Farm and non-farm labor decisions and household efficiency

Abstract

This paper develops a theoretical framework for modeling farm households’ joint production and consumption decisions in the presence of technical inefficiency. Following Lopez (1984), a household model where farmers display different preferences between on-farm and off-farm labor is adopted while their production activity can be subject to technical inefficiency. The presence of technical inefficiency does not only lead to the inability of farmers to achieve maximal output but it will also affect the consumption allocation and the household’s labor supply decisions through its effect on both income and on the shadow price of on-farm labor, leading to overall household inefficiency. An application to a panel dataset of 296 farms in the UK illustrates the basic concepts introduced in the theoretical model. The results show that households in our sample are technically inefficient but their efficiency scores are very close. However there is a big dispersion in the household efficiency scores and some households can adapt better their consumption and labor supply decisions when production is technically inefficient.

Appendices

Appendix A: Elasticity calculation

1.1 Profit function

• Uncompensated Variable-Input Demand Elasticities

  $$\begin{array}{lll}\,{\text{Own}} {\hbox {-}} {\text{Price}}\!:\quad {\varepsilon }_{jj}^{v}={S}_{j}^{v}-1+\frac{{\beta }_{jj}^{vv}}{{S}_{j}^{v}}\\ \,{\text{Cross}} \hbox{-} {\text{Price}}\!:\quad {\varepsilon }_{jk}^{v}={S}_{k}^{v}+\frac{{\beta }_{jk}^{vv}}{{S}_{j}^{v}}\\ \,{\text{Crop}}\, {\text{Price}}\!:\quad {\varepsilon }_{jp}^{v}={S}^{y}+\frac{\sum _{k}{\beta }_{jk}^{vv}}{{S}_{j}^{v}}\end{array}$$

  where j, k = S, F, I are the three variable inputs used (i.e., seeds, fertilizers and intermediate inputs) and p is the crop price.

• Crop Supply Elasticities

  $$\begin{array}{ll}\,{\text{Crop}} \hbox{-} {\text{Price}}\!:\quad {\varepsilon }_{p}^{y}=-{\sum \limits _{j}}{S}_{j}^{v}+\frac{\sum _{j}\sum _{k}{\beta }_{jk}^{vv}}{{S}^{y}}\\ \,{\text{Variable}}\hbox{-}{\text{Input}}\, {\text{Price}}\!: \quad {\varepsilon}_{j}^{y}={S}_{j}^{v} + \frac{\sum _{k}{\beta }_{jk}^{vv}}{{S}^{y}}\end{array}$$

• The matrix of Compensated Variable-Input Demand Elasticities is obtained from:

  $$\left[\begin{array}{lll}{\epsilon }_{SS}^{v}&{\epsilon }_{SF}^{v}&{\epsilon }_{SI}^{v}\\ {\epsilon }_{FS}^{v}&{\epsilon }_{FF}^{v}&{\epsilon }_{FI}^{v}\\ {\epsilon }_{IS}^{v}&{\epsilon }_{IF}^{v}&{\epsilon }_{II}^{v}\end{array}\right]=\left[\begin{array}{lll}{\varepsilon }_{SS}^{v}&{\varepsilon }_{SF}^{v}&{\varepsilon }_{SI}^{v}\\ {\varepsilon }_{FS}^{v}&{\varepsilon }_{FF}^{v}&{\varepsilon }_{FI}^{v}\\ {\varepsilon }_{IS}^{v}&{\varepsilon }_{IF}^{v}&{\varepsilon }_{II}^{v}\end{array}\right]-\left[\begin{array}{l}{\varepsilon }_{Sp}^{v}\\ {\varepsilon }_{Fp}^{v}\\ {\varepsilon }_{Ip}^{v}\end{array}\right]{\left[{\varepsilon }_{p}^{y}\right]}^{-1}\left[{\varepsilon }_{S}^{y}\ {\varepsilon }_{F}^{y}\ {\varepsilon }_{I}^{y}\right]$$

1.2 Indirect utility function

• Uncompensated Leisure and Aggregate Marketed Good Demand Elasticities

  $$\begin{array}{lll}\,{\text{Own}} \hbox{-} {\text{Price}}\!:\quad {\varepsilon }_{jj}^{d} = \frac{{\alpha }_{jj}}{{Z}_{j}}-\frac{{\alpha }_{mj}}{Q}-1\\ \,{\text{Cross}} \hbox{-} {\text{Price}}\!:\quad {\varepsilon }_{jk}^{d} = \frac{0.5{\alpha }_{jk}}{{Z}_{j}}-\frac{{\alpha }_{mk}}{Q}\\ \,{\text{Income}}\!:\quad {\varepsilon }_{jm}^{d} = -\sum _{k}{\varepsilon }_{jk}^{d}\end{array}$$

  where j, k = ℓ^f, ℓ^o, c are the two leisures and aggregate marketed good and Z_j and Q are the numerator and the denominator, respectively, of the corresponding budget shares.

