Presentation.pdf

Incentive separability?
Paweł Doligalski Piotr Dworczak Joanna Krysta Filip Tokarski
October 5, 2023
Theory Brown Bag seminar, Northwestern
?
Co-funded by the European Union (ERC, IMD-101040122). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the European Research Council.

Motivation
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?

Motivation
Natural step: Interaction between IMD and optimal taxation?

Motivation
Fundamentally: Equity-efficiency trade-off (under incentive
constraints)

Motivation
constraints)
Famous results identifying cases when trade-off can be avoided:

Motivation
constraints)
Diamond Mirrlees ’71: distortionary taxes on firms are
redundant (when only consumers have private information)

Motivation
constraints)
Atkinson Stiglitz ’76: distortionary consumption taxes are
redundant (when consumers’ preferences over commodities are
homogeneous and weakly separable from labor supply)

Motivation
constraints)
Atkinson Stiglitz ’76: distortionary consumption taxes are
redundant (when consumers’ preferences over commodities are
homogeneous and weakly separable from labor supply)
’No distortion at the top’: the most productive agent should face
zero marginal tax rate (if the support of types is bounded)

What we do
Explore the logic behind these results using a mechanism
design approach.

What we do
design approach.
Introduce an abstract framework with no a priori restrictions on
preferences and the form of incentive constraints.

What we do
design approach.
Study the notion of incentive separability:
A set of decisions is incentive-separable if perturbing these
decisions along agents’ indifference curves preserves all
incentive constraints.

What we do
design approach.
Study the notion of incentive separability:
A set of decisions is incentive-separable if perturbing these
decisions along agents’ indifference curves preserves all
incentive constraints.
Main result: The optimal mechanism allows agents to make
unrestricted choices over incentive-separable decisions, given
some prices and budgets.

What we do
The main result implies a generalization of the
Atkinson-Stiglitz and Diamond-Mirrlees theorems.

What we do
Simple proof consisting of two steps:

What we do
Removing distortions in incentive-separable decisions
improves the mechanism;

What we do
Undistorted decisions can be decentralized.

What we do
We also study a new application: a novel justification for in-kind
programs such as food stamps programs in the US.

What we do
We also study a new application: a novel justification for in-kind
programs such as food stamps programs in the US.
Remark: Mechanism design useful in deriving properties of the
optimal mechanism (even if full optimum cannot be found)

Related literature
Redundancy (or not) of commodity taxes with income tax
Atkinson Stiglitz (1976), Ordover Phelps (1979), Christiansen (1981),
Cremer, Gahvari Ladoux (1998), Gauthier Laroque (2009), Cremer
Gahvari (1995), Cremer, Pestieau Rochet (2001), Saez (2002),
da Costa Werning (2002), Golosov, Kocherlakota Tsyvinski (2003), ...
Optimal in-kind redistribution
Nichols Zeckhauser (1982), Currie Gahvari (2008),
Condorelli (2012), Akbarpour r
⃝
Dworczak r
⃝
Kominers (2023), ...
Optimality (or not) of production efficiency
Diamond Mirrlees (1971), Stiglitz Dasgupta (1971), Naito (1999),
Hammond (2000),...
Welfare-improving tax reforms
Laroque (2005), Kaplow (2006), ...

Framework
A unit mass of agents with types 2 [0;1] Θ, uniformly
distributed (wlog);
Agents’ utility over a vector of K decisions x 2 RK
+
U (x; ) ;
U continuous in x, measurable in .
x could include consumption of goods, labor supply, effort, ...

Framework
A unit mass of agents with types 2 [0;1] Θ, uniformly
distributed (wlog);
Agents’ utility over a vector of K decisions x 2 RK
+
U (x; ) ;
U continuous in x, measurable in .
x could include consumption of goods, labor supply, effort, ...
An allocation rule x : Θ ! RK
+
associated aggregate allocation: x =
R
x() d
associated utility profile Ux : Ux () = U(x(); )

Framework
Social planner with objective function
W(Ux ;x);
continuous and non-increasing in x.

