LECTURE 3 Consumer Theory 24/27 August, 2021 Consumer Theory Roadmap • Consumer choice – Utility maximization problem – Expenditure minimization problem • Demand – Demand function and indirect utility function – Compensated demand function and expenditure function • Revealed preference • Slutsky equation • Consumer welfare LECTURE 3 ECA5101 Semester 1 AY 2021/2022 2 CONSUMER CHOICE: EXPENDITURE MINIMIZATION Part 1 ECA5101 Semester 1 AY 2021/2022LECTURE 3 3 Alternative Way to Think of Consumer Choice LECTURE 3 ECA5101 Semester 1 AY 2021/2022 4 F C 0 U1 A B E2 E1 E3 Utility Maximization vs. Expenditure Minimization • Utility maximization – Given a certain level of income/expenditure – Consumer chooses the basket that generates maximum utility • Expenditure minimization – Given a certain level of utility – Consumer chooses the basket that requires minimum expenditure LECTURE 3 ECA5101 Semester 1 AY 2021/2022 5 Example: Expenditure Minimization • Suppose the utility function is • The optimal basket is F=15, C=7.5 • Given the prices and an income of 600, consumer’s maximum utility is 15*7.5=112.5 LECTURE 3 ECA5101 Semester 1 AY 2021/2022 6 max F,C FC s.t. 20F + 40C = 600 U(F,C) = FC Example: Expenditure Minimization Cont’ • We can rewrite the problem as • Solving the expenditure minimization problem we still get F=15, C=7.5 • Given the prices and a utility level of 112.5, consumer’s minimum expenditure is 20*15+40*7.5=600 LECTURE 3 ECA5101 Semester 1 AY 2021/2022 7 min F,C 20F + 40C s.t. FC =112.5 Example: Expenditure Minimization Cont’ • For the consumer with utility function and when the price of food is 20, the price of clothing is 40 • F=15, C=7.5 is the utility maximizing basket given an income of 600 – And at this basket the utility is 112.5 • F=15, C=7.5 is also the expenditure minimizing basket given a utility of 112.5 – And at this basket the expenditure is 600 LECTURE 3 ECA5101 Semester 1 AY 2021/2022 8 U(F,C) = FC COMPENSATED DEMAND FUNCTION AND EXPENDITURE FUNCTION Part 2 ECA5101 Semester 1 AY 2021/2022LECTURE 3 9 What changes along a demand curve? LECTURE 3 ECA5101 Semester 1 AY 2021/2022 10 F F C PF PF =10PF = 20PF = 30 30 20 10 D 20 28 50 20 28 50 A B C Demand curve for food Compensated Demand Curve • Demand curve tells us how the utility maximizing quantity of a good changes with its price • Along a demand curve – Other prices are held constant – Income is held constant – But utility is not • Compensated demand curve shows the relationship between the price of a good and the quantity demanded assuming that other prices and utility are held constant LECTURE 3 ECA5101 Semester 1 AY 2021/2022 11 Compensated Demand Curve in Graph LECTURE 3 ECA5101 Semester 1 AY 2021/2022 12 F F C PF PF =10 PF = 20 PF = 30 30 20 10 20 35 60 20 35 60 A B Compensated demand curve for food C Notes on Compensated Demand Curve • Compensated demand curve tells us how the expenditure minimizing quantity of a good changes with its price • Along the compensated demand curve, utility does not change – But income does! – E.g., when price decreases, to keep the consumer’s utility level constant, consumer’s income is reduced • Only substitution effect is captured by the compensated demand curve LECTURE 3 ECA5101 Semester 1 AY 2021/2022 13 Example: Deriving Compensated Demand Curve LECTURE 