Chapter 7 MARKET DEMAND ELASTICITY MICROECONOMIC THEORY BASIC

Chapter 7 MARKET DEMAND 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 • e. Q, P will generally be negative – except in cases of Giffen’s paradox

Distinguishing Values of e. Q, P Classification of Value of e. Q, P at a Point Elasticity at This Point e. Q, P < -1 Elastic e. Q, P = -1 Unit Elastic e. Q, 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 e. Q, P is greater or less than -1 – If e. Q, P > -1, demand is inelastic and price and total expenditures move in the same direction – If e. Q, 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 Rises Falls

Income Elasticity of Demand • The income elasticity of demand (e. Q, I) measures the relationship between income changes and quantity changes • Normal goods e. Q, I > 0 – Luxury goods e. Q, I > 1 • Inferior goods e. Q, I < 0

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

Relationships Among Elasticities • Suppose that there are only two goods (X and Y) so that the budget constraint is given by P XX + P YY = I • The individual’s demand functions are X = d. X(PX, PY, I) Y = d. Y(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, s. Xe. X, I + s. Ye. Y, 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 (s. X)

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 + b. P + c. I + d. P’ 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 + b. P + c. I + d. P’ • 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’ + b. P where a’ = a + c. I* + d. P’* – 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 becomes more elastic at higher prices P -a’/b e. Q, P < -1 e. Q, P = -1 e. Q, P > -1 a’ Q

Constant Elasticity Functions • If one wanted elasticities that were constant over a range of prices, this demand function can be used Q = a. Pb. Ic. P’d where a > 0, b 0, c 0, and d ⋛ 0. • For particular values of I and P’, Q = a’Pb where a’ = a. Ic. P’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 (e. Q, 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 > e. Q, P > -1) , price and total expenditures move in the same direction – if demand is elastic (e. Q, 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 (e. Q, I) measures the effect of changes in income on quantity demanded – the cross-price elasticity (e. Q, P’) measures the effect of changes in another good’s price on quantity demanded

Important Points to Note: • There 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