Advanced Microeconomics

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Ethan Randell Houston
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1 Welfare measures and aggregation October 30, 2012

2 The plan: 1 Welfare measures 2

3 Example: 1 Our consumer has initial wealth w and is facing the initial set of market prices p 0. 2 Now he is faced with another set of market prices p 1. 3 How to evaluate changes in consumer's situation? 4 Combinations of prices and incomes will be called projects. 5 For now - let us consider projects with only prices changing (income xed).

4 The (likely) solution For both projects (p 0, w) and (p 1, w) we could compute the indirect utility function if we knew the form of v: v(p 0, w) and v(p 1, w) We could then compare the levels of utility and compare the situations. But maybe we do not know much about utility and prefer to only talk about money? Let us take an arbitrary price vector p - a reference/base price vector. We can convert the the problem to monetary terms by computing expenditure function:e( p, v(p 0, w)) and e( p, v(p 1, w)). This, in fact, is a money-metric utility function: e( p, v(p 1, w)) e( p, v(p 0, w)) - strictly increasing in v. Let us take either p 1 or p 0 to be our reference price vector.

5 The welfare measure will be either: Equivalent Variation (how much would I have to pay at old prices to attain new level of utility)? EV (p 0, p 1, w) = e(p 0, v(p 1, w)) e(p 0, v(p 0, w)) = e(p 0, v(p 1, w)) w = p 0 h(p 0, v(p 1, w)) w or Compensating Variation (how much would I have to pay at new prices to attain old level of utility) or how much compensation I need to attain the old level of welfare. CV (p 0, p 1, w) = e(p 1, v(p 1, w)) e(p 1, v(p 0, w)) = w e(p 1, v(p 0, w)) = w p 1 h(p 1, v(p 0, w))

6 Or: How much the consumer has to be paid to be exactly as well-o with old prices as in the new situation (a transfer that is equivalent in terms of welfare to the price change). v(p 0, w + EV ) = v(p 1, w) How much the consumer has to be paid to be exactly as well-o with new prices as in the old situation(the net revenue of a planner who must compensate the consumer for the price change). v(p 0, w) = v(p 1, w CV )

8 Demand functions and Example: Only one price changes (good 1): p 0 1 p1 1 and p0 l = p 1 l = p l l 1. We know that: w = e(p 0, u 0 ) = e(p 1, u 1 ) and h 1 (p, u) = e(p,u) p 1, so: EV (p 0, p 1, w) = e(p 0, u 1 ) w = e(p 0, u 1 ) e(p 1, u 1 ) = and similarly: ˆ p0 1 p 1 1 h 1 (p 1, p 1, u 1 )dp 1 CV (p 0, p 1, w) = ˆ p0 1 p1 h 1 (p 1, p 1, u 0 )dp 1

9 EV, CV and welfare evaluation They will dier They will provide the correct ranking They will be dierent from so-called consumer surplus (CS) CS(p 0, p 1, w) = ˆ p0 1 p 1 1 x 1 (p 1, p 1, w)dp 1 EV>CS>CV for normal goods. Opposite ordering for inferior goods (income eects are negative and in that case Hicksian demands are atter than Walrasian demand).

10 Demand functions and and CS

11 A complication What if 3 projects are to be compared: eg: 1,2 with 0? Can we use EV? EV (p 0, p 1, w) = e(p 0, u 1 ) w and EV (p 0, p 2, w) = e(p 0, u 2 ) w so basically, we will have 2 1 if e(p 0, u 2 ) > e(p 0, u 1 ). EV is OK Can we use CV? EV uses the new prices as base CV (p 0, p 1, w) = w e(p 1, u 0 ) and CV (p 0, p 2, w) = w e(p 2, u 0 ). So: CV (p 0, p 1, w) CV (p 0, p 2, w) = e(p 2, u 0 ) e(p 1, u 0 ) We cannot use CV, due to dierent price vectors in the expenditure function. If prices are dierent, higher expenditure does not have to mean higher utility. EV is a money-metric utility function

12 Consumer surplus is given by: CS(p 0, p 1, w) = L l=1 ˆ p1 l p 0 l x l (p 1 1, p 1 2,..., p 1 l 1, τ, p 0 l+1..., p 0 L, w)dτ If more than one price changes, the problem may be path dependent (the sequence of integration may matter) - not if preferences are homothethic.

13 When income is also changing EV: EV ((p 0, w 0 ), (p 1, w 1 )) = e(p 0, v(p 1, w 1 )) w 0 = p 0 h(p 0, v(p 1, w 1 )) w 0 CV: CV ((p 0, w 0 ), (p 1, w 1 )) = w 1 e(p 1, v(p 0, w 0 )) = w 1 p 1 h(p 1, v(p 0, w 0 )) CS CS((p 0, w 0 ), (p 1, w 1 )) = ˆ w1 w 0 dτ + L l=1 ˆ p1 l p 0 l x l (p 1 1, p 1 2,..., p 1 l 1, τ, p 0 l+1..., p 0 L, w)dτ

14 Under what conditions use CS? From Chipman and Moore, Consider a triple of projects, a base and two new projects (note that income diers in those projects, not only prices). 1 Even if preferences are homothetic, CS((p 0, w 0 ), (p 1, w 1 )) > 0 does not guarantee that (p 1, w 1 ) is better than (p 0, w 1 ). 2 Fix w 0. Consumer surplus correctly ranks the projects for every triple of projects such that (p, w) : w = w 0 if and only if consumer preferences are homothetic (rescale the prices with income prior to calculating CS). 3 Fix pi 0. Consumer surplus correctly ranks the projects for every triple of projects such that (p, w) : p i = pi 0 if and only if consumer preferences are quasi-linear with respect to commodity i.

