ARTICLE IN PRESS JID:YJETH

AID:3672 /FLA

YJETH:3672

[m1+; v 1.108; Prn:15/12/2008; 20:07] P.1 (1-16)

Journal of Economic Theory ••• (••••) •••–••• www.elsevier.com/locate/jet

On the robustness of laissez-faire Narayana Kocherlakota a,b,c , Christopher Phelan a,b,∗ a Department of Economics, University of Minnesota, 1925 4th Street South, 4-101 Hanson Hall,

Minneapolis, MN 55455-0462, United States b Research Department, Federal Reserve Bank of Minneapolis, 90 Hennepin Ave.,

Minneapolis, MN 55480, United States c NBER, United States Received 18 August 2007; final version received 11 August 2008; accepted 22 September 2008

Abstract This paper considers a model economy in which agents are privately informed about their type: their endowments of various goods and their preferences over these goods. While preference orderings over observable choices are allowed to be correlated with an agent’s private type, we assume that the planner/government is both uncertain about the nature of this joint distribution and unable to choose among multiple equilibria of any given social mechanism. We model the planner/government as having a maxmin objective in the face of this uncertainty. Our main theorem is as follows: Once we allow for this kind of uncertainty and assume no wealth effects in preferences, the uniquely optimal social contract is laissez-faire, in which agents trade in unfettered markets with no government intervention of any kind. © 2008 Published by Elsevier Inc. JEL classification: D02; D30; D63; D82; H21; P00; P51 Keywords: Mechanism design; Robustness

* Corresponding author at: Department of Economics, University of Minnesota, 1925 4th Street South, 4-101 Hanson Hall, Minneapolis, MN 55455-0462, United States. E-mail address: [email protected] (C. Phelan).

0022-0531/$ – see front matter © 2008 Published by Elsevier Inc. doi:10.1016/j.jet.2008.09.004 Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 2

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.2 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

“Primum non nocere”—medical aphorism. 1. Introduction In economies with privately observed effort levels or privately observed endowments, skills, or preferences, decentralization is problematic. To achieve efficiency in such economies, every observable aspect of an agent’s life must generally be monitored or controlled. That is, one lesson of modern information economics is that the optimal system in the presence of information problems appears as centrally planned as one can imagine. For instance, suppose society wishes to provide insurance against wealth shocks, and a hidden source of wealth is apples. Then, by taxing the selling of apples, economies can achieve at least partial insurance. Similarly, Townsend [9], Green [6], and Atkeson and Lucas [2] describe how a social planner can exploit observable choices over current and future consumption to provide social insurance. In this paper, we re-examine this result. We consider a model economy in which agents are privately informed about their wealths and over their preferences over goods. There is a social planner who would like to transfer goods from some agents (say, the wealthy) to other agents (say, the less wealthy). We assume that agents’ preferences exhibit no intrinsic wealth effects. However, the cross-sectional distribution of tastes and wealths may be such that they are correlated with one another in the population. As argued above, if this distribution of tastes and wealths is known to the planner/government, it is generically possible to design an interventionist mechanism that improves, from the perspective of the planner, on laissez-faire. We instead assume that, while agents know this joint distribution, the planner/government is uncertain about its nature. Here, by uncertain, we mean that the planner/government is unable to form a Bayesian prior over what this correlation might be. We also mean that, if there are multiple equilibrium outcomes to a mechanism, the planner/government cannot form a Bayesian prior over these outcomes. We model the planner/government as having a maxmin objective in the face of this uncertainty.1 Thus, the social welfare function is minimized over possible joint distributions and possible equilibrium outcomes. Our main theorem is that once we allow for this kind of uncertainty, the uniquely optimal social contract is laissez-faire, in which agents trade in unfettered markets with no government intervention of any kind. The logic behind our result is as follows. Suppose the planner wishes to transfer resources from the wealthy to the less wealthy, and consider an environment (Environment 1) such that preference orderings are correlated with hidden wealths. Then, there is an interventionist mechanism that achieves this kind of socially desirable redistribution. Now take another environment (Environment 2), in which the marginal distribution of tastes across agents is the same as in the original environment, Environment 1. However, in Environment 2, wealths and preferences are independent across agents. We prove that in Environment 2, the laissez-faire outcome is the best possible, and that it provides as much social welfare as the laissez-faire outcome in Environment 1. We then prove that under the interventionist mechanism, there is an equilibrium outcome in Environment 2 that is essentially equivalent to the one in Environment 1. But since there is no correlation between 1 Segal [8] uses a similar objective when analyzing the informational complexity of various mechanisms.

