Pareto improving structural reforms Gilles Saint-Paul (PSE & NYUAD)
March 3, 2019
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
1 / 48
Introduction
Standard view: If a reform is e¢ cient, side transfers can make everybody better-o¤ Example: Delpla-Wyplosz recommendation for rent buyback But side transfers are costly to the budget And taxes are distortionary Does this result hold under distortionary taxation?
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
2 / 48
A negative intuition
Aggregate gains from reform are small relative to their distributive e¤ects "triangle" vs "rectangle"
Welfare losses from distortionary taxation of the same order of magnitude as losses from regulation
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
3 / 48
An example
Price pegged above equilibrium market price. p (x ) = inverse demand function, c (x ) = marginal cost function. Îť = welfare cost of public funds.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
4 / 48
Reform Reform reduces price. by dp, Change in consumer surplus dCS = p 0 (x )xdx > 0. Change in producer surplus dPS = (p (x ) c (x ) + p 0 (x )x )dx. Need to transfer dPS to producers Additional taxes must be such that dT = λdT dPS = 1dPSλ . Total e¤ect on social welfare dCS + dPS
λdT =
p
c + λxp 0 . 1 λ
For this to work we need that λ where η D = markup.
p xp 0
Gilles Saint-Paul (PSE & NYUAD) ()
p c µ 1 = ηD , 0 xp µ
(1)
is the elasticity of demand and µ = p/c is the Pareto improving structural reforms
March 3, 2019
5 / 48
Replacing regulation by taxes
Consider a tax d τ on the good itself Revenues xd τ Output loss dx = d τ/p 0 < 0. Welfare gain (p
c )d τ/p 0 < 0, i.e.
p c xp 0
per unit of tax collected.
Hence total cost of public funds cannot exceed η D
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
µ 1 µ .
March 3, 2019
6 / 48
Basic setup
There is a continuum of consumers-producers over [0, 1], indexed by i. Consumers with index i are endowed with a speci…c labor input which only allows them to produce good i. All consumers have the same utility function, given by U (fcij g, li ) =
Gilles Saint-Paul (PSE & NYUAD) ()
Z 1 0
1/α
cijα dj
Pareto improving structural reforms
γ
li , γ
March 3, 2019
7 / 48
Production
Each good j is produced with a linear technology, yj = lj .
=)Wages coincide with prices
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
8 / 48
Regulation
A fraction r of the goods are regulated. Goods such that i > r are unregulated. Their (common by symmetry) price is normalized to 1. Goods such that i < r are regulated at pR > 1. People such that i < r cannot supply more than an amount of labor l̄
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
9 / 48
The structure of individual choice Indirect utility function V (R, l, p ) = where p = (rpR 1
R p
lγ , γ
α α
+1
r)
1 α α
.
Demand for nonregulated goodscN (R, p ) = Rp 1 Demand for regulated goods
(2)
α α
1
α
cR (R, p ) = Rp 1 α pR 1
α
Labor supply (constrained if i < r ): l (w /p ) = Gilles Saint-Paul (PSE & NYUAD) ()
w p
1 γ 1
Pareto improving structural reforms
. March 3, 2019
10 / 48
Non regulated households Labor supply l = lN = p
1 γ 1
.
Consumption of nonregulated goods cNN = p 1
α α
1 γ 1
(3)
Consumption of regulated goods cNR = p 1
α α
1 γ 1
1
pR 1
α
Utility uN =
Gilles Saint-Paul (PSE & NYUAD) ()
1
1 γ
p
Pareto improving structural reforms
γ γ 1
March 3, 2019
11 / 48
Regulated households’consumption
Nonregulated goods: α
cRN = p 1 α pR l̄
(4)
Regulated goods: α
α
cRR = p 1 α pR 1 α l̄
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
12 / 48
Equilibrium regulated labor supply Write down that supply equals demand for any nonregulated good. Walras’s law, demand then consistent with l̄ in regulated sectors. lN = rcRN + (1
r )cNN .
