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Weekly UpdatesRisk and uncertainty are prominent features of agricultural production and marketing. Not surprisingly, numerous studies in agricultural economics have focused on producer behavior under uncertainty (Just and Pope 2002). One of the most popular topics of study in this field has been the estimation of decision makers' risk preferences, both by direct elicitation from experimental data or responses to hypothetical questions (e.g., Binswanger 1980), or by analyzing observed production and/or investment choices (e.g., Brink and McCarl 1978; Love and Buccola 1991; Saha, Shumway, and Talpaz 1994; Chavas and Holt 1996; Kumbhakar 2002a).
The seminal studies estimating risk preferences from actual production and/or investment decisions have focused on the level of risk aversion, by estimating risk preferences separately from technology (e.g., Brink and McCarl 1978) and assuming restrictive-utility functions (e.g., mean variance analysis). Such studies have been superseded by work where risk preferences are estimated simultaneously with technology (e.g., Love and Buccola 1991; Coyle 1999), as doing so can increase estimation efficiency and may avoid inconsistency problems, even though Antle (1989) argued that there are some advantages in separating the estimation of technology and risk preferences. In addition, starting with Saha, Shumway, and Talpaz (1994), the literature has emphasized the estimation of decision makers' structure of risk aversion (i.e., the changes in absolute or relative risk aversion associated with changes in wealth) by allowing for more flexible-utility functions (Chavas and Holt 1996; Saha 1997; Bar-Shira, Just, and Zilberman 1997; Kumbhakar 2001, 2002a, 2002b; Kumbhakar and Tveteras 2003; Isik and Khanna 2003; Abdulkadri, Langemeier, and Featherstone 2003).
Knowledge about the structure of risk aversion is of interest because it determines, among other things, decision makers' responses to background risk, whether risky assets can be considered normal goods, and whether agents save for precautionary purposes (Gollier 2001). Importantly, however, Kallberg and Ziemba (1983, p. 1257) concluded that "... utility functions having different functional forms and parameter values but 'similar' absolute risk aversion indices have 'similar' optimal portfolios." They defined the index of absolute risk aversion of agent i under end-of-period wealth distribution d as the expected coefficient of absolute risk aversion corresponding to i under d. Kallberg and Ziemba's (1983) conclusions must be qualified by the fact that their study assumed multivariate normally distributed returns, a case relatively favorable to finding similar choices across alternative utility functions (Cerny 2004). More recently, Cerny (2004) argued that, except for investments involving very large and skewed risks, agents with similar values of relative risk aversion, evaluated at their initial wealth levels make almost identical portfolio decisions, regardless of their risk-aversion structures.
Estimation of the structure of risk aversion in production models is based on the premise that such structure affects optimal input choices under uncertainty. However, the aforementioned studies by Kallberg and Ziemba (1983) and Cerny (2004) suggest that, given the same level of risk aversion, differences in optimal input decisions induced by different structures of risk aversion are negligible, except for very large and skewed risks. This implies that, unless risks are very large and skewed, identification of the structure of risk aversion in production models may rely on sources of information too weak to allow for reliable econometric estimation.
The purpose of this study is to investigate whether it is indeed feasible to estimate the structure of risk aversion given the risks underlying the data usually employed by researchers in empirical production analysis. To this end, a thought experiment is performed with risks calibrated using historical farm data. Importantly, the experiment is designed to favor the likelihood of obtaining good estimates of the risk-aversion structure, so that failure to get reasonable estimates provides strong evidence against the hypothesis that the structure of risk aversion can be recovered from production data.
The study contributes to the literature by providing evidence against the hypothesis that typical production data contain enough information to allow identification of the structure of risk aversion. If anything, a flexible-utility parameterization slightly worsens the estimates of technology parameters. Overall, our findings greatly undermine the case for estimating the structure of risk aversion in studies of production. More generally, the method employed here may be useful in other situations where identification of the parameters and/or models of interest is suspected to be too weak to be useful, by allowing researchers to discard doomed-to-fail estimation projects at an early stage.