15 Example of problems with CS (from W. Novshek): Consider a consumer with demand functions x 1 = w 2p 1 and x 2 = w The corresponding indirect utility is: v = w2 4p 1p 2 (not that this is Cobb-Douglas with α s = 1/2). Show that using CS with projects (p1 0, p0 2, w 0 ) : (1, 1, 1) and (p1 1, p1 2, w 1 ) : (e, e, 2.2) will lead to a ranking opposite to that stemming directly from utility. 2p 2. Correct ranking: v 0 = 12 = 1 > v 1 = e 2 = , so project 0 is in fact better than project 1. CS: CS = p1 1 p1 0 w0 ( 2p 1 )dp 1 + p2 1 p2 0 w0 ( 2p 2 )dp 2 + w 1 w 0 dτ = = ( w0 2 ln p1 1 w0 2 ln p0 1 + w0 2 ln p1 2 w0 2 ln p0 2 ) + w 1 w 0 = w0 ln( p1 1 p1 2 2 p1 0 ) + w p0 1 w 0 = 1 ln( p1 1 p p1 0 ) + w p0 1 w 0 = 2 1 e2 ln( ) =.2 > 0, so according to CS, project 1 is better 2 1 than 0.

16 Example continued The function is homothethic, so we could get rid of the income change by normalizing prices by income. In that case, the equivalent projects would be (p 0 1, p0 2, w 0 ) : (1, 1, 1) and (p 1 1, p1 2, w 1 ) : (e/2.2, e/2.2, 1) CS = p1 1 p1 0 w0 ( 2p 1 )dp 1 + p 1 2 p 0 2 ( w0 2p 2 )dp 2 = = w0 ln( p1 1 p1 2 2 p1 0 ) = 1 e2 ln( p ) 0.21 < 0. So after normalizing prices with income, CV provides the correct ranking.

17 A special case (UMP) Quasi-linear preferences: u(x) = x 0 + φ(x 1,..., x L ) FOC: L L = x 0 + φ(x 1,..., x L ) + λ( p l x l w) L=0 1 = λp 0 φ (x l ) = λp l so:φ (x l ) = p l /p 0 x l = (φ ) 1 (p l /p 0 ). Walrasian demand DOES NOT depend on wealth.

18 A special case (EMP) L L = p l x l λ(x 0 + φ(x 1,..., x L ) u) L=0 FOC: λ = p 0 λφ (x l ) = p l so:φ (x l ) = p l /p 0 x l = (φ ) 1 (p l /p 0 ) = h l (p, u). Hicksian demand is THE SAME as Walrasian demand.

19 Denitions Wealth eects The Gorman form Can we use the techniques from previous classes to derive aggregate demand? Can we aggregate demands of individual consumers? Can we only look at aggregate demand ignoring the underlying consumer optimization?

20 Denitions Denitions Wealth eects The Gorman form Dene the distribution rule w 1 (p, w),..., w I (p, w) that for every level of aggregate wealth w R assigns individual wealths to all consumers 1,..., I. We assume that: w i (p, w) = w p, w i and that w i (, ) is continuous and homogeneous of degree 1. function: x(p, w) = i x i (p, w i (p, w)) is just a sum of Walrasian demands as described in previous sections (continuous, homogeneous of degree zero, Walras law).

21 Wealth eects Denitions Wealth eects The Gorman form Take x(p, w) = i x i(p, w i (p, w)) and assume that i dw i = 0. Question: under what conditions the aggregate demand will be irrespective of the distribution of income? Lets take a derivative: x(p, w)/ w i = x i(p, w i (p, w)) dw i x(p, w) w w dw i = x i(p, w i (p, w)) dw i w i dw i x(p, w) w dw i = i i x i (p, w i (p, w)) dw i dw i 0 = i x i (p, w i (p, w)) dw i dw i The wealth eects have to cancel-out.

22 Wealth eects Denitions Wealth eects The Gorman form It is equivalent to saying that: x li (p, w i ) w i = x lj(p, w j ) w j for every l, any two individuals i and j, and all (w 1,..., w I ). We need parallel and straight wealth expansion paths.

23 Gorman form of preferences Denitions Wealth eects The Gorman form A necessary and sucient condition for the set of consumers to exhibit parallel, straight wealth expansion paths at any price vector p is that preferences admit indirect utility function of the Gorman form with the coecients on w i the same for every consumer i. That is: v i (p, w i ) = a i (p) + b(p)w i Proof: Use Roys identity for the general case with b i (p): x ij = a i(p)/ p j + w i b i (p)/ p j b i (p) = a i(p)/ p j b i (p) } {{ } shift +w i b i (p)/ p j b i (p) }{{} slope

24 Examples Denitions Wealth eects The Gorman form Special cases: preferences need to be homothetic or preferences need to be quasi-linear.

Advanced Microeconomics

Advanced Microeconomics Welfare measures and aggregation October 17, 2010 The plan: 1 Welfare measures 2 Example: 1 Our consumer has initial wealth w and is facing the initial set of market prices p 0. 2 Now he is faced with