Substituting, l̄ = p
1 γ 1
1
pR 1 α ,
(5)
Yields equilibrium utility
uR
= V (pR l̄, l̄, p ) = p
γ γ 1
pR 1
α α
1 1 γα p γ R
= JuN
Gilles Saint-Paul (PSE & NYUAD) ()
(6) (7)
Pareto improving structural reforms
March 3, 2019
13 / 48
Extensive reform
Fraction of regulated sectors reduced to r 0 < r . Size of reform ∆r = r
r0
Lumpsum transfer T to each household i such that r 0 < i
r.
General proportional tax on labor income Ď„.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
14 / 48
Nonregulated households Let p 0 = new price level. Income = l (1
τ)
Labor supply: 1 γ 1
l = lN0 = p 0
1
(1
τ) γ 1 .
(8)
Utility uN0
=
1 γ
1
They bene…t from the i¤ uN0
τ p0
γ γ 1
.
(9)
uN , i.e. 1
Gilles Saint-Paul (PSE & NYUAD) ()
1
τ
p0 . p
Pareto improving structural reforms
(10)
March 3, 2019
15 / 48
Deregulated households By linearity of indirect utility function, their labor supply is the same as for nonregulated, lN0 = (p 0 (1
1 γ 1
τ ))
.
Disposable income RD0 = T + p 0
1 γ 1
(1
τ) γ
γ 1
Utility uD0 = They gain i¤ uD0 T
T + 1 p0
1 γ
p0
γ γ 1
(1
γ
τ) γ 1 .
(11)
uR , i.e. p
Gilles Saint-Paul (PSE & NYUAD) ()
0
1 γ
1
γ 1 γ
"
p0 p
γ γ 1
J
Pareto improving structural reforms
(1
τ) γ
γ 1
#
.
March 3, 2019
(12) 16 / 48
Nonderegulated households Income RR0 = l̄ 0 pR (1 Utility is uR0 =
pR l̄ 0 (1 p0
τ) l̄ 0γ . γ
τ)
(13)
Computing l̄ 0 : r 0 l̄ 0 pR (1
τ) = p0
1 γ 1
(1
τ) γ
1 1
Government’s BC T ∆r = τ (1
r 0 )p 0
h
1 γ 1
r 0 pR 1
(1
α α
+ τ (1
τ) γ
1 1
r 0)
i
T ∆r .
+ τr 0 pR l̄ 0 .
(14)
(15)
Allows to compute 1
l̄ 0 = pR 1 α p 0 Gilles Saint-Paul (PSE & NYUAD) ()
1 γ 1
(1
τ) γ
Pareto improving structural reforms
1
(16)
1 March 3, 2019
17 / 48
Condition for reform to be Pareto-improving LEMMA 1 – For any r 0 < r , the reform can be implemented in a Pareto-improving way if and only if there exists some τ 2 [0, 1] such that 1 and, T
p
0
1 γ
1
γ 1 γ
for T =
τ (1
"
p0 , p
τ
γ γ 1
p0 p
J 1
τ) γ 1 p0 ∆r
1 γ 1
(1
τ) γ
γ 1
#
.
(19)
α 1 α
.
(20)
If so, the reform is called viable.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
18 / 48
The La¤er curve PROPOSITION 1 – (i) There exists a unique tax rate τ max 2 (0, 1) which maximizes the utility of the deregulated agents, uD0 , which is hump-shaped in τ. (ii) This tax rate is equal to α
τ max
(γ 1)(p 0 1 α ∆r ) . = α γp 0 1 α (γ 1)∆r
(21)
(iii) Assume p
γ
r (γ
1),
(22)
then τ max
1
p 0 /p for all r 0
r.
(iv) Assume (22) is violated. Then there exists a unique r̃ 2 (0, r ) such that r 0 r̃ , τ max 1 p 0 /p.
.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
19 / 48
A possibility result
PROPOSITION 2 – (i) All reforms are viable (ii) If τ max 1 p 0 /p then there exists τ c 2 (0, 1 p 0 /p ] such that a reform is viable if and only if τ c τ 1 p 0 /p, 0 (iii) If τ max < 1 p /p then there exists τ c 2 (0, τ max ] such that a reform is viable if and only if τ c τ 1 p 0 /p, furthermore all reforms such that τ max < τ are Pareto-dominated by τ = τ max .
