Modelling pasture-based beef production costs using panel data from farms with different soil quality

Abstract

The objective of this paper is to analyse the cost structure of pasture-based beef production in Ireland. Specifically, the paper assesses (i) farmers’ capacity to respond to price changes by substituting inputs; (ii) the optimality of the scale of production; and (iii) the optimal utilization of land. As differences in soil quality may alter the size of utilized land and affect farmers’ dependency on purchased or home-produced feed, a short-run translog cost model was estimated separately for three groups of farms with differing soil quality. For empirical implementation, Irish beef farm data from 2000 to 2011, obtained from the Teagasc National Farm Survey (NFS), were used. Results suggest that substitution possibilities are limited in Irish beef farms. The lack or limited substitution possibilities between types of feed suggests that beef farms are vulnerable to feed price increases. We find statistically significant evidence of allocative inefficiency irrespective of the quality of soil, which takes the form of over-utilization of purchased cattle relative to other inputs. Moreover, cost advantages can also be achieved if Irish farmers decrease beef production. Given that increases in methane emissions from higher cattle numbers could jeopardise the achievement of climate neutrality by 2050, the implications of results are of interest to agricultural and broader policy makers, industry stakeholders and society as a whole.

Appendix

Appendix A. Estimated dual cost function parameters

Tables 5, 6, 7, 8, 9, 10, 11, 12 and 13

Table 5 Parameter estimates (farms with good soil quality, n = 3,653)

Table 6 Parameter estimates accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (farms with good soil quality, n = 3,652)

Table 7 Parameter estimates accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (farms with good soil quality, n = 3,652)

Table 8 Parameter estimates (farms with average soil quality, n = 2,752)

Table 9 Parameter estimates accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (farms with average soil quality, n = 2,747)

Table 10 Parameter estimates accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (farms with average soil quality, n = 2,746)

Table 11 Parameter estimates (farms with poor soil quality, n = 621)

Table 12 Parameter estimates accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (farms with poor soil quality, n = 621)

Table 13 Parameter estimates accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (farms with poor soil quality, n = 621)

Appendix B. Estimated input distance function parameters

Tables 14, 15, 16, 17, 18, 19, 20, 21 and 22

Table 14 Parameter estimates (farms with good soil quality, n = 2,424)

Table 15 Parameter estimates accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (farms with good soil quality, n = 2,424)

Table 16 Parameter estimates accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (farms with good soil quality, n = 2,424)

Table 17 Parameter estimates (farms with average soil quality, n = 1,680)

Table 18 Parameter estimates accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (farms with average soil quality, n = 1,680)

Table 19 Parameter estimates accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (farms with average soil quality, n = 1,680)

Table 20 Parameter estimates (farms with poor soil quality, n = 356)

Table 21 Parameter estimates accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (farms with poor soil quality, n = 356)

Table 22 Parameter estimates accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (farms with poor soil quality, n = 356)

Appendix C. Tests on properties of technology based on dual approach (system of Eqs. 8 and 9)

Table 23

Table 23 Restrictions on production technology

Appendix D. Allocative (In)efficiency

Table 24

Table 24 Estimated coefficients κ_ij at sample means (standard errors in parentheses) accounting for risk preferences

Appendix E. Estimated own and cross-price elasticities based on dual approach (system of Eqs. 8 and 9) and evaluated at sample means

Table 25

Table 25 Own- and cross-price elasticities (standard errors in parentheses)

Appendix F. Estimated Morishima elasticities based on dual approach (system of Eqs. 8 and 9) and evaluated at sample means

Tables 26, 27 and 28

Table 26 Morishima elasticities of substitution (standard errors in parentheses)

Table 27 Morishima elasticities of substitution accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (standard errors in parentheses)

Table 28 Morishima elasticities of substitution accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (standard errors in parentheses)

Appendix G. Estimated indirect Morishima elasticities based on primal approach (system of Eqs. 34 and 35) and evaluated at sample means

Tables 29 and 30

Table 29 Indirect Morishima elasticities of substitution accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (standard errors in parentheses)

