c 2009 Je rey A. Miron So far, we have derived and characterized individual demand curves.

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1 Lecture 12: Market Demand c 2009 Je rey A. Miron Outline 1. Introduction 2. From Individual to Market Demands 3. The Inverse Demand Curve 4. Extensive Versus Intensive Margins 5. Elasticity 6. Elasticity and Demand 7. Elasticity and Revenue 8. Constant Elasticity Demands 9. Elasticity and Marginal Revenue 10. Marginal Revenue Curves 11. The Income Elasticity of Demand 1 Introduction So far, we have derived and characterized individual demand curves. Now we want to aggregate these to form market demand curves. We also will de ne and characterize the concets of elasticity, revenue, and marginal revenue. This will be useful for later analysis of market euilibrium. 1

2 2 From Individual to Market Demands Let x 1 i ( 1 ; 2 ; m i ) reresent consumer i s demand function for good 1. good 1 is de ned as Then the market demand for X 1 ( 1 ; 2 ; m 1 ; : : : ; m n ) = nx x 1 i ( 1 ; 2 ; m i ) i=1 That is, the market demand for good 1 is the sum of the individual demands. Note that in general market demand deends on the distribution of income. In articular, transferring income from one erson to another, with the total xed, could change overall demand. Most of the time, however, it is a reasonable aroximation to think in terms of reresentative consumers, each earning the average income. In this case, we can just sum all the incomes and write the market demand curve as X 1 ( 1 ; 2 ; M) where M is the sum of all the incomes. that of an individual with income M. In this case, the demand curve is just like We can illustrate the market demand curve in the standard way: 2

3 Grah: A Market Demand Curve y = 10 x Geometrically, the market demand curve is the horizontal sum of the underlying individual demand curves: 3

4 Grah: Summing u Individual Demand Curves y = 9 3x y = 12 2x A = B + C A Market Demand B C D1 D The market demand holds other rices and income xed, so changes in these shift the curve. For examle, an increase in 2, assuming goods 1 and 2 are substitutes, shifts demand for good 1 outward. An increase in income shifts the demand in or out in the same direction, assuming good 1 is a normal good. 3 The Inverse Demand Curve The standard demand curve relates the uantity demanded to rice. Then, annoyingly, standard resentations lot the demand curve with rice on the vertical axis and uantity on the horizontal axis - the oosite of the standard convention for lotting y = f(x): 4

5 In many instances, however, it is useful to talk about rice as a function of uantity. That is, for a given uantity, we can ask what rice would be reuired for that uantity to be demanded. This second version is called the inverse demand function, written as (X). Deending on the alication, it can be more convenient to talk about the demand curve or the inverse demand curve. The key thing is that for a monotonic function, we can always go from one to the other. Consider an examle. If X = a b is the demand curve for good X, then = (a=b) (1=b)X is the inverse demand curve. It tells us the rice at which X units will be demanded. 4 Extensive and Intensive Margins Another small iece of language / notation is the following: When the rice changes, two sorts of adjustments can occur. Someone who is buying the good can increase or decrease the uantity urchased, but still be a urchaser. This is called the intensive margin; existing consumers of the good adjust how intensively they consume the good. In other situations, changes in rice can cause some consumers to enter or leave the market; that is, the change in rice can determine whether a consumer urchases the good or not. 5

6 For examle, if a consumer has references such that two goods are erfect substitutes, we know from earlier derivations of demand curves that the consumer will consume only one good over a certain rice range, and then switch to the other good when the rice ratio hits a threshold. This kind of adjustment is called the extensive margin. The sloe of a market demand curve is a ected by both kinds of adjustments. 5 Elasticity It is often useful to have a measure of how resonsive demand is to a change in rice. One ossibility is to use the sloe of the demand curve, dx() d For examle, a linear demand curve X() = a b has a derivative of b. This aroach is reasonable, but it su ers from a key limitation: the measure of resonsiveness deends on the units in which we measure the uantities. For instance, changing from ounds to ounces would change the resonsiveness by a factor of 16. We want a unit-free measure. The standard measure is elasticity, usually denoted by. follows: The de nition is as The elasticity of demand with resect to rice is the ercentage change in uantity demanded divided by the ercentage change in rice. This measure is unit free since it uses ercentages. Without calculus, this would be de ned as 6

