = P
1
Q1 Q0 Q1 +Q0 P 1 P 0 P 1 +P 0
=
: What is the own-price arc elasticity for a
price increase from $1 to $2? A price decrease from $2 to $1?
"Q;P =
Q1 Q0 Q1 +Q0 P 1 P 0 P 1 +P 0
"Q;P =
Q1 Q0 Q1 +Q0 P 1 P 0 P 1 +P 0
=
1=2 1 1=2+1 2 1 2+1
=
1 1=2 1=2+1 1 2 2+1
=
14
1=2 3=2 1 3
=
1=2 3=2 1 3
=
=
3=2
3=2
3=2
3=2
= 1
= 1
Using Logarithms to Compute Elasticities Recall from log rules that d log(x) dx
=
d log(x) =
1 x dx x
this looks a lot like an elasticity, with a little rearranging:
d log(Q) = d log(P ) = d log(Q) d log(P )
=
dQ Q dP P dQ Q dP P
= "Q;P
Remark 13 The change in the logs is equivalent to percentage changes. If you plot the demand curve on
log-log paper, the slope at any point is equal to the own-price elasticity. Example 14 Constant Elasticity Demand Function: X i = aP x
b
log(X i ) = log(aP x b ) = log(a) + log(P x b ) = log(a) b log(P x )
d log(X i ) d log(P x )
= b = "X;P x
In practice, elasticities are commonly measured by estimating the function log(X i ) = log(a) b log(P x ) using linear regression techniques (regressing log(X i ) on log(P x )): The coe¢cient on log(P x ) is the elasticity. One problem with estimating the elasticity using real world data is that observations are just a cloud of (Price, Quantity) combinations. All we know is that this is where quantity supplied equals quantity demanded. We don’t know whether these points are from shifting demand or supply. This is known as the identi…cation problem . To get identi…cation, you need to …nd a time in the market where you can argue that
demand shifted or supply shifted while the other did not (natural experiments). Much easier in a market with a constant supply (housing) or a market like electricity where there is a monopolist who can supply the whole market at any price and you can observe how demand changes as price is varied.
Identifying Demand Curve Using Elasticity If we make an assumption about the shape of the demand curve, we can estimate the parameters from an elasticity estimate. 15
Example 15 Suppose we know that at P x = 20; Q = 20; "Q;P = 1: We further assume that demand is
linear: Q = a bP x : What can we say about a; b?
@Q P P = b = 1 @P Q Q Q b = = 1 P
"X;P x
=
b = 1 20 = a 20 a = 40
What Determines the Elasticity Is there a way we could tell what products would be own-price elastic or own price inelastic? How should we think about elasticity intuitively? Substitutes/Complements: Demand for a good is more own-price elastic when there are good substi-
tutes for the good. If there are good substitutes, even a small increase in price might lead many people to buy the substitute. Example: Butter and "I Can’t Believe It’s Not Butter!". Demand for a good is more own-price inelastic when there are complements to the good. Example: Iphone apps. High Need/Addiction: Demand is less own-price elastic when the good is addictive or when there is
critical need for the good. Examples: Cocaine and Vaccines. Portion of Budget: Demand is less own-price elastic when a consumers expenditure on the good is
small relative to income. Example: Cereal. Time Horizon: Demand is less own-price elastic in the short run compared to the long run. Example:
Gasoline. Others: You might think of other good characteristics that a¤ect elasticity. For example, there are
some goods, like software programs, that take a large investment to learn. It is costly to switch to a substitute so the good will be less own-price elastic.
16
Example 16
Cigarettes
"Q;P = 0:107
Breakfast Cereal
"Q;P = 0:031
Cigars
"Q;P = 0:756
Vacation Airline Travel
"Q;P = 1:52
Bus Travel to Work
"Q;P = 0:04
Oil, U.S.
"Q;P = 0:061
Price Elasticity and Expenditure There is a link between own-price elasticity of demand and expenditure on goods. Suppose we wanted to see the sensitivity of expenditure to changing prices. Expenditure on good X by agent i equals P x X i = E which also equals seller revenue. Recall that X i (P x ) is a function of the price.
