A General-Equilibrium Model of Labor-Saving Technology Adoption: Theory and Evidences from Robotic Milking Systems in Idaho

Abstract

Automatic milking systems (AMSs) have become increasingly common in the US in the past few years. Recent surveys from Idaho, one of the largest dairy-producing states, as well as from other states and countries, suggest that: 1. among farms adopting robotic milking systems, few are reporting less labor usage after adopting this labor-saving technology; 2. small farms rather than large farms are adopting (or more interested in adopting) robotic milking systems. In this article, we propose a series of new modeling strategies, which introduces the role of general-equilibrium effects to explain these new stylized facts. We show that: first, farms adopting labor-saving technology may, in fact, use more labor to compensate for the loss in the value of labor; second, when smaller farms experience more labor efficiency gains or value their leisure time (or off-farm income) more, they are more likely than larger farms to adopt the new technology. We contribute to the technology-adoption literature in two important ways. First, to our knowledge, this is the first article that introduces general-equilibrium effects to the technology-adoption literature. Second, this is also the first article that provides a theoretical perspective to explain the stylized facts in the adoption of robotic milking systems.

1. Introduction

Vast price fluctuations, laboravailability issues, and trade-agreement uncertainties threaten the livelihood of dairy farms, especially small- and mid-sized dairies. The number of licensed dairy herds in the US decreased by 2731 in 2018 alone (USDA, 2019). Western states experienced a 10% to 40% loss in the number of dairy farms between 2002 and 2017 (Figure 1), where small- and mid-sized dairy farms have been disproportionally more affected than larger ones (USDA Agricultural Census, 2002 and 2017).

One response to such a risky and adverse business climate is the adoption of a labor-saving technology: automatic milking systems (AMS). Under an AMS, the milking operation is performed by milking robots that are equipped with various sensors and automated arms, replacing the need of milking labor. The first AMS system was developed in the Netherlands in the 1980s, with commercial production becoming available in 1992 [1]; it was subsequently introduced in the US by 2000 [2]. The vast majority of AMSs are located in north-western Europe (90%), and the rest are in North America [1]. The preponderance of European dairy farms have less than 500 milking cows, and the cost of labor for dairy farmers is high [3]. With an AMS, a dairy farm faces less uncertainty around labor availability and fewer fluctuations in milk yield. An AMS, in its essence, is a new labor-saving technology. A labor-saving technology is a technology that reduces the marginal value product of labor [4] or a device that reduces labor input [5]. There are variety of labor-saving techniques, which are widely used in different industries that range from agriculture to automobile manufacturing and other fields. For instance, labor-saving technology is integrated into all aspects of modern agricultural production from input to output: the seeder machine used for sowing in the initial period of production, the circle-pivot irrigation system used for irrigation, and the harvester machine used for crop harvesting. Similarly, robots are frequently used as a type of labor-saving technology in industrial production to substitute for or save labor. The application of robots used for welding manipulators and rotating drums in an automobile assembly is a representative example of a labor-saving technology. In the relatively labor-intensive service industry, labor-saving technology has gradually become a trend: in recent years, many companies have begun to adopt customer-service robots combined with manual customer services. Even in management positions that require human participation, labor-saving technology combined with artificial intelligence can assist a manager with their management and reduce the workload of workers. It is predicted that such a kind of technology improves the efficiency of grid management and reduces human participation [6].

Based on a recent survey from Idaho (see [7] for details) and the recent literature on surveys from other states and countries, we summarize two main puzzling, stylized facts in the adoption of robotic milking systems.

First, among farms that are adopting robotic milking systems, few are reporting less labor usage after adopting a labor-saving technology. That is, adoption of an input-saving technology could lead to a more intensive use of that input. The previous literature ([8]) has successfully explained, in the context of irrigation-technology adoption, that a farm may adopt water-saving technology and end up using more water. However, it is unclear how farms adopting labor-saving technology behave in a similar fashion.

Second, it is small farms rather than large farms that are adopting (or are interested in adopting) robotic milking systems. The previous literature (see [8] for a comprehensive review) suggests that there is a size threshold for adopting new technologies. However, it is well documented that large farms, rather than small farms, are typically more likely to adopt a new technology. It is unclear why the threshold is not in favor of large farms.

