Optimal depletion of nonrenewable energy added to the neoclassical model dampens economic growth as its price rises at the rate of the capital return. The present paper introduces sustainable energy requiring its own capital with the Inada condition requiring both energy inputs are necessary for production. Production is developed in terms of factor shares and price elasticities. Investment in sustainable energy offers relief from the rising nonrenewable price but lowers consumption or capital growth. Simulations of the U.S. economy foresee a smooth transition with growth paths depending on the two saving rates and factor price substitution.

We consider the physical constraints of the production process by connecting the actual amount (volume) of physical output to its mass. To achieve this, we introduce a function that evaluates the mass of physical objects such as resources and output. Through the mass function, we demonstrate that the marginal product of resource inputs is less than the ratio of the mass density of resource inputs to that of the final products. This ratio's boundedness implies that the marginal resource product has an upper limit, meaning that the Inada condition for resources is not valid. This restriction on the marginal product of resource input has significant implications for long-term growth and environmental sustainability. It indicates that the growth of total physical output is only feasible with more resource input and a potentially higher level of pollution.

In this study, we propose a mathematical theory for building neoclassical production functions with homogeneous inputs in both aggregate and per capita terms. This theory is based on two concepts: Euler's equation and Cauchy's condition for first-order partial differential equations. The analysis is restricted to functions that exhibit constant returns to scale (CRS). For the function to meet the law of diminishing marginal returns, we present the necessary and sufficient conditions to be satisfied by the curve that defines Cauchy's condition. In this context, we also discuss the Inada conditions. We first present functions that depend on two inputs and then extend and discuss the results for functions that depend on several inputs. The main result of our research is the provision of a clean and clear theory for constructing neo-classical production functions. We believe that this result may contribute to closing the huge methodological gaps that separate schools of economic thought that defend or reject the use of production functions in economics.

This article proposes two conformal Solow models (with and without migration), accompanied by simulations for six Organisation for Economic Co-operation and Development economies. The models are proposed by employing suitable Inada conditions on the Cobb–Douglas function and making use of the truncated M-derivative for the Mittag–Leffler function. In the exact solutions derived in this manuscript, two new parameters play an important role in the convergence towards, or the divergence from, the steady state of capital and per capita product. The economical dynamics of these nations are influenced by the intensity of the capital and labor factors, as well as the level of depreciation, the labor force rate and the level of saving.

We explore martingale and convex duality techniques to maximize expected risk-averse utility from consumption in a general multi-dimensional (non-Markovian) semimartingale market model with jumps and non-linear wealth dynamics. The model allows to incorporate additional cash flows via non-linear margin payment functions in the drift term that depend on the allocation proportion. These can be used to cast frictions such as the impact of the portfolio choices of a ‘large’ investor on the expected assets’ returns, funding costs arising from differential borrowing and lending rates, and the cash inflow of a firm in a neoclassical economy with constant return-to-scale Cobb–Douglas technology subject to exogenous aggregate shocks. We provide a general verification theorem for random utility fields satisfying the usual Inada conditions, find conditions under which jumps in our model lead to precautionary saving, and present an explicit characterization for CRRA. We report two-fund separation-type results which assert that optimal allocations move along one-dimensional segments, and illustrate our results numerically for various margin payment functions and bounded variation tempered $$\alpha $$ -stable jumps.

Inada conditions asymptotically transform production function into the Cobb–Douglas

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, cumulative prospect theory (CPT), was originally introduced by [A. Tversky & D. Kahneman (1992) Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (4), 297–323]. The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by [H. Jin & X. Y. Zhou (2008) Behavioral portfolio selection in continuous time, Mathematical Finance 18 (3), 385–426]. Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, Cumulative Prospect Theory (CPT), was originally introduced by Tversky and Kahneman (1992). The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by Jin and Zhou (2008). Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

General existence of competitive equilibrium in the growth model with an endogenous labor–leisure choice

The present article presents novel results on the Ramsey–Cass–Koopmans growth model. It is shown that the shadow price of capital goes to infinity as the capital stock goes to zero even if all functions are bounded with finite derivatives and that imposing the Inada condition of infinite derivative of the per capita production function at zero stock is irrelevant. It is also shown that unless marginal utility at zero consumption is infinity, there will be a non-empty interval where the Keynes–Ramsey rule does not hold. The paper also shows that the stable saddle path in a phase diagram with the state variable and the shadow price has an unrecognized economic interpretation that enables us to illustrate the value function as the integral of the stable saddle path.

We consider a mechanism design setting in which agents can acquire costly information on their preferences as well as others’. The choice of the mechanism generates informational incentives as it affects what information is acquired before play begins. A mechanism is informationally simple if agents have no incentive to learn about others’ preferences. This property is of interest for two reasons: First, the extended game has an equilibrium in dominant strategies only if it is informationally simple. Second, this endogenizes an independent private value property of the interim information structure. Our main result is that a mechanism is informationally simple only if it is de facto dictatorial. This holds for generic environments and any smooth cost function which satisfies an Inada condition. Hence even interim strategy-proof mechanisms incentivize agents to learn about others, and do not admit dominant strategies at the ex-ante stage. Moreover, the Independent Private Value assumption is unlikely to arise endogenously, though we show that full surplus extraction is infeasible.

