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### Chapter Fifteen

Market Demand

From Individual to Market Demand Functions

- Think of an economy containing n consumers, denoted by i = 1, … ,n.
- Consumer i’s ordinary demand function for commodity j is

From Individual to Market Demand Functions

- When all consumers are price-takers, the market demand function for commodity j is
- If all consumers are identical then where M = nm.

From Individual to Market Demand Functions

- The market demand curve is the “horizontal sum” of the demand curves of the individual consumers.
- For example, suppose there are only two consumers; i = A,B.

From Individual to Market Demand Functions

p1

p1

p1’

p1’

p1”

p1”

20

15

p1

The “horizontal sum”of the demand curvesof individuals A and B.

p1’

p1”

35

Elasticities

- Elasticity is a measure of the “sensitivity” of one variable with respect to another.
- The elasticity of a variable X with respect to a variable Y is defined as

Economic Applications of Elasticity

- Economists use elasticities to measure the sensitivity of
- quantity demanded of commodity i with respect to the price of commodity i (own-price elasticity of demand)
- demand for commodity i with respect to the price of commodity j (cross-price elasticity of demand).

Economic Applications of Elasticity

- demand for commodity i with respect to income (income elasticity of demand)
- quantity supplied of commodity i with respect to the price of commodity i (own-price elasticity of supply

Own-Price Elasticity of Demand

- Q: Why not just use the slope of a demand curve to measure the sensitivity of quantity demanded to a change in a commodity’s own price?
- A: Because then the value of sensitivity depends upon the chosen units of measurement.

Own-Price Elasticity of Demand

is a ratio of percentages and so has nounits of measurement. Hence own-priceelasticity of demand is a measure of the sensitivity of quantity demanded to changes in own-price which is independent of the choice of units of measurement.

Arc and Point Elasticities

- If we measure the “average” own-price elasticity of demand for commodity i over an interval of values for pi then we compute an arc-elasticity, usually by using a mid-point formula.
- Elasticity computed for a single value of pi is called a point elasticity.

Point Own-Price Elasticity

An example of computing a point own-priceelasticity of demand.Suppose pi = a - bXi. Then Xi = (a-pi)/b and

Therefore,

Point Own-Price Elasticity

pi = a - bXi*

pi

a

own-price elastic

(own-price unit elastic)

a/2

own-price inelastic

a/2b

a/b

Xi*

Point Own-Price Elasticity

A second example of computing a pointown-price elasticity of demand.

Suppose

Then

so

Revenue and Own-Price Elasticity of Demand

- If raising a commodity’s price causes almost no decrease in quantity demanded then sellers’ revenues will rise.
- Hence own-price inelastic demand will cause sellers’ revenues to rise as price rises.

Revenue and Own-Price Elasticity of Demand

- If raising a commodity’s price causes a very large decrease in quantity demanded then sellers’ revenues will fall.
- Hence own-price elastic demand will cause sellers’ revenues to fall as price rises.

Revenue and Own-Price Elasticity of Demand

so if

then

and a change to price does not altersellers’ revenue.

Revenue and Own-Price Elasticity of Demand

In summary:

Own-price inelastic demand;price rise causes rise in sellers’ revenue.

Own-price unit elastic demand;price rise causes no change in sellers’revenue.

Own-price elastic demand;price rise causes fall in sellers’ revenue.

Marginal Revenue and Own-Price Elasticity of Demand

- A seller’s marginal revenue is the rate at which revenue changes with the number of units sold by the seller.

Marginal Revenue and Own-Price Elasticity of Demand

Let p(q) denote the seller’s inverse demand function; that is, the price at which the seller can sell q units. Then

so

Marginal Revenue and Own-Price Elasticity of Demand

says that the rate

at which a seller’s revenue changeswith the number of units it sellsdepends on the sensitivity of quantitydemanded to price; i.e., upon theof the own-price elasticity of demand.

Marginal Revenue and Own-Price Elasticity of Demand

Selling onemore unit does not change the seller’srevenue.

If

then

Selling onemore unit reduces the seller’s revenue.

If

then

If

then

Selling onemore unit raises the seller’s revenue.

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