The goal of this article is to fill the research gap by reconciling these stylized facts in the recent surveys and empirical literature with the conceptual frameworks in the technology-adoption literature. To do so, we propose a series of new modeling strategies, which introduce the role of general-equilibrium effects to the technology-adoption literature ([8]). In particular, we ask two related research questions: first, under what conditions would a labor-saving technology-adopting farm choose to use more labor? Second, what is the role of the heterogeneity of farm size in a labor-saving technology-adoption decision?

First, we introduce a conceptual framework that allows for analyzing the general-equilibrium effects of technology adoption. The technology-adoption literature has always been emphasizing the production aspects of the economy (see literature review [8,9]). However, a close look at the stylized facts suggests that automation, a labor-saving technology, makes labor relatively less valuable. When a farm household values leisure time, which affects the household labor supply in the production process, focusing solely on the production aspects of technology adoption overlooks the adopter’s preference over consumption and leisure. In this article, we combine classic technology-adoption models and general-equilibrium models in the labor–leisure literature to illustrate that at the adoption threshold, farms adopting labor-saving technology may, in fact, use more labor to compensate for the loss in the value of labor.

Second, we also consider farm heterogeneity in our general-equilibrium framework. This allows us to explore why smaller farms are more likely to adopt costly technology. We show that when smaller farms experience more labor-efficiency gains or value their leisure time more (or off-farm income), they are more likely to adopt new technology than larger farms.

In sum, we contribute to the technology-adoption literature in two important ways. First, to our knowledge, this is the first article that introduces general-equilibrium effects to the technology-adoption literature. Second, this is also the first article that provides a theoretical perspective to explain the stylized facts in the adoption of robotic milking systems.

The rest of the paper is arranged as follows: in Section 2, we briefly review the recent literature on the cause and impact of adopting labor-saving technologies. In Section 3, we provide the main general-equilibrium model of technology adoption. In Section 4, we discuss potential extensions to the model. Finally, we give conclusions in Section 5.

2. Literature Review

Recent years have seen a growing amount of the literature regarding the adoption of different new technologies by small and medium-sized enterprises (SMEs). For instance, one study from [10] discusses the adoption of Information and Communication Technologies (ICTs) in rural areas, which analyzes adoption barriers and also proposes policy recommendations. The later empirical research by [11] also studied the barriers to adopting new technology but includes various kinds of technologies (like organic farming or renewable energy) in eight European countries (Bulgaria, Czech Republic, Greece, Hungary, Italy, Latvia, Slovenia, and the United Kingdom) and their result shows that financial resources are one of the biggest barriers to SMEs to adopt new technology.

In the following sections, we provide a review of the recent literature, focusing on labor-saving technology adoption. Ref. [9] classifies the recent literature in economics addressing the topic of the adoption of labor-saving technology into the time at which adoption decisions are made: ex ante (before adoption) or ex post (after adoption). Different from their classification, the literature review here divides the previous literature into two parts: factors that affect the adoption of labor-saving technologies and its impact after adoption. Such classification (Table 1) can help us not only better understand the factors that correlate to the adoption of labor-saving technologies in ex ante but also compare their different effects in ex post.

2.1. Factors Affecting Adoption of Labor-Saving Technology

Numerous recent empirical studies have established some key factors relating to the decision-making of labor-saving-technology adoption and its measurement regarding to labor-market factors as well as natural factors and other factors. Below we will briefly introduce these factors.

2.1.1. Labor-Market Factors

The important elements from the literature on the adoption of labor-saving technology are variables such as labor supply, human capital, and wages in the labor market. For labor-supply analysis, changes in the labor-supply structure will affect the adoption of labor-saving technologies [24]. The early contribution on the labor-supply structure comes from the popular Habakkuk hypothesis in economic history, proposed by H. J. Habakkuk [25], which claims that technological progress because of labor scarcity acted as one of the driving forces in the adoption of labor-saving technologies. Recent studies have shown that labor scarcity is the main reason for the adoption of labor-saving technologies [26,27,28,29]. When labor is scarce, producers are more likely to adopt labor-saving techniques.

Wages are another essential factor that stimulate labor-saving technology. David and Otsuka [30] summarize the long-term factors influencing the adoption of labor-saving technologies by farmers, through case studies in the Philippines. These authors claim that the relative wages, such as threshing wages, help decide whether to choose a labor-saving technology. Articles by Naylor [31], Marquetti [32] and Zuleta and Alberico [33] argue that the adoption of labor-saving technology is driven by an increase in real wages. Lommerud and Straume [34] validates this argument and their paper shows that increasing ‘flexicurity’, interpreted as less employment protection and higher reservation wages for workers, increases firms’ incentives to adopt labor-saving technology. As wages goe up, it would force profit-seeking capitalists to invest in labor-saving technology. More recent work by de Souza [35], who uses two disaggregated datasets of manufacturing industries (EU-Klems and Unido), obtains similar result.

