Let O ⊂ R d be a bounded domain of class C 2 . In the Hilbert space L2(O; C n ), we consider a matrix elliptic second order differential operator AD,ε with the Dirichlet boundary condition. Here ε > 0 is the small parameter. The coefficients of the operator are periodic and depend on x/ε. We find approximation of the operator A −1 D,ε in the norm of operators acting from L2(O; C n ) to the Sobolev space. This approximation is given by the sum of the operator (A 0 D ) −1 and the first order corrector, where A 0 D is the effective operator with constant coefficients and with the Dirichlet boundary condition.2000 Mathematics Subject Classification. Primary 35B27. Key words and phrases. Periodic differential operators, homogenization, effective operator, operator error estimates.Supported by RFBR (grant no. 11-01-00458-a) and the Program of support of the leading scientific schools. d 2d−2 for d ≥ 3 and of order ε| log ε| for d = 2 was obtained. The proof essentially relies on the maximum principle which is specific for scalar elliptic equations.

Let O ⊂ R d be a bounded domain with boundary of class C 1,1 . In L 2 (O; C n ), consider a matrix elliptic second-order differential operator A D,ε with Dirichlet boundary condition. Here ε > 0 is a small parameter; the coefficients of A D,ε are periodic and depend on x/ε. The operator A −1 D,ε in the norm of operators acting from L 2 (O; C n ) to the Sobolev space H 1 (O; C n ) is approximated with an error of order ε 1/2 . The approximation is given by the sum of the operator (A 0 D ) −1 and a first-order corrector. Here A 0 D is an effective operator with constant coefficients and Dirichlet boundary condition.Key words: homogenization of periodic differential operators, effective operator, corrector, operator error estimates.An extensive literature is devoted to problems of homogenization in the small-period limit; see the books [1]-[3].1. Operator error estimates. Our considerations are based on an operator-theoretic approach developed in the papers [4] and [5] by Birman and Suslina. With the use of this approach, operator error estimates in homogenization problems for a wide class of matrix elliptic differential operators (DOs) have been obtained.Let Γ be a lattice in R d , and let Ω be the elementary cell of this lattice. For any Γ-periodic function ϕ(x), we set ϕ ε (x) = ϕ(x/ε), ε > 0. In [4] and [5], matrix elliptic DOs acting on L 2 (R d ; C n ) and admitting a factorization of the formwere considered. Here g(x) is an m × m matrix-valued function with complex entries. We assume that g(x) is bounded, uniformly positive definite, and Γ-periodic. Next, b(D) = d l=1 b l D l is an m × n matrix first-order DO and the b l are constant matrices with complex entries. With the operator b(D) the symbol b(ξ) = d l=1 b l ξ l , ξ ∈ R d , is associated. It is assumed that m n and that rank b(ξ) = n for ξ = 0. This is equivalent to the inequalities α 0 1 n b(θ) * b(θ) α 1 1 n for θ ∈ S d−1 with some positive constants α 0 and α 1 . A precise definition of the operator A ε is given in terms of the corresponding quadratic form. Examples of DOs of this type are discussed in [4]; they include, in particular, the scalar operator A ε = − div g(x/ε)∇ and the operator of elasticity theory.In [4] and [5], the behavior of the solution of the equation A ε u ε + u ε = F for small ε was studied; here F ∈ L 2 (R d ; C n ). As ε → 0, the solution u ε converges in L 2 (R d ; C n ) to the solution u 0 of the "homogenized" equationis the effective operator with constant effective matrix g 0 (see Section 3 below). In [4] the estimate u ε − u 0 L 2 (R d ;C n ) Cε F L 2 (R d ;C n ) was proved. In operator terms, this means that(1)In [5], the following approximation of the resolvent (A ε + I) −1 in the norm of operators acting from L 2 (R d ; C n ) to the Sobolev space H 1 (R d ; C n ) was obtained (this corresponds to an approximation * Supported by the RFBR (grant no. 11-01-00458-a) and by the program "Leading scientific schools" (grant no.

In dynamic resource allocation models, the non-existence of voting equilibria is a generic phenomenon due to the multi-dimensionality of the choice space even with agents heterogeneous only in their discount factors. Nevertheless, at each point of time there may exist a "median voter" whose preferred instantaneous consumption rate is supported by a majority of agents. Based on this observation, we propose an institutional setup ("intertemporal majority voting") in a Ramsey-type growth model with common consumption and heterogeneous agents, and show that it provides a microfoundation of the choice of the optimal consumption stream of the median agent. While the corresponding intertemporal consumption stream is in general not a Condorcet winner among all feasible paths, its induced instantaneous consumption rate receives a majority at each point in time in the proposed intertemporal majority voting procedure. We also provide a characterization of balanced-growth and steadystate voting equilibria in the case in which agents may differ not only in their time preference, but also in their instantaneous utility functions.JEL Classification: D11, D71, D91, O13, O43.

We consider two models of economic growth with exhaustible natural resources and agents heterogeneous in their time preferences. In the first model, we assume private ownership of natural resources and show that every competitive equilibrium converges to a balanced-growth equilibrium with the long-run rate of growth being determined by the discount factor of the most patient agents. In the second model, natural resources are public property and the resource extraction rate is determined by majority voting. For this model we define an intertemporal voting equilibrium and prove that it also converges to a balanced-growth equilibrium. In this scenario the long-run rate of growth is determined by the median discount factor. Our results suggest that if the most patient agents do not constitute a majority of the population, private ownership of natural resources results in a higher rate of growth than public ownership. At the same time, private ownership leads to higher inequality than public ownership, and if inequality impedes growth, then the public property regime is likely to result in a higher long-run rate of growth. However, an appropriate redistributive policy can eliminate the negative impact of inequality on growth.

In dynamic resource allocation models, the non-existence of voting equilibria is a generic phenomenon due to the multi-dimensionality of the choice space even with agents heterogeneous only in their discount factors. Nevertheless, at each point of time there may exist a "median voter" whose preferred instantaneous consumption rate is supported by a majority of agents. Based on this observation, we propose an institutional setup ("intertemporal majority voting") in a Ramsey-type growth model with common consumption and heterogeneous agents, and show that it provides a microfoundation of the choice of the optimal consumption stream of the median agent. While the corresponding intertemporal consumption stream is in general not a Condorcet winner among all feasible paths, its induced instantaneous consumption rate receives a majority at each point in time in the proposed intertemporal majority voting procedure. We also provide a characterization of balanced-growth and steadystate voting equilibria in the case in which agents may differ not only in their time preference, but also in their instantaneous utility functions.JEL Classification: D11, D71, D91, O13, O43.