This document discusses incentive separability in mechanism design. It introduces a framework that allows for complex incentive constraints like private information, moral hazard, and voluntary participation. It defines a new concept of incentive separability, where perturbing certain decisions along agents' indifference curves preserves incentive constraints. The main theorem states that for incentive-separable decisions, it is optimal to allow agents to make unrestricted choices over those decisions given prices and budgets. This can extend and unify previous results showing no distortions are needed for certain decisions. Applications discussed include the Atkinson-Stiglitz theorem on redundant commodity taxes and optimality of removing production distortions.
We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs are the unique solution to a system of integral equations, which reduce to a linear matrix equation under suitable representations of the underlying probabilities. Even when implied volatilities are all higher than historical volatilities, it can be optimal to sell options on some assets while buying options on others, as hedging demand outweighs demand for asset-specific returns.
We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs are the unique solution to a system of integral equations, which reduce to a linear matrix equation under suitable representations of the underlying probabilities. Even when implied volatilities are all higher than historical volatilities, it can be optimal to sell options on some assets while buying options on others, as hedging demand outweighs demand for asset-specific returns.
Inference for stochastic differential equations via approximate Bayesian comp...Umberto Picchini
Despite the title the methods are appropriate for more general dynamical models (including state-space models). Presentation given at Nordstat 2012, Umeå. Relevant research paper at http://arxiv.org/abs/1204.5459 and software code at https://sourceforge.net/projects/abc-sde/
Certainly! "Modeling and simulation of energy efficiency measures in industrial processes" is a fascinating topic to explore in your research paper. Here are some key points and areas you can cover:
Research internship on optimal stochastic theory with financial application u...Asma Ben Slimene
This is a presntation of my second year intership on optimal stochastic theory and how we can apply it on some financial application then how we can solve such problems using finite differences methods!
Enjoy it !
Presentation on stochastic control problem with financial applications (Merto...Asma Ben Slimene
This is an introductory to optimal stochastic control theory with two applications in finance: Merton portfolio problem and Investement/consumption problem with numerical results using finite differences approach
We consider the problem of model estimation in episodic Block MDPs. In these MDPs, the decision maker has access to rich observations or contexts generated from a small number of latent states. We are interested in estimating the latent state decoding function (the mapping from the observations to latent states) based on data generated under a fixed behavior policy. We derive an information-theoretical lower bound on the error rate for estimating this function and present an algorithm approaching this fundamental limit. In turn, our algorithm also provides estimates of all the components of the MDP.
We apply our results to the problem of learning near-optimal policies in the reward-free setting. Based on our efficient model estimation algorithm, we show that we can infer a policy converging (as the number of collected samples grows large) to the optimal policy at the best possible asymptotic rate. Our analysis provides necessary and sufficient conditions under which exploiting the block structure yields improvements in the sample complexity for identifying near-optimal policies. When these conditions are met, the sample complexity in the minimax reward-free setting is improved by a multiplicative factor $n$, where $n$ is the number of contexts.
Abstract:
"We study different possibilities to apply the principles of rough-paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of rough-paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelli's controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy."
René Schott
Econometrics is a tough subject so is its homework and assignments in general. In case you are looking for econometrics homework help, you can rely on Economicshelpdesk. Our tutors are expert and we honour our client’s need as well as privacy.
Seminar: Gender Board Diversity through Ownership NetworksGRAPE
Seminar on gender diversity spillovers through ownership networks at FAME|GRAPE. Presenting novel research. Studies in economics and management using econometrics methods.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
Inference for stochastic differential equations via approximate Bayesian comp...Umberto Picchini
Despite the title the methods are appropriate for more general dynamical models (including state-space models). Presentation given at Nordstat 2012, Umeå. Relevant research paper at http://arxiv.org/abs/1204.5459 and software code at https://sourceforge.net/projects/abc-sde/
Certainly! "Modeling and simulation of energy efficiency measures in industrial processes" is a fascinating topic to explore in your research paper. Here are some key points and areas you can cover:
Research internship on optimal stochastic theory with financial application u...Asma Ben Slimene
This is a presntation of my second year intership on optimal stochastic theory and how we can apply it on some financial application then how we can solve such problems using finite differences methods!
Enjoy it !
Presentation on stochastic control problem with financial applications (Merto...Asma Ben Slimene
This is an introductory to optimal stochastic control theory with two applications in finance: Merton portfolio problem and Investement/consumption problem with numerical results using finite differences approach
We consider the problem of model estimation in episodic Block MDPs. In these MDPs, the decision maker has access to rich observations or contexts generated from a small number of latent states. We are interested in estimating the latent state decoding function (the mapping from the observations to latent states) based on data generated under a fixed behavior policy. We derive an information-theoretical lower bound on the error rate for estimating this function and present an algorithm approaching this fundamental limit. In turn, our algorithm also provides estimates of all the components of the MDP.