Table 30 Indirect Morishima elasticities of substitution accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (standard errors in parentheses)

Appendix H. Estimated own- and cross-price elasticities based on primal approach and evaluated at sample means

Tables 31, 32, 33 and 34

Table 31 Estimated own-price elasticities (standard errors in parentheses)

Table 32 Own- and cross-price elasticities (standard errors in parentheses)

Table 33 Own- and cross-price elasticities accounting for farmer risk preferences by using the adjusted Herfindahl index as proxy variable (standard errors in parentheses)

Table 34 Own- and cross-price elasticities accounting for farmer risk preferences by using the ratio of total expenditures on insurance over total farm expenses as proxy variable (standard errors in parentheses)

Appendix I. Density histograms

Figures 3 and 4

Fig. 3

Density histogram of time trend across soil groups

Fig. 4

Density histogram of beef system across soil groups (d1)

Appendix J. The input distance function

To define an input distance function we begin by defining the production technology of a farm using the input set L(Q), which represents all input vectors N, $X \in {R}_{+}^{N}$, which can produce output Q:

$$L\left(Q\right)=\left\{{\varvec{X}} \in {R}_{+}^{N}:x\; can\; produce\; Q\}\right..$$

(21)

A short-run primal input distance function dual of the variable cost function (Eq. 2) is then defined on the input set, L(Q), as:

$$D\left({\varvec{X}}, Q\right)=\mathrm{max }\left\{\rho :\left(\frac{{\varvec{X}}}{\rho }\right)\in L\left(Q\right)\right\}.$$

(22)

The input distance function, D(X, Q), measures the largest factor of proportionality ρ, by which the input vector X can be scaled down to produce output Q. The distance function takes a value no smaller than unity, while a value of unity indicates technical efficiency. The shadow cost function can be derived by solving the problem of shadow cost-minimization subject to the input distance function constraint:

$$C(Q, {{\varvec{w}}}^{{\varvec{s}}})={}_{{\varvec{X}}}{}^{{\varvec{m}}{\varvec{i}}{\varvec{n}}}\{{{\varvec{w}}}^{{\varvec{s}}}{\varvec{X}}:D({\varvec{X}}, Q)\ge 1\}.$$

(23)

According to Eq. 23, farmers are assumed to minimize the shadow cost of producing a given level of output, Q, for some shadow input price vector w^s. Although both functions fully describe technology (Eqs. 2 and 23), the distance function has the advantage of requiring not assumptions of cost minimization behaviour or variable input price exogeneity (Baños-Pino et al., 1999).

By applying the dual Shephard’s lemma

$${D}_{i}\left(.\right)=\frac{\partial D(.)}{\partial {X}_{i}}=\frac{{w}_{i}^{s}(.)}{C(.)}$$

(24)

to the shadow cost function (Eq. 23), the following shadow price ratios are derived (Färe & Primont, 1995):

$$\frac{\partial {\varvec{D}}(.)/\partial {{\varvec{X}}}_{{\varvec{i}}}}{\partial {\varvec{D}}(.)/\partial {{\varvec{X}}}_{{\varvec{j}}}}=\frac{{w}_{i}^{s}(.)}{{w}_{j}^{s}(.)},$$

(25)

where D_i (.) is the first derivative of D(.) and is interpreted as the marginal product (shadow price) of the i^th variable input. If the cost-minimization assumption is valid, the normalized shadow price ratio (Eq. 25) should be equal to the input market price ratio, P_i /P_j. However, if farmers are allocatively inefficient, i.e. inputs i and j are not selected in the optimal proportion, the shadow and market price ratios will differ. To estimate the magnitude of such deviations, a relationship between the shadow and input market prices is introduced by means of a parametric price correction (Grosskopf et al., 1995):

$$\bigcup_a^b{\kappa_{ij}=\left(\frac{D_i\left(.\right)}{D_j\left(.\right)}\right)/\left(\frac{P_i}{P_j}\right)}.$$

(26)

When κ_ij ≠ 1, inputs i and j are not allocated efficiently in production with κ_ij > ( <) 1 implying that input i is under-utilized (over-utilized) relative to input j.

