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### Market Demand

Chapter 5

Introduction

- Demand for a commodity is differentiated from a want
- In terms of society’s willingness and ability to pay for satisfying the want
- This chapter determines total amount demanded for a commodity by all households
- Called market demand or aggregate demand
- Sum of individual household demands
- Assuming individual household demands are independent of each other
- Will explore network externalities
- Independence assumption does not hold

Introduction

- Major determinants of market demand for a commodity are
- Its own price
- Price of related commodities
- Households’ incomes
- Elasticity of demand is a measure of influence each parameter has on market demand
- We investigate own-price elasticity of demand
- Classify market demand as elastic, unitary, or inelastic
- Depending on its degree of responsiveness to a price change
- Relating own-price elasticity to households’ total expenditures for a commodity
- Demonstrate how this determines whether total expenditures for a commodity will increase, remain unchanged, or decline, given a price change

Introduction

- Define income elasticity of demand and elasticity of demand for the price of related commodities
- Called cross-price elasticity
- Applied economists are active in estimating elasticities for all determinants (parameters) of market demand
- Knowledge of the influence these determinants provides information for firms’ decisions
- Government policymakers also use estimates of elasticities
- With reliable estimates of elasticities based on economic models
- Economists can explain, predict, and control agents’ market behavior

Market Demand

- One arm of Marshallian cross
- Conveys individual household preferences for a commodity given a budget constraint
- Sum of all individual households’ demands for a single commodity
- Consider only two households—Robinson, R, and Friday, F—and two commodities

- xR1 and xF1 are household Robinson’s and household Friday’s demand for commodity x1, respectively
- Both households face same per-unit prices for the two commodities
- Each household is a price taker
- Both households are bound by budget constraints
- IR and IF represents income for Robinson and Friday, respectively

Market Demand

- Market demand, Q1, is sum of amounts demanded by the two households

- Holding p2, IR, and IF constant, we obtain market demand curve for x1 in Figure 5.1
- For a private good, market demand curve is horizontal summation of individual household demand curves

Market Demand

- At p*1 Robinson demands 5 units of Q1 and Friday demands 3
- For a total market demand of 8 units
- Varying price will result in other associated levels of market demand
- Will trace out market demand curve for Q1
- Will have a negative slope
- For market demand to have a positive slope, a large portion of households would have to consider x1 a Giffen good
- Assume market demand for a commodity is inversely related to its own price
- Shifts in market demand will occur if there is a change in household preferences, income, price of another commodity, or population
- As illustrated in Figure 5.2, market demand curve will shift outward for an increase in
- Income
- Population
- Price of substitute commodities, or
- A decrease in prices of complement commodities

Network Externalities

- Horizontal summation of households’ demand functions assumes individual demands are independent of each other
- For some commodities, one household’s demand does depends on other households’ demands
- Example of network externalities
- Exist when a household’s demand is affected by other households’ consumption of the commodity
- Positive network externalities result when
- Value one household places on a commodity increases as other households purchase the item
- Negative network externalities exist if household’s demand decreases as a result of other households’ actions

Bandwagon Effect

- Specific type of positive network externality
- Individual demand is influenced by number of other households consuming a commodity
- The greater the number of households consuming a commodity
- More desirable commodity becomes for an individual household
- Key to marketing most toys and clothing is to create a bandwagon effect
- Results in market demand curve shifting outward
- Individual household demand increases in response to increased demand by numerous other households

Market Effect

- If positive network externalities exist, summation of individual household demands does not take into account households’ increase in demand when other households increase their demand for the commodity
- Will underestimate true market demand
- Shown in Figure 5.3
- Individual household demand curves are positively influenced by other households’ level of demand for commodity
- Results in a further outward shift of individual household demand curves, and market demand curve
- Instead of market demand being sum of 5 plus 3 units at p*1
- Positive network externalities result in a market demand of 7 plus 6 units

Market Effect

- If negative network externalities exist
- Summation of individual household demands will overestimate true market demand
- Shown in Figure 5.4
- Results in inward shift of individual household demand curves
- With a corresponding inward shift in market demand curve

Elasticity

- Market demand function provides a relationship between price and quantity demanded
- Quantity demanded is inversely related to price
- Of greater interest to firms and government policymakers is how responsive quantity demanded is to a change in price
- Downward-sloping demand curve indicates
- If a firm increases its price, quantity demanded will decline
- Does not show magnitude of decline
- To measure magnitude of responsiveness use derivative or slope of curve
- The larger the partial derivative, the more responsive is y

Units Of Measurement

- One problem in using derivative is units of measure
- By changing units of measure—say from dollars to cents or pounds to kilograms
- Cause magnitude of change or value of derivative to vary
- For example, if y is measured in pounds, x in dollars, and ∂y/∂x = 2
- Measurement is 2 pounds per dollar
- For each $1 increase in x, y will increase by 2 units
- However, if change scale used to measure y to ounces, then ∂y/∂x = 32
- For each $1 increase in x, y will increase by 32 units
- Just changing scale makes it appear that y is more responsive to a given change in x

