“The Fundamental Equation of the 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? It describes the key predictions from our economic model. It forms the foundation for modern “marginalist” economics. If you ever want to go on to do advanced economics – either theoretical or empirical – you need to know the Slutsky equation! It is really the heart of many things. 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…

Total (observed) effect! 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? Key intuition is that price changes also effect income. In these examples, income is explicitly effected, but in all examples our “purchasing power” changes when prices change. Substitution effect + Income effect = Total (observed) effect! What if your wage went from $10 to $100 Would you work more or less?

Total effect = substitution effect – income effect The Slutsky Equation: Total effect = substitution effect – income effect Key is that even though law of demand says sum of effects must be negative, we cannot make definitive predictions about price and income effects. Often wealth effects are assumed away! Read more: http://homepage.newschool.edu/het//profiles/slutsky.htm

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 Rate of exchange = marginal rate of substitution NOTE: we only observe the total effect! We are inferring the other two from our theory. - Income effect: The change in demand due to a change in purchasing power

Why is this so important? 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? The change in income is the quantity of the good times the change in price Now there are two things changing: price and income: “total derivative” If you buy 5 slices of pizza a week, and the price of pizza goes from $2 per slice to $3 per slice, how much more money will you need to still buy 5 slices per week? $5 This is the response to a compensated price change! Divide through by Δp

A Numerical Illustration of Slutsky Substitution From Hal Varian Intermediate Microeconomics (2003) p. 140 - 141 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? What is the change in quantity demanded if the price drops to $2 per gallon? 2 is the total observed change in quantity demanded for a change in price

A Numerical Illustration Continued… Old price: $3/gallon New price: $2/gallon Old demand: 14 New demand: 16 Change in quantity demanded: 2 How much less money do you need to buy your original 14 gallons per week? What would be your demand at the new price if you reduce your income by $14 per week? Even when we take away income, the relative price of milk is still cheaper than other goods, so demand goes up! The substitution effect holding utility constant is: Note: 14 is demand with utility constant; 16 is demand with income constant, which is the reality – we don’t really get “compensated” for price changes. The Income effect is: Total (observed) effect: Test yourself: do the example when price increases from $2 to $3

Slutsky Illustration with Pictures: the “pivot-shift” method original budget line -$14 “compensated budget line” “true new budget line” milk 14 15.3 16

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” milk 14 15.3 16

demand falls when price declines! 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? C = 1/3B 1.5(1/3B) + 2B = 6 B = 2.4, C = .8 What is the total effect of the price change? 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: Draw the graphs for this problem. Compute the effects and draw the graphs if coffee and bagels are perfect complements at a ratio of 2 cups to 1 bagel. 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… 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 These are called “general restrictions” and have been critical for empirical microeconomics! George Box “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