We apply our results to the problem of learning near-optimal policies in the reward-free setting. Based on our efficient model estimation algorithm, we show that we can infer a policy converging (as the number of collected samples grows large) to the optimal policy at the best possible asymptotic rate. Our analysis provides necessary and sufficient conditions under which exploiting the block structure yields improvements in the sample complexity for identifying near-optimal policies. When these conditions are met, the sample complexity in the minimax reward-free setting is improved by a multiplicative factor $n$, where $n$ is the number of contexts.
Abstract:
"We study different possibilities to apply the principles of rough-paths theory in a non-commutative probability setting. First, we extend previous results obtained by Capitaine, Donati-Martin and Victoir in Lyons' original formulation of rough-paths theory. Then we settle the bases of an alternative non-commutative integration procedure, in the spirit of Gubinelli's controlled paths theory, and which allows us to revisit the constructions of Biane and Speicher in the free Brownian case. New approximation results are also derived from the strategy."
René Schott
Econometrics is a tough subject so is its homework and assignments in general. In case you are looking for econometrics homework help, you can rely on Economicshelpdesk. Our tutors are expert and we honour our client’s need as well as privacy.
Seminar: Gender Board Diversity through Ownership NetworksGRAPE
Seminar on gender diversity spillovers through ownership networks at FAME|GRAPE. Presenting novel research. Studies in economics and management using econometrics methods.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
Revisiting gender board diversity and firm performanceGRAPE
Cel: oszacować wpływ inkluzywności władz spółek na ich wyniki.
Co wiemy?
• Większość firm nie ma równosci płci w organach (ILO, 2015)
• Większość firm nie ma w ogóle kobiet we władzach
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We study the contribution of rising longevity to the rise of wealth inequality in the U.S. over the last seventy years. We construct an OLG model with multiple sources of inequality, closely calibrated to the data. Our main finding is that improvements in old-age longevity explain about 30% of the observed rise in wealth inequality. This magnitude is similar to previously emphasized channels associated with income inequality and the tax system. The contribution of demographics is bound to raise wealth inequality further in the decades to come.
(Gender) tone at the top: the effect of board diversity on gender inequalityGRAPE
The research explores to what extent the presence of women on board affects gender inequality downstream. We find that increasing presence reduces gender inequality. To avoid reverse causality, we propose a new instrument: the share of household consumption in total output. We extend the analysis to recover the effect of a single woman on board (tokenism(
Gender board diversity spillovers and the public eyeGRAPE
A range of policy recommendations mandating gender board quotas is based on the idea that "women help women". We analyze potential gender diversity spillovers from supervisory to top managerial positions over three decades in Europe. Contrary to previous studies which worked with stock listed firms or were region locked, we use a large data base of roughly 2 000 000 firms. We find evidence that women do not help women in corporate Europe, unless the firm is stock listed. Only within public firms, going from no woman to at least one woman on supervisory position is associated with a 10-15% higher probability of appointing at least one woman to the executive position. This pattern aligns with various managerial theories, suggesting that external visibility influences corporate gender diversity practices. The study implies that diversity policies, while impactful in public firms, have limited
effectiveness in promoting gender diversity in corporate Europe.
Tone at the top: the effects of gender board diversity on gender wage inequal...GRAPE
We address the gender wage gap in Europe, focusing on the impact of female representation in executive and non-executive boards. We use a novel dataset to identify gender board diversity across European firms, which covers a comprehensive sample of private firms in addition to publicly listed ones. Our study spans three waves of the Structure of Earnings Survey, covering 26 countries and multiple industries. Despite low prevalence of female representation and the complex nature of gender wage inequality, our findings reveal a robust causal link: increased gender diversity significantly decreases the adjusted gender wage gap. We also demonstrate that to meaningfully impact gender wage gaps, the presence of a single female representative in leadership is insufficient.
