Abstract This chapter derives two cardinal function representations of the technology set. An input distance function is shown to characterize T under the assumption of free disposability of inputs. An output distance function is shown to characterize T under the assumption of free disposability of inputs. The distance function is shown to generalize the production function, the input-requirement function, Allais’s disposable surplus measure, and the transformation. The consequences of nonempty T, free disposability of inputs and outputs, convexity, no free lunch, and no fixed cost for the distance functions are derived. The chapter closes with a discussion of distance function representations of T that require weaker disposability restrictions. The notion of free disposability of a single input is discussed.

Abstract This chapter initiates the economic analysis of producers. It introduces the concepts of profit, cost, and revenue and depicts their geometric illustrations. It then analyzes the optimizing behavior of a profit-maximizing producer facing a simple noncanonical technology. It is shown that profit-maximizing (rational) producers always minimize cost and maximize revenue. Rational producers minimize the cost of producing their optimal output and maximize the revenue from their optimal input vector. That demonstration validates the use of the profit, cost, and revenue functions defined in this chapter for a deeper analysis of producer behavior.

Abstract Two function representations of a single-input, single-output technology, the production function, and the input-requirement function are defined. The consequences for each function of imposing each of the five individual assumptions on the technology set are discussed verbally and mathematically, and are illustrated graphically. Each is shown to be a function representation of the technology under appropriate assumptions. Their role as potential mutual inverses is discussed. Generalizations beyond the single-input, single-output technology for each technology are introduced and discussed. The concepts of average product, marginal product, marginal changes, average productivities, and marginal productivities are defined and discussed.

Abstract This chapter presents three separate, but equivalent, derivations of the profit function introduced in Chapter 5. First, the profit function is derived directly from the technology set. Then, following Chapter 5, it is derived using the cost function, and finally using the revenue function. Regardless of the form T takes, the profit function is nondecreasing in output prices, nonincreasing input prices, positively homogeneous in input and output prices, and convex in input and output prices. It satisfies Hotelling’s Lemma so that the normals of hyperplanes supporting the profit function in price space are profit-maximizing supplies and demands.

Abstract This text shows how producers, confronted with existing technical possibilities and markets make production decisions. The book first discusses the roles that models, abstraction, and mathematics play in economic analyses. It then introduces a “canonical model” of “The Technology” as a set of inputs and output that satisfies five basic assumptions. Successive chapters build on this foundation to develop representations of technical possibilities that include production functions, input-requirement functions, input sets, and output sets. A “primer” on the optimal behaviour of price-taking, profit maximizing producers follows. It describes in a rigorous, but accessible, form the optimal producer behaviour using verbal, graphical, and mathematical arguments. Following chapters cover cost functions, revenue functions, and profit functions. These chapters treat the theories of cost-minimizing, revenue-maximizing, and profit-maximizing producers. A chapter on duality then shows that the existence of well-behaved profit function implies the existence of a canonical technology. Distance function representations are developed, and the text shows how to use distance functions to derive cost and revenue functions, how to use cost and revenue functions to construct “dual distance functions”, the role that distance functions play in calculating shadow prices, the use of distance functions to measure efficiency, and the use of distance functions to measure relative performance. The final chapter examines the consequences of relaxing the assumptions of the “canonical model” and price-taking producers.

We study efficiency measurement using a partial ordering for the S-dimensional reals that generalizes the canonical less than or equal to partial ordering. We seek measures that judge outcomes as favorably as possible using a dual normalization strategy that generalizes those used in the minimum-norm and efficiency-measurement literatures.We characterize the efficient frontier using dual methods and use that representation to identify a dual Nerlovian inefficiency measure. The Paretian inefficiency measure is defined as the minimal Nerlovian measure while constraining dual variates to fall in a predetermined closed convex set. We show that the Paretian inefficiency measure forms a dual conjugate pair with a restricted Nerlovian efficiency measure. We use those results to develop conditions that ensure that the Paretian inefficiency measure is an exhaustive function (cardinal) representation of the feasible set. We present a series of composition rules for different restrictions on the feasible set and dual-variate normalization that include generalizations of existing inefficiency measures. An empirical illustration of the concepts developed that is based on Catalan farming data closes the substantive part of the paper.

