Our proposed methodology is based on two main steps. First, we perform an econometric analysis of regional agricultural production functions. Secondly, we perform predictions of the main trends and variation in agricultural yields, due to (assumed) variations in the main drivers/production inputs and to selected climate change scenarios.
Agricultural regional production functions
We model agricultural production functions (both Cobb-Douglas and translog) in order to tackle and measure the marginal product of each factor used in the production of the selected, agricultural yields.Footnote 1 We use production functions that, in our setting, simply describe the technical relationship between the (physical) amount of agricultural inputs and the total agricultural yields, in a pure microeconomic fashion. The general empirical form of the agricultural regional production functions can be synthesized as in Eq. (1):
$$ {y}_i={x}_i^{\prime}\upbeta +{\varepsilon}_i, $$
(1)
where yi (yields) and xi (agricultural production inputs) are observable variables, and εi is unobserved and referred to as an error term or disturbance term.
We aim at estimating the technological efficiency through the measurement of each selected input marginal product, and therefore the marginal impact on agricultural production. In our framework production functions are regional, since technologies vary according to the selected socio-economic applications and they are also money independent, (no input/output price is included in the regression line). Finally, production functions are temperature and climate invariant since, by embodying Mendelshon et al. (1994) criticism, we do not “insert” variables like temperature and precipitation levels in the production function specification. The first step of analysis aims at getting a technological picture of the marginal product of each input used for producing selected agricultural output, given a determined production technology in a given period of time and in a given geographical region.
The economic literature presents a large number of theoretical and empirical production functions. In our suggested methodological framework, Eq. (1) is operationalized through the use of two types of production functions. The first one is a Cobb-Douglas, as described in Eq. (2). The regional agricultural total output q (in logs) depends on a log-linear combination of n production inputs and an error term.
$$ \ln\ q=\beta {(1)}_0+\sum \limits_{n=1}^N{\beta}_n\ln\ {x}_n+u. $$
(2)
The regional Cobb-Douglas empirical model is easy to estimate and interpret and requires estimation of a few parameters. Main disadvantages are the (stringent) assumptions that firms operate in a setting of perfect competition, with all firms having the same production elasticities (and that substitution elasticities equal 1).
The regional translog production function is modeled in Eq. (3). In this case, the regional agricultural total output q (in logs) depends on a log-linear combination of n production inputs, the m interactions of n production inputs, and an error term.
$$ \ln\;q={\beta}_0+\sum \limits_{n=1}^N{\beta}_n\;\ln\;{x}_n+\frac{1}{2}\;\sum \limits_{n=1}^N\sum \limits_{m=1}^N{\beta}_{nm}\;\ln\;{x}_n\ln\;{x}_m+u $$
(3)
The translog function is very commonly used in the literature, and it is a generalization of the Cobb-Douglas function. It is a flexible functional form providing a second-order approximation and less restriction on production elasticities and substitution elasticities. However, the translog model is more difficult to interpret and requires estimation of many parameters. In addition, it can suffer from curvature violations. Finally, both Cobb-Douglas and translog functions are linear in parameters and can be easily estimated.
Prediction methodology and assumptions
We attempt to perform prediction using selected production function-estimated coefficients in the regression model. We want to predict the value for the dependent variable (produced quantity/agricultural yields), given the potential climate change-induced variations of the two main drivers of agricultural productivity: land and labor, at a given value for the selected explanatory variables. Since the model is assumed to hold for all potential observations, it will also hold that:
$$ {y}_0={x}_0^{\prime }\ \upbeta +\varepsilon 0, $$
(4)
where ε0 satisfies the same properties as all other error terms. The obvious predictor for y0 is
$$ \overline{y}\ 0={x}_0^{\prime }\ b $$
(5)
As E {b} = β, it is easily verified that this is an unbiased predictor.Footnote 2
The prediction exercise requires some further qualitative speculations and assumptions, since in the future, the world (and every selected agricultural region) might have changed in ways that are difficult to imagine, as difficult as it would have been at the end of the nineteenth century to imagine the changes of the last 100 years. Nonetheless, our key assumptions that drive the variations of the \( {x}_0^{\prime } \) are the following:
- 1.
We assume that there is no technological shock in agricultural production, in terms of dramatic changes in machinery that adopt very different technologies with respect to the current ones, e.g., the drones.
- 2.
We assume a future with climate change. We assume that climate change will affect the agricultural sector in different ways, which are summarized by the IPCC-predicted scenarios (see IPCC; 2000, 2007). Each scenario is constructed according to a storyline. Each storyline assumes a distinctly different direction for future developments, such that the four storylines differ in increasingly irreversible ways. Together, they describe divergent futures that encompass a significant portion of the underlying uncertainties in the main driving forces. They cover a wide range of key “future” characteristics such as demographic change, economic development, and technological changes.
- 3.
According to the (probabilistic, future) occurrence of each scenario, we assume that land (cultivated hectares) and agricultural labor (labor force employed in the sector) will vary. The four main scenarios and storylines are summarized in Table 1.
The quantification of the value/percentage variations of \( {x}_0^{\prime } \) under different climate change scenarios is relatively discretionary and needs assistance from other experts. It is important to understand how the world will vary and how the scenarios will implement in order to define a range of possible values that \( {x}_0^{\prime } \) can assume, given the occurrence of each IPCC scenarios. Following Ansuategi et al. (2015), we organized focus groups and meetings with experts (for more clarity, see following sections for the applications) that clarify the possible effects of climate change-induced IPCC scenarios on the agricultural sector and the possible range of values that of \( {x}_0^{\prime } \) can assume under each scenario. For instance, given all above, it might (probabilistically) be very unlikely to occur that regional agriculture might spur and develop under a B1 scenario in a certain region. Therefore, land and labor will be used in smaller units, with a percentage variations that cover a range that researchers and scientists determine according to the state of the arts of knowledge on climate change and common sense.
Finally, on the methodological point of view, the different assumed values for \( {x}_0^{\prime } \) (according to the selected IPCC scenario) are then multiplied by b, the translog-estimated coefficients for land and labor marginal products (as in Eq. 5), in order to compute the climate change-induced variations in agricultural yields.