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.3 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–••• 3

Fig. 1.

wealths and preferences, this equilibrium is necessarily worse than the laissez-faire outcome. It follows that the maxmin objective of the planner is higher with laissez-faire. We depict this logic graphically in Fig. 1. The interventionist mechanism dominates laissezfaire in Environment 1. Laissez-faire gives the same welfare in both environments. But laissezfaire dominates the interventionist mechanism in Environment 2. It follows that the social planner prefers laissez-faire: its worst performance is better than that of the interventionist mechanism. We assume agents know the nature of the joint distribution of tastes and wealths. Given this, why cannot the planner elicit this information from them, and tailor allocations accordingly? The key is that agents’ wealths have no impact on their preferences over the choices that a planner can offer them. Hence, any information that the planner receives from an agent about wealths is necessarily in the nature of cheap talk. This cheapness of talk generates a crucial implementation problem in Environment 2. Suppose the planner asks each agent to report his wealth, along with his preferences. Agents may well send reports that reveal (correctly) that tastes and wealths are independent. Then, the planner knows that agents should be allowed to trade on their own. However, agents are indifferent about what they report their wealths to be. Hence, agents may send reports that indicate (incorrectly) that tastes and wealths are correlated as in Environment 1. If this happens, and the planner is planning to intervene in Environment 1, he will end up also intervening in Environment 2. This (mistaken) intervention creates a welfare loss in Environment 2. Suppose Environment 2 is an environment in which getting good outcomes matters, according to the maxmin objective of the planner. Then, the only robust response to this implementation problem is to use the laissezfaire mechanism, regardless of what agents report about their wealths.2 2 Jackson [7] provides a set of sufficient conditions for the implementation of a social choice correspondence in Bayesian–Nash equilibria. One of these conditions is Bayesian monotonicity. It is not satisfied by our setup because of the absence of wealth effects in preferences.

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 4

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.4 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

Our analysis builds on the recent literature on robust mechanism design. Chung and Ely [4] use a maxmin criterion with respect to environments. We differ from them by extending the maxmin criterion to allow for the possibility of multiple equilibrium outcomes (as occurs in Bassetto and Phelan [3]). Earlier, we mentioned that Townsend [9] and a large succeeding literature have discussed how interventionist mechanisms are optimal when it is known that agents’ wealths are correlated with their intertemporal endowment profiles. However, Allen [1] shows any mechanism in these settings induces a laissez-faire outcome if agents can engage in hidden borrowing and lending. Cole and Kocherlakota [5] prove in this same setting that laissez-faire is the best mechanism if agents can engage in hidden lending. In these papers, laissez-faire is optimal because the agents can undo attempts on the part of the planner to improve on laissez-faire. Our argument is quite different. Laissez-faire is optimal in our setting because its unique equilibrium outcome is independent of the, by assumption unknown, correlation between an agent’s wealth and his preferences over observable choices. 2. Setup Consider an open economy with a unit measure of agents and N goods. The goods can be N , so traded by the society with an outside world at price vector p ∈ ΔN (the unit simplex in R+  n pn = 1), where for all n, pn > 0. An agent’s type i ∈ T describes his endowment bundle and preferences over consumption bundles. (The set of possible types, T, is assumed to be finite.) In particular, an agent of type i has an endowment vector Yi = (Yi,1 , . . . , Yi,N ) and a vector of preference parameters Ωi = (Ωi,1 , . . . , Ωi,N ). The utility function of a type i agent is assumed to be 1 βn eΩi,n e−αcn , ui (c) = − α N

n=1

where the parameters α and β = (β1 , . . . , βN ) are common across types, and where, without loss  p of generality, N n=1 n ln(βn /pn ) = 0. Let Π be the set of probability measures over the type space T. An environment is an element π in Π ; in an environment π , agents’ types are determined by i.i.d. draws from π . The agents’ realized types are private information and the probability that a given agent is of type i, πi , is assumed to equal the fraction of agents of type i. Let M be a finite message space with elements m ∈ M. Assume T ⊂ M (so it is always feasible for an agent’s message to be “I am of type i”). Let Φ be the space of probability measures over M with elements φ ∈ Φ. A mechanism is a mapping μ : M × Φ → R N . The outcome μ(m, φ) = (μ1 (m, φ), . . . , μN (m, φ)) describes the transfer of goods made to an agent who sends message m ∈ M, when the cross-sectional distribution of messages is φ ∈ Φ. A mechanism is resourcefeasible if for all φ in Φ:  m∈M

φ(m)

N 

pn μn (m, φ)  0

n=1

so that the society can afford the transfers being made regardless of the cross-sectional distribution of messages. A strategy σ is a mapping from the type space T into the set of probability measures over messages Φ (this allows the agent to mix over elements of M in sending messages). Given a Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.5 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–••• 5

strategy σ and an environment π , define φ ∗ (π, σ ) ∈ Φ to be the cross-sectional distribution of messages induced by σ :  φ ∗ (π, σ )(m) = πi σi (m). i∈T

Next let

       BRi (μ, φ) = m  ui Yi + μ(m, φ)  ui Yi + μ(m , φ) for all m ∈ M .

