An Equivalence Theorem between geometric structures and utility functions allows new methods for understanding preferences. Our classification of valuations into “Demand Types” incorporates existing definitions (substitutes, complements, “strong substitutes,” etc.) and permits new ones. Our Unimodularity Theorem generalizes previous results about when competitive equilibrium exists for any set of agents whose valuations are all of a “demand type.” Contrary to popular belief, equilibrium is guaranteed for more classes of purely‐complements than of purely‐substitutes, preferences. Our Intersection Count Theorem checks equilibrium existence for combinations of agents with specific valuations by counting the intersection points of geometric objects. Applications include matching and coalition‐formation, and the “Product‐Mix Auction” introduced by the Bank of England in response to the financial crisis.

The paradigm for providing affordable electricity for the world's poor-power for development-has begun to change. Historically, centralized governments built large consolidated power plants and distribution and transmission lines with the ultimate goal of providing electricity to all of their citizens. It has become increasingly common in recent decades, however, for donors, nongovernmental organizations (NGOs), firms, and communities to collaborate with governments to develop small-scale localized energy systems known as distributed generation (DG) either as complements or alternatives to centralized operations. DG programs have been implemented around the world but with a mixed record of success. Based on an analysis of the existing case study literature, we examine DG program goals and outcomes, identifying major factors that affect these outcomes, including appropriately chosen technology, adequate financing and payment arrangements, ongoing end users' involvement, and supportive national policies. We highlight the importance of institutions for collaborative governance in the pursuit of these factors.

We construct the moduli spaces of stable maps, Mg,n(P r , d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, Mg,n; this is valid over Spec Z. In another paper by the first author, a small part of the argument is replaced, making the result valid in far greater generality. Our method follows that used in the case n = 0 by Gieseker in [7], to construct Mg , though our proof that the semistable set is nonempty is entirely different.Proof. IfJ ss (L) = J, thenJ// L SL(W ) is a categorical quotient of J, and if J ss (L) =J s (L), the quotient is an orbit space. The result follows from Proposition 3.4 and Proposition 2.3.In the following GIT construction, we will first seek a linearisation L such that J ss (L) ⊂ J. This has many useful implications, which we shall explore now. In particular, it gives us the first half of the desired equality:J ss (L) =J s (L).Proposition 3.6. Suppose there exists a linearisation L such thatJ ss (L) ⊆ J. ThenJ ss (L) =J s (L).Proof. Every point of J corresponds to a moduli stable map, and so has finite stabiliser. The result follows from Corollary 2.6. This leaves us with the second required equality, thatJ ss = J. In this paper, we shall prove it using the existing construction of M g,n (P r , d) over C, given by Fulton and Pandharipande in [6]. This is not necessary (see [2]), but for brevity we take this shortcut for now.Corollary 3.7. There exists a map j : J → M g,n (P r , d), which is an orbit space for the SL(W ) action, and in particular a categorical quotient. The morphism j is universally closed.

We investigate how irreversibility in "dirty" and "clean" capital stocks affects optimal climate policy, from both theoretical and numerical perspectives. An increasing carbon tax will reduce investments in assets that pollute, and so reduce emissions in the short term: our "irreversibility effect". As such the "Green Paradox" has a converse if we focus on demand side capital stock effects. We also show that the optimal subsidy increases with the deployment rate: our "acceleration effect". Considering second-best settings, we show that, although carbon taxes achieve stringent targets more efficiently, in fact renewable subsidies deliver higher welfare when policy is more mild.

We propose new techniques for understanding agents' valuations. Our classification into "demand types", incorporates existing definitions (substitutes, complements, "strong substitutes", etc.) and permits new ones. Our Unimodularity Theorem generalises previous results about when competitive equilibrium exists for any set of agents whose valuations are all of a "demand type" for indivisible goods. Contrary to popular belief, equilibrium is guaranteed for more classes of purely-complements, than of purely-substitutes, preferences. Our Intersection Count Theorem checks equilibrium existence for combinations of agents with specific valuations by counting the intersection points of geometric objects. Applications include matching and coalition-formation; and the Product-Mix Auction, introduced by the Bank of England in response to the financial crisis.

Despite the importance of electrification for economic and social development, over one billion people globally lack access to electricity, primarily in rural areas of developing countries. Alongside the traditional means of expanding access, large-scale grid electrification, there exists another option for rural electrification: small-scale and localized distributed generation (DG), often powered by renewable energy sources. DG systems can be grid-connected or off-grid and can range in scale from less than 10 W solar lanterns at the small end to 60 MW biomass generation. Although DG has enabled some level of access to electricity for millions of people, little or no research has analyzed how the scale or level of access to electricity has shaped the ways that programs are financed or are viewed by governments, or the developmental impacts that various levels have on end-users. This article reviews the literature on DG in developing countries and finds that large-scale DG systems require a different set of approaches to finance, end-user training, and public policy support than do small-scale DG systems. Our review also reveals that even the smallest scale DG systems improve users' quality of life, yet access to electricity alone is not sufficient to achieve desired economic and social development goals. Policy makers, donors, non-governmental organizations (NGOs), and other actors engaged in rural development must both (1) make decisions considering the scale of DG programs, and (2) reflect on end-users' needs and productive uses for electricity if rural electrification projects are to result in long-term development benefits.

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This paper develops algorithms to solve strong-substitutes product-mix auctions. That is, it finds competitive equilibrium prices and quantities for agents who use this auction's bidding language to truthfully express their strong-substitutes preferences over an arbitrary number of goods, each of which is available in multiple discrete units. (Strong substitutes preferences are also known, in other literatures, as M -concave, matroidal and well-layered maps, and valuated matroids). Our use of the bidding language, and the information it provides, contrasts with existing algorithms that rely on access to a valuation or demand oracle to find equilibrium.We compute market-clearing prices using algorithms that apply existing submodular minimisation methods. Allocating the supply among the bidders at these prices then requires solving a novel constrained matching problem. Our algorithm iteratively simplifies the allocation problem, perturbing bids and prices in a way that resolves tie-breaking choices created by bids that can be accepted on more than one good. We provide practical running time bounds on both price-finding and allocation, and illustrate experimentally that our allocation mechanism is practical.