Gender board diversity spillovers and the public eyeGRAPE
A range of policy recommendations mandating gender board quotas is based on the idea that "women help women". We analyze potential gender diversity spillovers from supervisory to top managerial positions over three decades in Europe. Contrary to previous studies which worked with stock listed firms or were region locked, we use a large data base of roughly 2 000 000 firms. We find evidence that women do not help women in corporate Europe, unless the firm is stock listed. Only within public firms, going from no woman to at least one woman on supervisory position is associated with a 10-15\% higher probability of appointing at least one woman to the executive position. This pattern aligns with the Public Eye Managerial Theory, suggesting that external visibility influences corporate gender diversity practices. The study implies that diversity policies, while impactful in public firms, have limited effectiveness in promoting gender diversity in corporate Europe.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large New Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economies, we use this model to provide comparative statics across past and contemporaneous age structures of the working population. Thus, we quantify the extent to which the response of labor markets to adverse TFP shocks and monetary policy shocks becomes muted with the aging of the working population. Our findings have important policy implications for European labor markets and beyond. For example, the working population is expected to further age in Europe, whereas the share of young workers will remain robust in the US. Our results suggest a partial reversal of the European-US unemployment puzzle. Furthermore, with the aging population, lowering inflation volatility is less costly in terms of higher unemployment volatility. It suggests that optimal monetary policy should be more hawkish in the older society.
Evidence concerning inequality in ability to realize aspirations is prevalent: overall, in specialized segments of the labor market, in self-employment and high-aspirations environments. Empirical literature and public debate are full of case studies and comprehensive empirical studies documenting the paramount gap between successful individuals (typically ethnic majority men) and those who are less likely to “make it” (typically ethnic minority and women). So far the drivers of these disparities and their consequences have been studied much less intensively, due to methodological constraints and shortage of appropriate data. This project proposes significant innovations to overcome both types of barriers and push the frontier of the research agenda on equality in reaching aspirations.
Overall, project is interdisciplinary, combining four fields: management, economics, quantitative methods and psychology. An important feature of this project is that it offers a diversified methodological perspective, combining applied microeconometrics, as well as experimental methods.
The secret way to sell pi coins effortlessly.DOT TECH
Well as we all know pi isn't launched yet. But you can still sell your pi coins effortlessly because some whales in China are interested in holding massive pi coins. And they are willing to pay good money for it. If you are interested in selling I will leave a contact for you. Just telegram this number below. I sold about 3000 pi coins to him and he paid me immediately.
Telegram: @Pi_vendor_247
Yes of course, you can easily start mining pi network coin today and sell to legit pi vendors in the United States.
Here the telegram contact of my personal vendor.
@Pi_vendor_247
#pi network #pi coins #legit #passive income
#US
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
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Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
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#pinetwork
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
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A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
Abhay Bhutada Leads Poonawalla Fincorp To Record Low NPA And Unprecedented Gr...Vighnesh Shashtri
Under the leadership of Abhay Bhutada, Poonawalla Fincorp has achieved record-low Non-Performing Assets (NPA) and witnessed unprecedented growth. Bhutada's strategic vision and effective management have significantly enhanced the company's financial health, showcasing a robust performance in the financial sector. This achievement underscores the company's resilience and ability to thrive in a competitive market, setting a new benchmark for operational excellence in the industry.
What website can I sell pi coins securely.DOT TECH
Currently there are no website or exchange that allow buying or selling of pi coins..
But you can still easily sell pi coins, by reselling it to exchanges/crypto whales interested in holding thousands of pi coins before the mainnet launch.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and resell to these crypto whales and holders of pi..
This is because pi network is not doing any pre-sale. The only way exchanges can get pi is by buying from miners and pi merchants stands in between the miners and the exchanges.
How can I sell my pi coins?
Selling pi coins is really easy, but first you need to migrate to mainnet wallet before you can do that. I will leave the telegram contact of my personal pi merchant to trade with.
Tele-gram.
@Pi_vendor_247
The Evolution of Non-Banking Financial Companies (NBFCs) in India: Challenges...beulahfernandes8
Role in Financial System
NBFCs are critical in bridging the financial inclusion gap.
They provide specialized financial services that cater to segments often neglected by traditional banks.
Economic Impact
NBFCs contribute significantly to India's GDP.
They support sectors like micro, small, and medium enterprises (MSMEs), housing finance, and personal loans.
how to sell pi coins in South Korea profitably.DOT TECH
Yes. You can sell your pi network coins in South Korea or any other country, by finding a verified pi merchant
What is a verified pi merchant?
Since pi network is not launched yet on any exchange, the only way you can sell pi coins is by selling to a verified pi merchant, and this is because pi network is not launched yet on any exchange and no pre-sale or ico offerings Is done on pi.
Since there is no pre-sale, the only way exchanges can get pi is by buying from miners. So a pi merchant facilitates these transactions by acting as a bridge for both transactions.
How can i find a pi vendor/merchant?
Well for those who haven't traded with a pi merchant or who don't already have one. I will leave the telegram id of my personal pi merchant who i trade pi with.