Unit-free Measure Of Responsiveness

- Prior failure to convert from English to metric system of measurement caused loss of Mars Climate Orbiter
- To avoid making such errors in comparing responsiveness across different factors with different units of measurement in economics
- Use a standardized derivative, elasticity
- Removes scale effect
- Derivative is standardized (converted into an elasticity)
- By weighting it with levels of variables under consideration
- Results in percentage change in y given a percentage change in x
- Provides a unit-free measure of the responsiveness
- Partial derivative is not as useful as elasticity measurement

Logarithmic Representation

- As a percentage change measure, elasticity can be expressed in logarithmic form

Price Elasticity Of Demand

- For market quantity, Q is defined as
- Q,p(∂Q/∂p)(p/Q) = ∂ ln Q/∂ ln p
- Elasticity of demand indicates how Q changes (in percentage terms) in response to a percentage change in p
- Ordinary good: ∂Q/∂p < 0
- Implies Q,p < 0 given that p and Q are positive
- Examples of demand elasticities are provided in Table 5.1

Perfectly Inelastic Demand

- A change in price results in no change in quantity demanded
- Q,p = 0
- Represented in Figure 5.5
- Results in a vertical demand curve
- At every price level quantity demanded is the same
- Examples are difficult to find due to the lack of households with monomania preferences
- For example, alcoholics and drug addicts would have highly inelastic demands over a broad range of quantity

Perfectly Elastic Demand

- Smallest possible value of Q,p is for it to approach negative infinity
- If Q,p = - demand is perfectly elastic
- Very slight change in price corresponds to an infinitely large change in quantity demanded
- Illustrated in Figure 5.6
- Many examples of perfectly elastic demand curves
- Whenever a firm takes its output price as given it is facing a perfectly elastic demand curve
- For example, agriculture

Classification of Elasticity

- Between elasticity limits from - to 0, elasticity may be classified in terms of its responsiveness
- Q,p < -1, elastic, |∂Q/Q| > |∂p/p|
- Absolute percentage change in quantity is greater than absolute percentage change in price
- Quantity is relatively responsive to a price change
- Q,p = -1, unitary, |∂Q/Q| = |∂p/p|
- Absolute percentage change in quantity is equal to absolute percentage change in price
- Q,p > -1, inelastic, |∂Q/Q| < |∂p/p|
- Absolute percentage change in quantity is less than absolute percentage change in price
- Quantity is relatively unresponsive to a price change

Linear Demand

- Linear demand curve will exhibit all three elasticity classifications
- Consider linear demand function for commodity x1
- x1 = 120 – 2p1
- Plotted in Figure 5.7
- Elasticity of demand represented as
- 11 = (∂x1/∂p1)(p1/x1)
- Size of elasticity coefficient increases in absolute value for movements up this linear demand curve
- Because slope is remains constant while weight is increasing
- At point B
- 11 = (∂x1/∂p1)p1/x1 = -2(45/30) = -3, elastic
- At D
- 11 = (∂x1/∂p1)p1/x1 = -2(15/90) = -1/3, inelastic
- At point C (A) [E] elasticity of demand is unitary (-) [0]

Linear Demand

- General functional form for a linear market demand function
- Q1 = a + bp1, b<0
- Q1 denotes market demand for commodity 1
- p1 is associated price per unit
- Partial derivative is equal to constant b
- Elasticity of demand is not constant along a linear demand curve
- As p1/Q1 increases, demand curve becomes more elastic
- In the limit, as Q1 approaches zero, elasticity of demand approaches negative infinity, perfectly elastic
- p1 = 0 results in perfectly inelastic elasticity of demand

Linear Demand

- A straight-line (linear) demand curve is certainly the easiest to draw (Figure 5.8)
- However, such behavior is generally unrealistic
- Because linear demand curve assumes (∂Q1/∂p1) = constant
- Implies that a doubling of prices will have same effect on Q1 as a 5% increase

Proportionate Price Changes

- Assuming households respond to proportionate rather than absolute changes in prices
- May be more realistic to consider the demand function
- Q1 = apb1, a > 0, b > 0 or
- ln Q1 = ln a + b ln p1
- Elasticity of demand is
- 11 = (∂Q1/∂p1)(p1/Q1) = bap1b-1(p1/Q1) = b or
- 11 = (∂ ln Q1/∂ ln p1) = b
- Elasticity of this demand curve is constant along its entire length
- Constant elasticity of demand curve, with b = -1 is illustrated in Figure 5.9

Price Elasticity and Total Revenue

- Valuable use of elasticity of demand
- Predict what will happen to households’ total expenditures on a commodity or to producers’ total revenue when price changes
- Total revenue (TR) and total expendituresare defined as price times quantity (p1Q1)
- A change in price has two offsetting effects
- Reduction in price has direct effect
- Reduces total revenue for the commodity
- Results in an increase in quantity sold
- Increases total revenue
- Considering these two opposing effects, total revenue from a commodity price change may rise, fall, or remain the same
- Effect depends on how responsive quantity is to a change in price
- Measured by elasticity of demand

