8    Estimation when fixed costs cannot be ignored

8.1 Motivation

8.2 Theory

8.3 Statistical model

8.4 Results

8.5 Conclusions

Until now we have not said anything about fixed costs, yet these costs are known to have important implications for market entry. The same is true in the household setting and this chapter contains a contribution around the idea that these costs are important, should be accounted for and, indeed, where possible, should be estimated as part of the empirical investigation.

8.1 Motivation

The main component of this chapter is, once again, the methodology proposed in the seminal paper on Gibbs sampling Tobit regressions (Chib 1992). In that paper the censoring point is assumed to be known with certainty to be zero. This assumption is the one that is most often applied. But this assumption implies that the minimum economic quantity of marketable milk surplus is zero. Theory predicts otherwise.

When subsistence households make path-altering decisions such as the decision to enter a market they incur fixed costs. Usually, but not always, this cost is a cost of time. Because fixed costs are prevalent there is a non-negligible but positive level of surplus that must accrue before the household actually participates in the market. Estimation of this quantity is important for policy analysis in the developing countries settings where transactions costs are high and inhibit market participation. In this chapter we show how Gibbs sampling a non-zero censored Tobit regression leads to precise estimates of the surpluses that must be accrued and to estimate the levels of the household resources that are required for market participation.

In our application to peri-urban milk market participation in the Ethiopian highlands, we focus on seven covariates, namely, the level of a traditional production practice (the number of indigenous milking cows); the level of a modern production practice (the number of crossbreed milking cows); the level of the intellectual-capital stock (measured in terms of years of formal education of the household head and the number of visits by an extension agent discussing production and marketing activities during the twelve months prior to the survey date); the physical distance that the household resides from the market (measured in terms of return time to transport bucketed fluid milk to market); and two site specific dummy variables representing the two locations at which the data were collected. As the previous chapters have shown, these set of covariates are the ones bearing the strongest relationships to marketable surplus accrual in the household. For each non-participating household, and for the average across all non-participants, we estimate the levels of these covariates that precipitate entry into the market.

8.2 Theory

The development of a Tobit regression follows, essentially, the same steps as those outlined in equations (9)–(14) and the notion that non-participating households ‘distance’ from the market is motivated in three steps. First, we develop a Kuhn-Tucker representation of the household’s decision problem; second, we analyse the impact of relaxing a non-negativity restriction on marketable surplus; third, we formalise a Tobit regression that follows naturally from the first two steps. The difference here, however, is due to an alleged barrier impeding entry, leading to censoring of the Tobit regression at a point different from the traditional assumptions of zero.

Developments about the unknown-point-of-censoring model are now easily derived using the known-censoring formulation (equations 9–14) as a starting point.

With θ the assumed, random censoring point, the household’s problem is to select marketable surplus, v_i, to solve:

As we have done previously, assume for the moment that the constraint is non-binding (λ = 0) and focus on the interior solution φ_i(vi | xi) = 0, recognising that some households will have marketable surplus below the censoring point (that is, for some i, v_i ≤ θ).

8.3 Statistical model

Developments analogous to those for the standard Tobit model lead to the statistical model

where ε_i ~ N(0,σ²)and we observe y_i = max{z_i, 0}.

With θ unknown, the non-informative prior π(β, σ, z₂, θ) µσ^–1 can be combined with ideas in Albert and Chib (1993, p. 671, equation 3) to write the posterior density in the form:

where I(·) denotes the indicator function (Mood et al. 1974 p. 20). Following developments in Albert and Chib (1993, p. 671, equations 4–6), the likelihood can be augmented with the latent data in order to form a Gibbs sampling algorithm for the unobserved σ, β, z_i ε c, and the unknown censoring point, θ. Some simple manipulations reveal that s has an inverse-gamma distribution, that β is multivariate normal, and that each z_i (y_i = 0) is normal with mean x_i β and variance σ², truncated to the left by the condition zi ≤ θ. Regarding θ, it follows from equation (40) that (when σ, β and the z_i’s are known) the censoring point has the (fully) conditional distribution:

which is uniform on the interval [max{z_i, i ε c}, min{z_i, i ε c}]. (See Albert and Chib 1993, equation 18, for a similar development in the context of an ordered probit specification.) Consequently, the algorithm just outlined is adjusted by including an additional step to estimate θ. The distance estimates are calculated, as before, by inserting into the algorithm the additional computation:

where ε^{(s + 1)} denotes a draw from N(0, σ^{(s + 1)}). The algorithm in this section and the algorithm in chapter 5, together with the two measures of distance, (16) and (42), provide comparative measures with which to assess the robustness of the known, censoring assumption.

8.4 Results

Table 8 reports Gibbs sampling, data augmentation estimates of the Tobit regression based on a non-informative prior. Results from the zero-censoring formulation are presented in column 2 and results from the random censoring formulation are presented in column 3. Blockwise, the results consist of the parameter estimates, summary statistics and the distance-to-market estimates, respectively. The parameter estimates report means with 95% highest posterior density regions in parentheses; the distance estimates, report medians of the Gibbs sample.

Table 8. Response of marketable surplus (litre of milk per day) to covariates.

Note: Figures in parentheses are 95% confidence intervals (highest-posterior-density regions).

An important focal point between the two regressions is the true point of censoring in the marketable surplus data. This quantity is vitally important in policy calculations because it corresponds to a minimum efficient scale of operations for non-participants contemplating entry into milk markets.

The mean estimate of the censoring point, 0.98 (95% highest posterior density regions [0.93, 1.00]), leads to five observations that are important for policy. First, extension agents and policy planners aiming for increased market participation should target marketable surplus levels in non-participating households in the order of 0.98 litres of milk per household per day. Second (perhaps unsurprisingly, but in order to put this quantity in better perspective), 0.98 litres of milk per day lies close to the average level of milk sales among the market participating households at the two survey sites. Third, in view of this second observation, the estimated censoring point seems highly plausible. Fourth, with a 95% highest posterior density regions of [0.93, 1.00] the reported estimate is quite precise and leads, in turn, to precise policy calculations. Fifth, with 95% confidence, the true value of the censoring point is significantly different from zero—the censoring point that is typically assumed in the traditional formulation.

Focusing on the coefficient estimates, each of the parameters under the two formulations is significantly different from zero. Marketable surplus is declining with respect to increases in the amount of time that it takes to walk milk to the co-operative and is increasing with respect to the number of years of formal schooling of the household head, the number of crossbreed cows that the household milks, the number of local breed cows that it milks and the number of extension visits availed to it in the 12 months prior to the survey. In addition, the estimates of the coefficients of the site-specific dummy variables are quite different. Ceteris paribus, milk marketing potential appears to be significantly higher at the Ilu-Kura site. Whether this difference is due to climatic or to management factors remains an interesting, open question.

Comparing results across the two formulations, the differences in magnitudes of the parameter estimates are striking. Without exception, the absolute value of each of the regression estimates is greater in the zero-censoring formulation. Hence, in the (two-space) graphs of marketable surplus against the covariates, the Tobit regression is noticeably ‘steeper’ under the zero-censoring formulation. This point is important in itself, but also has a significant bearing on the distance-to-market estimates. In interpreting these estimates, the requisite calculations (equations 16 and 42) imply the following conventions. Households with a positive requirements level are resource deficient (lie outside the market boundary); households with a zero requirement lie on the market boundary; and households with a negative requirement are resource sufficient (lie within the market radius).