That is, BRi (μ, φ) is the set of messages m such that m is a best response given type i, mechanism μ, and distribution of messages by others, φ. Given a mechanism μ and an environment π , we define a strategy σ to be an equilibrium strategy if for all (i, m) ∈ T × M such that πi > 0 and σi (m) > 0, m ∈ BRi (μ, φ ∗ (π, σ )). Note that the agent is assumed to know the environment π when choosing his message. Let E(π, μ) denote the set of equilibrium strategies, given the environment π and mechanism μ. It is convenient to express an agent’s type in terms of four components, (yi , zi , θi , ωi ), instead of two, (Yi , Ωi ), where yi ≡

N 

pn Yi,n ,

zi,n ≡ Yi,n − yi ,

pn Ωi,n ,

ωi,n ≡ Ωi,n − θi .

n=1

θi ≡

N  n=1

In this notation, yi , a scalar, is the value of the endowment vector of type i, Yi , and zi = (zi,1 , . . . , zi,N ) has the convenient property that N n=1 pn zi,n = 0, and represents how type i’s wealth is distributed across goods. Likewise, θi is a scalar,  representing type i’s general urgency to consume, and ωi = (ωi,1 , . . . , ωi,N ) is such that N n=1 pn ωi,n = 0, and represents type i’s urgency to consume particular goods. Note that agents with different realizations of (y, θ ), but the same (z, ω), have the same preferences over transfer vectors of goods. In this sense, the agents’ preferences exhibit no wealth effects. This property plays a crucial role in what follows. We consider the problem of a planner who wants to choose a mechanism μ that maximizes his objective. For a given policy μ, environment π , and equilibrium σ , we formulate this objective as a social welfare function,      πi κ(yi , θi ) σi (m)ui Yi + μ m, φ ∗ (π, σ ) , V SP (μ, π, σ ) = i∈T

m∈M

where κ maps (y, θ ) pairs into the positive reals. If κ(y, θ ) = 1 for all (y, θ ), then the social welfare function V SP (μ, π, σ ) is simply the average utility in the population, and thus V SP is the standard utilitarian or ex-ante welfare criterion. By instead allowing κ(y, θ ) = 1, we allow for redistributive motives by the social planner. We do not allow κ to depend on (z, ω). In this way, we restrict the social welfare function to be symmetric across goods. That is, we do not allow the social planner to put high weight on those who enjoy a particular good (say, art), or those who have high endowments of one good or another, holding constant the overall value of their endowment bundle. Across environments and equilibria, we assume that the planner has two forms of ambiguity. First, the planner is unable to form a subjective prior over the set of possible environments, other than he is certain it falls in some exogenous set of possible environments X. Second, Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 6

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.6 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

given a mechanism and an environment, there may be multiple equilibria, and the planner cannot formulate a subjective prior over which of these equilibria will occur. Given these two forms of ambiguity, the planner wants to design a mechanism that works well regardless of the actual environment or the equilibrium that gets played. Hence, we define the payoff to the social planner from a given mechanism μ as: V∗SP (μ) ≡ inf

inf

π∈X σ ∈E(π,μ)

V SP (μ, π, σ )

3. Examples Here are two examples that fit into the above framework. Example 1 (Atkeson–Lucas). Let the number of goods, N , equal two, representing date 1 and date 2 consumption. Prices (p1 , p2 ) equal ( 12 , 12 ) and β1 = β2 = 12 (a zero interest rate and no discounting). Let the set of possible environments, X, be a singleton, π , such that half the agents are of each type. Let both types have an endowment of 1 unit in each period, or Yi = (1, 1) for 9 5 1 1 i ∈ {1, 2}. For type 1 agents, let Ω1 = ( 11 4 , 4 ) (which corresponds to θ1 = 2 and ω1 = ( 4 , − 4 )). 5 7 3 1 1 For type 2 agents, let Ω2 = ( 4 , 4 ) (which corresponds to θ2 = 2 and ω1 = (− 4 , 4 )). That is, those with a general urgency to consume (the high θ type 1 agents) prefer first to second period consumption, and those without a general urgency to consume (the low θ type 2 agents) prefer second to first period consumption. Example 2 (Mirrleesian). Set N = 2, and think of the goods as being consumption and effective time, where a negative transfer of effective time is providing labor to the market and a positive transfer is akin to receiving a service. This is like a (richer than usual) Mirrleesian setup. (As in the previous example, let prices (p1 , p2 ) equal ( 12 , 12 ) and β1 = β2 = 12 .) Again, assume X is a singleton, π , such that half the agents are of each type. Let Ωi = (0, 0) for both types i ∈ {1, 2}. For type 1 agents, let Y1 = (1, 0) (which corresponds to y1 = 12 and z1 = ( 12 , − 12 )). For type 2 agents, let Y2 = (1, 1) (which corresponds to y2 = 1 and z2 = (0, 0)). That is, each type has a unit endowment of the consumption good (good 1), but only type 2’s have a positive effective time endowment. 4. The laissez-faire mechanism In what follows, we will be especially interested in the properties of the laissez-faire mechanism. Define ciLF to be the solution to: ciLF = arg max ui (c) c

subject to

N 

pn c n  y i .