Another idea in this labor-market analysis is that other human-capital variables would determine the adoption of labor-saving technologies. These technologies may require skills of and knowledge by the workers [36]. In addition, many require basic skills in business analysis, such as monitoring and assessing production conditions [37]. Other factors, such as education, would have a positive effect on technology adoption. Uematsu and Mishra [38] used the 2006 Agricultural Resource Management Survey (ARMS) data in the US to estimate the marginal effect of education on technology adoption. These authors concluded that education could be a barrier to technology adoption.

2.1.2. Economic and Social Factors

Some of the literature has put forward views on factors affecting the adoption of labor-saving technology from economic and social perspectives, such as the application of other technologies, cultural and political context, gender difference, etc. First, many studies discuss the impact of other technology applications on labor-saving technology, but results are mixed. Caswell [39] analyzed agricultural production data in the US and Israel to predict the probability of adopting labor-saving technology decisions and found that irrigation equipment is the main factor that affects this probability. However, this result is inconsistent with David and Otsuka [30], who collected farm-level data to study the impact of modern rice varieties (MV) on the adoption of labor-saving technologies. Otsuka et al. found that although the widespread use of MV stimulated labor-demand growth, there was no evidence that MV promoted labor-saving techniques.

Second, in some recent studies, the adoption of labor-saving technology responds to the economic structure and/or the cultural or political context. Zhang et al. [40] summarizes the evidence that the cooperative-economic model in large-holder farming is more conducive to promoting labor-saving technologies than smal-holder farming. For cultural and political context, Luna [41] provides evidence by studying the residents of Burkina Faso and their reasons for the adoption of labor-saving technology. Luna concluded that the adoption decision is not because of their own needs, but due to government requirements and special local cultural context. Bhargava [42] contributes to this topic by applying a regression-discontinuity model with new Indian-agricultural-census data. Bhargava found that the adoption of the Mahatma Gandhi National Rural Employment Guarantee Act (NREGA) account for roughly a 20 percentage points difference for the adoption of the labor-saving technology, especially for small farmers.

Third, gender difference in determining technology adoption has long been recognized by economists [43]. Joshi et al. [44] found that women are more interested in the adoption of labor-saving technology and their willingness to pay (WTP) for direct-seeded rice (a kind of labor-saving technology) is higher than men.

2.1.3. Natural and Environment Factors

The natural environment determines the types of adoption of labor-saving technologies. Most of the natural environmental factors, which encompass geographical features that include fertility [45], soil textures [46], whether the location is in hilly and mountainous landscapes [40], and the chemical composition of the soil [47], are related to agricultural production. Among such natural environmental factors, the type of crops could be another reason that affects the adoption of labor-saving technology, which has been summarized in a substantial amount of the research literature. For example, robotic harvesting technology can be used for strawberries in greenhouses [48], while mechanized weeding can be used for rice planting [49]. Calvin and Martin [50] reported that 75% of vegetables and melons are harvested by machine, and 55% of fruit production is accomplished with processing products being more likely mechanically harvested than fresh. Therefore, to some extent different natural environmental conditions and types of crops determine the types of labor-saving technologies that can be adopted.

2.2. The Impact of Labor-Saving Technology

The second sub-section will summarize the literature on the impact of labor-saving technologies in different industries, seeking to present the recent research. This part will start with analyzing the general impact on efficiency or productivity after the adoption of labor-saving technology. Then, the focus will be on its impact on various topics, such as labor supply, labor demand, and wages. Finally, this section will summarize the literature that explores the impact of these technologies from the perspective of agriculture and other industries.

2.2.1. The Effect on General Production Efficiency

The first question that needs to be answered is what impact there will be after the application of labor-saving technology. Intuitively, labor-saving technology could help to improve production efficiency. Take labor-saving technology in agriculture, as an example: tractors and other farm machinery have continuously improved efficiency and productivity during the second half of the twentieth century [51]. For other industries, Van Biesebroeck [52] studied the impact of labor-saving technology on the efficiency of automobile assembly. By comparing the production efficiency of automobile-assembly plants in Japan and North America, they concluded that labor-saving technology has made automobile assembly plants more efficient. However, labor-saving technologies do not always bring positive effects. The evidence provided by Acemoglu and Restrepo [53] implies that further technological advances tend to reduce the marginal product of labor.