Whereas the degree of explanatory power in both specifications is quite comparable (although some slight improvement is apparent in the random-censoring formulation), the distance-to-market estimates appear to be markedly affected by the allowance of a random censoring point. First, and most significantly, the median levels of additional marketable surplus required for entry are 9.53 and 6.65 (litres per household per day) under the zero-censoring and the random-censoring formulations, respectively. This difference is substantial and prima facie suggests that fewer additional resources may be required when a departure in the censoring point (the minimum efficient scale of production) from zero is assumed. But this conclusion is false. In fact, due to the greater ‘slopes’ of the Tobit regression under the zero-censoring formulation, estimates of each median resource requirement are actually lower under zero-censoring. More importantly, with the exception of the distance (time to walk to market) covariate, all median requirements estimates are negative in the case of zero censoring and positive under random censoring. This observation adds weight to the view that imposing a zero point of censoring on the marketable surplus data is a stringent restriction that may lead to significant bias in regression and corresponding distance estimates and lead to less plausible estimates of policy-important quantities. Turning to the median reports themselves, the median additional levels of resources required for entry under the random censoring formulation are 1.08 years of education, 0.14 crossbred cows, 0.31 local breed cows, and 0.97 extension visits. These are small, but plausible levels of additional resources for the ‘representative household’.

One additional feature of the results that is not drawn out by these statistics is the wide range of estimates across the 1260 non-participating households. This feature is illustrated in greater detail by plotting the resource requirements. These plots are reported in Figures 11–16. In each case the households are ranked by their distance to market and then ordered so that the household that is farthest from the market (the household with the greatest resource deficiency) is the last household reported (household #1260) and the household with the greatest abundance of the resource is the first household in the ranking (household #1).

Figure 11 reports Gibbs-sample median requirements of additional milk across the households. By virtue of the fact that each household is a non-participant, each of the requirements estimates is positive. The reports range from a low of 1.5 litres of milk per day to a high of 18.6 and a significant divergence between the estimates from the two formulations is apparent—imposing zero censoring on the data leads to significantly greater estimates of additional milk requirements throughout the entire sample.

Figure 11. Distance to market estimates: Milk.

Figure 12 reports estimates of amounts by which time to walk to the milk group must be lowered in order to make marketing viable in the household. Unlike the reports corresponding to the remaining covariates, the time reports are bounded by the actual time required to walk to the milk group. For households who require greater time reductions than their observed travel time, this need can only be met through additions of other resources. Of the 1260 non-participants, there are 915 households for which travel time is not constraining and there are 345 households for which it is. Of those 345 households, there is no household for which the estimated reduction in travel time (under both formulations) is greater than the observed travel time. This point is important. It suggests that, for a significant proportion of the non-participating sample, a ceteris paribus reduction in travel time appears to be a potent policy alternative. The remaining observation of significance in Figure 12 is that the median reports under the two formulations are quite similar across the entire non-participating sample.

Figure 12. Distance to market estimates: Time to the milk group.

Unlike the travel-time estimates, the reports for education, crossbreed cows, local breed cows and visits by extension agents (Figures 13–16, respectively) are unbounded. For each of these covariates, the reports from the two formulations are quite similar. Also, for each of the covariates in question, the zero censoring model predicts that 604 households are resource deficient whereas the random censoring formulation reports 653 households are deficient. Small, but significant differences prevail across the reports from the two formulations, with the reports from the random censoring formulation generally greater than the reports from the zero-censoring formulation (there are 7 exceptions in the case of education, 471 in the case of crossbreed cows, 471 in the case of local breed cows and 0 in the case of extension).

Collectively, Figures 13–16 highlight the importance of three aspects of the distance-to-market estimates using the Tobit regression. First, the results suggest that random censoring leads to significantly different estimates of marketable surplus requirements. Second, despite the latter, differences in resource requirements estimates across the two formulations are, generally, quite small. Third, more generally, potentially important biases in estimates of policy quantities may arise from ignoring random censoring in market-participation studies using household production data.

Figure 13. Distance to market estimates: Education.

Figure 14. Distance to market estimates: Crossbred cows.

Figure 15. Distance to market estimates: Local breed cows.

Figure 16. Distance to market estimates: Visits by extension agents.

8.5 Conclusions

There are two key conclusions. First, small but significant differences arise from incorporating random censoring into the Tobit regression. Second, Markov chain Monte Carlo (MCMC) methodology has the ability to answer important research questions in a robust, appealing manner.