(1)

n=1

This is the optimal choice for an individual agent with type i, when confronted with the budget set defined by the price vector p. One can analytically solve (albeit with a bit of tedious algebra) LF as: for ci.n

1 βn LF ci,n = yi + + ωi,n . ln α pn Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.7 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–••• 7

Note that ciLF does not depend on zi , since, by definition, zi does not affect the wealth of type i. Further, ciLF does not depend on θi since, from the definition of θi , θi acts as a multiplicative shock on the entire utility function. We define the laissez-faire mechanism μLF as follows: for messages m = i ∈ T, the nth component of μLF (m) is given by: LF μLF n (i) = ci,n − Yi,n =

βn 1 ln + ωi,n − zi,n . α pn

LF (For messages m ∈ / T, μLF n (m) = 0 for all goods n.) Note that unlike general mechanisms, μ depends only on an agent’s announcement of his type, not on the joint distribution of messages, and thus dependence on this joint distribution is suppressed in the notation. Nevertheless, μLF is a resource-feasible mechanism, regardless of the distribution of messages. More subtly, because of the properties of exponential utility, μLF is independent of yi . Under the laissez-faire mechanism, the decision of what type to report is equivalent to deciding which zero-value transfer vector to purchase. Suppose (i, j ) is such that (zi , ωi ) = (zj , ωj ) (but (yi , θi ) is potentially not equal to (yj , θj )). Then, a type i agent will be indifferent between truthfully reporting he is of type i and falsely reporting he is of type j . Hence, it is straightforward to see that given the laissez-faire mechanism, there can be many equilibria, in which a type i agent mixes over true reports and false reports with the same (zi , ωi ). However, these equilibria are all consumption-equivalent, in the sense that in any of them, an agent of type i ends up consuming ciLF . If one substitutes ciLF into type i’s utility function (and simplifies), type i’s utility under the laissez-faire mechanism can be expressed as

1 ui (ciLF ) = − eθi e−αyi , α which is independent of zi (from the fact that zi does not show up at all in (1)) and independent of ωi (from the fact that the laissez-faire mechanism allows the agent to trade at prices p). Given an environment π, the planner’s welfare from the laissez-faire mechanism is equal to: V LF (π) =

 i∈T

1   πi κ(yi , θi )eθi e−αyi . πi κ(yi , θi )ui ciLF = − α i∈T

The last expression is the expectation of a random variable that depends on i only through (yi , θi ). Hence, if the environments π and πˆ are such that π(θi , yi ) =  π (yi , θi ) for all i, then V LF (π) = V LF (πˆ ). Intuitively, under laissez-faire, goods-specific shocks z and ω are irrelevant because agents’ trades undo their impact. 5. The social planner’s problem In this section, we provide sufficient conditions on the environment space X such that the laissez-faire mechanism is the uniquely optimal mechanism for the social planner. In the first subsection, we discuss the apparent shortcomings of the laissez-faire mechanism when we assume that the environment space is a singleton. In the second subsection, we show that this apparent problem disappears once we allow for sufficient uncertainty about the environment. Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 8

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.8 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

5.1. The seductiveness of intervention If types were observable, it is clear that, generically, a solution to the social planner’s problem would involve wealth transfers across types—a property not found in laissez-faire. When types are private, is it still possible to design a mechanism that improves upon the laissez-faire mechanism (from the perspective of the planner)? If π is known to the planner then, generically, the answer to this question is yes. Before proving this, it will be useful to have notation for marginal and conditional probabilities that relate to the four components of the agents’ types. Given an environment π, we define:  πj π(zi , ωi ) = {j |(zj ,ωj )=(zi ,ωi )}

to be the unconditional probability of a given (zi , ωi ). Somewhat imprecisely, we use similar notation for the unconditional probability of a given (yi , θi ) pair. Then, given an environment π, if π(zi , ωi ) > 0, we define: ⎧ ⎫ πj ⎨ if (zj , ωj ) = (zi , ωi ), ⎬ π(j |zi , ωi ) = π(zj , ωj ) ⎩ ⎭ 0 if (zj , ωj ) = (zi , ωi ) to be the probability of j , conditional on the event that (zj , ωj ) equals a given realization (zi , ωi ). We use a similar notation when we condition on a given realization of (y, θ ). Lemma 1. Assume that the environment space X = {π}. Suppose that there exists i and j in T such that:   κ(yk , θk )eθk −αyk π(k|zi , ωi ) > κ(yk , θk )eθk −αyk π(k|zj , ωj ). k∈T

k∈T

Then there exists a mechanism μ such that V∗SP (μ) > V∗SP (μLF ). Proof. Define the truth-telling strategy σ TT in the usual way as σiTT (i) = 1 and σiTT (j ) = 0 if i = j. We know that σ TT ∈ E(μLF , π). Pick  that maps (z, ω) pairs into R N and define a mechanism μ such that μ (i, φ) = μLF (i) for all φ = π and μ (i, π) = μLF (i) + (zi , ωi ) for  (i) = 0 if i not in T. Since μLF is feasible for all distributions of messages, all i in  T and μ  then if i∈T πi N n=1 pn n (zi , ωi ) = 0, μ is feasible as well. Further, if  is sufficiently small in absolute value for all (zi , ωi ) in the support of π , then E(μ , π) = {σ TT }. This statement is true because, under the laissez-faire mechanism (and thus μ ), each type strictly prefers to announce his true type, rather than a false type which engenders a different transfer, regardless of the announcements of all other players. Hence, if μLF is optimal, it is necessary that a choice of  = 0 solves  1 LF πi κ(yi , θi ) βn eθi +ωi,n e−α(yi +zi,n +μn (i)+n (zi ,ωi )) α N

max − 

i∈T

n=1

subject to  i∈T

πi

N 

pn n (zi , ωi ) = 0.

n=1

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.9 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–••• 9

βn 1 If one substitutes into the objective function μLF n (i) = α [ln pn + ωi,n ] − zi,n , the objective function simplifies to

 1 πi κ(yi , θi )eθi e−αyi pn e−αn (zi ,ωi ) . α N

max − 

i∈T

n=1

The necessary first order of this maximization problem (with respect to ) evaluated at  = 0 is, for all (zi , ωi ),  κ(yk , θk )eθk −αyk π(k|zi , ωi ), λ= k∈T

where λ is the Lagrange multiplier on the constraint. This proves the lemma.

