Mechanism design with multidimensional, continuous types and interdependent valuations

We consider the mechanism design problem when agents’ types are multidimensional and continuous, and their valuations are interdependent. If there are at least three agents whose types satisfy a weak correlation condition, then for any decision rule and any ?>0 there exist balanced transfers that render truthful revelation a Bayesian ?-equilibrium. A slightly stronger correlation condition ensures that there exist balanced transfers that induce a Bayesian Nash equilibrium in which agents’ strategies are nearly truthful.     

Tiebout equilibria in local public good economies with spillovers

This paper analyzes the effects of spillovers on the equilibrium population distribution across jurisdictions in a local public good economy with free mobility. Spillovers are parametrized by a matrix [α_ij] where α_ij ∈ [0, 1]. When spillovers are symmetric and close to 0 or 1 (pure local public goods and pure public goods), all equilibrium jurisdiction structures are symmetric. However, any population distribution can be sustained in equilibrium for some value of the spillover parameter α. In the class of utility functions with additive externalities, we identify the unique family of utility functions for which equilibria are symmetric except for an isolated value of α. This is a class of utility functions which are linear in the public good and a power function of the private good, u(c, γ) = − A(1 − c)^β + γ. With this specification of utility, we show that an increase in α results in a more fragmented equilibrium population distribution, and that when spillovers are asymmetric and large, a jurisdiction which is more centrally located (i.e. benefits more from the public goods provided in other jurisdictions) has a larger population in equilibrium.     

Algebra of Integrated Time Series: Evidence from Unit Root Analysis

It is argued if x_t ~ I(1) and y_t ~ I(1), then running a regression x_t on y_t would produce spurious results because e _t would generally be I(1). However, there may exist a ‘b’ such that e _t  = x _t - by _t is I(0), then running a regression x _t on y _t would not produce spurious results. This special case of two integrated time series is known in the literature as cointegration. In this particular case, x _t and y _t are said to be cointegrated. In our review of the development of the concept of cointegration, we identified that the underlying reason for this special case to arise is the proposition that if x _t  ~ I(d _x ), y _t  ~ I(d _y ), then z _t  = bx _t  + cy _t  ~ I(max(d _x ,d _y )). In this research, we offer evidence against this proposition.     

Asymptotic distribution of the sup-Wald statistic under specification errors

Consider a simple structural break model where y_t=α₁+β₁f(x_t)+u_t for t≤k₀ and y_t=α₂+β₂f(x_t)+u_t for t>k₀. The timing of break and the structural parameters are unknown. Suppose the true functional form of the regressor f(·) is misspecified as g(·). We do not place too many restrictions on the functional forms of f(·) and g(·). A frequently encountered example in economics is that the true model is measured in level, but we estimate a log-linear model, i.e. when f(x_t)=x_t and g(x_t)=log(x_t) For any f(·) and g(·), we derive a nonstandard limiting null distribution of the sup-Wald test statistic under some very general regularity conditions. Monte Carlo simulations support our findings.     

An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann-Morgenstern utility functions on a finite set of alternatives A. Each agent i's utility function is known to lie in the nonempty, convex, relatively open set Ui. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L no matter what the true utility functions are. Then, there exist utility functions ui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Existence of adaptively stable sunspot equilibria near an indeterminate steady state

We examine the nonlinear model x_t=E_tF(x_t+1). Markov stationary sunspot equilibria (SSEs) exist near an indeterminate steady state, , provided . Despite the importance of indeterminacy in macroeconomics, earlier results have not provided conditions for the existence of adaptively stable SSEs near an indeterminate steady state. We show that there exist Markov SSEs near x? that are E-stable, and therefore locally stable under adaptive learning, if .     

Implementation of individually rational social choice functions with guaranteed utilities

A simple mechanism is presented that allocates an indivisible object between two agents for almost any possible compensation rule. Furthermore, the equilibrium strategy guarantees a level of utility not less than −ε, where ε can be arbitrarily small.     

A class of multiattribute utility functions

A function u(z) is a utility function if u′(z) > 0. It is called risk averse if we also have u′′(z) < 0. Some authors, however, require that u ⁽ⁱ⁾(z) > 0 if i is odd and u ⁽ⁱ⁾(z) < 0 if i is even. The notion of a multiattribute utility function can be defined by requiring that it is increasing in each variable and concave as an s-variate function. A stronger condition, similar to the one in case of a univariate utility function, requires that, in addition, all partial derivatives of total order m should be positive if m is odd and negative if m is even. In this paper, we present a class of functions in analytic form such that each of them satisfies this stronger condition. We also give sharp lower and upper bounds for E[u(X ₁,... , X _s )] under moment information with respect to the joint probability distribution of the random variables X ₁,... , X _s assumed to be discrete and representing wealths. Partially supported by OTKA grants F-046309 and T-047340 in Hungary.     

