Week5

Economics 11 Week 5 Sungwoo Cho * April 2024 I Duality ¯ U = V ( p x , p y , I ) and E ( p x , p y , ¯ U ) = I Marshallian to Hicksian g x ( p x , p y , E ( p x ,p y , ¯ U ) ) = h x ( p x ,p y , ¯ U ) and g y ( p x , p y , E ( p x ,p y , ¯ U ) ) = h y ( p x ,p y , ¯ U ) Hicksian to Marshallian h x ( p x ,p y , V ( p x , p y , I ) ) = g x ( p x , p y ,I ) and h y ( p x ,p y , V ( p x , p y , I ) ) = g y ( p x , p y ,I ) * Contact: chosw13@ucla.edu . If you find any typo or error please send me an email. 1

II Income and Substitution Effects Remark : A change in price causes two effects known as the substitution and income effect. As the price of x decreases the consumer becomes richer, this is known as the income effect. Also as the price of x decreases x becomes relatively cheaper than y, this is known as the substitution effect. Large price changes Total effect The difference between the new consumption bundle and the old consumption bundle is called the total effect. The next graph shows that the budget constraint rotates to the right. The new consumption bundle is marked by point B. Total effect x = g x ( p 0 x ,p y ,I ) - g x ( p x ,p y ,I ) p 0 x - p x Total effect y = g y ( p 0 x ,p y ,I ) - g y ( p x ,p y ,I ) p 0 x - p x Substitution effect The substitution effect measures how much the consumer would substitute x for y given the price change holding utility constant. How do we compute demand while holding utility constant? The substitution effect is the movement from point A to point C. 2

Income effect x = Total effect x - Substitution effect x Income effect x = g x ( p 0 x ,p y ,I ) - g x ( p x ,p y ,I ) p 0 x - p x - h x ( p 0 x ,p y , ¯ U ) - h x ( p x ,p y , ¯ U ) p 0 x - p x Income effect y = Total effect y - Substitution effect y Income effect y = g y ( p 0 x ,p y ,I ) - g y ( p x ,p y ,I ) p 0 x - p x - h y ( p 0 x ,p y , ¯ U ) - h y ( p x ,p y , ¯ U ) p 0 x - p x Small price changes Slutsky Equation Start from the duality result: g x ( p x , p y , E ( p x ,p y , ¯ U )) = h x ( p x ,p y , ¯ U ) Let’s take the partial derivative with respect to p x and use the fact : I = E ( p x ,p y , ¯ U ) ∂g x ( p x , p y , I ) ∂p x + ∂g x ( p x , p y , I ) ∂I ∂E ( p x ,p y , ¯ U ) ∂p x = ∂h x ( p x ,p y , ¯ U ) ∂p x By Shephard’s Lemma: ∂g x ( p x , p y , I ) ∂p x + ∂g x ( p x , p y , I ) ∂I h x ( p x ,p y , ¯ U ) = ∂h x ( p x ,p y , ¯ U ) ∂p x By duality: ∂g x ( p x , p y , I ) ∂p x + ∂g x ( p x , p y , I ) ∂I g x ( p x ,p y ,I ) = ∂h x ( p x ,p y , ¯ U ) ∂p x Re-arrange terms: ∂g x ( p x , p y , I ) ∂p x | {z } Total Effect = ∂h x ( p x ,p y , ¯ U ) ∂p x | ¯ U = V | {z } Substitution Effect - ∂g x ( p x , p y , I ) ∂I g x ( p x ,p y ,I ) | {z } Income Effect 4

