 # MARKET DEMAND AND ELASTICITY

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MARKET DEMAND AND ELASTICITY
Chapter 7 MARKET DEMAND AND ELASTICITY MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

Elasticity Suppose that a particular variable (B) depends on another variable (A) B = f(A…) We define the elasticity of B with respect to A as The elasticity shows how B responds (ceteris paribus) to a 1 percent change in A

Price Elasticity of Demand
The most important elasticity is the price elasticity of demand measures the change in quantity demanded caused by a change in the price of the good eQ,P will generally be negative except in cases of Giffen’s paradox

Distinguishing Values of eQ,P
Value of eQ,P at a Point Classification of Elasticity at This Point eQ,P < -1 Elastic eQ,P = -1 Unit Elastic eQ,P > -1 Inelastic

Price Elasticity and Total Expenditure
Total expenditure on any good is equal to total expenditure = PQ Using elasticity, we can determine how total expenditure changes when the price of a good changes

Price Elasticity and Total Expenditure
Differentiating total expenditure with respect to P yields Dividing both sides by Q, we get

Price Elasticity and Total Expenditure
Note that the sign of PQ/P depends on whether eQ,P is greater or less than -1 If eQ,P > -1, demand is inelastic and price and total expenditures move in the same direction If eQ,P < -1, demand is elastic and price and total expenditures move in opposite directions

Price Elasticity and Total Expenditure
Responses of PQ Demand Price Increase Price Decrease Elastic Falls Rises Unit Elastic No Change Inelastic

Income Elasticity of Demand
The income elasticity of demand (eQ,I) measures the relationship between income changes and quantity changes Normal goods  eQ,I > 0 Luxury goods  eQ,I > 1 Inferior goods  eQ,I < 0

Cross-Price Elasticity of Demand
The cross-price elasticity of demand (eQ,P’) measures the relationship between changes in the price of one good and and quantity changes in another Gross substitutes  eQ,P’ > 0 Gross complements  eQ,P’ < 0

Relationships Among Elasticities
Suppose that there are only two goods (X and Y) so that the budget constraint is given by PXX + PYY = I The individual’s demand functions are X = dX(PX,PY,I) Y = dY(PX,PY,I)

Relationships Among Elasticities
Differentiation of the budget constraint with respect to I yields Multiplying each item by 1

Relationships Among Elasticities
Since (PX · X)/I is the proportion of income spent on X and (PY · Y)/I is the proportion of income spent on Y, sXeX,I + sYeY,I = 1 For every good that has an income elasticity of demand less than 1, there must be goods that have income elasticities greater than 1

Slutsky Equation in Elasticities
The Slutsky equation shows how an individual’s demand for a good responds to a change in price Multiplying by PX /X yields

Slutsky Equation in Elasticities
Multiplying the final term by I/I yields

Slutsky Equation in Elasticities
A substitution elasticity shows how the compensated demand for X responds to proportional compensated price changes it is the price elasticity of demand for movement along the compensated demand curve

Slutsky Equation in Elasticities
Thus, the Slutsky relationship can be shown in elasticity form It shows how the price elasticity of demand can be disaggregated into substitution and income components Note that the relative size of the income component depends on the proportion of total expenditures devoted to the good (sX)

Homogeneity Remember that demand functions are homogeneous of degree zero in all prices and income Euler’s theorem for homogenous functions shows that

Homogeneity Dividing by X, we get
Using our definitions, this means that An equal percentage change in all prices and income will leave the quantity of X demanded unchanged

Cobb-Douglas Elasticities
The Cobb-Douglas utility function is U(X,Y) = XY The demand functions for X and Y are The elasticities can be calculated

Cobb-Douglas Elasticities
Similar calculations show Note that

Cobb-Douglas Elasticities
Homogeneity can be shown for these elasticities The elasticity version of the Slutsky equation can also be used

Cobb-Douglas Elasticities
The price elasticity of demand for this compensated demand function is equal to (minus) the expenditure share of the other good More generally where  is the elasticity of substitution

Linear Demand Q = a + bP + cI + dP’ where: Q = quantity demanded
P = price of the good I = income P’ = price of other goods a, b, c, d = various demand parameters

Linear Demand Q = a + bP + cI + dP’ Assume that:
Q/P = b  0 (no Giffen’s paradox) Q/I = c  0 (the good is a normal good) Q/P’ = d ⋛ 0 (depending on whether the other good is a gross substitute or gross complement)

Linear Demand If I and P’ are held constant at I* and P’*, the demand function can be written Q = a’ + bP where a’ = a + cI* + dP’* Note that this implies a linear demand curve Changes in I or P’ will alter a’ and shift the demand curve

Linear Demand Along a linear demand curve, the slope (Q/P) is constant the price elasticity of demand will not be constant along the demand curve As price rises and quantity falls, the elasticity will become a larger negative number (b < 0)

Linear Demand Demand becomes more elastic at higher prices
eQ,P < -1 -a’/b eQ,P = -1 eQ,P > -1 Q a’

Constant Elasticity Functions
If one wanted elasticities that were constant over a range of prices, this demand function can be used Q = aPbIcP’d where a > 0, b  0, c  0, and d ⋛ 0. For particular values of I and P’, Q = a’Pb where a’ = aIcP’d

Constant Elasticity Functions
This equation can also be written as ln Q = ln a’ + b ln P Applying the definition of elasticity, The price elasticity of demand is equal to the exponent on P

Important Points to Note:
The market demand curve is negatively sloped on the assumption that most individuals will buy more of a good when the price falls it is assumed that Giffen’s paradox does not occur Effects of movements along the demand curve are measured by the price elasticity of demand (eQ,P) % change in quantity from a 1% change in price

Important Points to Note:
Changes in total expenditures on a good caused by changes in price can be predicted from the price elasticity of demand if demand is inelastic (0 > eQ,P > -1) , price and total expenditures move in the same direction if demand is elastic (eQ,P < -1) , price and total expenditures move in opposite directions

Important Points to Note:
If other factors that enter the demand function (prices of other goods, income, preferences) change, the market demand curve will shift the income elasticity (eQ,I) measures the effect of changes in income on quantity demanded the cross-price elasticity (eQ,P’) measures the effect of changes in another good’s price on quantity demanded

Important Points to Note:
There are a number of relationships among the various demand elasticities the Slutsky equation shows the relationship between uncompensated and compensated price elasticities homogeneity is reflected in the fact that the sum of the elasticities of demand for all of the arguments in the demand function is zero