From theory to tangibility: using 3D-printed prototypes to teach utility and production functions in economics

This paper explores dimensional analysis of production and utility functions in economics. As raised by Barnett, dimensional analysis is important in consistency checks of economics functions. However, unlike Barnett's dismissal of CES and Cobb-Douglas production functions, we will demonstrate that under constant return-to-scale and other assumptions, production function can indeed be justified dimensionally. And then we consider utility functions.

The production function explains a basic technological relationship between scarce resources, or inputs, and output. This paper offers a brief overview of the historical significance and operational role of the production function in business and economics. The origin and development of this function over time is initially explored. Several various production functions that have played an important historical role in economics are explained. These consist of some well known functions, such as the Cobb-Douglas, Constant Elasticity of Substitution (CES), and Generalized and Leontief production functions. This paper also covers some relatively newer production functions, such as the Arrow, Chenery, Minhas, and Solow (ACMS) functions, the transcendental logarithmic (translog), and other flexible forms of the production function. Several important characteristics of the production function are also explained in this paper. These would include, but are not limited to, items such as the returns to scale of the function, the separability of the function, the homogeneity of the function, the homotheticity of the function, the output elasticity of factors (inputs), and the degree of input substitutability that each function exhibits. Also explored are some of the duality issues that potentially exist between certain production and cost functions. The information contained in this paper could act as a pedagogical aide in any microeconomics-based course or in a production management class. It could also play a role in certain marketing courses, especially at the graduate level.

The dual identity of carbon sources and carbon sinks makes agriculture the focus of carbon neutralization-related research. Compared with traditional rural agriculture and urban industrial production, urban agriculture has its own particularities. It is of positive practical significance to explore the interaction and its evolution process between urban agricultural multifunctionality and carbon effects in seeking solutions to alleviate carbon pressure. Based on the changes in agricultural carbon emissions and carbon sequestration in Guangzhou from 2002 to 2020, we used the Granger causality analysis method to investigate the interaction between urban agricultural multifunctionality and carbon effects and then used the grey association model to analyse the evolution process of associative degrees between the two and divide the agricultural development stages. Finally, according to the practicalities of Guangzhou, we analyzed carbon effects generated in the multifunctional transformation of urban agriculture and put forward corresponding policy suggestions on how to solve the problem of excessive carbon dioxide emissions through agriculture in metropolitan areas. The results show that from 2002 to 2020 in Guangzhou, urban agricultural production decreased, the economic and social function increased, and the ecological function climbed and then declined. The carbon sequestration of urban agriculture in Guangzhou was approximately four times more than the carbon emissions. Carbon emissions experienced a process of first decreasing, then increasing, then remaining constant, and finally decreasing, while carbon sequestration first decreased and then increased. Second, the carbon emissions of urban agriculture in Guangzhou have a causal relationship with the production, social, and ecological functions. Carbon emissions are the Granger cause of the economic function but not the opposite. The carbon sequestration of urban agriculture in Guangzhou has a causal relationship with production and economic functions. Carbon sequestration is the Granger cause of the ecological function but not the opposite. There is no Granger causal relationship between carbon sequestration and the social function. Third, from 2002 to 2020, the interactive development process of urban agricultural multifunctionality and carbon effects in Guangzhou can be divided into three stages: production function oriented (2002–2006), economic and social function enhanced and production function weakened (2007–2015) and the economic and social function exceeded the production function (2016–2020). Fourth, the multifunctional transformation of urban agriculture has brought about carbon effects of reducing emissions and increasing sequestration. There is a long time lag between multifunctional transformation and carbon effects of urban agriculture.

