II. The Model

The California wine grape market is highly differentiated, with substantial regional variation in price, yield, and production practices. Prices are, on average, highest in Napa and Sonoma Counties (crush districts 3 and 4). The focus in these counties is on quality and yields are generally relatively low. In contrast, production in much of the Central Valley (crush districts 12–14) is focused on yield and prices are much lower. Therefore, we follow Fuller and Alston (Reference Fuller and Alston2012) and define a high-price region that includes crush districts 3 and 4, a low-price region that includes crush districts 12–14, and a medium-price region that contains the remaining crush districts. The California Department of Food and Agriculture maintains detailed production data at the level of crush districts. Thus, defining price regions in terms of crush districts allows us to calculate average prices and per-acre yields within the three price regions. We assume that the wine grape market is competitive within each of the price segments, and we model each of the three segments in a purely partial equilibrium setting to maintain an emphasis on model usefulness in applied settings.

A. Profit Maximization

The profit maximization problem for each producer is given by

(1)$$\max \; \pi = Pq(X,{\bf Z}) - xX - {\bf z} \cdot {\bf Z},$$

where P is the market price of the crop, q is the quantity harvested, X is the number of acres harvested in a given year, Z is a vector of the number of acres on which each method of pest control is used, x is the per-acre production costs excluding pest control costs, and z is a vector of the per-acre cost of each method. Because we assume that the market is competitive, producers maximize profit by choosing X and Z but must take price as a given. First-order conditions for the maximization problem are

(2)$$\displaystyle{{\partial \pi} \over {\partial x}} = P\displaystyle{{\partial q} \over {\partial x}} - x = 0\quad {\rm and}\quad \displaystyle{{\partial \pi} \over {\partial z_j}} = P\displaystyle{{\partial q} \over {\partial z_j}} - z_j = 0{\rm \;} \forall {\rm \;} j,$$

where j denotes a specific pest control method. Equation 2 implies that producers will apply pest control to an acre as long as the additional revenue earned by doing so is greater than the cost of application. Input demand functions are then given by

(3)$$X_1^{\rm *} = X(P,x,{\bf z})\quad {\rm and}\quad {\rm} {\bf Z}^{\rm *} = {\bf Z}(P,x,{\bf z}),$$

where X* ₁ and Z* are the optimal quantities of acres and pest control under current regulations.

Supply functions for individual producers are expressed as

(4)$$q_1^* = q_1 (X_1^*, {\bf Z}^* ) = q_1 (P,x,{\bf z}) = a + bP.$$

While the quantity produced is fundamentally a function of acreage and pest control use, for now we specify supply as a linear function of price and let the values of a and b reflect some particular level of x and z.

B. Regional Differences in Damage or Control

We assume that there are m+n producers that are differentiated due to the n California growers hypothetically altering pest control practices while the m non-California growers maintain current practices. Specifically, we assume that the California growers engage in no control. In Scenario 1, the California growers no longer face any threat of damage, and the result is that per-acre yield increases; in another scenario, the threat remains, and damage increases when growers do not manage bird damage. In both scenarios, the effect of no control is to restrict all Z _j to zero for the n California producers, which has two effects on those producers' marginal cost functions. The marginal cost function for each producer was

(5)$$MC_1 = {{ - a}\over b} + {q\over b}$$

and for the n growers now becomes

(6)$$MC_2 = {{ - a}\over b}} + k + {q\over b}(1 + L)}$}},$$

where

(7)$$k = \displaystyle{{\mathop \sum \nolimits \Delta\, z} \over {yield\;per\;acre}}$$

and

(8)$$L = \displaystyle{{\% \; yield\;change\;relative\;to\;current\;yield} \over {100}}. $

Equation (6) can be obtained by adding the change in per-unit control costs (k) to equation (5), solving for q and multiplying the resulting expression by (1+L), and then solving for MC. Recall that z represents the per-acre cost of the pest control method. We are considering the elimination of all bird control, which implies that the numerator of equation (7) is simply the negative of what is currently being spent per acre on bird control. Current yield reflects current control efforts and current damage. Thus, L is positive when considering the case of no control and no damage (Scenario 1) and it is negative when considering the case of no control and the resulting increase in damage (Scenario 2).