Baños-Pino et al. (1999) show what happens to the input distance function if the utilized level of the quasi-fixed input changes. Based on the envelope theorem and assuming that the quasi-fixed input L (land) varies, we have:

$$\frac{\partial D(.)}{\partial L}=\frac{\partial \mathcal{L}(.)}{\partial L},$$

(27)

with $\mathcal{L}$ being the Lagrangian associated with Eq. 23. From Eq. 27 we obtain:

$$\frac{\partial \mathcal{L}(.)}{\partial L}=\lambda \frac{\partial C(.)}{\partial L},$$

(28)

where λ is the Lagrange multiplier. The first-order condition corresponding to Eq. 23 is:

$$\frac{\partial \mathcal{L}(.)}{\partial {W}_{i}}={X}_{i}+\lambda \frac{\partial C(.)}{\partial {W}_{i}}=0.$$

(29)

Applying Shephard’s lemma, and assuming that D(.) = 1, the quantity of X_i and the quantity required to produce Q are the same, therefore λ = -1. We can then infer from Eqs. J7-8 that:

$$\frac{\partial D(.)}{\partial L}=-\frac{\partial C(.)}{\partial L}.$$

(30)

The Eq. 30 implies that the absolute value of $\frac{\partial D(.)}{\partial L}$ is equal to the shadow price of quasi-fixed input. Similar to Eq. 7 (CU measurement), the amount of land utilized is optimal, if the shadow price of land obtained by the estimated input distance function (${P}_{L-dist.}^{*}$) is equal to the market price of land (P_L). If land utilization is non-optimal, the degree of allocative inefficiency can be evaluated by using Eq. 31:

$${q}_{L}=\frac{{P}_{L-dist.}^{*}}{{P}_{L}}.$$

(31)

If q_L is greater than one, land is under-utilized, and vice versa.

In addition, it is possible to evaluate the degree of technical substitutability between variable inputs i and j by means of indirect Morishima elasticities of substitution. The indirect Morishima elasticity of substitution measures the degree of the relative change in the shadow prices required to induce substitution between X_i and X_j (Lee & Jin, 2012). The indirect Morishima elasticities are calculated as:

$${M}_{ij}=-\frac{dln\left(\frac{{D}_{i}\left(.\right)}{{D}_{j}\left(.\right)}\right)}{dln\left(\frac{{X}_{i}}{{X}_{j}}\right)}=\frac{{X}_{i}{D}_{ij}\left(.\right)}{{D}_{j}\left(.\right)}-\frac{{X}_{i}{D}_{ii}\left(.\right)}{{D}_{i}\left(.\right)}, i\ne j.$$

(32)

The indirect Morishima elasticity can be simplified as:

$${M}_{ij}={E}_{ij}\left(.\right)-{E}_{ii}(.),$$

(33)

where E_ij and E_ii are the cross- and own elasticities of (shadow) prices with respect to input quantities. The first component, E_ij, provides information on whether a pair of inputs are net complements or net substitutes, while the second component, E_ii, is the own-price elasticity of demand for the i^th input. Positive values of the indirect Morishima elasticities indicate complementarity between inputs, whereas negative values imply that inputs are substitutes (Rodríguez-Álvarez et al., 2007). Higher values of M_ij imply lower substitutability between inputs i and j. Note that the indirect Morishima elasticities of substitution are not symmetric (M_ij ≠ M_ji).