Although the representative theory of consumer/firm is widely used in particular forms of utility and production functions, is the optimal growth model adequately supported by a multi-regional microeconomic base in a “general form of utility and production functions”? Is the “optimization of macro-aggregate functions” consistent with the “summation of micro-optimal outcomes”? Economists have explored the problem of summation of macro-variables in a special form of production and utility function setting to find some theoretical foundations of macroeconomics in the case of heterogeneous representative consumers. However, heterogeneous representative producers have not been studied enough, and the conclusions are too dependent on the functional form. This study searches for the conditions for macroeconomics to reach micro-foundations in the framework of heterogeneous producers and general functions. Linking the entire macroeconomy through inter-regional capital flows, we distinguish between using and equity capital, construct a multi-regional dynamic optimal growth model in a homogeneous representative consumer and heterogeneous representative manufacturers, compare the summation of the regional dynamics with the aggregate macro-dynamics, and find that (A) the regional and macro-perspectives are generally inconsistent in their optimization results; (B) if the population growth rate is the same across regions, the macro-steady state is a weighted summation of the steady states; and (C) if the inverse of the absolute risk aversion is linearly additive to consumption, the model is reliable. In particular, CARA and CRRA-type utility functions are eligible. The eligible utility function can only be an exponential utility function or a power function utility function (the logarithmic utility function can be considered a degenerate power function utility).