The refoundation of the symmetric equilibrium in Schumpeterian growth models

We provide a refoundation of the symmetric growth equilibrium characterizing the research sector of vertical R&D-driven growth models. We argue that the usual assumptions made in this class of models leave the agents indifferent as to where targeting research: hence, the problem of the allocation of R&D investment across sectors is indeterminate. By introducing an “?-contamination of confidence” in the expected distribution of R&D investment, we prove that the symmetric structure of R&D investment is the unique rational expectations equilibrium compatible with ambiguity-averse agents adopting a maxmin strategy.     

Priorities in the location of multiple public facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfy anonymity, avoid the no-show paradox, strategy-proofness and population-monotonicity. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.     

Optimal control problems from second-order difference equations

We examine solutions x?l_∞(Rⁿ) of the equations $\text{F}{}_{\mathrm{t}}\text{(}\text{x}{}_{\mathrm{t?1}}\text{,}\text{x}{}_{\mathrm{t}}\text{,}\text{x}{}_{\mathrm{t+1}}\text{) = 0}$ and derive conditions for the existence of objective functions G_t so that $\text{x}$ solves max_x?l∞ΣG_t(x_t, x_t+1). We specialize the conditions to time autonomous equations and apply them to a competitive industry in temporary equilibrium. The objective function in the example is of the cost-benefit form: consumer surplus minus opportunity cost of labor minus industry-wide cost of investment.     

General equilibrium in economies with uncertain delivery

We consider a general equilibrium model of trade ex ante with differential information in which agents choose plans of state-contingent lists of bundles. Being unable to verify that the state of nature is s and not t, an agent has to accept the delivery of any bundle in the list for delivery in state s or in the list for delivery in state t. Under the assumption that each state of nature can be verified by at least one agent, we establish existence of equilibrium and we show that the equilibrium allocation satisfies a notion of coalitional incentive compatibility.     

Strategies as states

We define rationality and equilibrium when states specify agents’ actions and agents have arbitrary partitions over these states. Although some suggest that this natural modeling step leads to paradox, we show that Bayesian equilibrium is well defined and puzzles can be circumvented. The main problem arises when player j's partition informs j of i's move and i knows j's strategy. Then i's inference about j's move will vary with i's own move, and i may consequently play a dominated action. Plausible conditions on partitions rule out these scenarios. Equilibria exist under the same conditions, and more generally ε equilibria usually exist.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Optimal consumption and the timing of the resolution of uncertainty

As is well-known, consumers want to accumulate precautionary savings in the face of income risks when their marginal utility is convex (prudence). In this paper, we explore the effect of the timing of the resolution of income uncertainty on savings. An agent faces uncertainty about his income at date t+2. What is the effect of being informed that the uncertainty will be resolved at date t+1 on the consumption at date t? We show that the effect is positive, if and only if, marginal utility is convex (prudence), when either the risk free rate is equal to the rate of pure preference for the present, or when the utility function is HARA. The intuition is that an early resolution of uncertainty allows for time-diversifying the risk. It therefore plays a role similar to a reduction of the income risk, whose effect on savings is negative under prudence.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

Transitive measurable utility

This paper continues a study of theories of preferences under risk that do not use the independence axiom of the von Neumann-Morgenstern theory. Unlike its predecessor, it assumes that preferences are transitive. The effects of transitivity are noted in two representations of preferences. The first, which also uses continuity and dominance axioms, involves a function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. Although u might be nonlinear, it has other features of a von Neumann-Morgenstern linear utility function. The second representation has linear functions u and w on P, with w strictly positive except perhaps at preference-extreme measures—where it might vanish, such that u(p) w(q) > u(q) w(p) if and only if p is preferred to q. A symmetry axiom along with the axioms for the first representation are necessary and sufficient for the second representation.     

Sufficient conditions in free final time optimal control problems. A comment

By an application of sufficient conditions, assume that an optimal pair (χ_τ(·), u_τ(·)) and an adjoint function p_τ(·) were found in the control problem in question with the final time τ fixed but arbitrary. Then a sufficient condition for one of these pairs, say , to be optimal in the corresponding free final time problem is that the Hamiltonian, with (χ_τ(t), u_τ(τ?), p_τ(τ), τ) inserted, is nonnegative (nonpositive) to the left (right) of $\text{τ}{}^1$.     

Overlapping expectations and Hart's conditions for equilibrium in a securities model

Hart (J. Econ. Theory9 (1974), 293–311) gave conditions for equilibrium to exist in a securities model where each agent undertakes asset transactions to maximize expected utility of wealth. These conditions rule out agents wanting to undertake unbounded balanced transactions to reach a Pareto superior allocation given their expectations. With mild extra assumptions to make agents unwilling to risk incurring unbounded losses on their portfolios, Hart's conditions become equivalent to an assumption of “overlapping expectations,” which is comparable to a much weaker form of Green's “common expectations” (Econometrica41 (1973), 1103–1124).