Tele gram: @Pi_vendor_247
#pi #sell #nigeria #pinetwork #picoins #sellpi #Nigerian #tradepi #pinetworkcoins #sellmypi
how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
If you are looking for a pi coin investor. Then look no further because I have the right one he is a pi vendor (he buy and resell to whales in China). I met him on a crypto conference and ever since I and my friends have sold more than 10k pi coins to him And he bought all and still want more. I will drop his telegram handle below just send him a message.
@Pi_vendor_247
1. Incentive separability
Pawel Doligalski (Bristol & GRAPE)
Piotr Dworczak (Northwestern & GRAPE)
Joanna Krysta (Standford)
Filip Tokarski (Stanford GSB)
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (ERC Starting grant IMD-101040122)
1
3. Efficiency-equity trade-off: distortions are useful for redistribution
• central concept in Public Finance
However, sometimes the efficiency-equity trade-off can be avoided:
• Diamond-Mirrlees ’71: distortionary taxes on firms are redundant
• Atkinson-Stiglitz ’76: differentiated consumption taxes are redundant with
a non-linear income tax
• Sadka ’76, Seade ’77: there should be “no distortion at the top”
We study the general logic behind these results using Mechanism Design
2
4. What we do
We consider a general framework allowing for complex incentive constraints
- private information, moral hazard, voluntary participation etc
We introduce a new notion of incentive separability
- A set of decisions is incentive-separable if perturbing these decisions along
agents’ indifference curves preserves all incentive constraints.
3
5. What we do
We consider a general framework allowing for complex incentive constraints
- private information, moral hazard, voluntary participation etc
We introduce a new notion of incentive separability
- A set of decisions is incentive-separable if perturbing these decisions along
agents’ indifference curves preserves all incentive constraints.
Main theorem: It is optimal to allow agents to make unrestricted choices over
incentive-separable decisions, given some prices and budgets.
3
6. What we do
We consider a general framework allowing for complex incentive constraints
- private information, moral hazard, voluntary participation etc
We introduce a new notion of incentive separability
- A set of decisions is incentive-separable if perturbing these decisions along
agents’ indifference curves preserves all incentive constraints.
Main theorem: It is optimal to allow agents to make unrestricted choices over
incentive-separable decisions, given some prices and budgets.
The theorem allows us to
• Extend and unify Atkinson-Stiglitz and Diamond-Mirrlees theorems
- optimal to jointly remove consumption and production distortions
- harmonizing consumption taxes with distorted production—a bad idea
• Propose a novel argument for the optimality of food vouchers
3
7. Literature review
• Redundancy (or not) of commodity taxes with nonlinear income tax
Atkinson & Stiglitz (1976); Christiansen (1981); Cremer, Gahvari & Ladoux
(1998); Laroque (2005); Kaplow (2006); 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 of production efficiency
Diamond & Mirrlees (1971); Stiglitz & Dasgupta (1971), Naito (1999),
Hammond (2000)
4
8. Outline of the talk
1. Framework
2. General results
3. Applications
5
10. • A unit mass of agents with types θ ∈ [0, 1] ≡ Θ, uniformly distributed
• Agents’ utility over a vector of K decisions x ∈ RK
+
U (x, θ)
- x contains e.g. consumption of goods, labor supply, effort, ...
- U continuous in x, measurable in θ
- we make no ”single crossing” assumptions
6
11. • A unit mass of agents with types θ ∈ [0, 1] ≡ Θ, uniformly distributed
• Agents’ utility over a vector of K decisions x ∈ RK
+
U (x, θ)
- x contains e.g. consumption of goods, labor supply, effort, ...
- U continuous in x, measurable in θ
- we make no ”single crossing” assumptions
• An allocation rule x : Θ → RK
+
• associated aggregate allocation: x =
´
x(θ) dθ
• associated utility profile Ux , where Ux (θ) = U(x(θ), θ)
6
12. • Social planner with objective function
W (Ux , x),
- W continuous and weakly decreasing in x
- dependence on Ux : redistributive (welfarist) objective
- dependence on x: preferences over allocations beyond agents utilities
- e.g. (opportunity) cost of resources, preference for tax revenue, aggregate
constraints
7
13. • Social planner with objective function
W (Ux , x),
- W continuous and weakly decreasing in x
- dependence on Ux : redistributive (welfarist) objective
- dependence on x: preferences over allocations beyond agents utilities
- e.g. (opportunity) cost of resources, preference for tax revenue, aggregate
constraints
• Planner chooses allocation x : Θ → RK
+ subject to incentive constraints
x ∈ I ⊆ (RK
+)Θ
- no a priori restrictions on I
- I can represent IC constraints due to private info, moral hazard, etc.