Framework
W(Ux ;x);
Preferences over aggregate allocations x:
- opportunity cost of resources;
- preference for (tax) revenue;
- aggregate constraints, etc.

Framework
W(Ux ;x);
Preferences over aggregate allocations x:
- opportunity cost of resources;
- preference for (tax) revenue;
- aggregate constraints, etc.
Planner chooses x : Θ ! RK
+ subject to incentive constraints:
x 2 I

RK
+
Θ
- incentive-compatibility;
- moral-hazard constraints;
- voluntary participation, etc.

Incentive separability
Notation: For a subset of decisions S f1;:::;Kg:
xS = (xi)i2S; x S = (xi)i =
2S; x = (xS;x S):

2S; x = (xS;x S):
Definition
Decisions S are incentive-separable (at feasible allocation x0 2 I) if
f(xS; x S
0 ) : U(xS(); x S
0 (); ) = U(x0(); ); 8 2 Θg I:

2S; x = (xS;x S):
Definition
Decisions S are incentive-separable (at feasible allocation x0 2 I) if
f(xS; x S
0 ) : U(xS(); x S
0 (); ) = U(x0(); ); 8 2 Θg I:
Decisions S are incentive-separable (IS) if perturbations of xS that
keep all types indifferent satisfy incentive constraints.
IS is a joint restriction on preferences and incentives.

Examples of incentive-separable decisions
IS decisions: f(xS; x S
0 ) : U(xS(); x S
0 (); ) = U(x0(); ); 8 2 Θg I:
1. Voluntary participation:
I = f x : U(x(); ) U(); 8 2 Θ g;
All decisions are incentive-separable.

IS decisions: f(xS; x S
0 ) : U(xS(); x S
0 (); ) = U(x0(); ); 8 2 Θg I:
1. Voluntary participation:
I = f x : U(x(); ) U(); 8 2 Θ g;
All decisions are incentive-separable.
2. Private information:
I =

x : U(x(); ) max
0
2Θ
U(x(0); ); 8 2 Θ

;
Weak separability of decisions S,
U(x(); ) = e
U (v(xS()); x S(); ); 8 2 Θ;
implies incentive separability of S.

3. Moral hazard:
Hidden action a 2 A affects the distribution of observed state !:
U((a; y); (; !)) =
X
!
u(y; !)P(!ja; ) c(a; ); y 2 RL
+;
Set I: Agent with type reports truthfully, takes recommended
action a(), and consumes y(; !).
With = (; !) and x = (a; y), decisions y are incentive-separable.

3. Moral hazard:
Hidden action a 2 A affects the distribution of observed state !:
U((a; y); (; !)) =
X
!
u(y; !)P(!ja; ) c(a; ); y 2 RL
+;
Set I: Agent with type reports truthfully, takes recommended
action a(), and consumes y(; !).
With = (; !) and x = (a; y), decisions y are incentive-separable.
4. Easy to extend to combinations of incentive constraints, dynamic
private information, verifiable information, labels, ...

Preliminaries
We will keep x S
0 and Ux0
fixed, and re-optimize over the allocation of
incentive-separable goods xS:
U(xS()) := U(xS(); x S
0 (); ); 8 2 Θ;
R(xS) := W(Ux0
; xS + x S
0 );
where xS
k =
R
xk ()1k2Sd, 8k:

Preliminaries
We will keep x S
0 and Ux0
fixed, and re-optimize over the allocation of
incentive-separable goods xS:
U(xS()) := U(xS(); x S
0 (); ); 8 2 Θ;
R(xS) := W(Ux0
; xS + x S
0 );
where xS
k =
R
xk ()1k2Sd, 8k:
Assumptions:
U is locally nonsatiated, for all ;
U(xS
0 ()) U(0), for all ;
there exists an integrable x̄() such that
U(y) = U(xS
0 ()) =) y x̄().

S-undistorted allocations
Suppose that decisions S f1;:::;Kg are incentive-separable.

