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The Slutsky Equation or “The Fundamental Equation of the Theory of Value” Eugene Slutsky 1880 – 1948 Sulla teoria del bilancio del consumatore (1915) Sir John R. Hicks, 1904-1989 Value and Capital (1939) Why is this so important? 1. It describes the key predictions from our economic model. 2. It forms the foundation for modern “marginalist” economics. 3. Good practice with the fundamental skill of going from words to pictures to math! Read more: http://homepage.newschool.edu/het//profiles/hicks.htm and links from here… What happens to the demand for a good when the price changes? Would you buy more or fewer apples if the price goes up? What if you grew apples? What if your wage went from $10 to $100 Would you work more or less? Substitution effect + Income effect = Total (observed) effect! The Slutsky Equation: Total effect = substitution effect – income effect Read more: http://homepage.newschool.edu/het//profiles/slutsky.htm xi xiU xi xi pi pi M Total effect: The observed change in demand due to a change in price = Substitution effect: The change in demand due to a change in the rate of exchange between two goods holding purchasing power (utility) constant Income effect: The change in demand due to a change in purchasing power xi x xi Why is this so important? xi pi pi M U i The (own-price) substitution effect must be negative: If the price of a good increases consumers will substitute to other goods and the demand will go down. The Law of Demand: If the demand for a good increases when income increases (e.g. it is a normal good), then the demand for that good must decrease when its price increases. An Intuitive Derivation of the Slutsky Equation From Louis Phlips Applied Consumption Analysis (1990) p. 40 - 42 If price changes by Δp, how much do you need to change income so you could still buy the same amount? M xi pi M xi pi The change in income is the quantity of the good times the change in price Now there are two things changing: price and income: xi xi xi pi M pi M This is the response to a compensated price change! xi xi xi M pi pi M pi xi xiU xi xi pi pi M “total derivative” Divide through by Δp A Numerical Illustration of Slutsky Substitution From Hal Varian Intermediate Microeconomics (2003) p. 140 - 141 M x1 10 10 p1 Suppose your weekly demand for milk is: If you have $120 per week and milk is $3 per gallon, how many gallons per week do you buy? 120 x1 10 14 10 3 What is the change in quantity demanded if the price drops to $2 per gallon? 120 x1 10 16 10 2 2 is the total observed change in quantity demanded for a change in price 16 14 2 xi xiU xi xi pi pi M A Numerical Illustration Continued… How much less money do you need to buy your original 14 gallons per week? Old price: $3/gallon M x 10 1 New price: $2/gallon 10 p1 Old demand: 14 New demand: 16 Change in quantity demanded: 2 M xi pi 14 2 3 14 What would be your demand at the new price if you reduce your income by $14 per week? 120 14 106 x1 10 10 15.3 10 2 20 The substitution effect holding utility constant is: Even when we take away income, the relative price of milk is still cheaper than other goods, so demand goes up! 15.3 14 1.3 The Income effect is: 15.3 16 0.7 Total (observed) effect: 2 1.3 0.7 Test yourself: do the example when price increases from $2 to $3 Slutsky Illustration with Pictures: the “pivot-shift” method Y original budget line -$14 “compensated budget line” “true new budget line” 14 15.3 16 milk Slutsky Illustration with Pictures: the rotation method (Hicks Substitution) Y For small changes in price the Slutsky Substitution (pivot-shift) is EQUAL to the Hicksian Substitution (rotation). original budget line “compensated budget line” ROTATES AROUND THE ORIGINAL INDIFFERENCE CURVE “true new budget line” 14 15.3 16 milk Slutsky Illustration for an Inferior Good Demand decreases when income increases Y If the income effect outweighs the substitution effect then we have a Giffen Good: demand falls when price declines! This is very rare!! original budget line “compensated budget line” “true new budget line” X1 X2 XS X Test Yourself: Draw the Slutsky substitution lines for goods that are perfect complements. Is the total effect all substitution? All income? A little of both? Draw the Slutsky substitution lines for goods that are perfect substitutes. Is the total effect all substitution? All income? A little of both? Another Numerical Illustration Recall our example with coffee and bagels where: MUc = 1/2B, MUb = 2C Pc = $1, Pb = $2, M = $6 Optimal (2.4,1.2) What happens if the price of coffee increases to $1.50? What is the new optimal bundle? What is the total effect of the price change? C = 1/3B 1.5(1/3B) + 2B = 6 B = 2.4, C = .8 Old (2.4, 1.2) = new (2.4, .8) = (0, -.4) How much more money would you need to buy the old bundle at the new prices? $0.5 x 1.2 cups = $0.60 If you had this additional money what would be your new optimal bundle (at the new prices)? C = 1/3B 1.5(1/3B) + 2B = 6.6 B = 2.64, C = .88 Another Numerical Illustration Continued… Original bundle: (2.4, 1.2) New bundle: (2.4, 0.8) Compensated bundle: (2.64, 0.88) What is the total effect, substitution effect and income effect of the increase in coffee price? Total = Original - New: (2.4, 1.2) - (2.4, 0.8) = (0, -.4) Substitution = Original – Compensated: (2.4, 1.2) - (2.64, 0.88) = (.24, -.32) Income = Total – Substitution: (0, -.4) – (.24, -.32) = (-.24, -.08) Test yourself: 1. Draw the graphs for this problem. 2. Compute the effects and draw the graphs if coffee and bagels are perfect complements at a ratio of 2 cups to 1 bagel. 3. Compute the effects and draw the graphs if coffee and bagels are perfect substitutes at a ratio of 2 cups to 1 bagel. Why do we care? We can make predictions based on the theory! Remember: all models are wrong but some are useful… 1. 2. 3. 4. Own price substitution effects must be negative Income effects for normal goods must be positive Income effects for inferior goods must be negative Cross-price effects must be symmetrical George Box These are called “general restrictions” and have been critical for empirical microeconomics! “Do we mean to say that observed demand behavior does in fact always satisfy these conditions?...there is no reason why measured behavior should obey them, as theory is always a simplification of reality…All we can hope is that rough estimates, computed without imposing these constraints, will not be inconsistent with them.” -- Phlips (1990) p. 53 - 54