Arrow-Pratt Coefficient Research Articles

Relationship of the arrow-pratt coefficients with the tangent to the utility function

The article discusses a geometric approach for determining some properties of the utility function using the Arrow-Pratt coefficient. In particular, the geometric properties of the Arrow-Pratt coefficient are disclosed, it turned out that the Arrow-Pratt coefficient determines the distance from the origin to the straight line parallel to the tangent line at the point where the Arrow-Pratt coefficient is calculated. Therefore, an increase and decrease in the value of this coefficient determines the convexity or concavity of the utility function. With an increase in the value of the Arrow-Pratt coefficient, the tangent line moves away from the origin. This process describes the convexity of the utility function as a function of location from the origin. There are two concepts: the expected utility model and the utility function. Our approach is that due to a tangent line and a straight line parallel to the tangent with a free term, associated with the variance of variables and the Arrow-Pratt coefficient, we determine the behavior of the utility function: monotonicity, convexity, etc. The location of these lines relative to the origin gives us twenty-four different options, which are divided into four groups, depending on the signs of the first and second derivatives of the utility function at the point where the tangent line passes.

11. The Gender Wage Gap of Recent University Graduates

The purpose of this research was to question whether the behavioural tendencies of men and women could help explain the gender wage gap of recent university graduates. It was conducted after discovering that a 2013 study found that even once accounting for observable characteristics such as age, experience, industry, occupation, and field of study, female graduates were still earning 6-14% less than their male counterparts. Using the willingness of a graduate to gamble a current job offer for a potentially better job offer in the future as a proxy for risk, this research investigates the impact risk preferences have on the gender wage gap. More specifically, it attempts to calculate how much of the observed wage gap can be attributed to the greater risk aversion of women. 
 Our data was from the National Longitudinal Survey of Youth 1997. Using a McCall Job Search model and an MLE, we found that women take approximately 4.5 fewer weeks to accept a job, accept significantly lower starting salaries, and are systematically offered lower salaries than their male counterparts. Furthermore, we found that women have an Arrow-Pratt coefficient almost 1.25 times that of men. These results suggest that women are more willing to accept lower wage positions offered to them today because they are less willing to gamble that a higher wage position will come along tomorrow. Moreover, they propose that this female unwillingness to gamble can explain up to a quarter of the difference in the wages accepted by men and women.

Generalized Arrow-Pratt Theory

Given a utility defined on a Hilbert outcome space, we define at each outcome a generalized Arrow-Pratt (GAP) coefficient belonging to the Hilbert space. Comparing the risk aversion of such utilities using their GAP coefficients is equivalent to doing so in terms of other standard, decision-theoretically persuasive, criteria. The resulting GAP theory of risk aversion significantly expands the scope of the classical Arrow-Pratt theory by admitting a large class of risks with vector outcomes. This allows the theory to address risks that are embodied in a significant class of random processes. The GAP theory's implications are studied in five contexts with Hilbert outcome spaces, namely, the theories of portfolio choice, insurance, asset valuation, auctions and moral hazard in teams. We also show a duality between utility functions on Euclidean spaces and GAP coefficients. Finally, we provide a theoretically well-founded and computationally tractable method for estimating the expected GAP coefficient from empirical data.

Are Invisible Hands Good Hands in Health Care Markets? Extension

AbstractA previous study finds that increased competition in health care markets improves social welfare, although consumers use “too much” health care when they have health insurance. The analysis assumes that consumers have a constant Arrow-Pratt coefficient of absolute risk aversion. This note shows that this finding can be extended to the case where consumers are simply risk averse. Furthermore, if insurers offered insurance policies with slightly lower usage prices than the equilibrium level, social welfare would be improved.

Comparative statics under κ‐ambiguity for log‐Brownian asset prices

AbstractThis paper examines the comparative statics of optimal risky demand when economic agents are both risk averse and ambiguity averse. In a setting with log‐Brownian asset prices and ‐ambiguity but virtually for all utility functions, we show that the greater the Arrow–Pratt coefficient of absolute risk aversion under ambiguity, the less the optimal demand for risky assets. This monotonic property is demonstrated with varying risk‐free interest rate and allows for distinct estimated model parameters across agents.