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
20 / 48
Why?
Ď„ = 1 p 0 /p leaves real wage, labor supply and utility unchanged for NR and ND Aggregate gains from reform all go to the D rgoups But gains are positive Because relative price distortions are reduced Because employment goes up for the D groups
For distortions to prevent viability, compensatory taxes should reduce labor supply. Not true here.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
21 / 48
Intensive reforms Now assume prices of all regulated goods are uniformly lowered to pR0 < pR . LEMMA 2 – For any pR0 < p 0 , the reform can be implemented in a Pareto-improving way if and only if there exists some τ 2 [0, 1] such that p0 1 τ (23) p holds as well as
τ (1
1
τ) γ 1 p
0
α
1 α
r
1
γ γ
γ γ 1
p (1 τ ) p0
J
J
0
!
p0 p
γ γ 1
, (24)
with at least one strict inequality where 0
J = Gilles Saint-Paul (PSE & NYUAD) ()
0
γpR
α 1 α
γ
0
pR 1
γ 1 α
Pareto improving structural reforms
.
(25) March 3, 2019
22 / 48
Intensive reforms are viable too
PROPOSITION 3 – (i) All reforms are viable (ii) If τ max 1 p 0 /p then there exists τ c 2 (0, 1 p 0 /p ] such that a reform is viable if and only if τ c τ 1 p 0 /p, 0 (iii) If τ max < 1 p /p then there exists τ c 2 (0, τ max ] such that a reform is viable if and only if τ c τ 1 p 0 /p, furthermore all reforms such that τ max < τ are Pareto-dominated by τ = τ max .
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
23 / 48
Notations
There are q = n + r goods, indexed by i = 1, ...q. Vectors denoted by x, whose components xi . Matrices denoted by M and their components by mij . If f () : Rp ! Rn , its gradient rf is a matrix (∂fi /∂xj ).
If f () : Rp ! R, rf is a line vector to remain consistent with the preceding de…nition.
hx, yi = ∑i xi yi , x y =(xi yi ), x y =(xi /yi ) In particular hx, yi = hx z, y zi .
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
24 / 48
Preferences
Continuum of agents of total mass 1 Same utility function but di¤er by skills. For each i, a mass 1/q of agents have skills speci‌c to sector i Utility is U (c,l ) = u (c) v (l ), where v 0 , v 00 > 0 and u is homogeneous of degree one.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
25 / 48
The structure of demand
Let pi be the price of good i. Consumer demand is ci = RĎˆi (p), where R is expenditure. We have that
hp, Ďˆ(p)i = 1, 8p.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
(26)
March 3, 2019
26 / 48
Walrasian labor supply
Aggregate hedonic price level to 1, Hence utility can be rewritten as R
v (l ).
Labor supply is l = Îť (w ), where Îť = v 0
1
and w is the real wage.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
27 / 48
Production
Production yi in sector i only uses the speci‌c labor input li and unit labor requirement is equal to 1 yi = li .
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
28 / 48
Regulation
N = f1, ..., n g nonregulated sectors
R = fn + 1, ..., q g regulated sectors.
In deregulated sectors, the price adjusts to clear markets. In regulated sectors, the price is ‌xed by law above the competitive level, Output is determined by demand Supply is rationed.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
29 / 48
Pre-reform resouce allocation Output equals demand in all sectors l=Y ψ (p) Output equals supply in all nonregulated sectors lN = λ(pN ) Aggregate p normalized to 1 u (ψ(p)) = 1. Remarks: Unknows: Y , pN , l. By construction, total income always equals total expenditure, Y = hp, li . Distribution of income is R = p l. Income shares s = p ψ (p) Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
30 / 48
Distortion size
For prices to be above equilibrium in regulated sectors, we need that v 0 (li ) < pi for i 2 R. Let ωi = 1
v 0 (li ) 2 [0, 1). pi
ω i is a measure of distortions in sector i.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
31 / 48
De…ning reforms Pure structural reform (PSR) pR is replaced by pR + dpR . Structural reform with …scal adjustment (SRFA): 0 = p + dp , pR is replaced by pR R R income in each sector taxed at rate d τ i , (possibly negative) lump-sum transfer to sector i, dTi . Government budget constraint
hs, dτ i Y = h1, dTi
(27)
Structural reform with side transfers (SRST): SRFA such that dτ = 0. Budget constraint becomes
h1, dTi = 0.