• Uncompensated Leisure and Aggregate Marketed Good Demand Elasticities w.r.t. to variable-input and crop prices

  $$\begin{array}{l}\,{\text{Crop}} \, {\text{Price}}\!:\quad {e}_{jp}^{d} = \left({\varepsilon }_{\pi j}^{d}+{\varepsilon }_{\pi m}^{d}\frac{\bar{T}{\tilde{\pi }}^{f}}{M}\right){S}^{y}\\ \,{\text{Variable}} \hbox{-} {\text{Input}}\, {\text{Price}}\!:\quad {e}_{jq}^{d} =\left({\varepsilon }_{\pi j}^{d}+{\varepsilon }_{\pi m}^{d}\frac{\bar{T}{\tilde{\pi }}^{f}}{M}\right){S}_{\mathrm{q}}^{\mathrm{v}}\end{array}$$

  with q being the variable-inputs used (i.e., seeds, fertilizers and intermediate inputs).

• Compensated Leisure and Aggregate Marketed Good Demand Elasticities

  $$\begin{array}{l}\,{\text{Own}}\hbox{-}{\text{Price}}\!:\quad {\epsilon }_{jj}^{d}={\varepsilon }_{jj}^{d}+{\mathrm{S}}_{j}^{h}{\varepsilon }_{jm}^{d}\\ \,{\text{Cross}} \hbox{-} {\text{Price}}\!:\quad {\epsilon }_{jk}^{d}={\varepsilon }_{jk}^{d}+{\mathrm{S}}_{k}^{h}{\varepsilon }_{jm}^{d}\end{array}$$

Appendix B: Monte-Carlo simulation

In order to examine the possible bias in price elasticities obtained from the separable vis-a-vis the non-separable agricultural household model, we formulate a Monte-Carlo simulation where we generate a translog short-run profit function with one crop output, three variable inputs and one quasi-fixed input. For the indirect utility function we assume only off-farm labor together with aggregate marketed good. Profits obtained from farming were added to autonomous income. To keep the analysis simple we assume that rural households are fully efficient and thus no inefficiency terms exist in both the profit and indirect utility functions.

Linear homogeneity in crop and variable input prices was imposed in the profit function, while we ensure that all parameter restrictions and monotonicity conditions are satisfied in the simulated model. The same was done for the indirect utility function (homogeneous of degree zero, non-increasing in prices and non-decreasing in autonomous income). We draw variables from a normal distribution with $N\left(10,2\right)$ for all explanatory variables and $N\left(0,0.01\right)$ for all random terms.

We use the following parameter values for both functions during the Monte-Carlo simulation:²⁰β₀ = 0.070, β^y = 1.500, β^yy = − 0.166, ${\beta }_{S}^{yv}=0.070$, ${\beta }_{F}^{yv}=0.090$, ${\beta }_{L}^{yf}=0.087$, ${\beta }_{S}^{v}=-01.160$, ${\beta }_{F}^{v}=-0.199$, ${\beta }_{SS}^{v}=-0.010$, ${\beta }_{SF}^{v}=-0.030$, ${\beta }_{FF}^{v}=-0.100$, ${\beta }_{SL}^{vf}=-0.025$, ${\beta }_{FL}^{vf}=0.046$, ${\beta }_{L}^{f}=0.070$ and ${\beta }_{LL}^{f}=-0.025$ for the profit function and α_w = −0.220, α_ww = −0.700, α_c = −0.370, α_cc = −0.200 and α_wc = −0.110 for the indirect utility function. We run the simulation using 10 different sets of parameter values but the results were robust.

	Off-farm	Price of	Autonomous
	Wage Rate	Marketed Good	Income
Uncompensated Demand Elasticities
Off-Farm Leisure	0.752	0.227	0.525
Marketed Good	0.048	0.491	0.443
Compensated Demand Elasticities
Off-Farm Leisure	0.504	0.126	–
Marketed Good	0.063	0.191	–

We set the number of farms to be 300 and the time periods equal to 5. The number of replications was set to 1,000. For all iterations, we calculate the mean absolute bias of the estimated point elasticities in both the profit and the indirect utility function. Since in the short-run profit the structure remains the same in both models (separable vs non-separable) crop supply and variable input demand elasticity estimates do not deviate significantly. The maximum absolute mean bias obtained from the simulation is 0.003. Hence, we can assume that the separability hypothesis does not affect point elasticities obtained from the short-run profit function. The following Table present average values of the mean absolute bias over all iterations for the off-farm labor supply and aggregate marketed good elasticities. The highest bias is between off-farm leisure and autonomous income whereas the lowest for the demand elasticity of marketed good with respect to the off-farm wage rate. Generally, biases are significant implying that model formulation is important in determining household responses to changes in market conditions and autonomous income.