A simulation of the Heckscher–Ohlin theorem

Jeff Biddle's recent book, Progress through Regression: The Life Story of the Empirical Cobb-Douglas Production Function, is a clearly told story of a theory and its implementation from its first proposal as a log-linear empirical relation linking outputs to inputs by Charles Cobb and Paul Douglas (1928), resulting in a plethora of highly critical, constructive, and supportive reactions, through to its acceptance as a substantive production function relationship in a wide range of research areas. The developments, and the continued criticisms and responses, are carefully discussed, building on extensive archival research.The book is divided into three parts. Part 1 has three chapters surveying Douglas's published accounts of his production research commencing in 1927. Chapter 1 concerns time series; chapter 2, cross sections; and chapter 3, the many challenges he faced. The name Cobb-Douglas derives from Douglas's first published time-series study with Cobb (Cobb and Douglas 1928), after which Cobb ceased to participate. Peter Lloyd (2001: 4) describes the earlier history of the equation, noting that Johann von Thünen (attributed to 1863) developed what was “a linearly homogeneous Cobb-Douglas production function written in intensive form.” However, the Cobb-Douglas relation was not called a production function until the late 1930s. Moreover, although Francis Galton (1886) introduced the concept of regression for a conditional expectation—in his case for reversion to the mean in heights of sons given heights of their fathers—it was not until 1940 that the word was used to describe empirical production functions. Ordinary least squares (OLS) is not necessarily a regression—in teaching elementary econometrics, I use the example of having students write their signature on a data graph and then fit an OLS line to it, to demonstrate that it is not a conditional expectation.By the time of the conjunction describing his research as “production function regressions,” Douglas had essentially ceased undertaking his empirical research, first because of war service and then because in 1948 he was elected to the US Senate, serving for eighteen years. However, as part 2 describes, other econometricians took up the research baton, especially in agricultural economics, and indeed regression estimation of Cobb-Douglas production functions became ubiquitous. Calculating technical progress and its contribution to analyzing economic growth also became a major application, both still ongoing. Part 3 concludes with the author's views on the success of Douglas's enterprise, why that happened, and how it might be evaluated.The various advances and debates are set in their contexts with brief intellectual histories of the key protagonists. There is a wealth of detailed information about numerous applied studies across many areas of manufacturing and agriculture, numerous countries, micro and macro, time series, cross sections, and panels. Naturally, issues of data collection, measurements of variables like output, labor, and capital, and later technical progress, as well as index numbers all need to be addressed. The history also follows some of the many relevant developments in econometrics, focusing on those either initiated by or rapidly applied to estimating Cobb-Douglas production functions or their implications.To quote Professor Biddle: The economists in my narrative who sought to estimate production functions faced a variety of challenges, which in many cases were specific forms of generic challenges that hindered empirical research in economics in the mid-twentieth century. The procedure required measures of inputs and outputs, measures that often had to be constructed from imperfect and incomplete statistical data. Throughout the period I examine, linear regression was the statistical method used to estimate production functions, but the decision to use linear regression necessitated a number of subsidiary decisions, such as the form to be taken by the estimating equation. Theory gave, at best, uncertain guidance to the economist making these decisions; this, combined with the quality of the available data, ensured that any decision made was vulnerable to criticism. (4)One might say that little has changed. Indeed, the author highlights that many of the issues Paul Douglas faced in the mid-twentieth century remain, including accurately measuring both inputs and outputs, especially “quality” or technology adjusted, calculating relevant price indexes, and appropriate methods of estimation for cross-section, time-series, and panel data sets, to which we must now add how to handle nonstationary observational time series confronting stationary economic theory, an issue discussed in the next section.Part 1 provides careful and usually very detailed discussions of Douglas's many published papers, first from time-series then cross-section observations. There are useful discussions of the difficulties early empirical researchers faced from the lack of good, or sometimes any, data, especially of the effort involved in getting measures of capital and prices. At more than 120 pages, even a summary of Biddle's text in part 1 would be overly long, although the general theme is one of findings by Douglas being heavily debated, indeed often attacked, and Douglas responding to his critics partly by countering the most easily rebutted criticisms, modifying his approach somewhat to finesse others, and ignoring what he could not yet counter. Douglas tended to claim concordance across his empirical findings, though Biddle considers this rather a stretch. Often the debates concerned the implications of his findings for the then extant economic theory, concerning “equilibrium,” and marginal productivity theory and its relation to income distribution, although criticisms of lack of identification (particularly for cross-section studies) and parameter estimation biases also abounded. Some critics like David Durand (1937) also questioned the constancy over time of the coefficients of labor and capital, a problem I return to in section 3.Biddle remarks that “the original time-series regression of the 1928 Cobb-Douglas paper arose out of Douglas's prior research interests and represented cutting-edge work in empirical economics” (4). Mostly agreed, except in one important aspect not discussed by Biddle: Douglas, and many of his critics, seemed to show no awareness of the potential problems posed by unit-root nonstationarity leading to nonsense regressions in processes with stochastic trends. Before you react to this being anachronistic, Louis Bachelier (1900) introduced random walks for speculative prices and, contemporaneously, Reginald Hooker (1901) sought to deal with stochastic trends and regime shifts (see, e.g., Hendry and Mary Morgan 1995). Udny Yule (1926) provided an understanding of nonsense regressions in static formulations as due to unit roots in time-series variables, and Bradford Smith (1926) proposed nesting specifications in levels and differences, which would serendipitously have solved the nonsense regressions problem (see Terence Mills [2011] for the rediscovery of this “lost” contribution). All of these publications predate Douglas's first study, which nevertheless just postulated static relations. While linking outputs to inputs would hardly be “nonsense,” failing to address possible dynamics could lead to poor estimates and underestimated standard errors, illustrated in section 5 below. Since the 1980s, there have been massive advances in understanding and modeling unit-root nonstationarity through cointegration: Peter Phillips (1986) clarified the nonsense regressions problem, closely followed by the introduction of cointegration (see Rob Engle and Sir Clive Granger [1987] and Søren Johansen [1988]). Had Smith (1926) been followed up, static and dynamic representations could have been integrated, closing the circle by their link back to equilibrium correction.Biddle does note John Maurice Clark's (1928) criticism that the “Cobb-Douglas equation offered a good account of the ‘normal’ or long-run relationship between labor, capital, and output, but did a poor job of representing the impact of cyclical fluctuations in labor and capital utilization, which were governed by a ‘different law’” (31). However, Clark suggested cyclical adjustments rather than including dynamics, although the major successes in discovering the key features that affected the properties of economic times-series data by Yule (1927) (autoregressive processes) and Eugen Slutsky (1937) (moving averages: the Russian version was 1927) had appeared the previous year. In a similar vein, but rather later, Victor Smith (1945: 562) argued that “the statistical [production] function represents relationships that prevail in a dynamic, disequilibrium economy” but did not suggest adding dynamics to the relationship to account for that, perhaps not realizing that such an extended Cobb-Douglas equation could be solved for the “long-run” as I do in section 5 below.Frank Knight emphasized the conflict between static theory and dynamic (historical) data, but wrongly deduced that statistical methods could not quantify theoretical concepts, an argument Douglas rightly dismissed. Biddle further illustrates the (still ongoing) battle between the “supremacy” of theory versus evidence using Douglas's joint papers with Martin Bronfenbrenner and with Grace Gunn. The former tried to link the estimated coefficients with neoclassical economics whereas the latter sought better statistical methodology. An interesting aside for this reviewer is that in the 1930s and 1940s about half of Douglas's coauthors were female at a time when that was relatively rare, especially in economics, where half would still be a high proportion today in most English-speaking countries.The Cobb-Douglas story was also one of evolving technicality that Biddle calls the rhetoric of “mechanical rules” moving away from “expert judgement” rhetoric and trust, using as a case study confluence analysis (proposed by Ragnar Frisch [1934]) where his “bunch maps” mainly relied on expert judgment. That issue concerned the direction of minimization, namely, which variable to treat as dependent given measurement errors in all regressors. Shortly after this discussion (in footnote 28) Biddle derives the parameter biases in the Gunn and Douglas study and finds they made the correct decision about the direction of minimization. A formal analysis would have been useful, both to show that those earlier results could be reproduced—as Hendry and Morgan (1995) did for a number of historical empirical studies—as well as allowing modern misspecification tests to be calculated: see Deborah Mayo (2010) on passing severe testing as an essential basis for valid inference, applied in section 5. The analysis of errors in variables versus errors in equations by Tjalling Koopmans (1937) greatly clarified that particular debate.While the complete list of criticisms of production function estimation is vast, debates about empirical evidence and its role in economics have been ever present since Henry Moore (1911): essentially all key “economic functions” like consumption (as shown by Jim Thomas [1989]), investment (Dale Jorgenson [1967]; and q as debated following Dale in James Tobin 1969), Phillips curves (as criticized by Robert Lucas [1976]), money demand, and so forth, have been subject to great debates. Notwithstanding numerous criticisms, the Cobb-Douglas program continued—as did the empirical endeavors for other relationships.In his presidential address to the American Economic Association in 1947, Douglas claims a “scientific schizophrenia” between the economics of marginal productivity theory under perfect competition in economic theory and the presence of wage bargaining theory in labor economics. However, “the importance of studying the general relationships that exist at the level of society as a whole because of the conditioning influence they exert on individual industries, shows a consistency through the years on an