The indirect production function

In their factor allocation decisions, smallholder maize farmers in Burkina Faso face financial constraints. This calls for the use of the indirect production function to model their behavior. The producer’s objective is to maximize output, subject to the budget constraint. The problem can be written as,

(1) $$Max\,\,\,y = f\left( {x,z} \right)$$

(2) $${\rm subject \ to}\;C = \omega 'x$$

where y is the output, x denotes a vector of N variable inputs, z denotes the quasi-fixed input vector of order J, C represents the budget available for the purchase of variable inputs, $\omega $ denotes the vector of variable input prices, and $f\left( {x,z} \right)$ is a nondecreasing, twice continuously differentiable and quasi-concave function of ${\rm{\;}}x$ and z.

The Lagrangian for this problem can be written as,

(3) $$L = f\left(. \right) + \lambda \left( {C - \omega 'x} \right)$$

where $\lambda $ denotes the Lagrange multiplier.

Solving the first-order conditions for the problem ${f_j} = \lambda {\omega _j}$ , $\forall \;j$ and $C = \omega 'x$ , we obtain the solution of the endogenous variables in terms of the exogenous variables, that is,

(4) $${x_j} = {g_j}\left( {\omega ,C,z} \right),\;\;\forall \;j = 1, \ldots ,N$$

(5) $$\lambda = {g_j}\left( {\omega ,C,z} \right)$$

(4) represents the input demand function; according to economic theory, the demand function must be homogeneous of degree zero in input prices and expenditure, exhaustive, and symmetrical.

(5) represents the marginal productivity of the expenditure.

Substituting ${x_j} = {g_j}\left( {\omega ,C,z} \right)$ from equation (4) into equation (1) of the production function, we obtain the indirect production function (IPF) as

(6) $$y = \varphi \left( {\omega ,C,z} \right)$$

where $\varphi \left(. \right)$ represents the indirect production function.

The IPF expresses the maximum attainable output given the availability of funds, the price of variable inputs, and the amounts of quasi-fixed inputs. In addition to the usual symmetry restrictions on the coefficients, economic theory states that the IPF is homogeneous of degree zero in input prices and expenditure. The input demand function for the input variables can be derived from the IPF by using Roy’s identity (Chambers Reference Chambers1982):

(7) $${x_j}\left( {\omega ,C,z} \right) = \left( {\partial y/\partial {\omega _j}} \right)/\left( {\partial y/\partial C} \right),\;j = 1, \ldots ,J$$

The marginal productivity of the expenditure can be obtained from the partial derivative of the IPF with respect to the expenditure:

(8) $$\lambda = \partial y/\partial C$$

If the farm has no expenditure constraints, the value of $\lambda $ will be unity, given that the maize output is measured in value terms. That is, if the farm faces an expenditure constraint, the value of $\lambda $ will exceed unity. We can also test the hypothesis that a particular farm is expenditure constrained. That is, the null hypothesis of interest is ${H_0}:\lambda = 1$ , which can be tested against the alternative hypothesis ${H_a}:\lambda \gt 1$ .

The full econometric specification of the model of the indirect production and input demand functions is presented in the following section.

Econometric specification of the indirect production and input demand functions

The econometric model consists of the IPF and an inputs linear-approximate almost ideal demand system (AIDS). A quadratic functional translog form is chosen for the IPF to impose minimal a priori restrictions on the underlying production technology. This gives

(9) $$\eqalign{ln{y_i} = {\alpha _0} + \mathop \sum \limits_{j \,= 1}^J {\alpha _j}ln{\omega _{ji}} + \mathop \sum \limits_{m \,= \,1}^M {\alpha _m}ln{z_{mi}} + {\alpha _c}ln{C_i} \cr + {1 \over 2}\left\{ {\mathop \sum \limits_{k \,= \,1}^J \mathop \sum \limits_{j \,= 1}^J {\alpha _{jk}}ln{\omega _{ji}}ln{\omega _{ki}} + {\alpha _{cc}}{{\left( {ln{C_i}} \right)}^2} + \mathop \sum \limits_{L = \,1}^M \mathop \sum \limits_{m \,= \,1}^M {\alpha _{ml}}ln{z_{mi}}ln{z_i}} \right\} \cr + \mathop \sum \limits_{m \,= \,1}^M \mathop \sum \limits_{j \,= 1}^J {\alpha _{jm}}ln{w_{ji}}ln{z_{mi}} + \mathop \sum \limits_{j = 1}^J {\alpha _{jc}}ln{\omega _{ji}}ln{C_i} \cr + \mathop \sum \limits_{m \,= \,1}^F {\alpha _{mc}}ln{z_{mi}}ln{C_i} + {v_i}\;,\;i = 1, \ldots ,I}$$