• An allocation x ∈ I is called feasible
7
14. • Social planner with objective function
W (Ux , x),
- W continuous and weakly decreasing in x
- dependence on Ux : redistributive (welfarist) objective
- dependence on x: preferences over allocations beyond agents utilities
- e.g. (opportunity) cost of resources, preference for tax revenue, aggregate
constraints
• Planner chooses allocation x : Θ → RK
+ subject to incentive constraints
x ∈ I ⊆ (RK
+)Θ
- no a priori restrictions on I
- I can represent IC constraints due to private info, moral hazard, etc.
• An allocation x ∈ I is called feasible
• Notation: For a subset of decisions S ⊆ {1, ..., K} :
xS
= (xi )i∈S x−S
= (xi )i̸∈S x = (xS
, x−S
)
7
15. Incentive separability
Def: Decisions S ⊆ {1, ..., K} are incentive-separable (at feasible alloc x0) if
{(xS
, x−S
0 ) : U(xS
(θ), x−S
0 (θ), θ) = U(x0(θ), θ), ∀θ ∈ Θ} ⊆ I.
Decisions S are incentive-separable (IS) if all perturbations of xS
that keep all
types indifferent satisfy incentive constraints
8
16. Incentive separability
Def: Decisions S ⊆ {1, ..., K} are incentive-separable (at feasible alloc x0) if
{(xS
, x−S
0 ) : U(xS
(θ), x−S
0 (θ), θ) = U(x0(θ), θ), ∀θ ∈ Θ} ⊆ I.
Decisions S are incentive-separable (IS) if all perturbations of xS
that keep all
types indifferent satisfy incentive constraints
IS is a joint restriction on preferences and incentives. Examples:
1. Voluntary participation: all decisions are IS
2. Private information: decisions S are weakly-separable ⇒ S are IS
- S are weakly-separable when U(x, θ) = Ũ(v(xS ), x−S , θ)
3. Moral hazard: hidden action a affects the distribution of observed state ω
- U(x, θ) = −c(a) +
P
ω P(ω|a)uω(yω), yω ∈ RL
+
- Decisions within each state ω (namely, yω) are IS
4. Easy to extend to combinations of frictions, dynamic private info, aggregate
states, ...
8
17. Preliminaries
Consider a feasible allocation x0.
We will keep x−S
0 and Ux0 fixed, and re-optimize over the allocation of
incentive-separable goods xS
:
Uθ(xS
(θ)) := U(xS
(θ), x−S
0 (θ), θ)
R(xS
) := W (Ux0 , (xS
, x−S
0 ))
Assumptions:
1. Uθ is locally nonsatiated, for all θ
2. U(xS
0 (θ)) ≥ U(0), for all θ
3. For any x0, there exists integrable x̄(θ) such that
Uθ(y) = Uθ(xS
0 (θ)) =⇒ y ≤ x̄(θ)
9
19. Suppose that decisions S ⊆ {1, ..., K} are incentive-separable
Def: A feasible allocation x0 is S-undistorted if xS
0 solves
max
xS
R(xS
) s.t. Uθ(xS
(θ)) = Uθ(xS
0 (θ)), ∀θ ∈ Θ
10
20. Suppose that decisions S ⊆ {1, ..., K} are incentive-separable
Def: A feasible allocation x0 is S-undistorted if xS
0 solves
max
xS
R(xS
) s.t. Uθ(xS
(θ)) = Uθ(xS
0 (θ)), ∀θ ∈ Θ
Lemma 1 (Optimality)
Any feasible x0 that is not S-undistorted can be improved upon.
10
21. Suppose that decisions S ⊆ {1, ..., K} are incentive-separable
Def: A feasible allocation x0 is S-undistorted if xS
0 solves
max
xS
R(xS
) s.t. Uθ(xS
(θ)) = Uθ(xS
0 (θ)), ∀θ ∈ Θ
Lemma 1 (Optimality)
Any feasible x0 that is not S-undistorted can be improved upon.
Proof: Since x0 is not S-undistorted, there exists allocation x∗ = (xS
∗ , x−S
0 ) s.t.
1. Utility levels are unaffected: Uθ(xS
∗ (θ)) = Uθ(xS
0 (θ)),
2. Planner’s objective strictly improves under x∗ relative to x0.
Since goods S are incentive-separable, 1. implies that x∗ is also feasible. □
10
22. Simplifying Assumption: R(xS
) is linear in xS
with coefficients −λ ≪ 0
- Interpretation: constant marginal costs of producing xS
- Too strong, we will relax it in a moment
11
23. Simplifying Assumption: R(xS
) is linear in xS
with coefficients −λ ≪ 0
- Interpretation: constant marginal costs of producing xS
- Too strong, we will relax it in a moment
Lemma 2 (Decentralization)
Any feasible S−undistorted allocation x0 can be decentralized with prices λ.