important methodological issue,” although at odds with the individualistic approaches then dominant at Cowles and Chicago (22–23). This leads to part 2.Part 2, almost two hundred pages long, considers the diffusion of Cobb-Douglas regressions. Chapter 4 describes three case studies looking at why the popularity of Cobb-Douglas regressions grew after World War II. The first is its entry into econometrics textbooks by Gerhard Tintner (1952) (who had written about deriving production functions from farm records in Tintner 1944) and Lawrence Klein (1953); the second describes how the regression was adopted by researchers in agricultural economics (discussed below); and the third is the constant elasticity of substitution (CES) production function proposal by Kenneth Arrow et al. (1961) building on an earlier paper by Robert Solow (1956). This enabled estimating the elasticity of substitution between capital and labor—which was constrained to unity in Cobb-Douglas and to zero in Wassily Leontief (1951) functions—and so allowed testing the Cobb-Douglas specification. En route, Hendrik Houthakker (1956) (who references Horst Mendershausen's highly critical 1938 article on Douglas's production function research) had established that aggregating microlevel Leontief production functions could lead to macro Cobb-Douglas. Biddle suggests CES may have caught on following John Hicks's (1939) and Roy Allen's (1963) mathematical representation of the neoclassical theory of value and distribution in which the elasticity of substitution was a key parameter, highlighting the willingness of practitioners to use assumptions of perfect competition, optimization, and equilibrium to sustain estimates of CES.Biddle also discusses two Phelps Brown (1957) critiques of Douglas's time-series regressions: first, that it was “improbable at first sight that one unchanging production function should fit a growing, changing economy over a run of years” and, second, that the good fit was due to “constant growth trends” of output, capital, and labor leading to serious multicollinearity (548–49). Not only does the second contradict the first, it is surprising coming from an economist who had witnessed two world wars and the Great Depression, so we will focus on the first. (Douglas's cross-section results were also criticized by Phelps Brown, but he mainly reiterating earlier critiques by Jan Tinbergen [1942] and Jacob Marschak and William Andrews Jr. [1944].) Technical change has certainly not proceeded at a constant rate with constant impacts over the last 120 years, nor has financial innovation, nor have major shocks.The second main source of nonstationarity, namely, shifts of data distributions, have also been important historically, and remain so. Location shifts are abrupt changes in the means of distributions (e.g., levels of a nontrending time series), such as from major wars on unemployment, oil-crises jumps in oil prices, Great Depressions or Great Recessions, and pandemic lockdowns on outputs. Location shift nonstationarity was slowly unraveled from its precursor in Hooker 1901 and another “lost paper” by Smith (1929): it is remarkable that in discussing forecasting the outcome of 1929 earlier that year, Smith foresaw shifts as the main imponderable. Econometricians and statisticians have since developed a plethora of methods for detecting shifts and parameter changes, but fewer in handling location shifts, although see Jennifer Castle et al. 2015. A location shift can be represented by a step indicator, the first difference of which is an impulse indicator, whereas its cumulation is a trend indicator, several of which are applied in section 5 below.Chapter 5 discusses the take up of the Cobb-Douglas production function in agricultural economics, which kept associated research work alive while Douglas was in the Senate. This led to a refocusing on resource allocation by individual farmers, augmented by some experimental data, rather than tests of marginal productivity and perfect competition. Many of the practitioners were trained in statistics, especially of a Fisherian variety, with Earl Heady as a key player and communicator. Research often aimed to improve farmers' resource allocation efficiency, so many studies used individual farm data. Nevertheless, Zvi Griliches (1957) demonstrated potentially substantial specification biases from omitting relevant variables and aggregating across heterogeneous inputs, with possible solutions. sought to such biases his theoretical into a statistical using panel data methods that allowed over time and also across an approach extended by Biddle shows that such research an in panel data sets, an example where theory developments led to data to biases from use of experimental data was at the time to almost all of the statistical issues by critics of Douglas's methods of estimating production functions perhaps of the problem of estimated standard errors by Smith concludes part 2 in chapter which discusses the to which the regression was by economists to and economic growth where the Cobb-Douglas production function an important research This chapter has three sections, with over to and moving on to the period and concerns with the from production function as then growth using productivity (1957) had proposed an important role for Cobb-Douglas production function research in measuring growth technical progress, the first of several studies by That paper was also concerned to some of the assumptions of the Roy (1961) The was as progress, but the resulting estimated of capital on growth was Research from to trend developments is as are where technical progress is in of capital, as in Solow formal are not discussed, estimated capital Cobb-Douglas production functions. Since are to and run labor are also research, and as well as by especially concerning issues of and role of the Cobb-Douglas production function in growth is also with productivity discussing is as that of production functions are what productivity are although Biddle to with Robert that index numbers are better than Cobb-Douglas production functions for estimating out because econometrics was as as it allowed are a number of discussions concerning the many problems confronting empirical researchers measuring and data on output, capital, and data are essential to empirical research, a theme that in the many debates by by Griliches in his to the of economic observations. Econometricians are often in data I in Hendry of are on a but little on economic understanding by better data, though Douglas and his coauthors had to effort in and what was and are by Biddle as by Douglas, but their to data as are not discusses the of and as both the of production to a from the as well as estimating the with see and an of leading to the modern of Clark's greatly the of income accounts somewhat provides an discussion of Clark's World War there were other major developments in the in the of data, where James and with John income accounts to resource The of appropriate price was key to these The first adopted price on of price for of were proposed by and using later, and still and then and all made to improve including rather than changing the and are essential for calculating and, changes in their and so that using an can constant even when all the in the index are changing (see Hendry is often as a but this is as is that from the of modeling linear by (see Hendry 1995). While the of coefficients in may not to economic these would have to Cobb-Douglas is not discussed yet is essential to progress OLS applied to a variables on a of data. Louis that it took to a regression to data and Yule (1926) had to their studies empirical research was both in appropriate data and in calculating estimates of postulated relationships between Indeed, I the approach to empirical research of equations derives from that earlier where was time and associated were essential to the of empirical modeling though many modern publications still to the used (see Klein and Hendry and Charles for histories of of this section is to for data using modern and I Douglas would have to what estimates of a Cobb-Douglas regression like when account of dynamics, and shifts in combined with misspecification However, to the historical progress, I with the Cobb-Douglas is by the capital in the is by investment and a rate of and and by The role of should be by the of capital inputs account of and the of technology but such data are not There is historical data on which greatly over or which and even on from and which with and female labor all forms of technical progress, and must be by representing as 1 shows the relationship over time between and the in the and but not any long-run from constant to represented by the might be given such a estimating a static Cobb-Douglas specification with a constant trend is not very and standard errors are in is the standard the of tests proposed by tests conditional tests tests (see and and James tests The at 1 or all the misspecification tests their Moreover, the of at is highly from the of capital in A high combined with the massive were the Granger and Paul took to relations. The value of almost of suggests to be deal with the lack of dynamic adjustments to the many and the rate of all the of technical I a from a general with one of and using which for a potential trend shift at in see et al. for a and an to by the that can handle more variables than (see and Hendry which also the misspecification tests and their key role in The economics variables were when the trend were at trend for two and the economics variables at 1 a trend that time and is zero is an impulse to unit at time and zero and is the unit-root in and Hendry with critical in and James which the presence of a unit so the variables misspecification tests and the long-run solved of is with a labor of about The the and is when is the dependent trend show that growth was about prior to World War than but after both and the latter the of the productivity period the Great and records the and from and the which the of growth over time not by the economics a production how might they almost certainly to especially The estimated to the long-run after any disequilibrium is almost which given the potential need to and capital and with the appropriate and clearly use of inputs and outputs to but pandemic and have their the data on for period for which I lack capital shows the great in from the lockdowns in to the That and the of the on measuring labor inputs, will need an impulse to when the is the estimated Cobb-Douglas production function rather than chapter his Biddle considers the Cobb-Douglas regression was in the that it became used and the Cobb-Douglas Douglas was by a that relationships between the inputs to and outputs of production processes and could be through the of regression analysis to statistical data, and that of such relationships was relevant to important of economic theory and empirical can but from of major that may their and in at what they This in even the of the underestimated the of the by more than in conflict with because at the time no one about In Hendry I proposed for empirical failing that, be of be and to being a I would now from of to While Paul Douglas had both and he was also has been about the properties of economic time series at the time he his research, but these not to his faced serious and criticisms, many of which he or as he could not but he more in the of than why have Cobb-Douglas regressions Biddle that empirical estimates in line with views was for but not he suggests other then became that other are perhaps usually and their success or may for there are but not so the latter (see, e.g., more Charles theory of by was attacked, but over time as it so evidence about the in some like Thomas some and for many years being as to There is also that advances one at a perhaps as more and are having of their in earlier Had Douglas results like or even the might have early The of a relation between inputs and outputs was perhaps the most for the success of any to its That the Cobb-Douglas production function