(28)
Feasible structural reform (FSR) SRFA such that dT Gilles Saint-Paul (PSE & NYUAD) ()
0.
Pareto improving structural reforms
March 3, 2019
32 / 48
Post-reform equilibrium allocation of output and labor
l0 = Y ψ (p0 ) 0 lN
0 λ(pN
= u (ψ(p )) = 1,
(29)
(1N dτ N ))
(30)
0
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
(31)
March 3, 2019
33 / 48
Key observation: taxes not distortionary in regulated sectors
In regulated sectors, supply is constrained by demand. Therefore activity is not reduced by a marginal proportional tax. As a result, the feasibility constraint dTi regulated sectors.
0 is irrelevant for
Lump-sum tax can be replaced by proportional tax, with no e¤ect on equilibrium
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
34 / 48
Equivalence DEFINITION 1 – Let ρ0 = (dpR 0 , dτ 0 , dT0 ) and ρ1 = (dpR 1 , dτ 1 , dT1 ) be two SRFAs. They are equivalent [ρ0 ρ1 ] if and only if l00 = l10 , 0
R00 ψ(p0 )T
= R10 ψ(p10 )T .
DEFINITION 2 – Let ρ0 = (dpR 0 , dτ 0 , dT0 ) and ρ1 = (dpR 1 , dτ 1 , dT1 ) be two SRFAs. They are production-equivalent [ρ0 ρ1 ] if and only if l00 = l10 , p00 = p10 . Clearly, if two reforms are production-equivalent and deliver the same distribution of income (R00 =R10 ), then they are equivalent. Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
35 / 48
Distortions may limit feasibility only for N groups
Two reforms yielding the same net taxes for R agents are equivalent: LEMMA 3 – Let ρ0 = (dpR 0 , dτ 0 , dT0 ) and ρ1 = (dpR 1 , dτ 1 , dT1 ) be two SRFAs.. Assume that dτ N 0 = dτ N 1 , dTN 0 = dTN 1 , and dTR 0 Y sR 0 dτ R 0 = dTR 1 Y sR 1 dτ R 1 . Then ρ0 ρ1 . If reform is feasible for N agents, equivalent feasible reform can be implemented: LEMMA 4 – For any SRFA ρ such that dTN such that ρ̃ ρ.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
0, there exists an FSR ρ̃
March 3, 2019
36 / 48
Taxing away consumer gains
DEFINITION 3 – An SRFA is N-neutral if and only if dTN = 0 and dτ N = dpN
pN .
(32)
In an N-neutral reform, dlN = 0.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
37 / 48
N neutral reforms are viable if aggregate welfare goes up
The e¤ect on utilitarian social welfare is dW = hs ω, dl Y
li .
Consequently, aggregate social welfare goes up if and only if
hs ω, dl
li > 0.
(33)
Assume an SRFA ρ0 is N-neutral and satis…es (33). Then there exists an N-neutral FRS ρ1 such that (i) ρ0 ρ1 , and (ii) ρ1 is Pareto-improving. Hence Pareto and Utilitarian criteria coïncide for N-neutral reforms
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
38 / 48
Existence of Pareto-improving structural reform
PROPOSITION 6 – For any r-vector dz there exists an N-neutral SRFA such that dlR = dz. Corollary – There exists a Pareto-improving FSR. N-neutral reform can be constructed to generate any vector of employment growth in R sectors Such vector can be chosen to be positive By (33), social welfare goes up Hence Pareto-improving FSR exists.
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
39 / 48
But: Does FSR involve deregulation? PROPOSITION 7 – Let sR = hsR , 1i , pi σ = min σi = [rψ]ii > 0, i ψi and η = max
i ,j ,i 6=j
pj [rψ]ij ψi
0.