Price Elasticity and Total Revenue

- Relationship between total revenue and elasticity of demand may be established by differentiating total revenue (p1Q1) with respect to p1
- Using product rule of differentiation, dividing both sides by Q1 and multiplying left-hand-side by p1/p1 yields total revenue elasticity
- TR, p = 1 + 11
- Measures percentage change in total revenue for a percentage change in price
- Sign depends on whether 11 is > or < -1
- If 11 > -1, demand is inelastic and TR,p > 0
- Price and total revenue move in same direction
- If 11 < -1, demand is elastic, and TR,p < 0
- An increase in p1 is associated with a decrease in total revenue
- If elasticity of demand is unitary, Q,p = -1, then TR,p = 0

Price Elasticity And Total Revenue

- If elasticity of demand is elastic
- Quantity demanded will increase by a larger percentage than price decreases
- Total revenue will increase with a price decline
- Opposite occurs when demand is inelastic
- A price decline results in total revenue declining
- Because quantity demanded increases by a smaller percentage than price decreases
- In elastic portion of demand curve
- Price and total revenue move in opposite directions
- In inelastic portion
- Price and TR move in same direction

Price Elasticity and the Price Consumption Curve

- Setting p2 as numeraire price, p2 = 1
- Then p1x1 + x2 = I
- Solving for total revenue (expenditures) for x1 yields
- p1x1 = I – x2
- On vertical axis in Figure 5.7, at p1 = $45,
- Income I is initially allocated between total expenditures for x2, x2, and total expenditure on x1, I - x2
- Decreasing p1 from $45 to $30 results in a decline in total expenditure for x2 and an increase in total expenditure for x1
- Movement from B to C in indifference space results in a negatively sloping price consumption curve

Price Elasticity and the Price Consumption Curve

- Declining price consumption curve is associated with an increase in total expenditures on x1
- Indicating elastic demand
- Negatively sloping portion of price consumption curve is associated with elastic portion of demand curve
- Positively sloping price consumption curve is associated with inelastic portion of demand curve
- Decreasing p1 from $30 to $15 results in total expenditures for x2 increasing and total expenditures for x1 declining
- Indicating inelastic demand
- If price consumption curve has a zero slope, unitary elasticity exists

Price Elasticity and the Price Consumption Curve

- Slope of price consumption curve is determined by magnitude of income and substitution effects
- Total effect of a price change is sum of these two effects
- Closeness of substitutes for a commodity directly influences substitution effect
- The more closely related substitutes are to the commodity, the larger will be the substitution effect
- A relatively large substitution effect will decrease slope of price consumption curve
- Will make demand curve more elastic
- If a commodity has a close substitute and if price of substitute remains constant
- A rise in price of commodity will divert households’ expenditures away from product toward substitute

Price Elasticity and the Price Consumption Curve

- Other important determinants of slope of price consumption curve
- Proportion of income allocated for a commodity
- And whether commodity is normal or inferior
- The smaller the proportion of income allocated for a commodity, the larger the slope of price consumption curve
- The more inelastic the demand
- Income effect is relatively small for a commodity requiring a small fraction of income
- Results in a more inelastic demand

Price Elasticity and the Price Consumption Curve

- An inferior commodity will tend to result in a positively sloping price consumption curve
- Inelastic demand curve
- If inferior nature of a commodity results in a Giffen good, result is
- Backward-bending price consumption curve
- Positively sloping demand curve
- A final major determinant of demand elasticity is time allowed for adjusting to a price change
- Elasticities of demand tend to become more elastic as time for adjustment lengthens
- The longer the time interval after a price change, the easier it may become for households to substitute other commodities

Income Elasticity Of Demand

- Relationship between change in quantity demanded and change in income may be represented by the slope of an Engel curve
- Weighting this slope with income divided by quantity results in income elasticity
- ηQ = (∂Q/∂I)(I/Q)
- Measures percentage change in quantity to a percentage change in income
- Classified as follows

Cross-Price Elasticity of Demand

- Demand for a commodity such as an automobile will depend on its own price and income and
- Prices of other related commodities
- Measure responsiveness of demand to a price change in a related commodity by cross-price elasticity
- Cross-price elasticity of demand for commodities x1 and x2
- When Q1 is a gross substitute for Q2
- 12= (∂Q1/∂p2)(p2/Q1) = ∂ ln Q1/∂ ln p2> 0
- When Q1 is a gross complement for Q2
- 12= (∂Q1/∂p2)(p2/Q1) = ∂ ln Q1/∂ ln p2< 0
- Cross-price elasticity can be either positive or negative
- Depending on whether Q1 is a gross substitute or gross complement for Q2

Slutsky Equation in Elasticities

- Slutsky equation from Chapter 4

- Substitution elasticity

- Indicates how demand for x1 responds to proportional compensated price changes

Slutsky Equation in Elasticities

- Slutsky equation in elasticity form

- Where α1 = p1x1/I is proportion of income spent on x1
- Indicates how price elasticity of demand can be disaggregated into substitution and income components
- Relative size of income component depends on proportion of total expenditures devoted to commodity in question
- Given a normal good, the larger the income elasticity and proportion of income spent on the commodity, the more elastic is demand
- Income effect will be reinforced by substitution effect
- Larger the substitution effect, the more elastic is demand

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