The purpose of this paper is to investigate the quadratic production functions used by D. Gale Johnson [4] in his investigation of sharecropping and by H. Scott Gordon [3] in his classic analysis of a common property resource. Both Johnson and Gordon use linear average and marginal production functions in the geometric exposition of their respective theories, but neither author has explored the unusual properties of the quadratic production function which yields a linear marginal product and average product for one factor.The first part of this paper will investigate the quadratic production function used by Johnson in his sharecropping study, and the second part of this paper will explore the similar quadratic production function employed by Gordon in a subsequent paper on the economic theory of fisheries.In his seminal study of sharecropping, Johnson finds that when one maximizes with respect to labor, the tenant will rent additional land until the marginal physical product of land is zero. He demonstrates this remarkable result in two ways. First he does so with calculus by using a neoclassical production function, and second he does so with geometry in Figure 1 of this classic paper. [4].Although the calculus is correct, his geometric example raises interesting questions. He says linear functions are shown in Figure 1 of this paper for simplicity, but the example can be generalized for any type of average product function. However, he does not explain what type of production function can generate linear marginal and average product functions, and it is the intention of this paper to do so. In particular, one can ask whether such a production function is neoclassical, or one can ask what is the nature of the marginal and average product functions for the second factor of production? Also, one can ask whether such a production function is linear homogeneous?The properties of a neoclassical production function can be found in Burmeister and Dobell [1]. They list six properties of a neoclassical production function, Y = F(K,L):