Who is More Averse to Random Processes?

It is well-known that various criteria for comparing aversion to real-outcome risks are equivalent. Some of this theory has been extended to Euclidean-outcome risks. We extend it further by:(a) filling the conceptual gaps, most notably by providing a criterion using our generalized Arrow-Pratt (GAP) coefficient and, (b) admitting vector outcome spaces that can be used to model risks embodied in a large class of random processes. These extensions substantially expand the tool-kit and scope of the theory. We use it to predict the effect of risk aversion on:(a) money-metric valuations of financial assets and,(b) an asset portfolio when assets are specified by dividend processes. We also provide a theoretically well-founded and computationally tractable method for estimating the GAP coefficient from data.

Economic and Biological Values for Pasture-Based Dairy Cattle Production Systems and their Application in Genetic Improvement in the Tropics

Economic and biological values for milk yield (MY), milk butter fat (FY), daily gain (DG), weaning weight (WWT), mature live weight (MLW), calving interval (CI), pre-weaning survival rate (PreSR), post-weaning survival rate (PostSR), age at first calving (AFC), and productive life time (PLT) were estimated under fixed herd (FH) and pasture (FP) production circumstances assuming milk marketing based on volume, and volume and butter fat. Further, economic values were estimated involving risk using the Arrow Pratt coefficients at two levels. For the former economic values for the traits ranged from KSh. -17.246 to 100.536 while the biological values ranged between -1.29 to 0.791. Economic values with higher Arrow-Prat coefficient of absolute risk aversion (λ=0.02) were lower than those reported under λ=0.0001 indicating that the uncertainty of the future market is important and should be considered during the estimation of economic values. Genetic improvements targeting MY and growth traits would be recommended to production system with unlimited feed supply for profit maximization. However, since dairy production systems in the tropics are characterised by feed scarcity, fixing the herd and concentrating on genetically improved animals would result to more profitability than increasing animal populations.

Archimedean copulas derived from utility functions

The inverse of the (additive) generator of an Archimedean copula is a strictly decreasing and convex function, while utility functions (applying to risk averse decision makers) are nondecreasing and concave. This provides a basis for deriving an inverse generator of an Archimedean copula from a utility function. If we derive the inverse of the generator from the utility function, there is a link between the magnitude of measures of risk attitude (like the very common Arrow–Pratt coefficient of absolute risk aversion) and the strength of dependence featured by the corresponding Archimedean copula. Some new copula families are derived, and their properties are discussed. A numerical example about modeling dependence of coupled lives is included.

Economic values for production and functional traits of Small East African goat using profit functions

Economic values for production traits (milk yield, MY, g; 12-month live weight, yLW, kg; consumable meat percentage, CM, %) and functional traits (mature doe live weight, DoLW, kg; mature buck live weight, LWb, kg; kidding frequency, KF; pre-weaning survival rate, PrSR, %; post-weaning survival rate, PoSR,%; doe survival rate, DoSR, %; and residual feed intake, RFI, kg) were estimated using profit functions for the Small East African goat. The scenario evaluated was a fixed flock size, and the resultant economic values (Kes per doe per year) were 34.46 (MY), 62.35 (yLW), 40.69 (CM), 0.15 (DoLW), 2.84 (LWb), 8.69 (KF), 17.38 (PrSR), 16.60 (PoSR), 16.69 (DoSR) and -3.00 (RFI). Similarly, the economic values decreased by -14.7% (MY), -2.7% (yLW), -23.9% (CM), -6.6% (DoLW), -98% (LWb), -8.6% (KF), -8.2% (PrSR), -8.9% (PoSR), -8.1% (DoSR) and 0% (RFI) when they were risk rated. The economic values for production and functional traits, except RFI, were positive, which implies that genetic improvement of these traits would have a positive effect on the profitability in the pastoral production systems. The application of an Arrow-Pratt coefficient of absolute risk aversion (λ) at the level of 0.02 resulted in a decrease on the estimated economic values, implying that livestock keepers who were risk averse were willing to accept lower expected returns. The results indicate that there would be improvement in traits of economic importance, and, therefore, easy-to-manage genetic improvement programmes should be established.