3 ECA5101 Semester 1 AY 2021/2022 14 • Suppose the consumer has utility function • Suppose price of clothing is 2, utility is 8 • What is the compensated demand curve for food? • The expenditure minimization problem is U(F,C) = FC min F,C PFF + 2C s.t. FC = 8 Demand Curve vs. Compensated Demand Curve LECTURE 3 ECA5101 Semester 1 AY 2021/2022 15 F F C PF PF =10 PF = 20 20 10 35 60 35 60 A BC 100 100 D 80 80 Compensated Demand Function • Quantity demanded holding utility constant depends on – Price of the good – Utility level – Price of other goods • Suppose the consumer buys two goods, x and y • Compensated demand function for x is denoted as • While demand function for x is denoted as LECTURE 3 ECA5101 Semester 1 AY 2021/2022 16 xc (Px,Py,U) x(Px,Py, I ) Example: Deriving Compensated Demand Function • Consider the following utility maximization problem • The demand functions are • Demand functions are solutions to the utility maximization problem LECTURE 3 ECA5101 Semester 1 AY 2021/2022 17 max F,C FC s.t. PFF +PCC = I F = I2PF , C = I2PC Example: Deriving Compensated Demand Function Cont’ • The corresponding expenditure minimization problem is • The solution to this problem is • Compensated demand functions are solutions to the expenditure minimization problem LECTURE 3 ECA5101 Semester 1 AY 2021/2022 18 min F,C PFF +PCC s.t. FC =U Fc = PCPF U , Cc = PFPC U Expenditure Function • Compensated demand function tells us how expenditure- minimizing consumption of a good changes with prices and utility • How about consumer’s expenditure? – When prices or utility change, optimal consumption will change – Minimum expenditure will change • Minimum expenditure depends indirectly on prices and utility level • Expenditure function is minimum expenditure as a function of prices and utility level LECTURE 3 ECA5101 Semester 1 AY 2021/2022 19 E(Px,Py,U) = Pxxc (Px,Py,U)+Pyyc (Px,Py,U) Example: Expenditure Function • Given the utility function , the expenditure minimization problem is • The compensated demand functions are LECTURE 3 ECA5101 Semester 1 AY 2021/2022 20 min F,C PFF +PCC s.t. FC =U Fc = PCPF U , Cc = PFPC U U(F,C) = FC Example: Expenditure Function Cont’ • The expenditure function is • Recall that the indirect utility function is • What is the relationship between the two functions? LECTURE 3 ECA5101 Semester 1 AY 2021/2022 21 E(PF,PC,U) = PF PC PF U +PC PF PC U = 2 PFPCU V (PF,PC, I ) =U( I 2PF , I2PC ) = I 2 4PFPC Indirect Utility Function vs. Expenditure Function • They are inverse functions of each other • Indirect utility function – Maximum utility as a function of prices and income/expenditure • Expenditure function – Minimum expenditure as a function of prices and utility level LECTURE 3 ECA5101 Semester 1 AY 2021/2022 22 Compensated Demand Function and Expenditure Function • Demand function and Indirect utility function are linked by Roy’s identity • Shephard’s lemma – The rate at which minimum expenditure changes with the price of x is the compensated demand function of x LECTURE 3 ECA5101 Semester 1 AY 2021/2022 23 xc (Px,Py,U) = ∂E(Px,Py,U) ∂Px x(Px,Py, I ) = − ∂V (Px,Py, I ) /∂Px ∂V (Px,Py, I ) /∂I Proof of Shephard’s Lemma • The expenditure minimization problem is • The Lagrangian is • The solution to this problem gives rise to the compensated demand