8.1 From individual to market demand

Microeconomics I. Antonio Zabalza. University of Valencia 1 Lesson 8: Market demand This lesson covers practical questions related to the use of dem...
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Microeconomics I. Antonio Zabalza. University of Valencia

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Lesson 8: Market demand This lesson covers practical questions related to the use of demand curves as instruments of analysis. The ground will be covered very quickly as a large part of the contents was already treated last year.

8.1 From individual to market demand Consumer theory is a theory about individual behaviour. And yet, most economic issues of interest are of a collective nature. They concern a plurality of agents and their interaction. Market demand and market supply are supposed to represent not the behaviour of an individual agent, but that of a whole set of agents. Here we discuss how to go from the individual demand curve derived in previous lessons to the aggregate demand curve If we represent the individual’s i demand curve for good x as xi ( p x , p y , m i ) , the corresponding aggregate demand for this good is X ( px , p y , m1 ,..., mn ) = ∑ xi ( px , py , mi ) n

i =1

Observe the following characteristics of this demand curve:

Microeconomics I. Antonio Zabalza. University of Valencia

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a) Since prices are the same for everybody, market demand depends on the same prices as the individual demand. b) Income is in principle different for every individual. Therefore market demand depends on the distribution of income. The second characteristic means that to deal with market demand functions that are strictly derived from the aggregation of individual demands is not easy. In order to simplify this, in this course we are going to assume that we can think of market demand as the demand of some representative consumer, whose income M is the sum of all individual incomes. n

M = ∑ mi i =1

This means that we are going to ignore the distribution of income, although it should be kept in mind that this may be an important factor in the determination of demand. This assumption is very strong also on theoretical grounds because the conditions under which it can be applied are very restrictive. Under these conditions, the aggregate market demand curve is X = X ( px , p y , M ) ,

Microeconomics I. Antonio Zabalza. University of Valencia

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and its properties are the same as those of the demand for an individual consumer that faces prices px and py , and has income M. In particular, if we represent the demand curve in the ( px , x ) space, it means that along the curve we are holding constant py and M . Any change in these variables will originate a change in the position of the demand curve depending on whether the two goods are substitutes or complements and on whether good x is normal or inferior.

8.2 The inverse demand function A method of aggregation is trough the inverse demand function. This is the same as the demand function only that rather than interpreting it as giving us quantity as a function of price, we interpret it as giving us price as a function of quantity. If the usual way to present the demand function is x = x ( px , p y , m ) , the inverse demand function would be px = g ( x , py , m ) This function gives the marginal rate of substitution (MRS) between x and all other goods; the marginal willingness to pay for an extra unit of the good.

Microeconomics I. Antonio Zabalza. University of Valencia

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Aggregation in this case is somewhat different than in the previous case. Since all consumers face the same prices, all will have the same MRS at their optimal choices. Therefore prices cannot be added up. In fact, px represents the MRS of all consumers n   in the market when consuming X  X = ∑ xi  ,  i =1  facing p y , and with aggregate income M. Geometrically this is equivalent to adding the demand (or, for that case, supply) curves horizontally: for any given price, we add up the individuals’ quantities demanded, which, of course, are measured on the horizontal axis. Along this curve, we are holding constant the price of the other good and the aggregate level of income. Example Suppose a market is composed of two individuals with demand functions x1 = 10 − 0.5 p x2 = 10 − p What is the market demand function?

Assume that market demands are only defined for positive values of the goods; it would be absurd to suppose that at a given positive price the consumer demands a negative quantity of the good.

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The corresponding inverse forms are p = 20 − 2 x1 p = 10 − x2 And the horizontal summation would be represented as follows p

p

p

20

20

10

10

10

10

x1 + x2

x2

x1

5

At the price range 20 / 10 only the first consumer participates in the market. The slope of the market demand curve is the slope of the individual demand curve. At prices below 10, both consumers participate. The slope changes: now, similar price changes generate a larger demand response. At price 10 the market demand curve displays a kink. The aggregate demand curve is For 0 < X < 5 For 5 < X < 20

p = 20 − 2 X p = 13.3 − 0.66 X

20

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Check that you understand how the formula for the second segment of the market demand curve has been obtained.