This study derives space travel pricing by Walrasian Equilibrium, which is logical reasoning from the general relativity theory (GRT), the accounting equation, and economic supply and demand functions. The Cobb–Douglas functions embed the endogenous space factor as new capital to form the space travel firm’s production function, which is also transformed into the consumer’s utility function. Thus, the market equilibrium occurs at the equivalence of supply and demand functions, like the GRT, which presents the equivalence between the spatial geometric tensor and the energy–momentum tensor, explaining the principles of gravity and the motion of space matter in the spacetime framework. The mathematical axiomatic set theory of the accounting equation explains the equity premium effect that causes a short-term accounting equation inequality, then reaches the equivalence by suppliers’ incremental equity through the closing accounts process of the accounting cycle. On the demand side, the consumption of space travel can be assumed as a value at risk (VaR) investment to attain the specific spacetime curvature in an expected orbit. Spacetime market equilibrium is then achieved to construct the space travel pricing model. The methodology of econophysics and the analogy method was applied to infer space travel pricing with the model of profit maximization, single-mindedness, and envy-free pricing in unit-demand markets. A case study with simulation was conducted for empirical verification of the mathematical models and algorithm. The results showed that space travel pricing remains associated with the principle of market equilibrium, but needs to be extended to the spacetime tensor of GRT.

