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.

Market Demand Curves Assume that there are only two goods (X and Y) and two individuals (1 and 2) The first person’s demand for X is X1 = dX1(PX,PY,I1) The second person’s demand for X is X2 = dX2(PX,PY,I2)

Market Demand Curves Features of these demand curves: Both individuals are assumed to face the same prices Each buyer is assumed to be a price taker must accept the prices prevailing in the market Each person’s demand depends on his or her own income

total X = dX1(PX,PY,I1) + dX2(PX,PY,I2) Market Demand Curves The total demand for X is the sum of the amounts demanded by the two buyers The demand function will depend on PX, PY, I1, and I2 total X = X1 + X2 total X = dX1(PX,PY,I1) + dX2(PX,PY,I2) total X = DX(PX,PY,I1,I2)

Market Demand Curves To construct the market demand curve, PX is allowed to vary while PY, I1, and I2 are held constant If each individual’s demand for X is downward sloping, the market demand curve will also be downward sloping

Market Demand Curves To derive the market demand curve, we sum the quantities demanded at every price PX PX Individual 1’s demand curve Individual 2’s demand curve PX Market demand curve X* DX X1* + X2* = X* X1* X2* PX* dX1 dX2 X X X

Shifts in the Market Demand Curve The market demand summarizes the ceteris paribus relationship between X and PX Changes in PX result in movements along the curve (change in quantity demanded) Changes in other determinants of the demand for X cause the demand curve to shift to a new position (change in demand)

Shifts in Market Demand Suppose that individual 1’s demand for oranges is given by X1 = 10 – 2PX + 0.1I1 + 0.5PY and individual 2’s demand is X2 = 17 – PX + 0.05I2 + 0.5PY The market demand curve is X = X1 + X2 = 27 – 3PX + 0.1I1 + 0.05I2 + PY

Shifts in Market Demand To graph the demand curve, we must assume values for PY, I1, and I2 If PY = 4, I1 = 40, and I2 = 20, the market demand curve becomes X = 27 – 3PX + 4 + 1 + 4 = 36 – 3PX

Shifts in Market Demand If PY rises to 6, the market demand curve shifts outward to X = 27 – 3PX + 4 + 1 + 6 = 38 – 3PX Note that X and Y are substitutes If I1 fell to 30 while I2 rose to 30, the market demand would shift inward to X = 27 – 3PX + 3 + 1.5 + 4 = 35.5 – 3PX Note that X is a normal good for both buyers

Generalizations Suppose that there are n goods (Xi, i = 1,n) with prices Pi, i = 1,n. Assume that there are m individuals in the economy The j th’s demand for the i th good will depend on all prices and on Ij Xij = dij(P1,…,Pn, Ij)

Generalizations The market demand function for Xi is the sum of each individual’s demand for that good The market demand function depends on the prices of all goods and the incomes and preferences of all buyers

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