Equation (6) implies that the change in marginal cost is given by

(9)$$\Delta MC = k - \displaystyle{{qL} \over {b(1 + L)}}.$$

Thus, the change in marginal cost is the combined impact of two different effects: (1) a parallel shift of the original curve given determined by k and (2) a change in slope due to a change in the quantity of crop lost as production varies.

C. Market Supply

Market supply before the change is the horizontal summation of the (m+n) individual supply functions and is observed as Q _s=α+βP, where

(10)$$\alpha = a(m + n)\quad {\rm and}\quad \beta = b(m + n).$$

When the n growers hypothetically do nothing to control bird damage, the market supply function becomes

(11)$$Q_s = [n(a + bP - bk)(1 + L)] + [m(a + bP)],$$

where the two terms in brackets represent the aggregate supply functions for the n producers and m producers, respectively. It is useful to rewrite equation (11) in terms of the fraction of producers to which the change in pest control practices applies. Manipulating the relationships in equation (10) and substituting the results into equation (11), we obtain

(12)$$Q_s = f_n (\alpha + \beta P - \beta k)(1 + L) + f_m (\alpha + \beta P),$

where $f_n = \left( {\displaystyle{n \over {n + m}}} \right)$ and $f_m = \left( {\displaystyle{m \over {n + m}}} \right)$.

Then let market demand be given by Q _d=δ+γP, so that a change in pest control practices and damage reflected in (12) yields a new equilibrium given by

(13)$$P_2 = \displaystyle{{\delta - f_n \alpha + f_n \beta k - f_n \alpha L + f_n \beta kL - f_m \alpha} \over {\,f_n \beta + f_n \beta L + f_m \beta - \gamma}}. $$

D. Welfare Impacts

The market-wide change in consumer surplus is given by

(14)$$\Delta {\rm CS} = \mathop \int \limits_{P_2} ^{P_1} (\delta + \gamma P)dP.$$

It is useful to estimate the change in producer surplus for the n and m producers separately. For the n producers, the change is calculated as

(15)$$\Delta PS_n = \mathop \int \limits_{{\raise0.7ex\hbox{${ - \alpha} $} \!\mathord{\left/ {\vphantom {{ - \alpha} \beta}} \right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\beta $}} + k}^{P_2} (\,f_n \alpha + f_n \beta P - f_n \beta k)(1 + L)dP - \mathop \int \limits_{{\raise0.7ex\hbox{${ - \alpha} $} \!\mathord{\left/ {\vphantom {{ - \alpha} \beta}} \right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\beta $}}}^{P_1} (\,f_n \alpha + f_n \beta P)dP.$$

For the m producers, the change in surplus is given by

(16)$$\Delta PS_m = \mathop \int \limits_{P_1} ^{P_2} (\,f_m \alpha + f_m \beta P)dP.$$

E. Model Parameterization

Because of the linearity assumptions, deriving the initial supply and demand curves is straightforward, given that the price elasticities of supply and demand are known and the initial equilibrium is observed. Parameters in the initial market supply function can be estimated by

(17)$$\beta = \displaystyle{{E_s Q_1} \over {P_1}} \quad {\rm and}\quad \; \alpha = Q_1 - \beta P_1, $$

where E _s is price elasticity of supply. Parameters in the market demand function can be derived in a similar fashion.

Estimation of the new supply curve requires additional information. The model assumes that all producers (CA and non-CA) are identical, but in reality the scale of operations may vary considerably by region. As a result, although in theory f _n represents the fraction of all producers that alter pest control practices (the California producers), in practice it is desirable to estimate it based on the fraction of production that occurs within California. Estimates of yield per acre, the percentage of yield lost as a result of the pest control change, and the per-acre cost of the pest control method no longer being used are also required. The first of these is readily available from various sources, while the latter two must be estimated for the specific application.

