Pareto improving structural reforms Gilles Saint-Paul (PSE & NYUAD)
March 3, 2019
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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?
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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
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An example
Price pegged above equilibrium market price. p (x ) = inverse demand function, c (x ) = marginal cost function. Îť = welfare cost of public funds.
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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
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p c µ 1 = ηD , 0 xp µ
(1)
is the elasticity of demand and µ = p/c is the Pareto improving structural reforms
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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
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µ 1 µ .
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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 ) =
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Z 1 0
1/α
cijα dj
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γ
li , γ
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Production
Each good j is produced with a linear technology, yj = lj .
=)Wages coincide with prices
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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 lĚ&#x201E;
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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
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. March 3, 2019
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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 =
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1
1 γ
p
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γ γ 1
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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
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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
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(6) (7)
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Extensive reform
Fraction of regulated sectors reduced to r 0 < r . Size of reform â&#x2C6;&#x2020;r = r
r0
Lumpsum transfer T to each household i such that r 0 < i
r.
General proportional tax on labor income Ď&#x201E;.
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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
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1
τ
p0 . p
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(10)
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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
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0
1 γ
1
γ 1 γ
"
p0 p
γ γ 1
J
Pareto improving structural reforms
(1
τ) γ
γ 1
#
.
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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)
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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.
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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.
.
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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 .
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Why?
Ď&#x201E; = 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.
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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
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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 .
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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 .
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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 speciâ&#x20AC;Śc 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.
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The structure of demand
Let pi be the price of good i. Consumer demand is ci = RĎ&#x2C6;i (p), where R is expenditure. We have that
hp, Ď&#x2C6;(p)i = 1, 8p.
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(26)
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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.
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Production
Production yi in sector i only uses the speciâ&#x20AC;Śc labor input li and unit labor requirement is equal to 1 yi = li .
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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 â&#x20AC;Śxed by law above the competitive level, Output is determined by demand Supply is rationed.
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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) ()
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Distortion size
For prices to be above equilibrium in regulated sectors, we need that v 0 (li ) < pi for i 2 R. Let Ď&#x2030;i = 1
v 0 (li ) 2 [0, 1). pi
Ď&#x2030; i is a measure of distortions in sector i.
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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.
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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
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(31)
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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
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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) ()
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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.
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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.
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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
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Existence of Pareto-improving structural reform
PROPOSITION 6 â&#x20AC;&#x201C; For any r-vector dz there exists an N-neutral SRFA such that dlR = dz. Corollary â&#x20AC;&#x201C; 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.
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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) ()
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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
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If regulated prices indexed on aggregate price level the nonregulated have rents
ND no longer beneâ&#x20AC;Śts 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
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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 .
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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) ()
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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 .
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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&#x201E; (%) 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
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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 .
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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
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total 62.2 60.5 58.7 56.8 54.7 52.6 50.2 47.9 45.7 42.4
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