Definition
A feasible allocation x0 is S-undistorted if xS
0 solves
max
xS
R(xS) subject to U(xS()) = U(xS
0 ()); 8 2 Θ:

Definition
0 solves
max
xS
0 ()); 8 2 Θ:
Intuitively: The allocation of incentive-separable decisions is the
first-best way for the designer to deliver the target utility profile.

Definition
0 solves
max
xS
0 ()); 8 2 Θ:
Lemma (Optimality)
If x0 is not S-undistorted, then it can be improved upon.

Definition
0 solves
max
xS
0 ()); 8 2 Θ:
Lemma (Optimality)
If x0 is not S-undistorted, then it can be improved upon.
Proof: Replace xS
0 with S-undistorted xS
? :
1. Utility levels are unaffected: Ux? = Ux0
;
2. Incentive constraints are unaffected (by incentive separability);
3. Planner’s objective strictly improves.

S-undistorted allocations and Pareto efficiency
S-undistortedness is linked to Pareto efficiency.

Conditional Pareto efficiency of xS
0 : there does not exist an
alternative allocation xS of incentive-separable goods that
results in the same aggregate allocation;
makes a positive mass of agents strictly better off;
doesn’t make any agent worse off.

If R is strictly decreasing:
S-undistortedness =) conditional Pareto efficiency

If R is strictly decreasing:
S-undistortedness =) conditional Pareto efficiency
Intuition: If agents’ utilities could be increased, the planner would
scale agents’ allocations down to deliver initial utilities and
pocket the left-overs.

Decentralization
Definition
An allocation xS can be decentralized (with prices 2 RK
++) if there
exists a budget assignment m() such that, for all , xS() solves
max
y2RjSj
+
U(y) subject to S y m();

Decentralization
Definition
++) if there
max
y2RjSj
+
Lemma (Decentralization)
A feasible xS
0 can be decentralized with prices 2 RK
++ if and only if
x0 is regular: xS
0 solves
min
xS
xS subject to U(xS()) = U(xS
0 ()); 8 2 Θ:

Decentralization
Definition
++) if there
max
y2RjSj
+
Lemma (Decentralization)
A feasible xS
0 can be decentralized with prices 2 RK
++ if and only if
x0 is regular: xS
0 solves
min
xS
0 ()); 8 2 Θ:
Proof: Follows from consumer duality: Mas-Colell et al. 1995, Prop 3.E.1

Decentralization
Compare the definition of S-undistortedness:
max
xS
0 ()); 8 2 Θ;
with the definition of regularity
min
xS
0 ()); 8 2 Θ:

Decentralization
max
xS
0 ()); 8 2 Θ;
min
xS
0 ()); 8 2 Θ:
If R is affine, then all S-undistorted allocations are regular.

Decentralization
max
xS
0 ()); 8 2 Θ;
min
xS
0 ()); 8 2 Θ:
Much weaker conditions on R suffice, due to Separating
Hyperplane Theorem.

Decentralization
max
xS
0 ()); 8 2 Θ;
min
xS
0 ()); 8 2 Θ:
Much weaker conditions on R suffice, due to Separating
Hyperplane Theorem.
R has bounded marginals if there exist constants c̄ c 0
such that, for all y 2 RK , k 2 f1;:::; K}, and 0,
c̄ R(y) R(y + ek )
c:

Main result
Theorem
Starting at any feasible allocation x0, the planner’s objective can be
improved by allowing agents to purchase incentive-separable goods
at S undistorted prices subject to type-dependent budgets.

Main result
Theorem
Take-aways:

Main result
Theorem
Take-aways:
There should be no distortions among IS goods;

Main result
Theorem
Take-aways:
Agents can trade IS goods freely given prices and budgets;

Main result
Theorem
Take-aways:
With production, S-undistorted prices are equal to marginal costs;

Main result
Theorem
Take-aways:
Result silent about non-IS goods.

Main result
Theorem
Take-aways:
Proof: Mimics the logic of the second welfare theorem applied to
incentive-separable goods only.