Note: We evaluate the derivative of the Hicksian demand at the indirect utility function to make the equation consistent and to express everything in terms of prices and income. Derivation: cross-price Slutsky Equation Start from the duality result: g x ( p x , p y , E ( p x ,p y , ¯ U )) = h x ( p x ,p y , ¯ U ) Let’s take the partial derivative with respect to p y and use the fact : I = E ( p x ,p y , ¯ U ) ∂g x ( p x , p y , I ) ∂p y + ∂g x ( p x , p y , I ) ∂I ∂E ( p x ,p y , ¯ U ) ∂p y = ∂h x ( p x ,p y , ¯ U ) ∂p y By Shephard’s Lemma: ∂g x ( p x , p y , I ) ∂p y + ∂g x ( p x , p y , I ) ∂I h y ( p x ,p y , ¯ U ) = ∂h x ( p x ,p y , ¯ U ) ∂p y By duality: ∂g x ( p x , p y , I ) ∂p y + ∂g x ( p x , p y , I ) ∂I g y ( p x ,p y ,I ) = ∂h x ( p x ,p y , ¯ U ) ∂p x y Re-arrange terms: ∂g x ( p x , p y , I ) ∂p y | {z } Total Effect = ∂h x ( p x ,p y , ¯ U ) ∂p y | ¯ U = V | {z } Substitution Effect - ∂g x ( p x , p y , I ) ∂I g y ( p x ,p y ,I ) | {z } Income Effect Price changes for normal goods: If a good is normal, substitution and income effects reinforce one another: when the price falls, both effects lead to a rise in quantity demanded. Price changes for inferior goods: If a good is inferior, substitution and income effects move in opposite directions: the combined effect is indeterminate. When the price falls, the substitution effect leads to a rise in quantity demanded, but the income effect moves in the opposite direction Giffen’s paradox For an inferior good, if the income effect dominates the substitution effect an increase in price generates an increase in demand: • An increase in price leads to a drop in real income. • Since the good is inferior, a drop in income causes the quantity demanded to rise. • If the income effect is greater than the substitution effect consumption increases. 5

IV Proposed Exercises - Solutions 1. Suppose we have U ( x,y ) = x 1 4 y 3 4 . Thus, the utility maximization problem is: max x,y x 1 4 y 3 4 s.t. p x x + p y y = I (a) The Marshallian demands are: g x ( p x ,p y ,I ) = 1 4 I p x g y ( p x ,p y ,I ) = 3 4 I p y (b) The Hicksian demands are: h x ( p x ,p y , ¯ U ) = ¯ U 1 3 3 4 p y p x 3 4 h y ( p x ,p y , ¯ U ) = ¯ U (3) 1 4 p x p y 1 4 (c) Total effect: g x ( p 0 x ,p y ,I ) - g x ( p x ,p y ,I ) = 1 4 I p 0 x - 1 4 I p x = I 4 p x - p 0 x p 0 x p x g y ( p 0 x ,p y ,I ) - g y ( p x ,p y ,I ) = 3 4 I p y - 3 4 I p y = 0 Remember that p 0 x < p x , thus: p x - p 0 x > 0 and we can see that the numerator of the total effect is positive, that means when the price of x goes down, the purchased amount of x increases (price and quantity move in the opposite direction). In this particular example, the total change in y equals 0. 7

Substitution effect: h x ( p 0 x ,p y , ¯ U ) - h x ( p x ,p y , ¯ U ) = ¯ U 1 3 3 4 p y p 0 x 3 4 - ¯ U 1 3 3 4 p y p x 3 4 = ¯ U p y 3 3 4 " 1 ( p 0 x ) 3 4 - 1 ( p x ) 3 4 # = ¯ U p y 3 3 4 " ( p x ) 3 4 - ( p 0 x ) 3 4 ( p 0 x ) 3 4 ( p x ) 3 4 # = ¯ U p y 3 3 4    1 - p 0 x p x 3 4 ( p 0 x ) 3 4    From the previous expression we can see that numerator is positive as p 0 x < p x . h y ( p 0 x ,p y , ¯ U ) - h y ( p x ,p y , ¯ U ) = ¯ U (3) 1 4 p 0 x p y 1 4 - ¯ U (3) 1 4 p x p y 1 4 = ¯ U (3) 1 4 1 p y 1 4 p 0 x 1 4 - ( p x ) 1 4 From the previous expression we can see that h y ( p 0 x ,p y , ¯ U ) - h y ( p x ,p y , ¯ U ) is negative as p 0 x < p x . Note: to express the substitution effect in dollar amounts, we use the indirect utility function. Remember that we care about the “original” level of utility, i.e. the level of utility the consumer has before the price change. : V ( p x ,p y ,I ) = I 4 1 p x 1 / 4 3 p y 3 / 4 h x ( p 0 x ,p y , ¯ U ) - h x ( p x ,p y , ¯ U ) = ¯ U p y 3 3 4    1 - p 0 x p x 3 4 ( p 0 x ) 3 4    = I 4 1 p x 1 / 4    1 - p 0 x p x 3 4 ( p 0 x ) 3 4    8