III. Data

Much of the data used in this analysis was based on a survey of growers in California. We chose to rely on a survey rather than field studies for several reasons. First, field studies would have been impractical given that we needed data from across the state. Damage and control methods are likely to vary even within regions, and it would be difficult to implement a sufficient number of field studies to capture these differences. Growers also use many different combinations of control methods, and it would have been difficult to capture the variety of combinations in field studies. Furthermore, damage is likely to vary by year, and it is possible that growers' perceptions are based on their experiences over a number of recent years. This might provide us with data that are less subject to year-to-year variability than data from a field study. We do, however, recognize that survey data are subject to bias by growers and the survey design. Unfortunately, few previous studies have addressed the potential for bias in agricultural pest damage studies. A notable exception is Tzilkowski et al. (Reference Tzilkowski, Brittingham and Lovallo2002), who implement both a survey and field study to assess damage in corn production. The authors could not reject the hypothesis that the average damage estimates from the survey and field studies were equal. Additionally, our own study is part of a larger effort that also assesses the effectiveness of individual control methods using field studies. Although it is not possible to compare the field study results with the survey results directly, preliminary field study results do suggest that the estimates of damage from the survey are quite reasonable.

The survey instrument consisted of 21 questions that solicited information about the location and size of the grower's farm, growers' level of fruit production experience, acreage and yield data, bird damage, bird management methods, and estimated costs for bird damage management.Footnote ¹ All survey mailings were completed between March 5 and May 1, 2012, and the sample size was 1,319. Respondents received up to three mailings (i.e., an initial letter and questionnaire, a reminder letter, and a final reminder letter and replacement questionnaire within two weeks of the follow-up reminder letter). To encourage survey response, all mail survey implementation incorporated several characteristics of the Dillman (Reference Dillman2000) Total Design Method, including a brief, respondent-friendly questionnaire, multiple contacts by first-class mail, and cover letter elements that personalized correspondence. Non-response follow up was not possible because we were not given access to the necessary contact information.

We received a total of 237 completed questionnaires from wine grape growers.Footnote ² Responding growers were located in 29 different counties, with the most responses coming from San Joaquin and Sonoma Counties. All responses from San Joaquin were included in both the low-price and medium-price regions because crush districts 11 and 12 both overlap the county. Total responses for the three price regions were 63 (low price), 155 (medium price), and 69 (high price). Responding growers had been farming an average of about 30 years (low-price average experience=35.6, SD=18; medium-price=29.9, SD=20; high-price=29.7, SD=15) and average wine grape acreage was 307 (low-price average acreage=510, SD=967; medium-price=365, SD=818; high-price=69, SD=326).Footnote ³ Statewide, 75% of responding growers considered farming their primary occupation and 94% considered wine grapes their most important fruit crops. About 38% of the growers believed bird damage has a significant effect on their profits, 21% believed damage increased in the five years prior to the survey, and 68% believed that it remained stable over that time period. Growers were also asked to name the top-three most damaging bird species, and starlings were the most frequently named in all three price regions (Table 1).

Table 1 Top Five Most Commonly Named Species Causing Damage by Region

Current damage levels and control costs in each region reflect the use of the bird damage control methods that appear in Table 2. Statewide, a majority of respondents used auditory and visual scare devices and nearly half used netting. However, nearly one-third of respondents did nothing to limit bird damage. There were considerable differences in control methods across price regions. The use of chemical repellants appears to decrease with price, and, as expected, the use of netting increases with price. In general, growers in the medium-price region are considerably more likely to use some type of bird control.

Table 2 Bird Control Methods Used in Wine Grape Production with Percent of Respondents Using by Region (in %)

Survey responses indicate that statewide average bird control costs per acre were $15.94, and current damage is about 1.77% of per-acre yields, but considerable variation exists between regions (Table 3).Footnote ⁴ Damage levels any without control and without regional control were derived from responses to two separate questions. The first asked growers to forecast damage levels if they engaged in no bird control on their property, but other growers continued to use current practices; the other asked growers to forecast damage if no one in their region attempted to control bird damage. One of the objectives of this study was to understand the impact of bird control at a regional level, but it seems unlikely that the effects of no regional control could be estimated with any certainty. This motivated the use of two different questions to solicit this information, and our quantitative analyses rely on the midpoints of the resulting loss estimates within each price region. While the results of the two questions were quite similar within each region (unreported tests failed to reject the hypotheses of equality within regions), there were interesting differences in responses across the three regions.