Main result
Theorem
Take-aways:
Proof: Mimics the logic of the second welfare theorem applied to
incentive-separable goods only.
Bounded marginals of R: The planner optimally selects an
allocation that can be decentralized with strictly positive prices.

Extensions
We can allow the set of IS goods to depend on types , by
letting S : Θ ! 2f1; 2;:::; Kg, and
xS() xS()
(); xS
k
Z
xS
k ()1k2S()d; 8k:

Extensions
xS() xS()
(); xS
k
Z
xS
k ()1k2S()d; 8k:
All results go through verbatim.

Extensions
xS() xS()
(); xS
k
Z
xS
k ()1k2S()d; 8k:
We can add a constraint x 2 F to the definition of IS goods:
f(xS; x S
0 ) 2 F : U(xS(); x S
0 (); ) = U(x0(); ); 8 2 Θg I:

Extensions
xS() xS()
(); xS
k
Z
xS
k ()1k2S()d; 8k:
f(xS; x S
0 ) 2 F : U(xS(); x S
0 (); ) = U(x0(); ); 8 2 Θg I:
F-constrained S-undistorted allocations are optimal.

Extensions
xS() xS()
(); xS
k
Z
xS
k ()1k2S()d; 8k:
f(xS; x S
0 ) 2 F : U(xS(); x S
0 (); ) = U(x0(); ); 8 2 Θg I:
F-constrained S-undistorted allocations are optimal.
Then, ‘no distortion at the top’ is a corollary—under single
crossing, allocation to the highest type is incentive-separable.

Application 1: Atkinson-Stiglitz
Original setting:
Agents with hidden ability make a labor supply decision L 2 R+
and consumption decisions y 2 RL
+; x = (y; L).

Original setting:
+; x = (y; L).
Preferences are weakly separable: U(v(y); L; ).

Original setting:
+; x = (y; L).
All goods k are produced from labor at marginal cost k 0.

Original setting:
+; x = (y; L).
Planner’s objective is ( is the marginal value of public funds):
W(Ux ; x) = V(Ux ) + (L
K 1
X
k=1
k yk ):

Original setting:
+; x = (y; L).
Planner’s objective is ( is the marginal value of public funds):
W(Ux ; x) = V(Ux ) + (L
K 1
X
k=1
k yk ):
Corollary
Consider a feasible allocation x0. The planner’s objective can be
improved by allowing agents to freely purchase commodities at prices
proportional to marginal costs (with budgets determined by an
income tax).

Our main result is more general than existing extensions of the
Atkinson-Stiglitz theorem:
More complex settings and additional incentive constraints:

only a subset of commodities may be weakly separable;

there may be moral-hazard constraints on some decisions;

combine the redistributive (Mirrlees, 1971) and social-insurance
(Varian, 1981) strands of the taxation literature.

We rule out any distortionary mechanisms, not just distortionary
taxation, e.g., rationing, public provision. (IMD suboptimal!)

What is essential is not the availability of nonlinear income tax
but rather the ability to implement budgets for IS goods.

What is essential is not the availability of nonlinear income tax
but rather the ability to implement budgets for IS goods.
No assumptions needed for first-order analysis.

Application 2: food vouchers
Let F f1;:::;Kg denote the consumption of various food items;
F denotes other commodities and decisions.

Idiosyncratic food tastes more pronounced when food
consumption is high (Jensen Miller, 2010):
U(x; ) =
(
U1(v(xF ); x F ; ) if v(xF ) v;
U2(x; ) otherwise.

U(x; ) =
(
U1(v(xF ); x F ; ) if v(xF ) v;
U2(x; ) otherwise.
v(xF ) is the nutritional value of food.

