Slutsky Equation. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15


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1 Slutsky Equation M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Slutsky Equation 1 / 15
2 Effects of a Price Change: What happens when the price of a commodity decreases? 1 The commodity becomes relatively cheaper, so consumers substitute towards it and away from now relatively more expensive other commodities. 2 The consumers budget of m can purchase more than before. It is in a sense the consumer received and increased income. The consequent changes in quantities demanded are the substitution and income effect of the price change. M. Utku Ünver Micro Theory (BC) Slutsky Equation 2 / 15
3 Example: Original choice: x Original prices (p 1, p 2 ) p 1 decreased to p 1 While the initial income is m = p 1 x 1 + p 2 x 2, now only p 1 x 1 + p 2 x 2 = m is needed to buy the original choice. So it is as if the consumer s income by m m = (p 1 p 1 )x 1 M. Utku Ünver Micro Theory (BC) Slutsky Equation 3 / 15
4 Slutsky s insight was that the effects on quantities demanded of any price can always be decomposed into a pure substitution effect and an income effect. Slutsky asserted that if, at the new prices, * less income than before is needed to buy the original choice then real income has increased. * if more income than before is needed to buy the original choice then real income has decreased. M. Utku Ünver Micro Theory (BC) Slutsky Equation 4 / 15
5 M. Utku Ünver Micro Theory (BC) Slutsky Equation 5 / 15
6 Pure Substitution Effect and Income Effect Slutsky isolated the change in quantities demanded due only to the change in relative prices by asking: What is the change in quantities demanded when the consumer s income is adjusted so that, at the new prices, she can just buy the original bundle? The change from x to x is the pure substitution effect. The rest of the change (i. e. x to x ) is the pure income effect. M. Utku Ünver Micro Theory (BC) Slutsky Equation 6 / 15
7 Example: Suppose demand function for milk is: x 1 = 10 + m 10p 1 Initially m = 120 p 1 = 3 x 1 = = 14. Here 14 3 = 42 is spent on milk, 78 is spent on other goods. Now suppose p 1 = 2. Question: How much income is needed to buy the initial bundle. Answer: = 106 M. Utku Ünver Micro Theory (BC) Slutsky Equation 7 / 15
8 The consumer will buy x 1 money. = Therefore = 1.3 is the substitution effect. Question: What about the income effect? x 1 = = 16 = 15.3 units of milk with that will be the eventual consumption. There is a change of (16 14) = 2 units. 1.3 is due to substitution effect and 0.7 is due to the income effect. M. Utku Ünver Micro Theory (BC) Slutsky Equation 8 / 15
9 Signs and Relative Magnitudes of Substitution and Income Effects The substitution effect always moves opposite to the price movement. We say that the substitution effect is negative since the change in demand due to the substitution effect is opposite to the price change. On the other hand the income effect might be negative or positive depending on whether the good is inferior or not. M. Utku Ünver Micro Theory (BC) Slutsky Equation 9 / 15
10 Normal Good: Since both the substitution and income effects increase demand when own price decreases, the demand curve is downward sloping for normal goods. M. Utku Ünver Micro Theory (BC) Slutsky Equation 10 / 15
11 Inferior Good: With inferior goods the income effect and substitution effect are at opposite directions. M. Utku Ünver Micro Theory (BC) Slutsky Equation 11 / 15
12 Giffen Good: If the income effect and substitution effect are on opposite directions and if the income effect is larger than the substitution effect then a price decrease, decreases the demand. Such goods are called giffen goods. M. For Utkugiffen Ünver Micro good Theory we (BC) have upward Slutsky sloping Equation portions in demand curves. 12 / 15
13 For a given consumer who consumes multiple goods: M. Utku Ünver Micro Theory (BC) Slutsky Equation 13 / 15
14 Example: Perfect Complements. No substitution effect. Only income effect. M. Utku Ünver Micro Theory (BC) Slutsky Equation 14 / 15
15 Example: Cobb Douglas u = x1 0.3x Initial: p 1 = 2, p 2 = 1, m = 100 Final: p 1 = 1, p 2 = 1, m = 100 Initially x 1 = 0.3 m p 1 = = 15, x 2 = = 70 Finally x 1 = 0.3 m p 1 = = 30, x 2 = = 70 How much money is needed to buy the initial bundle with new prices? m = p 1 x 1 + p 2 x 2 = = 85 What is the demand with m and new prices? Initially x 1 m = 0.3 p 1 = = 25.5, x 2 = = 59.5 Substitution Effect= x 1 x 1 = = 10.5 Income Effect= x 1 x 1 = = 4.5 M. Utku Ünver Micro Theory (BC) Slutsky Equation 15 / 15
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