# Locationally Varying Production Technology and Productivity: The Case of Norwegian Farming

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## Abstract

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## 1. Introduction

“Spatial dimensions of input groupings may be particularly important in agriculture because the inputs must be tailored to the heterogeneity of firm resources, which differ substantially by climate and land quality (location).”

## 2. Locational Heterogeneity in Production

## 3. Methodology

#### 3.1. Proxy Variable Identification

**First step.**We first identify the material elasticity function ${\beta}_{M}\left({s}_{i}\right)$. To achieve this, we investigate the optimality condition of the firm regarding ${M}_{it}$ in Equation (4), which can be expressed in logarithmic form as:

**Second step.**To identify the remaining parameters of the production function, including latent firm productivity, we employ (7) to derive the explicit form of the conditional demand function for ${M}_{it}$. We then invert this function to serve as a proxy for the unobservable scalar ${\omega}_{it}$. In other words, by utilizing the inverted (log) material function ${\omega}_{it}=ln[{P}_{t}^{M}/{P}_{t}^{Y}]-{\beta}_{K}\left({s}_{i}\right){k}_{it}-{\beta}_{L}\left({s}_{i}\right){l}_{it}-{\beta}_{N}\left({s}_{i}\right){n}_{it}-ln\left[{\beta}_{M}\left({s}_{i}\right)\theta \right]+[1-{\beta}_{M}\left({s}_{i}\right)]{m}_{it}$, we substitute ${\omega}_{it-1}$ into Equation (12), to obtain:

#### 3.2. Semiparametric Estimation

**Inference.**Due to the multistep nature of our estimator and the presence of nonparametric components, calculating the asymptotic variance of the estimators is not straightforward.3 Therefore, we employ a bootstrap approach for statistical inference. Specifically, we utilize Efron’s (1987) bias-corrected bootstrap percentile confidence intervals, which correct for the finite-sample bias of the estimators. To approximate the sampling distributions of the estimators, we employ a wild residual block bootstrap method that takes into account the panel structure of the data. We perform the bootstrap resampling jointly for both stages, since the estimation in the second stage relies on the first-stage estimator. We performed $B=999$ bootstrap iterations to ensure reliable results.4

**Testing of Location Invariance.**The traditional fixed-parameter specification assumes that both the production function and the productivity evolution are invariant across locations. This fixed-coefficient assumption represents a nested and special case within our semiparametric spatially varying model. To formally assess whether our model is compatible with the fixed-coefficient alternative, we employ Ullah’s (1985) nonparametric goodness-of-fit test. This test involves comparing the restricted parametric model with the unrestricted semiparametric model. The null hypothesis suggests that the restricted parametric model, which assumes location-invariant coefficients, adequately fits the data. On the other hand, the alternative hypothesis posits that the unrestricted semiparametric model, which allows for location-varying coefficients, provides a good fit to the data. The residual-based test statistic is ${T}_{n}=(RS{S}_{0}-RS{S}_{1})/RS{S}_{1}$, where $RS{S}_{0}={\sum}_{i}{\sum}_{t}{\left(\tilde{{\zeta}_{it}+{\eta}_{it}}\right)}^{2}$ and $RS{S}_{1}={\sum}_{i}{\sum}_{t}{\left(\widehat{{\zeta}_{it}+{\eta}_{it}}\right)}^{2}$ are the sum of squared residuals under the null and (unrestricted semiparametric) alternative, respectively. The test statistic is expected to converge to zero under the null and to be positive under the alternative. The null distribution of the test statistic follows a chi-square distribution.

## 4. Data

## 5. Results

#### 5.1. Production Function

#### 5.2. Productivity Process

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | As noted by Just and Pope (2001), while the agricultural marketing literature frequently considers temporal and spatial distinctions, these aspects are often disregarded in the agricultural production economics literature. |

2 | The firm’s location ${s}_{i}$ is suppressed in the list of state variables because of its time-invariance. |

3 | The asymptotic property of the estimator is well-documented in the literature. For example, Li et al. (2002) proposed a local least squares method with a kernel weight function to estimate the smooth coefficient function (similar to what we do in the paper) and established the consistency of the estimator and its asymptotic normality. |

4 | Malikov et al. (2022) investigated the performance of the proposed bootstrap procedure in Monte Carlo simulations. Their simulations show satisfactory performance of the bootstrap confidence intervals in finite samples. |

5 | In this paper, we used a data-driven leave-one-location-out cross-validation method to choose the optimal bandwidth. These selected optimal bandwidths are capable of adapting to the local distribution of the data and yield the smallest sum of squared residuals. We also tried using fixed bandwidths, but the results remained robust. These additional results are available upon request. |

6 | The standard deviations of longitude and latitude in our sample are 0.6941 and 1.5162 decimal degrees, respectively. |

7 | As examples of exceptions that include spatial heterogeneity in their analysis of agricultural production, we mention Billé et al. (2018), Canello and Vidoli (2020), and Bai et al. (2021). Recently, several efficiency studies dealing with spatial aspects of agricultural production have also emerged, e.g., Fusco and Vidoli (2013) and Vidoli et al. (2016). |

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**Figure 1.**Spatial distribution of farms in the sample. The map covers the southern part of Norway, i.e., no grain farms in the sample in the northern part of Norway.