7 = = This is usually negative; we therefore often omit the minus sign. With calculus, the elasticity is given by = = d d d d = 0 () where () is the demand curve for the good in uestion. As an examle, consider the linear demand curve = a b Then = b = b a b Thus, for a linear demand curve, the elasticity is zero when = 0 - that is, where the demand curve intersects the horizontal axis. The elasticity aroaches negative in nity as the uantity aroaches zero. More generally, the elasticity declines in absolute value (becomes less and less elastic) as one moves down the demand curve from high rices to low rices. Another fact about linear demand is that the oint at which = 1 7

8 is where = a=2b This is exactly half way down the demand curve. This fact comes in handy later. 6 Elasticity and Demand If < 1 at a articular value of ; then we say the demand curve is elastic at that oint (remember that the elasticity can change as one moves along the demand curve). If > 1 at a articular value of, then we say the demand curve is inelastic at that oint. If = 1 at a articular value of, then we say the demand curve is unit elastic at that oint. Elasticity tells us how resonsive the demand is to rice. If good substitutes for a good are available, demand will be relatively elastic (resonsive); consumers will switch to the substitutes as rice rises. 8

9 7 Elasticity and Revenue A useful concet in many contexts is the amount of revenue generated in a articular market. By de nition, R = In articular, we often want to know how revenue changes as rice changes. This is a straightforward derivative that can be exressed in an intuitive way: dr d = R0 () = 0 () + (). The e ect of a rice increase on revenue can be ositive or negative because there are e ects in both directions. A higher rice means more revenue er unit, but it also means fewer units demanded given that demand is a decreasing function of rice. The e ect of rice on revenue is ositive if 0 () + () > 0 which is euivalent to 0 () > () or = 0 () > 1: That is, total revenue increases when rice increases if the demand curve is inelastic. This makes sense. If demand is inelastic, the uantity demanded does not resond much to the change in rice. If the uantity demanded does not change much, then the main e ect of an increase in rice is to rovide more revenue er unit sold. Thus, total revenue will go u. We can also see this geometrically. Consider the following diagrams: 9

10 Grah: The E ect of Demand Elasticity on the Resonse of Revenue to Price y = 24 4x demand is inelastic dr/d > y = 9 :5x demand is elastic dr/d <

11 Note that the revenue associated with a articular ; combination is the area of the rectangle associated with that combination. 8 Constant Elasticity Demands One useful demand curve is the following: = a Why? Let s derive the elasticity of this demand curve: = 0 () = a 1 a =. This is not always realistic, but it is sometimes convenient. logs of both side to get Note that we can take ln = ln a + ln Demand curves are often seci ed in this form for estimation uroses. 9 Elasticity and Marginal Revenue It is often of interest to see how revenue changes when we change uantity (e.g., in cases where a rm controls uantity and takes rice as determined by the demand curve). This concet is known as marginal revenue, and is useful in later analyses of market euilibrium. This is similar to the derivation above: Given that 11

12 R =, it follows that dr d = + d d = 1 + d d 2 3 = d 5 d = Let s think about the intuition behind this. If = 1, an increase in uantity has no e ect on revenue. If < 1, meaning demand is very resonsive to rice, you can cut the rice just a bit to increase uantity; this means revenue goes u. If > 1, meaning demand is not very resonsive to rice, you have to cut rice a lot to increase uantity; this means revenue will decline.. 10 Marginal Revenue Curves It will be useful later on to be familiar with grahical resentations of marginal revenue curves. Assume the inverse demand curve is = a b 12

13 Then R = a b 2 so dr d = a 2b The marginal revenue curve is also linear. It has the same intercet as the demand curve. Also, it is twice as stee. Grahically, this looks like: 13

14 Grah: Linear Demand and Marginal Revenue y = 4 x y = 4 2x MR D The constant elasticity demand curve is = a so the inverse demand curve is = a 1 1 and revenue is given by R = a The marginal revenue curve is therefore MR() = ( 1 + 1)a 1 1 If > 1, then marginal revenue is everywhere negative. So, this case does not yield lausible redictions. Instead, we will only want to consider cases where < 1. In that case, the marginal revenue curve is roortional to, but lower than, the demand curve. 14

15 Grah: Demand and Marginal Revenue in Constant Elasticity Case y = 2 1=2 x 1=2 y = (1=2) 1=2 x 1= D 1 MR The Income Elasticity of Demand If = (; m) then we can de ne an income elasticity of demand as d m dm 15