2

The assumption in Lemma 1 is that, relative to the laissez-faire outcome, there is a gain to the planner to transferring resources from one group of agents (those with (z, ω) = (zj , ωj )) to another group of agents (those with (z, ω) = (zi , ωi )). These agents differ in their preferences over goods. The lemma shows that the planner can exploit these differences in preferences to get the type (zj , ωj ) agents to give up resources, without violating incentive compatibility. Note that if (z, ω) and (y, θ ) are stochastically independent, then this condition of the lemma is necessarily violated. Does the assumption that the environment π is known matter? The usual answer would be no. The above mechanism is incentive-compatible. Hence, by the revelation principle, there is an equilibrium in which agents truthfully reveal their types. The planner can then condition transfers upon the distribution of announced types, which, under truth telling will reveal the environment. The transfer mechanism can then be chosen to solve the standard social planner’s problem environment by environment. Thus, under this view, societal welfare is the same, whether or not the planner can observe π . By conditioning transfers on the joint distribution of announcements, it is possible to design a better mechanism than the laissez-faire mechanism. But this analysis leaves out the possibility of multiple equilibria. The revelation principle guarantees only that there is an equilibrium in which agents truthfully report their types. There may be another equilibrium in which agents do not do so—and this equilibrium could be worse. In the next subsection, we discuss what happens once we allow for this possibility. 5.2. The main theorem In this subsection, we state and prove our main theorem: if the environment space X (and the corresponding ambiguity of the planner) is “sufficiently rich,” then μLF is the uniquely optimal mechanism. We know from Lemma 1 that if X is a singleton, then (generically) laissez-faire is not optimal. Our theorem provides a characterization of X that is sufficient to make laissez-faire optimal. We begin with two supporting lemmas. The first relies critically on exponential utility and multiplicative taste shocks. Lemma 2. Let π  , π be two environments in X such that π  (zi , ωi ) = π(zi , ωi ) for all i. Suppose σ ∈ E(μ, π) and define  π(j |zi , ωi )σj (m). σi (m) = j ∈T

Then,

φ ∗ (π, σ ) = φ ∗ (π  , σ  )

and σ  ∈ E(μ, π  ).

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 10

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.10 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

  Proof. Note that σi (m)  0 for all i, m and that m∈M σi (m) = 1 because m∈M σj (m) = 1  for all j. Hence, σi is a well-defined mixed reporting strategy. We next prove that the cross-sectional distribution of messages φ ∗ (π  , σ  ) is the same as ∗ φ (π, σ ). This step depends on the equal marginals across π and π  . Recall that:  πi σi (m). φ ∗ (π  , σ  )(m) ≡ i∈T

σi ,

Substituting in for we get:    πi σi (m) = πi π(j |zi , ωi )σj (m). i∈T

j ∈T

i∈T

Note that:  π(j |zi , ωi )σj (m) j ∈T

depends on i only through (zi , ωi ). Hence, if A = {(z, ω) | π  (z, ω) > 0}, we can write:    πi σi (m) = π  (zi , ωi ) π(j |zi , ωi )σj (m). i∈T

Since

j ∈T

(zi ,ωi )∈A

π  (zi , ωi ) = π(zi , ωi ), φ ∗ (π  , σ  )(m) =



we can use the Law of Iterated Expectations to conclude that:  π(zi , ωi ) π(j |zi , ωi )σj (m)

(zi ,ωi )∈A

=



j ∈T

πj σj (m)

j ∈T ∗

= φ (π, σ )(m). We now claim that σ  ∈ E(μ, π  ). For this it is sufficient to show m ∈ BRi (μ, φ ∗ (π  , σ  )) for all (i, m) such that πi > 0 and σi (m) > 0. Take such an (i, m) pair as given. By the construction of π  and σ  , that πi > 0 and σi (m) > 0 implies there exists a type j with zj = zi and ωj = ωi such that πj > 0 and σj (m) > 0. Since σ ∈ E(μ, π), 1 ∗ βn eθj +ωj n e−αyj −αzj n −αμn (m,φ (π,σ )) α n=1 1  ∗ − βn eθj +ωj n e−αyj −αzj n −αμn (m ,φ (π,σ )) α n N

−

for all m . Next, replace φ ∗ (π, σ ) with φ ∗ (π  , σ  ), zj with zi , and ωj with ωi (since they are equal), and multiply each side by e−αyi +θi /e−αyj +θj , giving 1 ∗   βn eθi +ωin e−αyi −αzin −αμn (m,φ (π ,σ )) α N

−

n=1

1  ∗   βn eθi +ωin e−αyi −αzin −αμn (m ,φ (π ,σ )) α N

−

n=1

for all

m .