Abstract This chapter studies producers who minimize the cost of producing their output. The core notion is the cost function, which is a money-metric generalization of the input-requirement function introduced Chapter 3. This analysis follows the approach developed in Chapter 5 that combines mathematical, verbal, and geometric arguments. The discussion first characterizes the cost function’s properties in input prices: nondecreasing, concave, positively homogeneous, and Shephard’s Lemma. These properties do not reflect restrictions placed upon T. Then the discussion turns to the properties of the cost function in output. Imposing free disposability of output upon T implies that cost is a nondecreasing function of output. And imposing convexity upon T ensures that the cost function is a convex function of output.

We develop a framework for regulated production systems where output generation and pollution abatement impose competing technological demands. Using a multi-ware technology, we model the production set as the intersection of two input requirement frontiers, one for production and one for abatement, each reflecting distinct trade-offs. We characterise the efficient set using gaps in efficiency distances and recover shadow prices for smooth and non-smooth frontiers, clarifying how subdifferentials govern shadow prices at switching points. To address unobserved heterogeneity in technological orientation, we estimate a full-information maximum likelihood model with endogenous regime-switching. Applying the framework to Chinese livestock farms, we recover regime-specific land costs, identify drivers of regime assignment and quantify inefficiency using multi-ware distances. Results reveal sharp asymmetries in spatial burdens between production and abatement, and systematic switching behaviour driven by inputs and weather. Our framework offers a widely applicable, policy-relevant tool for evaluating environmental and economic performance.JEL Classification: D24, Q12, Q52

Abstract This chapter discusses the role of economic models and introduces basic mathematical concepts. It elaborates the role that verbal, mathematical, and visual analyses play in economics. The principle of using abstraction to analyze problems is discussed. Thought experiments and their role in economic analysis are introduced and discussed. Mathematical concepts, including the notion of a set, a subset of a larger set, the intersection of two sets, and the empty set are presented. Mathematical symbols used in the text are listed and identified.

Abstract This chapter treats the revenue function introduced in Chapter 5 as the money-metric generalization of the single-output production function. It gives the maximal revenue obtainable from a given vector of inputs for a given set of output prices. As a function of output prices; it is nondecreasing, convex, positively homogeneous, and satisfies McFadden’s Lemma. Those properties require a mild restriction upon T and the assumption that producers are price-taking revenue maximizers. We then derive the consequences of free disposability of inputs, convexity, no free lunch, and no fixed cost for the behavior of the revenue function of inputs.

Abstract The input set is defined verbally and mathematically and illustrated visually. It is shown to provide an exhaustive characterization of the technology. The consequences of each of the five basic assumption for the input set are discussed and illustrated. The isoquant, represented as the radial boundary of the input set is defined, and the slope of its graph is identified as the marginal rate of substitution. The output set is defined. As the lower inverse for the input set, it provides an exhaustive representation of the technology. The consequences of each of the five basic assumptions for the output set are discussed and illustrated. The transformation curve is defined as the radial boundary of the output set, and the slope of the transformation curve is defined as the marginal rate of transformation. The chapter closes with a discussion of different versions of convexity that input and output sets may possess.

The Arrow-Savage-Debreu framework provides a theoretical foundation for analyzing productivity and efficiency under uncertainty. It treats uncertainty while preserving core economic principles and offering insights into producer decision-making. A key challenge is to reconcile ex ante conceptual models with ex post empirical data. We survey econometric and mathematical programming methods used to address this challenge. These methods include stochastic production functions, latent state models, auxiliary-variable methods, and data envelopment analysis techniques. We discuss the strengths and limitations of each method, highlighting how they handle the fundamental challenge of measuring efficiency when production decisions are made under uncertainty but only realized outcomes are observable. Our analysis demonstrates that such measurement complexity necessitates carefully designed empirical approaches to capture the true nature of production and environmental efficiency.

Increasing agricultural productivity is a gradual process with significant time lags between research and development (R&D) investment and the resulting gains. We estimate the response of US agricultural Total Factor Productivity to both R&D investment and weather and quantify the public R&D spending required to offset the emerging impacts of climate change. We find that offsetting the climate-induced productivity slowdown by 2050 will require R&D spending over 2021 to 2050 to grow at 5.2 to 7.8% per year under a fixed spending growth scenario or by an additional $2.2 to $3.8B per year under a fixed supplement spending scenario (in addition to the current spending of ~$5B per year). This amounts to an additional $208 to $434B or $65 to $113B over the period, respectively, and would be comparable in ambition to the public R&D spending growth that followed the two World Wars.