functions, the multiplier, and the expenditure function LECTURE 3 ECA5101 Semester 1 AY 2021/2022 24 min x,y Pxx +Pyy s.t. U(x, y) =U Λ = Pxx +Pyy+λ(U −U(x, y)) xc (Px,Py,U), yc (Px,Py,U), λ*(Px,Py,U), E(Px,Py,U) Proof of Shephard’s Lemma Cont’ • By the envelope theorem, – The derivative of the expenditure function w.r.t. the price of x is the derivative of the Lagrangian function w.r.t. the price of x when x, y, and the multiplier are held at the optimal level • Thus LECTURE 3 ECA5101 Semester 1 AY 2021/2022 25 ∂E(Px,Py,U) ∂Px = ∂Λ ∂Px x=x c ,y=yc ,λ=λ* ∂E(Px,Py,U) ∂Px = ∂Λ ∂Px x=x c ,y=yc,λ=λ* = xc (Px,Py,U) Example: Deriving Compensated Demand Function from Expenditure Function • For the expenditure minimization problem • The expenditure function we have derived is • Using Shephard’s lemma, LECTURE 3 ECA5101 Semester 1 AY 2021/2022 26 min F,C PFF +PCC s.t. FC =U E(PF,PC,U) = 2 PFPCU ∂2 PFPCU ∂PF = PC PF U ∂2 PFPCU ∂PC = PF PC U Properties of Expenditure Function • Expenditure function is non-decreasing in prices – E.g., holding utility and other prices fixed, when the price of food increases, consumer’s minimum expenditure cannot decrease • Expenditure function is concave in prices – For simplicity, consider expenditure as a function of Px only – A twice-differentiable function f(x) is concave if and only if f ’’(x) <= 0 LECTURE 3 ECA5101 Semester 1 AY 2021/2022 27 ∂2E(Px,Py,U) ∂Px2 ≤ 0 Proof: Expenditure Function Concave in Price • A useful result about concave functions – A concave function always lies below the tangents to it • Suppose when the prices are Px0 and Py0, to get a utility of U0, the expenditure-minimizing basket is (x0, y0) – The minimum expenditure is LECTURE 3 ECA5101 Semester 1 AY 2021/2022 28 Px0x0 +Py0y0 Proof: Expenditure Function Concave in Price Cont’ • Suppose the price of x increases to Px1 – While the price of y remains at Py0 • Suppose given prices Px1 and Py0, to get a utility of U0, the expenditure-minimizing basket is (x1, y1) – The minimum expenditure now is LECTURE 3 ECA5101 Semester 1 AY 2021/2022 29 Px1x1 +Py0y1 Proof: Expenditure Function Concave in Price Cont’ LECTURE 3 ECA5101 Semester 1 AY 2021/2022 30 Px E 0 A B Px0 Px0x0 +Py0y0 Px1 Px1x1 +Py0y1 E(Px,Py0,U0 ) Proof: Expenditure Function Concave in Price Cont’ • If given the new price, the consumer still buys the initial basket (x0, y0) – The consumer still gets a utility of U0 – The expenditure is • If the consumer still buys (x0, y0), expenditure increases linearly in price – From to LECTURE 3 ECA5101 Semester 1 AY 2021/2022 31 Px1x0 +Py0y0 Px0x0 +Py0y0 Px1x0 +Py0y0 Proof: Expenditure Function Concave in Price Cont’ LECTURE 3 ECA5101 Semester 1 AY 2021/2022 32 Px E 0 A B Px0 Px0x0 +Py0y0 Px1 Px1x1 +Py0y1 CPx1x0 +Py0y0 E(Px,Py0,U0 ) Proof: Expenditure Function Concave in Price Cont’ • We need to show that – Note that both (x0, y0) and (x1, y1) give the consumer the same level of utility, U0 • Since (x1, y1) is the expenditure-minimizing basket given Px1 – Any other basket that gives the consumer a utility of U0 will cost the consumer more (or the same) • Thus the expenditure function is concave LECTURE 3 ECA5101 Semester 1 AY 2021/2022 33 Px1x1 +Py0y1 ≤ Px1x0 +Py0y0 Slope of Compensated Demand Curve • Since expenditure function is concave in price • Using the Shephard’s