Thus, m ∈ BRi (μ, φ ∗ (π  , σ  )), implying σ  ∈ E(μ, π  ).

2

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.11 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–••• 11

Lemma 2 starts with an equilibrium σ of mechanism μ in environment π. It constructs a new strategy σ  by averaging out σ ’s dependence on (y, θ ). It then says that this new strategy σ  is an equilibrium of mechanism μ in any environment π  with the same marginal probabilities over (z, ω) as π. The intuition behind the result is simple. Because utility is exponential, all agents who have the same (z, ω) have the same ranking over transfer vectors. Any dependence of a strategy σ on (y, θ ) is just a way of creating a mixed strategy by conditioning on a privately observed payoffirrelevant variable. Hence, if σ is an equilibrium in an environment π given mechanism μ, we can create a new equilibrium σ  (given π and μ) by averaging out σ ’s dependence on (y, θ ). But, now that σ  depends only on (z, ω), changing the distribution of (y, θ ) has no impact on any of the agents’ decision problems. It follows that σ  is an equilibrium in any environment π  with the same marginal probabilities over (z, ω) as π. We next prove a second supporting lemma: If π is known and (y, θ ) is independent of (z, ω), then laissez-faire is the solution to the standard Bayesian mechanism design problem. In such a problem, the message space equals the type space (or M = T ) and the profile of messages φ is assumed to equal the true profile of types π . Given these restrictions, let BMDP be defined as:   N  1 θi +ωi,n −α(yi +zi,n +δi,n ) πi κ(yi , θi ) − βn e e (BMDP) max δ α i∈T

n=1

subject to  i∈T

πi

N 

pn δi,n  0

n=1

and for all i such that πi > 0, i ∈ BRi (δ, π). Lemma 3. Suppose that π is such that (y, θ ) is stochastically independent from (z, ω), so that for all i ∈ T: πi = π(zi , ωi )π(yi , θi ). Then, δ solves BMDP if and only if δi = μLF (i) for all i such that πi > 0. Proof. In this proof, we first show the resource constraint holds as an equality in any solution to BMDP. Next we show that BMDP has a unique solution. From there, we show that this solution does not depend on the agent’s announcement of y or θ . Finally, we use these results, along with the independence assumption in the statement of the lemma, to show that μLF solves BMDP. First note if δ solves BMDP, the resource constraint holds with equality. To see this,  that suppose i∈T πi N n=1 pn δi,n +  = 0 for scalar  > 0. Define δ such that δ i,n =  + δi,n . Mechanism δ has a strictly higher objective function value than δ, satisfies the resource constraint with equality, and satisfies the incentive constraints since mechanism δ is incentive compatible and (from the assumption of exponential utility)  cancels from each side of −

1 1 βn eθi +ωi,n e−α(yi +zi,n +δi,n +)  − βn eθi +ωi,n e−α(yi +zi,n +δj,n +), α α N

N

n=1

n=1

(2)

the incentive constraint associated with a type i agent not falsely claiming to be of type j . Next we show that if mechanisms δ and δ  each solve BMDP, then for all i such that πi > 0, δi = δi . (That is, in terms of outcomes, the solution to BMDP is unique.) To this end, let Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 12

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.12 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

ui,n = e−αδi,n and consider choosing u : T → R N instead of δ. Then the objective function and incentive constraints become linear in the choice variables and the resource constraint becomes 

πi

i∈T

N 

  pn − log(ui,n )/α  0.

n=1

Suppose u0 and u1 each solve BMDP and u0i = u1i for some i such that πi > 0. Let u = γ u0 + (1 − γ )u1 for some γ ∈ (0, 1). From the linearity of the objective function and the incentive constraints, u also solves BMDP if it satisfies the resource constraint. Since − log(u) is a strictly convex function of u, the resource constraint is satisfied as a strict inequality, contradicting the optimality of u0 and u1 . Next we establish that if δ solves BMDP, then there is a function δ ∗ such that δi = δ ∗ (ωi , zi ) for all i. (That is, in terms of outcomes, δ ∗ does not depend on yi or θi .) Again let the choice variable be u instead of δ and suppose u solves BMDP with ui = uj for some i and j such that zi = zj , ωi = ωj , πi > 0, and πj > 0. Assume without loss of generality that N 

N      pn − log(uj,n )/α  pn − log(ui,n )/α .

n=1

(3)

n=1

Next note that if one divides each side of (2) by exp(−αyi + θi ), (yi , θi ) falls from the incentive condition. Thus, if j ∈ BRj (δ, π), then i ∈ BRj (δ, π) as well. This implies −

1 1 βn eθj +ωj,n e−α(yj +zj,n ) uj,n = − βn eθj +ωj,n e−α(yj +zj,n ) ui,n . α α N

N

n=1

n=1

(4)

Define uˆ k = uk for k = j , and uˆ j = ui . From (4) and the definition of u, ˆ −

1 1 βn eθj +ωj,n e−α(yj +zj,n ) uj,n = − βn eθj +ωj,n e−α(yj +zj,n ) uˆ j,n α α N

N

n=1

n=1

and thus the objective function of BMDP is equal under u and u. ˆ Likewise, (4) implies that uˆ is incentive compatible. Finally, (3) implies that N 