Consider a nonzero r-vector of price reductions dx xm = max dx > 0, x̄Y = hsR , dxi /sR > 0, and σR = hsR , σ R dxi / hsR , dxi > 0. Assume η<
1 q
1
min(
1
sR 2
σ,
0. Let
x̄Y σR ) xm
(34)
Then (i) there exist a N-neutral SRFAs such that dpR pR = dx and (ii) These reforms are such that dlR > 0, implying that they satisfy (33). Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
40 / 48
Conditions for Pareto-improving deregulation
Price cross-elasticities of demand small enough relative to own price elasticities structural reform not too unbalanced: maximum reduction in a regulated price not too large relative to average.
If this holds reform can be implemented In a Pareto improving way So that employment grows in all regulated sectors
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
41 / 48
If regulated prices indexed on aggregate price level the nonregulated have rents
ND no longer bene‌ts from induced fall in p For them not to lose, they must pay lower taxes than NR For nobody to lose, NR must have rents This reduces T , making compensation less e¢ cient
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
42 / 48
ND rent falls, taxes must leave rents to NR Now uR0 = uN0 J 0 , where 0
J =
α 1 α
γ(pR /pN0 )
γ
(pR /pN0 ) 1
γ 1 α
< J.
ND viability condition becomes 1 uR0
τ
pN pN0
J J0
γ 1 γ
,
uR =) uN0 > uN .
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
43 / 48
Partial reform never viable if regulation not too strict PROPOSITION 8 – There exists p̄ > 1 such that for any pR 2 (1, p̄ ), the only viable reform is r 0 = 0. α 0.3
γ 1.2
r 0.2
0.3
1.2
0.8
1.2
0.5 0.5 0.2
-2
1.2
0.2
0.3
2
0.2
∆r 0 0.2 0 0.5 0 0.2 0 0.2 0 0.2
pRm 2.17 1.67 1.09 ∞ 1.5
pRc 1.18 1.2 1.06 1.07 1.05 1.05 1.42 1.47 1.15 1.17
Table 1 Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
44 / 48
Example 1
People only consume a subset of the goods which di¤ers across people in a random way
They di¤er by their type = proportion of regulated goods consumed. Type r
h (r )
r̄ = total proportion of regulated goods Under reform, r̄ falls to r̄ 0 = λr̄ , with λ 2 [0, 1).
Reform is assumed to a¤ect people proportionally: r 0 = λr .
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
45 / 48
Example 1 (Ctd) Îť 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Table
Ď„ (%) 7.6 6.7 5.8 4.9 4.1 3.3 2.6 1.9 1.3 0.7 2
fraction of winners (%) among nonregulated non deregulated 60.8 85.0 61.8 86.7 63.0 88.4 64.4 90.0 65.4 91.7 66.7 95.0 67.3 95.0 68.3 96.7 67.5 96.7 65.2 91.7
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
46 / 48
Example 2
Goods are on a circle, indexed by j mod .1. Agents are uniformly distributed over this circle They all consume a fraction ρ of the goods, located in an interval Agent located at x consumes goods [x, x + ρ mod 1]. The distribution of preferences is independent of the allocation of producers to sectors. The regulated goods are those such that j 2 [0, r̄ ]. We assume 1
r̄ > ρ > r̄ .
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
March 3, 2019
47 / 48
Now reform scope matters crucially r̄ 0 0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 Table 3
τ (%) 9.6 8.5 7.4 6.4 5.4 4.4 3.5 2.6 1.7 0.9 –ρ
fraction of winners (%) among nonregulated non deregulated 46 45 45.8 46 45.7 45 45.3 46 44.8 45 44.3 44 43.5 43 42.7 43 41.9 43 40.8 40 = 0.4, pR = 1.4, r̄ = 0.3
Gilles Saint-Paul (PSE & NYUAD) ()
Pareto improving structural reforms
total 62.2 60.5 58.7 56.8 54.7 52.6 50.2 47.9 45.7 42.4
March 3, 2019
48 / 48