U(x; ) =
(
U1(v(xF ); x F ; ) if v(xF ) v;
U2(x; ) otherwise.
) Heterogeneous preferences over food when v(xF ) v

U(x; ) =
(
U1(v(xF ); x F ; ) if v(xF ) v;
U2(x; ) otherwise.
A motive for differential taxes on food items (Saez, 2002)

U(x; ) =
(
U1(v(xF ); x F ; ) if v(xF ) v;
U2(x; ) otherwise.
) Weakly-separable preferences over food when v(xF ) v

U(x; ) =
(
U1(v(xF ); x F ; ) if v(xF ) v;
U2(x; ) otherwise.
A motive for no taxes on food items (Atkinson-Stiglitz thm)

U(x; ) =
(
U1(v(xF ); x F ; ) if v(xF ) v;
U2(x; ) otherwise.
A motive for no taxes on food items (Atkinson-Stiglitz thm)
is private ! food items are IS for types s.t. v(xF
0 ()) v.

Corollary
improved by assigning budgets to all agents with v(xF
0 ()) v;
and letting them spend these budgets on food items F (but no other
goods) priced at marginal cost.

Corollary
0 ()) v;
Role of voucher programs:

Corollary
0 ()) v;
- They isolate recipients from tax distortions.

Corollary
0 ()) v;
Consider the U.S. food stamps program (SNAP):

Corollary
0 ()) v;
- Purchases exempt from commodity taxes (state and local)

Corollary
0 ()) v;
Complementary to classical rationale for in-kind transfers
(Nichols and Zeckhauser, 1982)

Corollary
0 ()) v;
SNAP criteria mostly exclude unemployed (work requirement)

Corollary
0 ()) v;
SNAP criteria mostly exclude unemployed (work requirement)
- Restricting eligibility to strengthen work incentives is suboptimal

Application 3: Atkinson-Stiglitz meet Diamond-Mirrlees
Same consumption setup as in the Atkinson-Stiglitz framework.

There are J firms; zj 2 RK denotes firm j’s production vector;
Zj is the set of feasible production technologies for firm j.

z = (z1;:::;zJ) is a production plan and z =
PJ
j=1 zj is aggregate
production; Z is the Minkowski sum of Zj over j;

PJ
j=1 zj is aggregate
Allocation (x;z) is feasible if x 2 I, z 2 Z; and z x.

PJ
j=1 zj is aggregate
The planner maximizes W(Ux ;z x), non-decreasing in z x.

PJ
j=1 zj is aggregate
The planner maximizes W(Ux ;z x), non-decreasing in z x.
A feasible production plan z0 is efficient if there does not exist
z 2 Z such that z z0 and z 6= z0.

Corollary
For any feasible allocation, the planner’s objective can be improved
by choosing an S-undistorted allocation of incentive-separable goods
and an efficient production plan.

Corollary
Key insight: production decisions are (trivially)
incentive-separable.

Corollary
To obtain a decentralization result, we need additional
assumptions on feasible production plans.

Corollary
To obtain a decentralization result, we need additional
assumptions on feasible production plans.
Assumption
The set Z is closed, bounded from above and convex; for any z0 2 Z
and any nonempty proper subset A f1;:::;Kg, there exists z 2 Z
such that zA zA
0 , z A z A
0 and z A 6= z A
0 .

Theorem
For any feasible allocation, there exists a price vector 2 RK
++ such
that the planner’s objective can be improved by simultaneously:
(i) allowing agents to purchase incentive-separable goods at prices
subject to type-dependent budgets;
(ii) allowing firms to maximize profits and trade all goods taking
prices as given, and taxing their profits lump-sum.

Theorem
++ such
The assumption on Z guarantees strictly positive prices.

Theorem
++ such
The assumption on Z guarantees strictly positive prices.
We can impose stronger restrictions on preferences to relax the
assumptions on Z (e.g., to allow for fixed supply).

Concluding Remarks
Concluding Remarks

Concluding Remarks
Incentive separability: Useful notion to study optimality and
decentralization in complex incentive problems
We focused on applications within public finance.
The notion of incentive separability could be applied in other
contexts.
Is incentive separability (effectively) necessary for a set of
decisions to remain undistorted in the optimal mechanism
(regardless of redistributive preferences)?

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