Table 3 Acreage-weighted Mean Per-acre Yield Loss and Control Cost Estimates from Survey of California Growers (in %)

Growers in the low- and high-price regions expected that damage without any regional control would be worse than if they alone did nothing to control birds. This indicates that these growers believe bird damage control produces an external net benefit. That is, growers benefit from control by neighboring growers, perhaps through impacts on bird populations in the region. Conversely, growers in the medium-price region expected that lack of regional control would result in less damage than if they alone did not control birds. This has the opposite implication of an external net cost being associated with bird control. It seems plausible that growers in regions that rely heavily on lethal control methods would realize a benefit of control by other growers and growers in regions that rely heavily on exclusion and scare tactics would be harmed by control by other growers.

Examination of the types and intensity of control methods used in the three regions reveals several factors that might explain or the differences in expectations about damage without control. Although netting is implemented less frequently in the medium-price region than in the high-price region, medium-price growers are most likely to use auditory and visual scare devices and predator nest boxes. The overall high level of netting and use of scare tactics in the medium-price region supports the observation by medium-price growers that bird control involves an external net cost because these methods of control tend to displace birds to other properties, where they are not being used.Footnote ⁵ The use of scare tactics in the medium-price region may be an important explanation for another reason. Not only do scare tactics used by other growers displace birds to other properties, but their frequent and widespread use may habituate birds to their use and render them less effective. Finally, the expectation of an external net cost rather than an external net benefit in the medium-price region is consistent with the fact that, compared to growers in the other regions, they are much more likely to engage in some type of bird control. If growers in the other regions believe that they benefit from their neighbors' control efforts, their own incentive to engage in control decreases.

Estimates of California wine grape production and prices within each of the three price regions were based on the 2012 California grape crush report (CDFA, 2013b). Prices were calculated as the production-weighted average of the crush districts that made up each price region. Yield per acre in each region was calculated based on production estimates from the crush report and estimates of bearing acreage reported by CDFA (2013a). Table 4 summarizes the price and production estimates used in the analyses. We assumed that 90% of U.S. wine grape production occurs within California based on information from the Wine Grape Growers of America (n.d.) and the Wine Institute (2012). Data on the percentage of U.S. production of low-, medium-, and high-priced grapes produced within California were unavailable, so we assumed that 90% of the production of each of these categories occurs in California.

Table 4 Price, Production, and Yield Estimates in 2012 by Price Region in California

Sources: CDFA 2013a, 2013b.

Price elasticities of demand were assumed to be −2.6, −5.2, and −9.5 in the low-, medium-, and high-price regions, respectively, as estimated by Fuller and Alston (Reference Fuller and Alston2012). Price elasticities of supply were not readily available for the regions we used. We constructed estimates of supply elasticities by first taking the production-weighted average of supply elasticities for eight major varieties of wine grapes in four regions of California that were estimated by Volpe et al. (Reference Volpe, Green, Heien and Howitt2010). The four regions used by Volpe et al. (Reference Volpe, Green, Heien and Howitt2010) were the North Coast, Central Coast, North Central Valley, and South Central Valley. The North Coast region is roughly comparable to our high-price region, and we assume that the production-weighted average supply elasticity from the North Coast region applies in the high-price region. Similarly, the South Central Valley is roughly comparable to our low-price region, and we assume that the same supply elasticity applies in both. Our medium-price region is roughly equivalent to the combined Central Coast and North Central Valley regions, and we assume the average of supply elasticities from these two smaller regions applies in our medium-price region. The resulting supply elasticities in the low, medium, and high-price regions were 0.21, 0.45, and 0.77, respectively.

To account for uncertainty associated with the data and parameter estimates, Monte Carlo simulations were used to calculate all results and examine how uncertainty about parameter estimates affects the results. With one exception, we assumed that each parameter follows a triangular distribution, with minimum and maximum values set at ±50% of the point estimates.Footnote ⁶ Because the point estimate of the fraction of production that occurs in California was 0.90, we assumed that it was triangular, distributed with minimum and maximum values set at ±10% of 0.90. Random draws from a triangular distribution were simulated by generating a random number on the uniform interval [0,1] and evaluating the inverse of the triangular cumulative distribution function at that number. Each iteration of a simulation consisted of random draw of each parameter and the full set of model results based on those parameter values. Six simulations were performed. In each of the three regions, the impact of the elimination of the threat of bird damage (Scenario 1) and the impact of the elimination of bird control (Scenario 2) were simulated. Each of these simulations consisted of 100,000 iterations to sufficiently characterize the mean results.