One of the difficulties in estimating a distance function is that D(.) is unknown. Thus, it is assumed that its value is equal to one (D(.) = 1) which implies the assumption of technical efficiency. Since technical efficiency affects only the intercept of the distance function, and the estimates of allocative inefficiency that are obtained by means of a translog input distance function are independent of the degree of technical efficiency, the assumption of technical efficiency will not affect the estimates of allocative inefficiency (Baños-Pino et al., 2002). In order to assess if the land is efficiently utilized by farmers and to calculate the indirect Morishima elasticities, a twice differentiable and flexible short-run translog input distance function is specified as:

$${\mathrm{ln}}\left({\mathrm{D}}\right)={\mathrm{ln}}(1)={\beta }_{0}+{\upbeta }_{{\mathrm{Q}}}{\mathrm{lnQ}}+{~}^{1}\!\left/ \!{~}_{2}\right.{\upbeta }_{{\mathrm{QQ}}}{\left({\mathrm{lnQ}}\right)}^{2}+\sum_{{\mathrm{i}}}{\upbeta }_{{\mathrm{i}}}{\mathrm{ln}}{\mathrm{\rm X}}_{{\mathrm{i}}}+{~}^{1}\!\left/ \!{~}_{2}\right.\sum_{{\mathrm{i}}}\sum_{{\mathrm{j}}}{\upbeta }_{{\mathrm{ij}}}\mathrm{ ln}{\mathrm{\rm X}}_{{\mathrm{i}}}\mathrm{ ln}{\mathrm{\rm X}}_{{\mathrm{j}}}+{\upbeta }_{{\mathrm{L}}}{\mathrm{lnL}}+{~}^{1}\!\left/ \!{~}_{2}\right.{\upbeta }_{{\mathrm{LL}}}{\left({\mathrm{lnL}}\right)}^{2}+\sum_{{\mathrm{i}}}{\upbeta }_{{\mathrm{iQ}}}{\mathrm{ln}}{\mathrm{\rm X}}_{{\mathrm{i}}}{\mathrm{lnQ}}+\sum_{{\mathrm{i}}}{\upbeta }_{{\mathrm{iL}}}{\mathrm{ln}}{\mathrm{\rm X}}_{{\mathrm{i}}}{\mathrm{lnL}}+{\upbeta }_{{\mathrm{LQ}}}{\mathrm{lnLlnQ}}+{\upbeta }_{{\mathrm{T}}}\mathrm{ T}+{\upbeta }_{\mathrm{T{\rm T}}}{\mathrm{\rm T}}^{2}+\sum_{{\mathrm{i}}}{\upbeta }_{\mathrm{\rm T}{\mathrm{i}}}\mathrm{\rm T}{\mathrm{ln}}{\mathrm{\rm X}}_{{\mathrm{i}}}{+\upbeta }_{{\mathrm{TQ}}}\mathrm{ TlnQ}+{\upbeta }_{{\mathrm{TL}}}\mathrm{ TlnL}+{\upbeta }_{{\mathrm{D}}}\mathrm{ D}+\sum_{{\mathrm{i}}}{\upbeta }_{{\mathrm{Di}}}{\mathrm{Dln}}{\mathrm{\rm X}}_{{\mathrm{i}}}+{\upbeta }_{{\mathrm{DQ}}}{\mathrm{DlnQ}}+ {\upbeta }_{{\mathrm{DL}}}{\mathrm{DlnL}}+{\upbeta }_{{\mathrm{DT}}}\mathrm{D T}+\varepsilon,$$

(34)

where ε is a composed error term, ε = u + υ. The first term u (u ≥ 0) captures technical inefficiency and it is assumed to be the absolute value of a normal random variable with zero mean and variance ${\sigma }_{u}^{2}$. The term υ is assumed to capture random noise. The random noise component is assumed to be symmetrically distributed with zero mean and variance ${\sigma }_{\upsilon }^{2}$. The error components u and υ are assumed to be independently distributed. A well-behaved input distance function satisfies, at sample means, the regularity conditions of being non-decreasing and quasi-concave in inputs (β_i’s, β_L > 0), and decreasing in output(s) (β_Q < 0) (Rodríguez-Álvarez et al., 2007).