2. Suppose a consumer has the following utility function: u ( x,y ) = x 2 + xy . Therefore 2 Marshallian demands: g x ( p x ,p y ,I ) = I 2 1 ( p x - p y ) g y ( p x ,p y ,I ) = I 2 ( p x - 2 p y ) ( p x - p y ) p y Indirect Utility Function: V ( p x ,p y ,I ) = I 2 4 1 ( p x - p y ) p y Hicksian demands: h x ( p x ,p y , ¯ U ) = ¯ U 1 / 2 p 1 / 2 y ( p x - p y ) 1 / 2 h y ( p x ,p y , ¯ U ) = ¯ U 1 / 2 p x - 2 p y [( p x - p y ) p y ] 1 / 2 Expenditure function: E ( p x ,p y , ¯ U ) = 2 ¯ U 1 / 2 [( p x - p y ) p y ] 1 / 2 Note that for g x > 0 we need to impose p x > p y . (a) How does a small change in p x affect the demand for x ? i) What is the total effect? (easiest to compute it directly) Total effect x = ∂g x ( p x , p y , I ) ∂p x = - I 2 1 ( p x - p y ) 2 ii) What is the income effect? (easiest to compute it directly) Income effect x = - ∂g x ( p x , p y , I ) ∂I g x ( p x ,p y ,I ) = - 1 2 1 ( p x - p y ) I 2 1 ( p x - p y ) = - I 4 1 ( p x - p y ) 2 iii) What is the substitution effect? (two ways to compute this: directly or use the Slutsky equation) Direct way: 2 The detailed derivation is left as an exercise 10

Substitution effect x = ∂h x ( p x ,p y , ¯ U ) ∂p x | ¯ U = V = - 1 2 p y ( p x - p y ) 2 s ¯ U ( p x - p y ) p y = - 1 2 s ¯ Up y ( p x - p y ) 3 = - 1 2 s I 2 4 1 ( p x - p y ) p y p y ( p x - p y ) 3 = - 1 2 s I 2 4 1 ( p x - p y ) 4 = - I 4 1 ( p x - p y ) 2 Using Slutsky equation: Total effect x = Substitution effect x + Income effect x Substitution effect x = Total effect x - Income effect x = ∂g x ( p x , p y , I ) ∂p x - - ∂g x ( p x , p y , I ) ∂I g x ( p x ,p y ,I ) = - I 2 1 ( p x - p y ) 2 - - I 4 1 ( p x - p y ) 2 = - I 2 1 ( p x - p y ) 2 + I 4 1 ( p x - p y ) 2 = - I 4 1 ( p x - p y ) 2 (b) How does a small change in p y affect the demand for x ? i) What is the total effect? (easiest to compute it directly) Total effect x = ∂g x ( p x , p y , I ) ∂p y = I 2 1 ( p x - p y ) 2 11

Using Slutsky Equation: Total effect x = Substitution effect x + Income effect x Substitution effect x = Total effect x - Income effect x = ∂g x ( p x , p y , I ) ∂p y - - ∂g x ( p x , p y , I ) ∂I g y ( p x ,p y ,I ) = I 2 1 ( p x - p y ) 2 - - I 4 ( p x - 2 p y ) ( p x - p y ) 2 p y = I 2 1 ( p x - p y ) 2 + I 4 ( p x - 2 p y ) ( p x - p y ) 2 p y = I 4 2 p y ( p x - p y ) 2 p y + I 4 ( p x - 2 p y ) ( p x - p y ) 2 p y = I 4 2 p y + p x - 2 p y ( p x - p y ) 2 p y = I 4 p x p y 1 ( p x - p y ) 2 13