*3.2. Analytical Framework*

Decision makers' choices among alternatives in a choice situation can be analyzed using discrete choice models [34]. The decision makers in our case were sample households, and the alternatives represent hypothetical irrigation schemes characterized by different attributes and attribute levels. Assuming a utility-maximizing individual (*n*), the probability that a hypothetical irrigation scheme (*i*) in a choice situation (*Ct*) is chosen is equivalent to the probability that the expected utility from this alternative is higher than the utility from other alternatives in the choice set. Due to RUT, we can formulate this mathematically as:

$$P(\mathbb{C}\_{nt} = i) = P(\mathbb{U}\_{\text{nit}} > \mathbb{U}\_{\text{njt}}), \forall i \neq j \tag{1}$$

The utility function (*Uni*) has both deterministic and unobserved components. It can be written as:

$$
\mathcal{U}\_{nit} = V\_{nit} + \varepsilon\_{nit} \tag{2}
$$

where *Vnit* is an observable, and hence deterministic, component of the expected utility from alternative *i*, and  *nit* is the idiosyncratic random error term.

We assumed the utility function to be linear in the covariates and utility to be separable in price and non-price attributes to re-specify the utility function as:

$$
\Delta L\_{\rm nit} = -\alpha\_n p\_{n\dot{\jmath}t} + \beta\_n' \mathfrak{x}\_{n\dot{\jmath}t} + \mathfrak{e}\_{n\dot{\jmath}t} \tag{3}
$$

where *α<sup>n</sup>* and *β<sup>n</sup>* are individual specific parameter estimates, and  *njt* is the distributed extreme value type I with variance given by *η*<sup>2</sup> *n* Π<sup>2</sup> 6 , where *η<sup>n</sup>* is a scale parameter. Dividing Equation (3) by *η<sup>n</sup>* does not affect behavior and results in a new error term, which is an IID extreme value distributed with variance equal to Π2/6 [35,36]. Because of the division *Unit* = −(*αn*/*ηn*)*pnjt* + (*βn*/*ηn*) *xnjt* +  *njt*/*ηn*.

Therefore, the utility model in preference space can be written as:

$$
\mathcal{U}\_{n\dot{\jmath}t} = -\lambda\_n P\_{n\dot{\jmath}t} + \mathcal{c}\_n' \mathcal{x}\_{n\dot{\jmath}t} + \varepsilon\_{n\dot{\jmath}t} \tag{4}
$$

where the utility coefficients are defined as *λ<sup>n</sup>* = *αn*/*ηn*, *cn* = *βn*/*ηn*, and *εnjt* =  *njt*/*ηn*.

Equation (4) can be estimated using either conditional logit (CL) or random-parameters logit (RPL) models. CL, however, assumes the preferences for the attributes to be similar across individuals and requires the strong assumption of irrelevance of independent alternatives (IIA) to hold. RPL, on the other hand, is a flexible model that allows for random taste variation, unrestricted substitution patterns, and correlation in unobserved factors over time [34]. In this study, we report results of different specifications of the RPL model.

Our main interest is quantifying the relative implicit prices or the willingness-to-pay (WTP) values for the attributes of the irrigation services. The WTPs are ratios of two randomly distributed coefficients. Depending on the choice of distributions for the random coefficients of the RPL model, this can lead to WTP distributions that are heavily skewed and that may not even have defined moments [35,36]. Hence, the need to estimate RPL in WTP space arises [35].

The WTP for an attribute is the ratio of the attribute's estimated coefficient to the estimated coefficient of the annual payment, i.e., *wn* = *cn <sup>λ</sup><sup>n</sup>* = *βn ηn αn ηn* = *βn*/*αn*. Therefore, we can rewrite the utility function given in Equation (4) as:

$$
\delta L\_{n\dot{\jmath}t} = -\lambda\_n P\_{n\dot{\jmath}t} + (\lambda\_n w\_n)' \ge\_{n\dot{\jmath}t} + \mathbb{Z}\_{n\dot{\jmath}t}.\tag{5}
$$

We estimated Equation (5) with the assumption of correlated WTP coefficients as suggested by [35] and [36]. We are therefore reporting RPL models with and without correlated random coefficients estimated in WTP space.

We also analyzed sample individuals' choice-simplification strategies and the effect of the scale parameter on unobserved heterogeneity using latent class models (LCM). LCM is type of mixed logit (or RPL) model where the mixing density function of the coefficients to be estimated is of discrete nature, and hence the estimated coefficients take a finite set of distinct values [34]. We assumed that *β* takes *Z* possible values labeled *b*1, ... , *bZ*, with probability *sz* that *P*(*β* = *bz*) = *SZ*. In this case, the RPL becomes the latent class model, and the choice probability is given as:

$$P\_{nit} = \Sigma\_{z=1}^{Z} s\_z \left( \frac{e^{b\_z' x\_{nit}}}{\Sigma\_f e^{b\_z' x\_{nj}} t} \right) \tag{6}$$

We estimated constrained latent class models [37] to look into attribute non-attendance (ANA) patterns employed by the respondents—to simplify their decision making and scaleadjusted latent class models [38] to assess preference heterogeneity while considering response error.