To improve the precision of the estimated parameters, the input distance function is jointly estimated with the four input cost share equations:

$$\frac{{{\mathrm{P}}}_{{\mathrm{i}}}{{\mathrm{X}}}_{{\mathrm{i}}}}{{\mathrm{VC}}}=\frac{\partial {\mathrm{lnD}}(.)}{\partial {{\mathrm{lnX}}}_{{\mathrm{i}}}}=\frac{\partial {\mathrm{D}}(.)}{\partial {{\mathrm{X}}}_{{\mathrm{i}}}}\frac{{{\mathrm{X}}}_{{\mathrm{i}}}}{D(.)}={\upbeta }_{{\mathrm{i}}}+\sum_{{\mathrm{j}}}{\upbeta }_{{\mathrm{ij}}}{\mathrm{ln}}{\mathrm{\rm X}}_{{\mathrm{j}}}+{\upbeta }_{{\mathrm{iQ}}}{\mathrm{lnQ}}+{\upbeta }_{{\mathrm{iL}}}{\mathrm{lnL}}+{\upbeta }_{\mathrm{\rm T}{\mathrm{i}}}\mathrm{\rm T}+{\upbeta }_{{\mathrm{Di}}}{\mathrm{D}}+{\mu }_{i} (i=1,\dots ,4).$$

(35)

Under the assumption that allocative efficiency can be systematic and continuous in time, the error term μ_i (Eq. 35) has an additive structure of the form (Ferrier & Lovell, 1990; Rodrı́guez-Álvarez et al., 2004):

$${\mu }_{i}={\eta }_{i}+{A}_{i} (i=1,\dots ,4).$$

(36)

The independent and identically distributed (iid) random terms μ_i with zero mean and ${\sigma }_{\eta }^{2}$ variance, aim to capture the effects of random noise on efficient input shares. The terms A_i which can be positive or negative, capture persistent allocative inefficiency in the use of variable inputs.

The system of Eqs. 34 and 35 was estimated by applying the I3SLS procedure. The 2-year lagged values of all regressors except for trend, their squares, and their interactions were used as instruments to correct for potential endogeneity. Prior to econometric estimation, the appropriate restrictions on parameters were imposed to satisfy the conditions of symmetry (${\upbeta }_{{\mathrm{ij}}}={\upbeta }_{{\mathrm{ji}}}$) and homogeneity of degree one in inputs ($\sum_{{\mathrm{i}}=1}^{4}{\beta }_{{\mathrm{i}}}=1;\sum_{{\mathrm{i}}=1}^{4}{\upbeta }_{{\mathrm{ij}}}= \sum_{{\mathrm{i}}=1}^{4}{\upbeta }_{{\mathrm{iQ}}}= \sum_{{\mathrm{i}}=1}^{4}{\upbeta }_{{\mathrm{iL}}}=\sum_{{\mathrm{i}}=1}^{4}{\upbeta }_{\mathrm{\rm T}{\mathrm{i}}}=\sum_{{\mathrm{i}}=1}^{4}{\upbeta }_{{\mathrm{Di}}}=0$). Moreover, variables were normalized with their (arithmetic) sample mean values. Using the estimated parameters from the translog estimating equation, the relationship between the shadow and input market prices (Eq. 26) and the indirect Morishima elasticities of substitution (Eq. 32) take the following forms:

$${\kappa }_{ij}=\frac{{P}_{j}(\frac{1}{{X}_{i}})(\frac{\partial {\mathrm{lnD}}\left(.\right)}{\partial {{\mathrm{lnX}}}_{{\mathrm{i}}}})}{{P}_{i}(\frac{1}{{X}_{j}})(\frac{\partial {\mathrm{lnD}}\left(.\right)}{\partial {{\mathrm{lnX}}}_{{\mathrm{j}}}})},$$

(37)

$${M}_{ij}=\frac{{\beta }_{ij}}{{X}_{j}{D}_{j}(.)}-\frac{{\beta }_{ii}-{D}_{i}(.)}{{X}_{i}{D}_{i}(.)}.$$

(38)

To account for the effect of farmer risk preferences on the estimation of shadow factor prices, the system of Eqs. 34, 35 was also estimated by incorporating the adjusted Herfindahl index of farm activity diversification (H), and the ratio of total expenditures on insurance over total farm expenses (R) as proxy variables for farmer risk preferences.