The article deals with the scientific problem of clarifying the content of the concept of "digital economy". According to the authors, this problem is relevant for the theory of the digital economy, since its solution contributes to the formation of the theory's conceptual apparatus. It (the problem) is also relevant for improving the practice of public management of the economy, as it contributes to the correct assessment of the object of management, which is the digital economy. The purpose of the study is to justify the scientific tools, which would make it possible to get rid of a certain "blurring" of the concept of "digital economy", recognized by many researchers. The final goal of the study is to achieve a more accurate and concrete identification of the concept of "digital economy" for its relevant application in solving applied management and economic development tasks. To achieve the specified goal, a set of research methods with the tools inherent in these methods is applied, namely: sectoral models of economic analysis and production functions for individual sectors, identification of the core and so-called "belts" of technological changes in the digital economy, evaluation based on economic multipliers, comparative analysis the level of digital development of countries according to indicators of the contribution of certain sectors to the created GDP of countries. The application of the mentioned methods and the corresponding tools of analysis gave grounds for such identification of the digital economy. The digital economy is part of the national economy, which is formed in the process of interconnected movement of resources and products of three sub-sectors. Such sub-sectors are the "technological core" in the form of the ICT sector, the "consumer e-belt" sector and the "public administration infrastructure" sector, which is directly related to the creation of an enabling environment for the ICT sector and the "consumer e-belt ". The general type of production functions of each of the three sub-sectors of the digital economy is determined. Reasoned feasibility of quantitative identification of the digital economy using a set of indicators that reflect: the share of the digital economy in the GDP of the country, the share of the ICT sector in the GDP, the multiplier effect on the entire economy of the digital economy itself and its ICT sector (mde, mIKT/de, mIKT/Y).

Application of a combination production function model

In this study the production functions (Cobb-Douglas, Zener-Rivanker, and the transcendental production function) have been used to assess the profitability of insurance companies, by reformulating these nonlinear functions based on the introduction of a set of variables that contribute to increase the explanatory capacity of the model. Then the best production function commensurate with the nature of the variable representing the profitability of insurance companies was chosen, to use it to assess the efficiency of their profitability versus the use of different factors of production and thus the possibility of using it in forecasting. It was found that the proposed model of the production function "Zener-Rivanker" is the best production functions representing the profitability of the Tawuniya and Bupa Insurance Companies. The proposed model of the Cobb-Douglas production function is suitable for the results of both Enaya and Sanad Cooperative Insurance Companies. The explanatory capacity of the production functions was also increased when the proposed variables were added (net subscribed premiums-net claims incurred).

This paper presents sufficient conditions for the existence of a unique and globally stable steady state equilibrium for OLG economies with production. The conditions impose separate requirements on the utility and production functions. Moreover, the conditions do not require assumptions concerning the third order derivatives of the production and utility functions.