@E @P x
@X i (P x ) = P x + X i (P x ) @P x P x @X i (P x ) = X i (P x ) i + X i (P x ) (via product rule) X (P x ) @P x
|
{z
Multiply by
X X
}
= X i (P x )"X;P x + X i (P x ) = X i (P x )("X;P x + 1) 0 =
if "X;P x = 1
> 0 if "X;P x > 1 < 0 if "X;P x < 1
This can be easily represented by looking at the gains/losses on the demand chart in Figures 14 and 15. Notice that P x X i = Expenditure is the area of the rectangle that has coordinates (0; 0); (0; P ); (Q; P ); and (Q; 0): Notice that if we begin at P 0 ; Q0 and a price change leads us to P 1 ; Q1 :
Expenditure = Revenue = P 1 Q1 P 0 Q0 = P 1 Q1 P 0 Q1 + P 0 Q1
| {z } P Q = (P P )Q (Q Q )P | {z } | {z } 0
0
Add and Subtract P o Q1
1
0
Gain
1
0
1
0
Loss
The Rectangle (P 1 P 0 )Q1 is represented in yellow in Figures 14 and 15 as Revenue Gained. The rectangle (Q0 Q1 )P 0 is represented in red as Revenue Lost. In Figure 14, the price change is in the inelastic region, and in accordance with our theory, the revenue/expenditure gain is larger than the loss. In Figure 15, the 17
price change is in the elastic region, and in accordance with our theory, the revenue/expenditure gain is larger than the loss. What implications does this have for the seller? If the goal is to maximize revenue, the seller will set a price such that the quantity demanded is on the unit elastic portion of the demand curve, otherwise they could change their price and increase expenditure (raise price if in inelastic region, lower price if in elastic region). We will learn later that the pro…t maximizing monopolist will never price in the inelastic region of demand (since they could produce raise their price, thereby raising revenue and lowering costs), and where they set their price in the elastic region depends on the marginal revenue curve (if they lower their price, revenue will go up, but cost will go up as well). Figure 14: Change in Seller Revenue (Consumer Expenditure) with a Price Increase in the Inelastic Portion of the Demand Curve.
18
Figure 15: Change in Seller Revenue (Consumer Expenditure) with a Price Increase in the Elastic Portion of the Demand Curve.
Income and Cross-Price Elasticity of Demand The income elasticity of demand for good X = "X;I i =
%X i %I i
is the percentage change in demand for good
X per percentage change in income. We say that good X is a normal good if demand for X increases when
the income increases "X;I > 0 or
@X @I
> 0 , otherwise good X is an inferior good . If demand for X increases
even faster than income, "X;I > 1; then the good is a luxury good . The cross-price elasticity of demand between goods X and Y = "X;P y =
%X i %P y
is the percentage change
in demand for good X per percentage change in the price of good Y: If the demand for good X increases
when the price of good Y increases "X;P y > 0 or
@X @P y
> 0 , then we say that goods X and Y are gross
substitutes . Otherwise they are gross complements .
"X;I > 0 )
Normal Good
"X;I < 0 )
Inferior Good
"X;I > 1 )
Luxury Good
"X;P y > 0 )
Good X and Y are Gross Substitutes
"X;P y < 0 )
Good X and Y are Gross Complements
Example 17 Let Demand X i = (P x +P y I )
1
; What is the own-price elasticity as a function of the price?
19
Is good X a normal or an inferior good? Are goods X and Y gross complements or gross substitutes?
@X i @P x @X i @I "X;P x
=
@X i = (P x + P y I ) @P y
= (P x + P y I )
2
2
= (P x + P y I )
2
= (P x + P y I )
1
P x (P x + P y I )
1
P x
= X i P x < 0 "X;P y
= X i P y < 0 =) X and Y are gross complements
"X;I = IX i > 0 =) X is a normal good
20