Comment

We study optimal investment problems in hedge funds for a loss averse manager under the framework of cumulative prospect theory. We obtain explicit solutions for a general utility function satisfying the Inada conditions and a piece-wise exponential utility function. Through a sensitivity analysis, we find that the manager reduces the risk of the hedge fund when her/his loss aversion, risk aversion, ownership in the fund, or management fee ratio increases. However, the increase of incentive fee ratio drives the manager to seek more risk in order to achieve higher prospect utility.

A note on 2-input neoclassical production functions

This paper characterizes a specification for the utility of leisure that is based on the general equation for an ellipse. We show that this functional form has multiple benefits. The elliptical utility function provides Inada conditions at both the upper-bound and lower-bound constraints on labor supply, which is not the case for the two most common alternative functions. The presence of these two Inada conditions in the elliptical utility of leisure specification speeds up the computation by a factor between three and six times. We further show that the elliptical utility of leisure function is a close approximation to the constant relative risk aversion and constant Frisch elasticity functions in terms of marginal utilities, microeconomic outcomes in a life cycle model, and macroeconomic outcomes in a simple real business cycle model.

We study production functions with capital and labor as arguments that exhibit positive, yet diminishing marginal products and constant returns to scale. We show that such functions satisfy the Inada conditions if (i) both inputs are essential and (ii) an unbounded quantity of either input leads to unbounded output. This allows for an alternative characterization of the neoclassical production function that altogether dispenses with the Inada conditions. Although this proposition generalizes to the case ofn> 2 factors of production, its converse does not hold: 2nInada conditions do not imply that each factor is essential.

We consider an economy where agents' consumption sets are given by the cone $\mathbf{L}^0_+$ of non-negative measurable functions and whose preferences are defined by additive utilities satisfying the Inada conditions. We extend to this setting the results in \citet{Dana:93} on the existence and uniqueness of Arrow-Debreu equilibria. In the case of existence, our conditions are necessary and sufficient.

This paper proposes two approaches that quantify the exact relationship among the viability, the absence of arbitrage, and/or the existence of the num\'eraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence among the No-Unbounded-Profit-with-Bounded-Risk condition (NUPBR hereafter), the existence of the num\'eraire portfolio, and the existence of the optimal portfolio under an equivalent probability measure for any "nice" utility and positive initial capital. Herein, a 'nice" utility is any smooth von Neumann-Morgenstern utility satisfying Inada's conditions and the elasticity assumptions of Kramkov and Schachermayer. Furthermore, the equivalent probability measure ---under which the utility maximization problems have solutions--- can be chosen as close to the real-world probability measure as we want (but might not be equal). Without changing the underlying probability measure and under mild assumptions, our second contribution proves that the NUPBR is equivalent to the "{\it local}" existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities.

In this note we generalize the Negishi approach to equilibrium. We embed a standard one-period exchange economy into a two-period model, where agents' first-period utility functions can be any strictly increasing and concave functions satisfying the lower Inada condition, and prove the existence of equilibrium by working with the space of agents' first-period consumptions. The celebrated Negishi method is a special case of this approach where agents' first-period utility functions are all assumed to be logarithmic.

One of the most puzzling questions in economics is the existence of extremely high income differences between rich and poor countries. In neoclassical economics, we choose to start with the first order assumption, that people and institutions are rational, meaning that capital flow from places with low return to higher returns. This doesn’t mean that all countries should be alike, but any differences in income could only reflect a level effect from different institutional set-ups, because of the economic law of diminishing return. There cannot be any dynamic growth effect from differences in institutions. This was one of the main insights from the classical Solow model, which in turn lead to the conclusion that institutions was important, but actually not that important. In this paper I challenge this mainstream view. My hypothesis is that maybe our institutions actually have some “secrets” as Hayek once put it. Maybe small initial changes in those institutions could eventually lead to success or totally disaster, simply because the time preferences in the economy in it selves is not an exogenous variable. My suggestion is that the initial condition for man, is “stealing” from the capital base (which we normally call a “crime”) which in turn is actually a dominant strategy. Institutions securing private property are very fragile, meaning that a certain degree of punishment, breaking these institutions, must be in place, before any growth take-off. From an individual point of view, there seems to be a clear trade-off between generating income from productive work and stealing from the capital base (“from the rich”). It seems therefore logical to think that the punishment from breaking the institutions of private property must somehow reflect the overall accumulation of capital, meaning there could be a abstract or natural level of punishment, securing doing crime becomes relatively “bad business.” But if punishment is lower than the “natural level” crime, crime in itself could deteriorating the capital base, leading to lower income and then give birth to even more crime. This asks the question. If the capital base is lower, shouldn't that generate a higher and higher return of capital at the margin, then simply establish a new equilibrium at a lower income level? I will show that it is possible that the destructive forces from crime is enough to overcome the forces from the Inada condition, then established a negative growth path and the destruction of society.