Economic theory tells us that the IPF is homogeneous of degree zero in input prices and expenditure. This gives rise to the following set of restrictions on the parameters of the model:

$$\mathop \sum \limits_{j \,= 1}^J {\alpha _j} + {\alpha _c} = 0\;\;;\;\;\;\;\;\;\;\;\;\mathop \sum \limits_{j = 1}^J {\alpha _{jk}} + {\alpha _{jc}} = 0,\;\;\;\forall j = 1,\; \ldots ,J\;;$$

$$\mathop \sum \limits_{j \,= 1}^J {\alpha _{jm}} + {\alpha _{mc}} = 0,\;\;\forall m = 1,\; \ldots ,M\;\;;\;\;\;\;\;\;\;\;\mathop \sum \limits_{j = 1}^J {\alpha _{jc}} + {\alpha _{cc}} = 0$$

The input linear-approximate AIDS is derived in budget share form as:

(10) $${W_i} = {\alpha _i} + \sum\limits_j {{\gamma _{ij}}} ln{\omega _j} + {\beta _i}ln\left( {{C \over \omega }} \right) + \sum\limits_k {{\delta _{ik}}} ln{Z_k}\;,\;i = 1, \ldots .,I$$

where ${W_i}$ is the budget share of input i, and $\omega $ , a Stone geometric price index suggested by Deaton and Muellbauer (Reference Deaton and Muellbauer1980), is defined as:

$$ln\left( \omega \right) = \mathop \sum \limits_j {W_j}ln{\omega _j}$$

Economic theory indicates that the demand function must be homogeneous of degree zero in input prices and expenditure, exhaustive, and symmetrical. This gives rise to the following set of restrictions on the parameters:

1. i. exhaustive ( $\mathop \sum \nolimits_{\bf{i}} \boldsymbol{\alpha _i} = 1$ ; $\mathop \sum \nolimits_{\bf{i}} \boldsymbol{\beta _i} = {\bf0}$ ; $\mathop \sum \nolimits_{\bf{i}} \boldsymbol{\gamma _{ij}} = {\bf0};$ );

2. ii. homogeneous ( $\mathop \sum \nolimits_j {\gamma _{ij}} = 0$ );

3. iii. symmetrical ( ${\gamma _{ij}} = {\gamma _{ji}}$ ).

The AIDS model has the advantage of automatically incorporating all these properties into the estimation. The price and income elasticities can be derived from the parameter estimates as:

${\varepsilon _{ii}} = - 1 + \displaystyle{{{\gamma _{ij}}} \over {{\omega _i}}} - {\beta _i},$ own price elasticity

${\varepsilon _{ij}} = \displaystyle{{{\gamma _{ij}}} \over {{\omega _i}}} - {{{\beta _i}} \over {{\omega _i}}}{\omega _j},$ cross-price elasticity

${\varepsilon _{ir}} = 1 + \displaystyle{{{\beta _i}} \over {{\omega _i}}},$ income elasticity

Our interest is to jointly estimate the IPF and the linear-approximate AIDS as a system of equations with the abovementioned restrictions on the parameter estimates by using the seemingly unrelated regressions (SUR) method. To this end, the demand for the other inputs of production was not considered to avoid a problem of multicollinearity. The use of input budget shares as an explained variable has the advantage of reducing the risks of heteroskedasticity (Deaton Reference Deaton1986). To estimate the model proposed, we employ data obtained from household surveys from rural areas in Burkina Faso.

Method of data collection

The data used were collected in 2017 from 2,000 rural households as part of the evaluation of the National Land Management Program. The survey is nationally representative and covers all households residing in rural areas. The sample was drawn using a three-stage random sampling process. In the first stage, 90 communes were drawn from all 302 rural communes. In the second stage, 270 villages were chosen from the selected rural communes. In the third stage, 2,160 households were selected from the 270 villages. The data collected cover all the sociodemographic and economic characteristics and the institutional framework of rural households and their farms. After excluding rural households that do not produce maize, a sample of 969 observations was retained for the estimation of the model.

Descriptive analysis of data

Maize production depends on the level of quasi-fixed inputs, price of variable inputs, and input expenditures. Maize production is measured as the maize revenue of harvest, expressed in CFA francs using the farm-gate price. The maize quasi-fixed inputs include family labor and farmland. Family labor is estimated as the amount of labor measured in person-days that family members spent working on the maize farm. Farmland is the maize cultivated land in hectares for a growing season. A positive effect of fixed inputs on maize revenue is expected.

The main variable inputs include seed, fertilizer, and manure. The price of seed is measured by the price per kilogram of seed at the beginning of the rainy season. The price of fertilizer is captured by the average weighted price of a kilogram of NPK and urea. The price of manure is approximated by the average price of a cartload of manure. It is expected that the price of the inputs will have a positive effect on maize revenue. The maize input expenditure accounts for the overall operational costs incurred by the maize production process.

Table 1 presents the average values of the main variables used in the econometric estimates. The data indicate that in maize production, a farm household commits 1.4 hectares of land and spends approximately 90,450 CFA francs in variable inputs to achieve a revenue of 222,780 CFA francs at the end of the rainy season.

Table 1. Average values for inputs involved in maize production

Source: Authors’ estimates from survey data.