Advanced Microeconomics

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Aldous McGee
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1 Welfare measures and aggregation October 17, 2010

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 The (likely) solution For both situations (p 0, w) and (p 1, w) if we knew the utility function we could compute the indirect utility function: v(p 0, w) and v(p 1, w) We could then compare the utilities and compare the situations. But maybe we do not have a notion of utility and prefer to only talk about money? 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)). What is p - let us take either p 1 or p 0.

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 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))

6 Or: How much the consumer has to be paid to be exactly as well-o 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 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), so: p1 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 1 p h 1 (p 1, p 1, u 0 )dp 1

9 Demand functions and

10 EV, CV and welfare evaluation They will dier because of price eects They will provide the correct ranking They will be dierent from so-called consumer surplus (CS). EV>CV for normal good. What about inferior good?

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 ) thus we cannot use EV, due to dierent price vectors in the expenditure function (unable to compare monetary amounts expressed in dierent prices).

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)dτ If more than one price changes, the problem may be path dependent (the sequence of integration may matter).

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

14 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.

15 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.

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

17 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 homeogeneous 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).

18 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. Lets take a derivative: x(p, w)/ w i x(p, w) w w w i dw i = x i(p, w i (p, w)) dw i = x i(p, w i (p, w)) dw i dw i x(p, w) w i dw 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.

19 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 straigh wealth expansion paths.

20 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)/ p j +w i b i (p) b i (p) }{{}}{{} shift slope

21 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 30, 2012 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