We study a programming approach to inducing inefficiency measures for convex technologies. It takes the technology and the numeraire as given and uses variational arguments to isolate shadow prices that make a decision maker’s observed behavior as efficient as possible. The focus is on how the numeraire determines the element of the efficient frontier to which a decision maker’s performance is compared, the resulting technical inefficiency measure, and whether that measure offers a cardinal representation of the technology. We use the results to study an inefficiency measure, the polyhedral measure, that generalizes an array of existing measures.

Context: In-season nitrogen (N) management tools are essential for optimizing N application rates, maximizing farmers' economic returns and minimizing adverse environmental impacts. The primary limitation to developing such tools is the risk associated with uncertainties in weather forecasts and crop price projections required to estimate yields and returns for different N rates. Therefore, characterizing the risk associated with these uncertainties is crucial for determining optimum N rates in-season. Objective: This study investigated the N application decision-making process for farmers, accounting for risks associated with weather and crop price uncertainties through crop modeling and economic analysis. Methods: We used field trial data for winter wheat in Kansas to examine how optimal nitrogen rates and economic returns vary over sites, years, and differing farmers' risk attitudes. First, the Environmental Policy Integrated Climate (EPIC) agroecosystem model was used to simulate the distribution of final yields under different N applications during early spring. Then, an autoregressive moving average (ARMA) model estimated the wheat price distribution at harvest based on historical prices. Finally, optimal N application rates for farmers with different risk appetites were estimated using two risk decision models: the constant-absolute-risk-averse (CARA) expected utility model, which treats upside (higher-than-expected returns) and downside (lower-than-expected returns) deviations equally, and the invariant-preference, generalized-deviation (IPGD) model, which focuses on downside risk. Results: We found that optimal N rates vary greatly between sites and years, as well as across farmers with different risk preferences. Due to the positive skewness of economic return distribution, farmers tend to apply lower N rates when considering downside risk. On average, the optimal N rate for farmers with a CARA coefficient of 0.002 is 77 kg/ha in the CARA model and 67 kg/ha in the IPGD model. Compared to the outcome of risk-neutral N usage, risk-averse N usage for a farmer with a CARA coefficient of 0.008 could reduce the uncertainty (standard deviation) of return by 6.2 %, on average, while the expected return decreased by only 1.2 %. Conclusions: By lowering the N rate, risk-averse farmers would reduce the uncertainty of returns and incur a minor return loss, suggesting the possibility of improving agricultural resilience while also improving N use efficiency. Our analysis also underscores the importance of yearly site-specific N management, given the substantial variation in optimal rates across years and locations. Significance: This study provides the foundation for an N application decision framework that considers both weather and price uncertainty. The analysis also demonstrates the potential co-benefit of enhancing agriculture's climate and market resilience while potentially lowering N losses.

Path-based cardinal characterizations of closed and nonempty sets are defined, and their basic properties are detailed. Differential properties and applications to performance measurement are considered.

We show that the weighted additive DEA score (WA) for the additive DEA model is simultaneously the (dual) support function for a translation of the DEA technology and the gauge of the dual polar set of the translated technology. Those results are used: to show that WA and the indicator function for the translated technology form a dual conjugate pair; to show that WA and the translated technology's gauge function form a dual polar pair; to develop a simple but exact link between WA and the profit function for the DEA technology; to show that WA and the directional distance function form a dual polar pair; and to develop an exact decomposition of profit inefficiency that extends the Cooper, Pastor, Aparicio, and Borras (2011) and Aparicio et al. (2016) decomposition.& COPY; 2022 Elsevier B.V. All rights reserved.

To emphasize the nexus between the theory and the empirics of production, this chapter is split into two parts. The first presents a brief overview of the state of neoclassical production theory as it exists in the third decade of the twenty-first century. The second part presents an overview of the history of the development of functional forms for the production function.

This chapter describes a formal model of a stochastic production technology. Alternative axioms and different structural restrictions are presented, and producer decision-making under uncertainty is examined. The presentation emphasizes the formal similarities between the stochastic production environment and more traditional models of a nonstochastic technology and producer behavior under certainty. The nonstochastic multiple-output technology is shown to be special case of the more general stochastic production structure.

Competitive equilibria are studied in both partial-equilibrium and general-equilibrium settings for economies characterized by consumers with incomplete preference structures. Market equilibrium determination is developed as solving a zero-maximum problem for a supremal convolution whose dual, by Fenchel's Duality Theorem, coincides with a zero-minimum for an infimal convolution that characterizes Pareto optima. The First and Second Welfare Theorems are natural consequences. The maximization of the sum of consumer surplus and producer surplus is studied in this analytic setting, and the implications of nonsmooth preference structures or technologies for equilibrium determination are discussed.