N      pn − log(uj,n )/α  pn − log(uˆ j,n )/α ,

n=1

n=1

which implies uˆ is resource feasible. This contradicts that u uniquely solves BMDP. Next, consider the full information planning problem, with the restriction that there exists a function δ ∗ (zi , ωi ) such that δi = δ ∗ (zi , ωi ) for all i. (This restriction implies that δ is independent of (y, θ ).) This full information problem can be written as   N  1 θi +ωin −α(yi +zin +δn∗ (zi ,ωi )) (FIP) max πi κ(yi , θi ) − βn e e δ∗ α i∈T

n=1

subject to  i∈T

πi

N 

pn δn∗ (zi , ωi )  0.

n=1

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.13 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–••• 13

Let A = {(z, ω) | π(z, ω) > 0}. The stochastic independence of (z, ω) and (y, θ ) implies that δ ∗ solves FIP if and only if it solves:

max − ∗ δ

N  1  ∗ π(z, ω) βn eωn e−α(zn +δn (z,ω)) α n=1

(z,ω)∈A

subject to  (z,ω)∈A

π(z, ω)

N 

pn δn∗ (z, ω)  0.

n=1

This maximization problem can be analytically solved to establish that the solution, δ ∗ , satisfies δ ∗ (zi , ωi ) = μLF (i) for all i ∈ T . Since μLF solves the full information problem subject to a restriction that δ not depend on yi or θi , it solves BMDP, since the incentive conditions of BMDP incorporate this restriction. 2 We can now use these two lemmas to construct our notion of sufficient ambiguity. Suppose σ is an equilibrium given a non-laissez-faire mechanism μ and environment π. If π is the only environment in X, then, from Lemma 1, we know that μ might well be optimal. But suppose that there is another environment π  such that (y, θ ) is stochastically independent from (z, ω), and this environment π  has the same marginal probabilities over (z, ω) as does π. From Lemma 2, we know that σ  , an averaged version of σ, is an equilibrium in environment π  , given mechanism μ. From Lemma 3, we know that unless μ equals μLF , σ  must induce a worse outcome from the planner’s point of view. In other words, the mechanism μ, even if it does well in environment π, must necessarily induce a worse outcome than the laissez-faire mechanism in environment π  . At this stage, there is one issue that remains. The planner is interested in maximizing performance in the worst-case scenario. We need to impose one more condition on X so as to ensure that the failure of μ to perform well in π  matters in this regard. The following definition uses one simple condition that works: it requires that laissez-faire perform poorly in environment π  , relative to other environments. Definition 1. An environment space X is complete if for any π ∈ X, there exists some π  ∈ X such that (1) for all i ∈ T, π(zi , ωi ) = π  (zi , ωi ), (2) for all i ∈ T, πi = π  (yi , θi )π  (zi , ωi ), and π ) for all  π in X. (3) V LF (π  )  V LF ( Here is a systematic way to think about how to construct complete environment spaces. Let X be a compact environment space. Each π in X implies marginal probabilities over (y, θ ); let F be the set of all of such implied marginal probability functions. Recall that V LF ranks environments π only according to their implied marginal probabilities over (y, θ ). The set F is compact because X is. Hence, V LF attains its minimum in F ; let f ∗ denote this minimal element of F . Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 14

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.14 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

Each π implies marginal probabilities over (z, ω); let G be the set of all of such implied marginal probability functions. Now consider the set X ∗ of joint probabilities over (y, θ, z, ω) that can be formed by multiplying f ∗ by all of the elements of G; that is, π ∈ X ∗ if and only if πi = f ∗ (yi , θi )g(zi , ωi ) for some g in G. We claim that X ∪ X ∗ is complete. Take any π in X. The construction of X ∗ ensures that there is some element π  of X ∗ with the same marginal probabilities over (z, ω). This π  exhibits stochastic independence between (y, θ ) and (z, ω). Finally, V LF attains its minimal value (over X ∪ X ∗ ) at any element of X ∗ , including π  . It follows that X ∪ X ∗ is complete. (In fact, it is readily shown that X ∪ X ∗ is the smallest complete environment space that contains X.) We can now state our main theorem. Theorem 1. Suppose the environment space X is complete. Suppose μ is a mechanism that differs from the laissez-faire mechanism μLF . (There exists an environment π ∈ X, strategy σ ∈ E(μ, π), type i, and message m, such that πi > 0, σi (m) > 0, and μ(m, φ ∗ (π, σ )) = μLF (i).) Then V∗SP (μLF ) > V∗SP (μ). Proof. X is complete. Hence, given π, there exists another π  in X that satisfies the three conditions of completeness. Define σ  such that  π(j |zi , ωi )σj (m) for all i in T. σi (m) = j ∈T

From Lemma 2, we know σ  ∈ E(μ, π  ). From the statement of the theorem, we know that there exists i such that πi > 0, σi (m) > 0, and μi (m, φ ∗ (π, σ )) = μLF (i). By construction, this implies there exists a type j such that πj > 0, σj (m) > 0, and μj (m, φ ∗ (π  , σ  )) = μLF (j ). Lemma 3 then implies that V LF (π  ) > V SP (μ, σ  , π  ). Hence:   V∗SP μLF = V LF (π  ) > V SP (μ, σ  , π  ) 

inf

σ  ∈E(μ,π  )

V SP (μ, σ  , π  )

 V∗SP (μ). The theorem follows.