That is, there exists a budget assignment m : Θ → R+ such that, for all
θ ∈ Θ, xS
0 (θ) solves
max
xS ∈R
|S|
+
Uθ(xS
) subject to λ · xS
≤ m(θ).
11
24. Simplifying Assumption: R(xS
) is linear in xS
with coefficients −λ ≪ 0
- Interpretation: constant marginal costs of producing xS
- Too strong, we will relax it in a moment
Lemma 2 (Decentralization)
Any feasible S−undistorted allocation x0 can be decentralized with prices λ.
That is, there exists a budget assignment m : Θ → R+ such that, for all
θ ∈ Θ, xS
0 (θ) solves
max
xS ∈R
|S|
+
Uθ(xS
) subject to λ · xS
≤ m(θ).
Proof: x0 is S-undistorted if xS
0 solves
max
xS
−λ · xS
s.t. Uθ(xS
(θ)) = Uθ(xS
0 (θ)), ∀θ ∈ Θ
It is a collection of expenditure minimization problems. The claim follows from
consumer duality (Mas-Colell et al. 1995, Prop 3.E.1). □
11
25. Main theorem (simplified)
Combining Lemmas 1 and 2, we arrive at:
Theorem
Starting at any feasible allocation x0, the planner’s objective can be (weakly)
improved by allowing agents to purchase incentive-separable goods at prices λ
subject to type-dependent budgets.
• There should be no distortions between IS goods
• Agents can trade IS goods freely given prices and budgets
• With production, prices of IS goods should be equal to marginal costs
• Result silent about non-IS goods
12
26. Main theorem (general)
Assumption (R has bounded marginals)
There exist scalars c̄ > c > 0 such that, for all y ∈ RK
, k ∈ {1, ..., K}, ϵ > 0
c̄ ≥
|R(y + ϵek ) − R(y)|
ϵ
≥ c.
Theorem 1
Starting at any feasible allocation x0, the planner’s objective can be (weakly)
improved by allowing agents to purchase incentive-separable goods at
undistorted prices subject to type-dependent budgets.
• Undistorted prices: prices that decentralize an S-undistorted allocation
• R has bounded marginals → the planner will always select an allocation
implying strictly positive prices
13
27. Main theorem (general)
Assumption (R has bounded marginals)
There exist scalars c̄ > c > 0 such that, for all y ∈ RK
, k ∈ {1, ..., K}, ϵ > 0
c̄ ≥
|R(y + ϵek ) − R(y)|
ϵ
≥ c.
Theorem 1
Starting at any feasible allocation x0, the planner’s objective can be (weakly)
improved by allowing agents to purchase incentive-separable goods at
undistorted prices subject to type-dependent budgets.
• Undistorted prices: prices that decentralize an S-undistorted allocation
• R has bounded marginals → the planner will always select an allocation
implying strictly positive prices
The results can be easily extended to allow for:
• the set of IS goods to depend on type θ and initial allocation x0
• additional constraints faced by the planner: x ∈ F
13
29. APPLICATION 1: ATKINSON-STIGLITZ
Atkinson-Stiglitz model:
- θ: privately observed productivity
- c ∈ RK−1
+ : vector of consumption of goods
- produced with const marginal costs λ
- L: labor in efficiency units
- consumption is weakly separable: U(v(c), L, θ)
→ consumption goods c are incentive-separable
14
30. APPLICATION 1: ATKINSON-STIGLITZ
Atkinson-Stiglitz model:
- θ: privately observed productivity
- c ∈ RK−1
+ : vector of consumption of goods
- produced with const marginal costs λ
- L: labor in efficiency units
- consumption is weakly separable: U(v(c), L, θ)
→ consumption goods c are incentive-separable
- Planner’s objective is (α is the marginal value of public funds):
W (Ux , x) = V (Ux ) + α(L −
K−1
X
k=1
λk ck ).
14
31. APPLICATION 1: ATKINSON-STIGLITZ
Atkinson-Stiglitz model:
- θ: privately observed productivity
- c ∈ RK−1
+ : vector of consumption of goods
- produced with const marginal costs λ
- L: labor in efficiency units
- consumption is weakly separable: U(v(c), L, θ)
→ consumption goods c are incentive-separable
- Planner’s objective is (α is the marginal value of public funds):
W (Ux , x) = V (Ux ) + α(L −
K−1
X
k=1
λk ck ).