lemma, we have • Compensated demand curve is never upward sloping LECTURE 3 ECA5101 Semester 1 AY 2021/2022 34 ∂2E(Px,Py,U) ∂Px2 ≤ 0 ∂2E(Px,Py,U) ∂Px2 = ∂xc (Px,Py,U) ∂Px ≤ 0 THE SLUTSKY EQUATION Part 3 ECA5101 Semester 1 AY 2021/2022LECTURE 3 35 What is the Slutsky Equation? • A mathematical way to represent income and substitution effects – For continuous price changes – The graphical way works for discrete price changes • Given a demand function • We are interested in LECTURE 3 ECA5101 Semester 1 AY 2021/2022 36 x(Px,Py, I ) ∂x(Px,Py, I ) ∂Px Demand vs. Compensated Demand • Recall that a utility-maximizing basket is also an expenditure minimizing basket • If the utility maximizing basket is • The utility at this basket is • This basket is also the expenditure minimizing basket given the same level of utility LECTURE 3 ECA5101 Semester 1 AY 2021/2022 37 x(Px,Py, I ) = xc (Px,Py,V (Px,Py, I )) x(Px,Py, I ), y(Px,Py, I ) V (Px,Py, I ) y(Px,Py, I ) = yc (Px,Py,V (Px,Py, I )) Demand vs. Compensated Demand Cont’ • Similarly, if the expenditure minimizing basket is • The expenditure/income at this basket is • This basket is also the utility maximizing basket given the same amount of income LECTURE 3 ECA5101 Semester 1 AY 2021/2022 38 xc (Px,Py,U), yc (Px,Py,U) E(Px,Py,U) xc (Px,Py,U) = x(Px,Py,E(Px,Py,U)) yc (Px,Py,U) = y(Px,Py,E(Px,Py,U)) Deriving the Slutsky Equation • Since • Differentiating the compensated demand, • Rearranging, • By Shephard’s lemma, LECTURE 3 ECA5101 Semester 1 AY 2021/2022 39 xc (Px,Py,U) = x(Px,Py,E(Px,Py,U)) ∂xc ∂Px = ∂x ∂Px + ∂x ∂E ⋅ ∂E ∂Px ∂x ∂Px = ∂xc ∂Px − ∂x ∂E ⋅ ∂E ∂Px ∂x ∂Px = ∂xc ∂Px − xc ∂x ∂E The Slutsky Equation and Substitution Effect • Since x=xc and E=I, the Slutsky equation is • Substitution effect is the first term – It is the slope of the compensated demand curve – Tells us how consumption changes holding utility constant – And we already know LECTURE 3 ECA5101 Semester 1 AY 2021/2022 40 ∂x ∂Px = ∂xc ∂Px − x ∂x ∂I ∂xc ∂Px ∂xc ∂Px ≤ 0 The Slutsky Equation and Income Effect • The Slutsky equation is • Income effect is the second term – It measures the change in consumption through the change in purchasing power – The sign of the income effect depend on whether the good is normal or inferior LECTURE 3 ECA5101 Semester 1 AY 2021/2022 41 ∂x ∂Px = ∂xc ∂Px − x ∂x ∂I −x ∂x ∂I Example: Verifying Slutsky Equation • Consider the utility function • We have derived the demand functions • And also the compensated demand functions LECTURE 3 ECA5101 Semester 1 AY 2021/2022 42 U(x, y) = xy x = I2Px , y = I2Py xc = PyPx U , yc = PxPy U Example: Verifying Slutsky Equation Cont’ • From the demand function we have • We now decompose the effect of a price change into substitution effect and income effect • By the Slutsky equation, the substitution effect is LECTURE 3 ECA5101 Semester 1 AY 2021/2022 43 ∂x ∂Px = − I 2Px2 ∂xc ∂Px = − PyU 2Px1.5 Example: Verifying Slutsky Equation Cont’ • The indirect utility function is • Thus the substitution effect is • The income effect is • The sum of the two effects is exactly LECTURE 3 ECA5101 Semester 1 AY 2021/2022 44 V (x, y, I ) = I 2 4PxPy ∂xc ∂Px = − PyU 2Px1.5 = − IPx−0.5 4Px1.5 = − I 4Px2 −x ∂x ∂I = − I 2Px ⋅ 1 2Px = − I 4Px2 − I 2Px2 CONSUMER WELFARE Part 4 ECA5101 Semester 1 AY 2021/2022LECTURE 3 45 How