2

Throughout our analysis, we have used an exogenous message space M. This exogenous message space needed to be rich enough to include the laissez-faire mechanism, and so we required M to be larger than the type space T. However, we can readily extend our results if we generalize our notion of a mechanism to be a pair (M, μ), where M is a message space of any size. In particular, the proof of (the critical) Lemma 2 never uses the assumption that M is larger than T. The key multiplicity problem that makes laissez-faire optimal cannot be eliminated by using small message spaces. 6. Conclusion The main result of this paper—that laissez-faire delivers the uniquely optimal maxmin policy—depends on two key characteristics of our model. Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.15 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–••• 15

(1) There is sufficient uncertainty about the correlation structure of shocks such that the actual environment might be one where laissez-faire is strictly optimal. (2) If a government implements a non-laissez-faire policy and the actual environment turns out to be the one where laissez-faire is optimal, a non-laissez-faire outcome is implemented. That is, the government cannot implement a policy that delivers the laissez-faire outcome when laissez-faire is optimal and a non-laissez-faire outcome when a non-laissez-faire outcome is optimal. The assumptions of exponential utility and multiplicative taste shocks are used to deliver the second of these characteristics. In our opinion, these assumptions capture the real constraints which disallow governments from finely tuning mechanisms.3 It is the first characteristic—that laissez-faire might be optimal—which appears to us as the more strict, depending on the application. Suppose that the social planner wishes to transfer resources from the wealthy to the less wealthy. Recall that even with these redistributive motives, laissez-faire is optimal when hidden wealth is independent of preferences over observable choices. The hidden source of wealth may be one’s ability to turn time into money, such as in a Mirrlees model. Then, it seems clear that wealths will be correlated with agents’ preferences over leisure and consumption. This correlation means that laissez-faire will not be optimal. On the other hand, if the hidden source of wealth is a shock to endowments, such as in the model of Green [6], then whether the wealthy are more or less willing to save than the poor will depend on the timing of the shocks. If high endowment agents learn of their high endowments when they receive them, they will be more willing to save than low endowment agents. On the other hand, if agents learn of their endowments in advance, the wealthy (those who know they will have a high endowment next period) will be less willing to save than the poor. That there may be no correlation between wealth and desire to save or borrow appears, at least to us, entirely possible. Acknowledgments The authors thank Robert Lucas for discussions on this paper and over two decades of discussions as our teacher, colleague, and friend. We also thank participants at the University of Chicago Symposium on Dynamic General Equilibrium honoring Robert Lucas, the editor Karl Shell, an anonymous associate editor, an anonymous referee, and our discussant Ivan Werning for helpful comments. We thank Roozbeh Hosseini and Kenichi Fukushima for excellent research assistance. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. References [1] Allen Franklin, Repeated principal–agent relationships with lending and borrowing, Econ. Letters 17 (1–2) (1985) 27–31. [2] Andrew Atkeson, Robert E. Lucas Jr., On efficient distribution with private information, Rev. Econ. Stud. 59 (3) (1992) 427–453.

3 We also conjecture that we could extend our main theorem to a class of environments in which preferences do potentially exhibit wealth effects. This extension would require us to add one more condition to our definition of completeness: that the environment π  is such that preferences do not exhibit wealth effects.

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004

ARTICLE IN PRESS JID:YJETH 16

YJETH:3672

AID:3672 /FLA [m1+; v 1.108; Prn:15/12/2008; 20:07] P.16 (1-16) N. Kocherlakota, C. Phelan / Journal of Economic Theory ••• (••••) •••–•••

[3] Marco Bassetto, Christopher Phelan, Tax riots, Rev. Econ. Stud. 75 (3) (2008) 649–669. [4] Kim-Sau Chung, Jeffrey C. Ely, Foundations of dominant strategy mechanisms, Rev. Econ. Stud. 74 (2) (2006) 447–476. [5] Harold Cole, Narayana R. Kocherlakota, Efficient allocations with hidden income and hidden storage, Rev. Econ. Stud. 68 (3) (2001) 523–542. [6] Edward J. Green, Lending and the smoothing of uninsurable income, in: Edward C. Prescott, Neil Wallace (Eds.), Contractual Arrangements for Intertemporal Trade, University of Minnesota Press, Minneapolis, 1987, pp. 3–25. [7] Matthew Jackson, Bayesian implementation, Econometrica 59 (2) (1991) 461–477. [8] Ilya Segal, Communication in economic mechanisms, in: Richard Blundell, Whitney K. Newey, Torsten Persson (Eds.), Advances in Economics and Econometrics: Theory and Application, The Ninth World Congress, in: Econometric Soc. Monogr., Cambridge University Press, 2006. [9] Robert M. Townsend, Optimal multiperiod contracts and the gain from enduring relationships under private information, J. Polit. Economy 90 (6) (1982) 1166–1186.

Please cite this article in press as: N. Kocherlakota, C. Phelan, On the robustness of laissez-faire, J. Econ. Theory (2009), doi:10.1016/j.jet.2008.09.004