By Theorem 1, it is welfare-improving to let agents trade goods c freely given
prices proportional to marginal costs and type-dependent budgets
Theorem 1 implies the Atkinson-Stiglitz theorem:
- prices proportional to marginal costs → no distortionary taxes on c
- type-dependent budgets → implemented with non-linear income tax
14
32. APPLICATION 1: ATKINSON-STIGLITZ
Theorem 1 is much more general than the Atkinson-Stiglitz theorem:
1. We allow for more complex settings and incentives
- only a subset of commodities may be incentive separable
- there may be moral hazard (or other incentive problems)
- combine redistributive (Mirrlees ’71) and social-insurance (Varian ’80)
strands of optimal income tax literature
2. We rule out any distortionary mechanisms
- e.g. quotas, public provision,...
3. What is essential is the ability to implement budgets for S goods,
rather than the availability of nonlinear income tax
- Theorem 1 applies even if earnings are unobserved (e.g. due to tax evasion)
but total expenditure on IS goods is observed
15
33. APPLICATION 2: FOOD VOUCHERS
Let E ⊆ {1, ..., K} denote consumption of various food items, −E : other
commodities and decisions
Jensen & Miller (2010): idiosyncratic food tastes more pronounced when food
consumption is high.
16
34. APPLICATION 2: FOOD VOUCHERS
Let E ⊆ {1, ..., K} denote consumption of various food items, −E : other
commodities and decisions
Jensen & Miller (2010): idiosyncratic food tastes more pronounced when food
consumption is high.
• Let v(xE
) denote the nutritional value of food (common to all agents)
• Utility function
U(x, θ) =
UL(v(xE
), x−E
, θ) if v(xE
) ≤ v (only nutrition matters)
UH (x, θ) if v(xE
) > v (nutrition and tastes matter)
- v: nutritional threshold above which idiosync. food tastes become active
16
35. APPLICATION 2: FOOD VOUCHERS
Let E ⊆ {1, ..., K} denote consumption of various food items, −E : other
commodities and decisions
Jensen & Miller (2010): idiosyncratic food tastes more pronounced when food
consumption is high.
• Let v(xE
) denote the nutritional value of food (common to all agents)
• Utility function
U(x, θ) =
UL(v(xE
), x−E
, θ) if v(xE
) ≤ v (only nutrition matters)
UH (x, θ) if v(xE
) > v (nutrition and tastes matter)
- v: nutritional threshold above which idiosync. food tastes become active
• θ is privately observed → food items are IS for types θ s.t. v(xE
(θ)) ≤ v
• Food items produced with const marginal cost
16
36. APPLICATION 2: FOOD VOUCHERS
Conflicting motives for taxes on food:
- food choices of the poor are IS → suboptimal to distort
- the rich exhibit heterogeneous tastes for food → optimal to distort their
food choices (Saez, 2002)
17
37. APPLICATION 2: FOOD VOUCHERS
Conflicting motives for taxes on food:
- food choices of the poor are IS → suboptimal to distort
- the rich exhibit heterogeneous tastes for food → optimal to distort their
food choices (Saez, 2002)
Solution? A food vouchers program:
Corollary 1
Consider a feasible allocation x0. The planner’s objective can be improved by
assigning budgets to all agents with θ such that v(xE
(θ)) ≤ v, and letting
them spend these budgets on food items E (but no other goods) priced at
marginal cost.
- means-tested: allocated only to ”poor” types {θ : v(xE
(θ)) ≤ v}
- allows to buy food at producer prices: isolates from tax distortions
17
38. APPLICATION 2: FOOD VOUCHERS
Conflicting motives for taxes on food:
- food choices of the poor are IS → suboptimal to distort
- the rich exhibit heterogeneous tastes for food → optimal to distort their
food choices (Saez, 2002)
Solution? A food vouchers program:
Corollary 1
Consider a feasible allocation x0. The planner’s objective can be improved by
assigning budgets to all agents with θ such that v(xE
(θ)) ≤ v, and letting
them spend these budgets on food items E (but no other goods) priced at
marginal cost.