8.3 The concept of elasticity This concept was extensively discussed last year in the course of Introduction to Economic Analysis. Here we will review quickly the main formal issues involved. If the demand function is x = x ( p x, p y, m ),

(

)

the own price elasticity of demand ε x , px is defined as δ x px ε x , px = δ px x For discrete changes ε x , px

∆x px ∆x x Percentage change in quantity = = = ∆px x ∆px px Percentage change in price

The elasticity is a unit-free measure. Its value is the same irrespective of the units in which the quantity of x is expressed.

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Depending on the variable with respect to which we want to measure the response of demand we also have: The cross price elasticity of demand δ x py ε x , py = δ py x The income elasticity of demand δx m ε x ,m = δm x The own price elasticity will normally take a negative value due to the negative slope of the demand curve. The cross price elasticity will be positive if the goods are substitutes and negative if they are complements (recall the definition of substitutes and complements), and the income elasticity will be positive for normal goods and negative for inferior goods. Example: The elasticity of the linear demand curve Suppose the demand curve is q = a − bp where a and b are positive parameters. Applying the above definition of own price elasticity, which for simplicity we will denote simply ε , we have

Microeconomics I. Antonio Zabalza. University of Valencia

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−bp −bp ε= = q a − bp

Clearly, the elasticity of the linear demand curve is not constant; it depends on the price level at which it is measured. The price level goes from 0 to a/b. Measuring the elasticity in absolute terms, so that we can ignore the negative sign, we have: If p = 0



ε =0

If p = a b



ε →∞

If p = a 2b



ε =1

p

ε →∞

ε >1

a/b

ε =1

ε 0) dR If ε > 1 < 0 ∴ R decreases dp dR If ε = 1 = 0 ∴ R constant dp dR If ε < 1 > 0 ∴ R increases dp

Price decrease (dp 1 < 0 ∴ R increases dp dR If ε = 1 = 0 ∴ R constant dp dR If ε < 1 > 0 ∴ R decreases dp

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8.5 Elasticity and marginal revenue In the previous section we have examined how revenue changes when price changes. Now we examine how revenue changes when quantity changes. We begin, as above, with the definition of revenue R = pq

Then we find the derivative of revenue with respect to quantity. dR dp = p+q dq dq

Observe that dR dq is marginal revenue (MR). It can also be written as     q dp  dR 1  = p 1 + = p  1+ =  p dq dq  p dq     q dp 

 1 p 1 +   ε

Expressing the elasticity in absolute terms, we have

Microeconomics I. Antonio Zabalza. University of Valencia

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 1 dR = p 1 −  dq  ε  1 MR = p 1 −   ε 

The behaviour of revenue when the quantity is changed depends also on the elasticity. For instance, if the elasticity of demand is –1, then marginal revenue is zero (revenue does not change when output increases). dR > 0 ∴ MR positive dq dR If ε = 1 = 0 ∴ MR zero dq dR If ε < 1 > 0 ∴ MR negative dp If ε > 1

Example: Suppose you have the following linear inverse demand curve p=a-bq The slope of this demand curve is –b. What is the MR curve, and how is it placed with respect to the demand curve (the average revenue curve)?

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Given the above demand function, marginal revenue is dR dp = p+q dq dq But from the demand function, we have dp = −b dq Therefore dR = p − bq dq = a − bq − bq = a − 2bq The marginal revenue curve is then MR = a − 2bq

This is a straight line with a vertical intercept a and a slope –2b. The MR curve starts at the same point as the average curve (the inverse demand curve), but then descends with a slope twice steeper than the demand curve. [Remember that this is what we saw when we discussed monopoly last year].

Microeconomics I. Antonio Zabalza. University of Valencia

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p

Slope= -b

Slope= –2b

a

ε =1 a/2

Demand = Average revenue

a/2b

a/b Marginal revenue

q