Variable Name | Var. | Mean | First Quartile | Median | Third Quartile |
---|---|---|---|---|---|

Production function variables | |||||

Output | Y | 468,669.14 | 218,259.31 | 340,533.00 | 588,818.06 |

Capital | K | 1,957,916.73 | 901,384.75 | 1,545,676.80 | 2,586,060.80 |

Labor | L | 853.70 | 400.00 | 700.00 | 1100.00 |

Land | N | 35.23 | 19.20 | 29.00 | 41.00 |

Materials | M | 188,155.60 | 92,480.62 | 145,776.42 | 228,719.84 |

Productivity determinants | |||||

Subsidy/return ratio | ${X}_{1}$ | 0.30 | 0.22 | 0.28 | 0.36 |

Off-farm income share | ${X}_{2}$ | 0.80 | 0.74 | 0.86 | 0.92 |

Debt/asset ratio | ${X}_{3}$ | 0.46 | 0.30 | 0.50 | 0.64 |

Location variables | |||||

Longitude | ${s}_{1}$ | 10.77 | 10.22 | 10.89 | 11.33 |

Latitude | ${s}_{2}$ | 60.55 | 59.51 | 60.01 | 60.81 |

Locationally Varying | Location-Invariant | ||||
---|---|---|---|---|---|

Mean | 1st Qu. | Median | 3rd Qu. | Point Estimate | |

Capital | 0.183 | 0.119 | 0.201 | 0.237 | 0.139 |

(0.109, 0.291) | (0.020, 0.246) | (0.125, 0.320) | (0.140, 0.390) | (0.095, 0.185) | |

Labor | 0.079 | 0.062 | 0.090 | 0.130 | 0.110 |

(0.041, 0.140) | (0.020, 0.140) | (0.026, 0.166) | (0.070, 0.231) | (0.051, 0.173) | |

Land | 0.263 | 0.214 | 0.243 | 0.264 | 0.447 |

(0.036, 0.373) | (0.021, 0.396) | (0.009, 0.359) | (0.031, 0.316) | (0.364, 0.521) | |

Materials | 0.38 | 0.367 | 0.38 | 0.398 | 0.378 |

(0.371, 0.395) | (0.35, 0.386) | (0.367, 0.399) | (0.392, 0.416) | (0.360, 0.398) |

Mean | 1st Qu. | Median | 3rd Qu. | <1 | =1 | >1 | |
---|---|---|---|---|---|---|---|

RTS | 0.926 | 0.833 | 0.911 | 1.011 | 26.29 | 74.86 | 4.57 |

(0.762, 1.110) | (0.655, 0.995) | (0.700, 1.079) | (0.830, 1.206) |

Locationally Varying | Location-Invariant | ||||||
---|---|---|---|---|---|---|---|

Variables | Mean | 1st Qu. | Median | 3rd Qu. | >0 | <0 | Point Estimate |

Lagged | 0.789 | 0.734 | 0.835 | 0.877 | 100 | 0 | 0.088 |

productivity | (0.522, 0.976) | (0.609, 0.983) | (0.592, 1.057) | (0.471, 1.078) | (−0.037, 0.200) | ||

Subsidy/return | −0.640 | −0.803 | −0.688 | −0.475 | 0 | 73.71 | −0.789 |

ratio | (−0.714, −0.301) | (−0.886, −0.502) | (−0.775, −0.411) | (−0.593, −0.043) | (–1.086, −0.517) | ||

Off-farm | −0.098 | −0.217 | −0.085 | 0.057 | 1.14 | 32 | 0.070 |

income share | (−0.307, 0.012) | (−0.431, −0.022) | (−0.296, 0.000) | (−0.198, 0.150) | (−0.089, 0.303) | ||

Debt/asset | −0.222 | −0.292 | −0.216 | −0.104 | 0 | 63.43 | −0.143 |

ratio | (−0.334, −0.072) | (−0.403, −0.109) | (−0.342, −0.079) | (−0.264, 0.019) | (−0.264, −0.013) |

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## Share and Cite

**MDPI and ACS Style**

Kumbhakar, S.C.; Zhang, J.; Lien, G.
Locationally Varying Production Technology and Productivity: The Case of Norwegian Farming. *Econometrics* **2023**, *11*, 20.
https://doi.org/10.3390/econometrics11030020

**AMA Style**

Kumbhakar SC, Zhang J, Lien G.
Locationally Varying Production Technology and Productivity: The Case of Norwegian Farming. *Econometrics*. 2023; 11(3):20.
https://doi.org/10.3390/econometrics11030020

**Chicago/Turabian Style**

Kumbhakar, Subal C., Jingfang Zhang, and Gudbrand Lien.
2023. "Locationally Varying Production Technology and Productivity: The Case of Norwegian Farming" *Econometrics* 11, no. 3: 20.
https://doi.org/10.3390/econometrics11030020