- means-tested: allocated only to ”poor” types {θ : v(xE
(θ)) ≤ v}
- allows to buy food at producer prices: isolates from tax distortions
Compare to the U.S. food stamps program (SNAP)
- means-tested, purchases exempt from commodity taxes (state and local)
- however, restricted eligibility of unemployed—suboptimal in our setting
17
40. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES
x: allocation of consumption goods and other households decisions
Production sector consists of J firms. For a firm j ∈ {1, ..., J}:
• zj ∈ RK
is the production vector (negative entries represent inputs)
• Zj is the set of possible production vectors
z = (z1, ..., zJ ) is a production plan, z =
PJ
j=1 zj is aggregate production vector
Z is the Minkowski sum of Zj over j
18
41. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES
x: allocation of consumption goods and other households decisions
Production sector consists of J firms. For a firm j ∈ {1, ..., J}:
• zj ∈ RK
is the production vector (negative entries represent inputs)
• Zj is the set of possible production vectors
z = (z1, ..., zJ ) is a production plan, z =
PJ
j=1 zj is aggregate production vector
Z is the Minkowski sum of Zj over j
Allocation (x, z) is feasible if x ∈ I, z ∈ Z and z ≥ x
The planner maximizes W (Ux , z − x), non-decreasing in z − x
18
42. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES
x: allocation of consumption goods and other households decisions
Production sector consists of J firms. For a firm j ∈ {1, ..., J}:
• zj ∈ RK
is the production vector (negative entries represent inputs)
• Zj is the set of possible production vectors
z = (z1, ..., zJ ) is a production plan, z =
PJ
j=1 zj is aggregate production vector
Z is the Minkowski sum of Zj over j
Allocation (x, z) is feasible if x ∈ I, z ∈ Z and z ≥ x
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 ∈ Z such that
z ≥ z0 and z ̸= z0
18
43. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES
Corollary 2
For any feasible allocation, the planner’s objective can be (weakly) improved
by choosing an S-undistorted allocation of incentive-separable goods and an
efficient production plan.
Key insight: production decisions are (trivially) incentive-separable.
19
44. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES
Corollary 2
For any feasible allocation, the planner’s objective can be (weakly) improved
by choosing an S-undistorted allocation of incentive-separable goods and an
efficient production plan.
Key insight: production decisions are (trivially) incentive-separable.
To ensure positive decentralizing prices, we need additional assumptions on
feasible production plans:
Assumption
The aggregate production set Z is closed, bounded from above and convex.
Furthermore, for any z0 ∈ Z and any nonempty proper subset A ⊂ {1, ..., K},
there exists z ∈ Z such that zA
≤ zA
0 , z−A
≥ z−A
0 and z−A
̸= z−A
0 .
19
45. APPLICATION 3: ATKINSON-STIGLITZ MEET DIAMOND-MIRRLEES
Theorem 2
For any feasible allocation, there exists a price vector λ ∈ RK
++ such that the
planner’s objective can be (weakly) improved by simultaneously:
(i) allowing agents to purchase incentive-separable goods at prices λ subject
to type-dependent budgets;
(ii) allowing firms to maximize profits and trade all goods taking prices λ as
given, and taxing their profits lump-sum.
1. Beneficial to jointly remove distortions in production and IS-consumption
2. Achieved by allowing hhs and firms to trade goods at common prices
However, harmonizing consumption taxes w/out addressing production
distortions could reduce welfare:
• with a production distortions, the producer prices are distorted
20
46. CONCLUSIONS
• Incentive separability: a useful notion to study optimality and
decentralization in complex incentive problems
• We focused on applications within Public Finance
- Atkinson-Stiglitz theorem
- food stamps
- Atkinson-Stiglitz meet Diamond-Mirrlees
• Potential applications in other contexts
- employee compensation / principal-agent problems within organizations
- monopoly pricing with multiple goods / attributes
- ...
21
48. Examples of incentive-separable decisions
IS decisions: {(xS , x−S
0 ) : U(xS (θ), x−S
0 (θ), θ) = U(x0(θ), θ), ∀θ ∈ Θ} ⊆ I.
1. Voluntary participation:
I = { x : U(x(θ), θ) ≥ U(θ), ∀θ ∈ Θ } ;
All decisions are incentive-separable.
2. Private information:
I =
x : U(x(θ), θ) ≥ max
θ′∈Θ
U(x(θ′
), θ), ∀θ ∈ Θ
;
Weak separability of decisions S,
U(x(θ), θ) = e
U (v(xS
(θ)), x−S
(θ), θ), ∀θ ∈ Θ,
implies incentive separability of S.
22
49. Examples of incentive-separable decisions
3. Moral hazard:
Hidden action a ∈ A affects the distribution of observed state ω:
U((a, y), (τ, ω)) =
X
ω
u(y; ω)P(ω|a, τ) − c(a; τ), y ∈ RL
+;
Set I: Agent with type τ reports τ truthfully, takes recommended action a(τ),
and consumes y(τ, ω).
With θ = (τ, ω) and x = (a, y), decisions y are incentive-separable.
4. Easy to extend to combinations of incentive constraints, dynamic private
information, verifiable information, labels, ...
23