Model
We adopt standard assumptions in the production literature by postulating that, at decision time t = 0 a competitive producer chooses the amounts of input (x) that maximize the expected utility of end-of-period random wealth,
(1) [x.sup.*] = [argmax.sub.x][E.sub.w]{U[??](x)]}
where [x.sup.*] denotes the vector of optimal input amounts, [E.sub.w](*) is the expectation operator with respect to random variable [??], U(*) is the producer's utility function, and [??](x) is his/her end-of-period random wealth. The latter is defined to be the agent's initial wealth ([W.sub.0]) plus random profits from production,
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [??] is random end-of-period output price, [??](x) is random output, and r is the vector of variable input prices.
For present purposes, model (1)-(2) is too general to be operational. To be able to make headway from an empirical standpoint, it is necessary to be more specific about the utility function U(*), the technology [??](x), and the nature of randomness in price and output. Such issues are addressed in the following subsections.
The Decision Maker's Utility Function
The producer is assumed to be characterized by the hyperbolic absolute risk aversion (HARA) utility function,
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
which is defined on the domain of W satisfying ([[gamma].sub.0] + W) > 0. The negative of parameter [[gamma].sub.0] represents the agent's lowest admissible wealth. Parameter [[gamma].sub.1] is the agent's "baseline" risk aversion (Cerny 2004), and must be strictly positive if (3) is to represent risk-averse preferences. HARA utility is adopted here because it comprises the most popular functional forms used in expected utility analysis (i.e., the exponential, quadratic, and power utilities) (Gollier 2001).
Importantly, quite different structures of risk aversion can be obtained under appropriate parameterizations of (3). To see this, note that the HARA coefficient of relative risk aversion is
(4) R(W) = [[gamma].sub.1] W[([[gamma].sub.0] + W).sup.-1]
so that [partial derivative]R(W)/[partial derivative]W = [[gamma].sub.1][[gamma]sub.0][([[gamma].sub.0] + W).sup.-2]. Because the sign of [partial derivative]R(W)/[partial derivative]W equals the sign of [[gamma].sub.0], the HARA agent is characterized by decreasing, constant, or increasing relative risk aversion (DRRA, CRRA, and IRRA) if and only if parameter [[gamma].sub.0] is negative, zero, or positive, respectively. (1) Furthermore, as shown later, it is straightforward to parameterize (3) so as to test Cerny's (2004) claim that optimal decisions are essentially the same, regardless of whether the agent's utility is characterized by DRRA, CRRA, or IRRA, except for decisions involving very large and skewed risks. More specifically, Cerny (2004) labels R(W) as the agent's "local" relative risk aversion, and argues that the key determinant for optimal portfolio decisions is R([W.sub.0]) (i.e., the local risk aversion evaluated at the "safe" wealth level [W.sub.0]). Loosely speaking, this means that individuals characterized by similar values of R([W.sub.0]) will behave similarly toward risk.
Production Technology
The production technology [??](x) is assumed to be Cobb-Douglas
(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [[alpha].sub.0], [[alpha].sub.A], and [[alpha].sub.B] are technology parameters and [[??].sub.y] is a random variable whose distribution is discussed in the next section. Major reasons for adopting technology (5) are its simplicity and the fact that the Cobb-Douglas technology is arguably the most widely used production function in economic analysis. Examples of studies employing two variable inputs are Saha, Shumway, and Talpaz (1994) and Saha (1997), who analyzed wheat farms in Kansas with capital and materials as inputs, and Kumbhakar and Tveteras (2003), who studied salmon farms in Norway using feed and labor as inputs. (2) More complex technologies, or a Cobb-Douglas production function involving additional inputs, would require the estimation of additional technology parameters, thereby posing stronger challenges for the estimation of utility parameters [[gamma].sub.0] and [[gamma].sub.1].
The present analysis was also performed using the Just-Pope production function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Just and Pope 1978) instead of (5), parameterized with [[alpha].sub.0] = 3, [[alpha].sub.A] = 0.2, [[alpha].sub.B] = 0.6, [[beta].sub.A] = 0.06, and [[beta].sub.B] = -0.03 following Saha, Shumway, and Talpaz (1994), and with log([[??].sub.y]) distributed as described in Lence (2009). (Note that [[beta].sub.B] < 0 means that input B is risk reducing.) To save space, results for the Just-Pope specification are omitted, as they led to the same conclusions as the results for the simpler production function (5). Importantly, technology (5) also seems better suited than the Just-Pope setup for the present purposes. This is true because the latter is more complex and therefore more prone to the critique that it hinders the chances for risk preference identification. (3)
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