to measure the change in utility when price changes? LECTURE 3 ECA5101 Semester 1 AY 2021/2022 46 • When the price of a good decreases – The consumer is better off (higher utility) • When the price of a good increases – The consumer is worse off (lower utility) • How to quantify the benefit or loss due to a change in price? – In particular, can we put a $ value on the utility gain or loss? Why is measuring consumer welfare important? LECTURE 3 ECA5101 Semester 1 AY 2021/2022 47 • Consider the merger between Grab and Uber • CCCS concluded the merger was anti-competitive • There may be some benefits – E.g., merger may reduce the cost of production • There may be some costs – E.g., the new firm may be able to set higher prices – Need to estimate the potential damage to consumers due to higher prices Consumer Surplus • Consumer surplus (CS) for an individual consumer is the difference between the consumer’s willingness to pay for a good and the cost of purchasing the good – E.g., the consumer is willing to pay 1 million to buy a house – The consumer actually paid 0.8 million – CS is 0.2 million • CS is the area below the demand curve and above the price LECTURE 3 ECA5101 Semester 1 AY 2021/2022 48 Consumer Surplus in Graph LECTURE 3 ECA5101 Semester 1 AY 2021/2022 49 P 0 Q 20 80 A Suppose the consumer’s demand is Q=100-P The current price is $20100 CS=0.5*80*(100-20)=$3,200 Change in Consumer Surplus LECTURE 3 ECA5101 Semester 1 AY 2021/2022 50 P 0 Q 20 80 A 100 30 70 B CS decreases by $750: utility loss of the consumer is equivalent to $750 Suppose the price rises from $20 to $30 Expenditure Function and Consumer Welfare • Suppose initially prices are Px0 and Py0 • To get a utility of U0, the consumer needs a minimum expenditure of • If the price of x increases to Px1 • To receive the same level of utility, the consumer needs a minimum expenditure of LECTURE 3 ECA5101 Semester 1 AY 2021/2022 51 E(Px0,Py0,U0 ) E(Px1,Py0,U0 ) Compensating Variation • To maintain the same level of utility, the consumer needs an additional income of – Recall that expenditure function is non-decreasing in prices • This is called the compensating variation – Intuitively, after a price increase, to prevent any utility loss, the consumer needs to be compensated by the amount of LECTURE 3 ECA5101 Semester 1 AY 2021/2022 52 E(Px1,Py0,U0 )−E(Px0,Py0,U0 ) ≥ 0 E(Px1,Py0,U0 )−E(Px0,Py0,U0 ) CV and Compensated Demand Curve • Recall that compensated demand function is the derivative of the expenditure function w.r.t. price • When the price of x increases from Px0 to Px1, CV is the area under the compensated demand curve over the interval [Px0, Px1] LECTURE 3 ECA5101 Semester 1 AY 2021/2022 53 ∂E(Px,Py,U) ∂Px = xc (Px,Py,U) E(Px1,Py0,U0 )−E(Px0,Py0,U0 ) = ∂E(Px,Py,U0 ) ∂PxPx0 Px1 ∫ dPx = xc (Px,Py,U0 ) Px0 Px1 ∫ dPx CS vs. CV • In general consumer surplus and compensated variation are different • For the same price change from Px0 to Px1 – Change in CS is the area under the demand curve over the interval [Px0, Px1] – CV is the area under the compensated demand curve over the interval [Px0, Px1] LECTURE 3 ECA5101 Semester 1 AY 2021/2022 54 CS vs. CV in Graph LECTURE 3 ECA5101 Semester 1 AY 2021/2022 55 P 0 x 20 80 A 100 30 40 B CS: AB3020 xc (Px,Py0,U 0 ) xc (Px,Py0,U1) C D 7055 CV: AC3020 or DB3020