ANA refers to the simplification strategy respondents employ by disregarding one or more attributes characterizing the alternatives in the choice sets. ANA can be stated or inferred. Stated ANA occurs when sample respondents state the attribute/attributes they disregarded or ignored in choosing between alternatives in a choice set, and inferred ANA is implied from the relative weights of the estimated random coefficients of the utility model. We are presenting inferred ANA patterns, as we did not generate data on stated ANA. The latent class models were gradually estimated with constraints on the coefficients of the attributes assumed to be ignored at every step, following earlier studies [39–41].

#### **4. Results and Discussion**

#### *4.1. The Sample Population*

This section is based on the socioeconomic survey on 200 farm households that preceded the DCE survey. As summarized below in Table 2, the sample is entirely of smallholder farmers (with cultivable farmland of 0.1 hectare per household). Most (68%) of our sample respondents were men. The sample had an average age of 43 years and 19 years

of farming experience. Only 37.5% of the sample households depend on farming for their livelihoods, while the rest of the households complement it with one or more income generating activities. Yet, two-thirds of the annual income a typical household generates is from agriculture. Most of the households (~96%) were either in secondary or in professional/vocational school. Most of the respondents (88%) indicated that agricultural water shortage happens sometimes, while 11% of them indicated that it happens all the time. The average number of months with a serious agricultural water shortage was three.


**Table 2.** Characteristics of the sample households.

† Frequencies calculated for each pump separately (*n* = 200).

A given sample household was, on average, 2 km far from the nearest agricultural water source. Expectedly, almost all (96%) of the respondents consider the quality of the agricultural water to be good, as the primary source water is a perennial river. Households use different types of pump for irrigation. Most of the respondents (94%) use their own irrigation pumps, whereas 82.50% of the respondents use pumps rented from neighbors. Only 2% of the respondents were found to be using the Sayyod pump station that provides water for several villages. It is important to note that farmers use more than one pump whenever they afford to do it.

Direct elicitation of the amount farmers are willing to pay for irrigation water showed that most of the farmers (~87%) are willing to pay up to UZS 10,000 per year. Almost 13% are willing to pay even more than that.

#### *4.2. Willingness to Pay*

The WTP estimation was based on the DCE conducted on 300 farm households after the socioeconomic survey discussed above. We report the results of the RPL models estimated in WTP space over 1000 Halton random draws (Table 3). Our discussion will be based on the RPL model with correlated coefficient estimates (Model 2). We also presented the model estimated with the assumption of uncorrelated random coefficients (Model 1) to show the consistency of the relative weights farmers attach to the different aspects of irrigation water.


**Table 3.** Willingness to pay for irrigation schemes.

Note: \* *p* < 0.10, \*\* *p* < 0.05, \*\*\* *p* < 0.01. Model 1 is RPL with independent random coefficients, and Model 2 is RPL with correlated random coefficients. LL stands for log likelihood; AIC stands for Akaike Information Criterion; and BIC stands for Bayesian Information Criterion.

The first attribute of irrigation service is the availability of canal water in the dry season (mainly May to October). There is a very high WTP for this component in the study area. The mean of the marginal WTP for one more month of water in between May and October was UZS 150,000. This implies that farmers have a high effective demand for irrigation facilities meant for making water available in the dry seasons—especially during production of key crops. The key crops were a mix of vegetables, legumes, and wheat for farmers' own consumption; and wheat; and cotton produced for commercial purposes. The Sayyod pumping station provides water through an irrigation network, and, although canal water is usually available, its distribution to consumers in different parts of the irrigation scheme is the key and is managed by water authorities.

The second attribute of irrigation facilities is crop-watering frequency per month. The watering frequency each farmer enjoys is determined by the water demand of the crops and, more importantly, water availability and the actual distribution determined by the water-user association (if functional) or water authorities that manage the distribution of water. Farmers have little control over the frequency, and yet this is an attribute that determines the level of production and the productivity of crops grown by farmers. One more watering per month has an implicit price of UZS 177,000. This is slightly higher than the implicit price for canal water in the dry season component of irrigation schemes.

Another important attribute of irrigation is water quality. This is usually the case when there is water scarcity, consumers revert to groundwater resources, and its quality is affected by high salinity, making it subsequently detrimental to crop production. Our model was specified in such a way that we could compare WTP for medium compared to bad quality and for high-quality compared to bad-quality irrigation water.

Farmers have a clear preference for high water quality over bad irrigation water quality. Farmers are willing to pay UZS 121,000 for high-quality irrigation water over low-quality irrigation water, everything else held constant. The model also shows that farmers are not interested in slight improvement of the quality; rather, they are keen on considerable improvement in the quality.

The results also show that farmers were not interested in water sharing with downstream users. This is not unexpected behavior of human beings whenever they are dealing with scarce resources, and irrigation water is very scarce in this desert.

The results also show that, for farmers, the most-important feature of an irrigation scheme is watering frequency (Table 3). The higher the number of times they obtain water, the better. Similar results were reported for Ethiopian smallholder farmers [12]. Farmers are willing to pay more for irrigation water in the dry seasons than for improvements in irrigation water quality. The priority is therefore more water in the irrigation schemes.

The lower half of Table 3 shows that there is unobserved heterogeneity around the mean WTP values for the different irrigation scheme components. The heterogeneity is very strong and significant in all attributes, except medium water quality (cf. bad water quality). Particularly, there is significant variability around the marginal WTP values for water sharing with downstream and high water quality (cf. bad water quality). We further disentangle the unobserved heterogeneity to see if there are any latent classes of preference among the respondents. We also look into heuristics that respondents might have applied to simplify the choice decisions.