An alternative explanation for the variation in reported estimates of risk aversion

An Alternative Explanation for the Variation in Reported Estimates of Risk Aversion There is a large literature estimating Arrow-Pratt coefficients of absolute and relative risk aversion. A striking feature of this literature is the very wide variation in the reported estimates of the coefficients. While there are often legitimate reasons for these differences in the estimates, there is another source of variation that has not been considered to date. The Arrow-Pratt coefficients are properties of the utility functions, but a number of estimates are obtained by equating these to risk aversion measures defined in a mean-variance framework. This paper shows that while the legitimacy of the mean-variance approach may hold under general conditions the additional assumptions invoked when estimating the risk aversion parameter hold only in very restricted circumstances and that serious under or over estimation can easily arise as a result. JEL Classification: C91, D81, G11

Archimedean Copulas Derived from Morgenstern Utility Functions

The (additive) generator of an Archimedean copula - as well as the inverse of the generator - is a strictly decreasing and convex function, while Morgenstern utility functions (applying to risk averse decision makers) are nondecreasing and concave. This provides a basis for deriving either a generator of Archimedean copulas, or its inverse, from a Morgenstern utility function. If we derive the generator in this way, dependence properties of an Archimedean copula that are often taken to be desirable, match with generally sought after properties of the corresponding utility function. It is shown how well known copula families are derived from established utility functions. Also, some new copula families are derived, and their properties are discussed. If, on the other hand, we instead derive the inverse of the generator from the utility function, there is a link between the magnitude of measures of risk attitude (like the very common Arrow-Pratt coefficient of absolute risk aversion) and the strength of dependence featured by the corresponding Archimedean copula.

Groundwater pumping by heterogeneous users

Farm size is a significant determinant of both groundwater-irrigated farm acreage and groundwater-irrigation-application rates per unit land area. This paper analyzes the patterns of groundwater exploitation when resource users in the area overlying a common aquifer are heterogeneous. In the presence of user heterogeneity, the common resource problem consists of inefficient dynamic and spatial allocation of groundwater because it impacts income distribution not only across periods but also across farmers. Under competitive allocation, smaller farmers pump groundwater faster if farmers have a constant marginal periodic utility of income. However, it is possible that larger farmers pump faster if the Arrow-Pratt coefficient of relative risk-aversion is sufficiently decreasing in income. A greater farm-size inequality may either moderate or amplify income inequality among farmers. Its effect on welfare depends on the curvature properties of the agricultural output function and the farmer utility of income. Also, it is shown that a flat-rate quota policy that limits the quantity of groundwater extraction per unit land area may have unintended consequences for the income distribution among farmers.

An Alternative Explanation for the Variation in Reported Estimates of Risk Aversion

There is a large literature estimating Arrow-Pratt coefficients of absolute and relative risk aversion. A striking feature of this literature is the very wide variation in the reported estimates of the coefficients. While there are often legitimate reasons for these differences in the estimates, there is another source of variation that has not been considered to date. The Arrow-Pratt coefficients are properties of the utility functions, but a number of estimates are obtained by equating these to risk aversion measures defined in a mean-variance framework. This paper shows that while the legitimacy of the mean-variance approach may hold under general conditions the additional assumptions invoked when estimating the risk aversion parameter hold only in very restricted circumstances and that serious under or over estimation can easily arise as a result.

Traveling wave solutions to the nonlinear evolution equation for the risk preference

A singular nonlinear partial differential equation (PDE) is introduced, which can be interpreted as the evolution of the risk preference in the optimal investment problem under the random risk process. The unknown quantity is related to the Arrow-Pratt coefficient of relative risk aversion with respect to the optimal value function. We show the existence of monotone traveling wave solutions and the nonexistence of non-monotone such solutions, which are

Expected Future Earnings, Taxation, and University Enrollment: A Microeconometric Model with Uncertainty