In economic analysis, the importance of the homotheticity of production functions (or utility functions), which is due to Shephard (1953), has been well recognized. Its important feature lies in the fact that every expansion path is a ray from the origin and the underlying production (or utility) function is homothetic. Although the proof of the part of this statement is easy, the proof of the only if part, at least as it appears in the literature, is not easy, requiring a few pages for the proof. Lau (1969) proved it by way of partial differential equations (cf. pp. 379-81), and Fare and Shephard (1977a, b) used a set-theoretic approach, whereas Sandler and Swimmer (1978) proved it for the two-input case. F0rsund (1975) provides a simple proof for a closely related proposition (cf. his Proposition 2), and yet his proof requires the use of partial differential equations. It would thus be desirable to obtain an elementary, short alternative proof of the above important proposition. The purpose of this note is to offer such a proof. In this paper we prove both the only if and the part simultaneously. Although our proof is carried out in terms of production theory, the same proof obviously applies to the theory of consumption. Let f(x) be the production function of a firm which produces a single output, where x is an n-dimensional input vector. The firm minimizes its

The purpose of this study is to investigate the Industrial structure of the Canadian domestic telecommunications satellite relay system. In doing so, the hypothesis is developed that such domestic or regional systems are natural monopolies in supplying long-distance message and broadcasting services, it is also maintained that their market structures are contestable, but sustainable against potential entry from firms serving the same market using either the same technology or from competitors in other parts of the telecommunications industry with different technology.This hypothesis is tested empirically by specifying a set of production and cost functions that are representative of the neoclassical production technology, and then applying the statistical data supplied by Canada's domestic telecommunications satellite authority - Telesat Canada. Drawing upon the theories of consistent aggregation, duality, and regulation, the model is constructed, the data base is established, and the analysis is developed in five parts.Part One includes the introduction, a chapter on the theory of natural monopoly, and another on the institutional framework of the Canadian telecommunications industry. Its major theme is to establish the necessary and sufficient conditions required to establish natural monopoly status and to apply these conditions to the extra-terrestrial segment of the Canadian telecommunications industry - Telesat Canada - as part of the total system. Further, a brief discussion on Telesat Canada's pricing and investment behaviour is analyzed to ascertain if the organisation pursues monopolistic practices.Part Two concentrates on the theory of the multi-product firm and the attendant problems of jointness in production for the variable product proportions case. By drawing on the thereoms of consistent aggregation and by adopting the simplying assumption that all the individual output production functions are linearly homogeneous and that the outputs are produced in fixed proportions, the multiple output production technology reduces to a single output production specification. The nature of output is perceived and defined as a function of the earth segment configuration; denoting it in terms of the number of received circuit days (RCDs) of equivalent one-way 4KHZ voice circuits received at each earth station.Parts Three and Four cover the theoretical model building and empirical investigation for testing the hypothesis that Telesat Canada is a natural monopoly. Part Three develops a statement on the nature of the production technology, the underlying properties of a neoclassical production function, establishes a data base on the inputs side, and tests a number of production function specifications to ascertain the most appropriate for describing the production technology of Telesat Canada and domestic satellite telecommunication networks in general. Estimates for scale economies and elasticities of factor substitution are derived for both homogeneous and non-homogeneous specifications.Part Four, on the other hand , draws heavily on duality theory to construct a set of cost functions using the physical unit of output measure RCDs for the purposes of estimating global scale economies for the total extra-terrestrial system and the 6/4 GHZ system only. A log-linear and polynomial specification were applied and scale estimates ranging between 2.03 and 2.54 were derived for the total system and 6/4 GHZ service only system respectively using the former specification, while a more modest estimate of only 1.54 (at the mean) was calculated for the 6/4 GHZ service using the polynomial form. These estimates compared well with those produced for the international satellite telecommunications industry as well as with those estimated for the Canadian terrestrial telecommunications sector.Part Five summarizes the results and draws some conclusions. It is shown that under certain conditions the Baumol cost subadditivity sufficiency conditions are satisfied, giving evidence to the hypothesis that Telesat Canada is a natural monopoly with continuously decreasing costs over the relevant output range. However, it is contended that the industry is contestable in providing long-haul telecommunication services from potential entrants using existing or developing new terrestrial telecommunication technologies. Hence, despite the lack of apparent sunk costs, natural monopoly status in the extra-terrestrial segment of the domestic market is conferred by the mode of transmission, which could be usurped by evolving technology. Noting this technological advantage in long-haul telecommunication services, the author concludes that future developments are likely to lead to specialisation in service provision with satellites providing the bulk of long-haul services while their terrestrial counterparts concentrate on local service distribution.

Towards cultivated land multifunction assessment in China: Applying the “influencing factors-functions-products-demands” integrated framework