Taxation changes the expectations of prospective university students about their future level and uncertainty of after-tax income. To estimate the impact of taxes on university enrollment, we develop and estimate a structural microeconometric model, in which a high-school graduate decides to enter university studies if expected lifetime utility from this choice is greater than that anticipated from starting to work right away. We estimate the ex-ante future paths of the expectation and variance of net income for German high-school graduates, using only information available to those graduates at the time of the enrollment decision, accounting for multiple nonrandom selection and employing a microsimulation model to account for taxation. In addition to income uncertainty, the enrollment model takes into account university dropout and unemployment risks, as well as potential credit constraints. The estimation results are consistent with expectations. First, higher risk-adjusted returns to an academic education increase the probability of university enrollment. Second, high-school graduates are moderately risk averse, as indicated by the Arrow-Pratt coefficient of risk aversion estimated within the model. Thus, higher uncertainty among academics decreases enrollment rates. A simulation based on the estimated structural model indicates that a revenue-neutral, flat-rate tax reform with an unchanged basic tax allowance would increase enrollment rates for men in Germany because of the higher expected net income in the higher income range.

Production decisions under joint price and production uncertainty

Abstract While production decisions in the presence of price uncertainty have been extensively studied, this is not so for the case in which the level of production is itself uncertain. In this paper,we provide a decision analysis under multiplicative production uncertainty, both with and without price uncertainty. We depict equilibrium and obtain comparative statics results with the aid of a diagram based on the difference between expected price and marginal cost. Comparative statics results are obtained for the model with production uncertainty alone and also for simultaneous price and production uncertainty (including two special cases). We first derive results based on the Arrow–Pratt coefficients of risk aversion, and then supplement these with the Ross measure of relative risk aversion, since this proves useful in the presence of multiple sources of uncertainty. We find that increases in risk (both price and production) or input prices reduce expected output. However, expected output supply is an increasing function of (expected) price only for “low” levels of risk aversion, and in general the relationship is ambiguous.

Dual Representations of Cardinal Preferences

Given a set of vector outcomes and the set of lotteries over it, we define sets of (a) von Neumann-Morgenstern representations of cardinal preferences over the lotteries, (b) mappings that yield the certainty equivalent outcomes corresponding to a lottery, (c) mappings that yield the risk premia corresponding to a lottery, (d) mappings that yield the acceptance set of lotteries corresponding to an outcome, and (e) vector valued functions that yield generalized Arrow-Pratt coefficients corresponding to an outcome. Our main results establish bijections between these sets of mappings for very general specifications of outcome spaces, lotteries and preferences. As corollaries of these results, we derive analogous dual representations of risk averse preferences. Some applications to financial theory illustrate the potential uses of our results. Finally, we provide criteria for comparing the risk aversion of preferences in terms of the dual representations.

Comparative Risk Aversion with Vector Outcomes

The classical theory of comparative risk aversion shows the equivalence of various criteria for comparing the aversion of cardinal preferences to risks with real outcomes. Parts of this theory have been extended to outcomes in Euclidean spaces. We complete, unify and generalize this theory. Our general equivalence result admits outcomes in locally convex topological vector spaces. Our special equivalence result, which involves our generalized Arrow-Pratt (GAP) coefficient, admits outcomes in Hilbert spaces. These results significantly expand the range of applications of the theory by making admissible risks that are embodied in a large class of random processes, including widely-used Wiener-related processes and second-order processes. We use our results to predict the effects of differential risk aversion on the subjective valuations of financial assets with continuous random dividend processes. We also provide a theoretically well-grounded, yet computationally tractable, method for estimating the GAP coefficient.

Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem

We are concerned with the solvablity of certain nonlinear partial differential equation (PDE), which is derived from the optimal investment problem under the random risk process. The equation describes the evolution of the Arrow-Pratt coefficient of absolute risk aversion with respect to the optimal value function. Employing the fixed point approach combined with the convergence argument we show the existence of solutions.

Comparing risk attitudes of organic and non-organic farmers with a Bayesian random coefficient model

Organic farming is usually considered to be more risky than conventional farming, but the risk aversion of organic farmers compared with that of conventional farmers has not been studied. Using a non-structural approach to risk estimation, a Bayesian random coefficient model is used to obtain individual Arrow–Pratt coefficients of absolute risk aversion for a sample of Dutch organic and non-organic arable farmers. The model is estimated using Gibbs sampling. The results indicate that organic farmers are significantly less risk averse than their non-organic colleagues.