2.2.2. The Effect on the Labor Market

The most direct impact of labor-saving technology on the labor market is manifested in unemployment and wages. This has been the subject of research since the Great Depression and is regarded as a cause of unemployment [54]. Research conducted by Gregory et al. [55] shows that direct labor-saving technology has been massively job-destroying, eliminating 6.6 million jobs. Its impact has been particularly severe on intermediate and low-skill jobs. In a more recent article, Morss [56] analyzed recent US Bureau of Labor Statistics data and concludes that 4.8 million jobs have been lost in the manufacturing sector, with 1.4 million of these losses occurring since 2007, as result of labor-saving technologies. Morss also claims that automation and globalization, since the late 1970s and through today, in the US have led certain manufacturing industries to disappear, and wages have been flat since the 1970s. Similar evidence from the banking industry shows labor-saving technologies have replaced some lower-skilled workers, leading to a decline in labor demand [57]. In recent years, the development of robotic technology and its impact on the labor market has heightened economists’ interest. For example, Acemoglu and Restrepo [53] estimate one more robot per thousand workers reduces the employment to population ratio by about 0.18–0.34 percentage points and wages by 0.25–0.5%, except for industrial production. In agricultural production, the results from Charlton et al. [14] suggest that compared to the upward trend of wages caused by the increase in the production of labor-intensive crops, the adoption of labor-saving technologies would put downward pressure on wages. Another example is the share of labor payments for MV (modern rice varieties). The production declined from 55% in 1996 to 40% in 2003, as a result of the widespread use of labor-saving technologies [58].

2.2.3. The Effect on Agriculture

The prosperity and high efficiency of modern agricultural industry is the result of labor-saving technologies [51]. A substantial body of the recent literature shows labor-saving technology will help to significantly reduce production costs and improve productivity to deal with labor shortages and high labor costs [15,16,17,18]. A wealth of the literature studies the impact of specific technology used for agricultural production.Kaur et al. [59], for example, analyzed the yield and cost of production towards the adoption of the direct sowing of rice (DSR) and mechanical transplanting (MT). These authors conclude that these technologies, when adopted by large farmers, bring higher off-farm income. In addition, they found that MT adopter would produce a higher yield at a lower cost. Moreover, labor-saving technological change enables more farm labor to migrate to the non-farm sector [60]. However these technologies would lead to net reductions in agricultural-labor use [12].

2.2.4. The Effect on other Industries

In addition to agricultural production, labor-saving techniques are common in other industries. However, the impacts are diverse. For instance, Fung [22] studied whether the adoption of labor-saving technologies increases the efficiency of human resources. The results show that the benefits of using labor-saving technologies far outweigh the costs of using them, and the spillovers of these technologies are associated with higher firm-level employment. However, Van Reenen [61] finds a weak positive or insignificant relation within the same context. A later article by Evangelista and Savona [62] find the relation to be negative. Labor-saving techniques are widespread in high-tech industries in the US. Although the adoption of these technologies leads to a decline in short-term labor demand, in the long run, such technology applications bring positive effects: rationally optimizing the labor-demand–supply structure and guiding surplus labor to other industries [23].

2.2.5. The Effect on Social Development

In addition to saving time and labor, labor-saving technology reshapes society in a general sense. First, labor-saving technology has subtly changed ideology, from the perspective of social consciousness. One qualitative study by de Schweinitz Jr [63] took the local resistance to the New Deal’s Rural Electrification Act program in the US before World War II, as an example that indicates the resistance of conservatism to new technology. The later development of ideologies, such as liberalism, are all related to the adoption of labor-saving technologies [64]. Second, from an anthropological point of view, there is compelling evidence that labor-saving technologies have affected human-reproduction patterns. This evidence is explained by the result in Kramer and McMillan [65] and discussed from a broader view in Gibson and Mace [66]. Gibson and Mace point out that labor-saving technologies reduces women’s time in dealing with household chores, increases their probability of pregnancy, changes the time when women get married, and makes women more willing to give birth at a younger age. Furthermore, these technologies shorten the length of first-birth intervals. Third, labor-saving technologies even affect the development of societal structure. One key idea is that labor-saving technology will lead to labor migration. Barkley [67] studied agricultural labor migration in the US from 1940 to 1985 and showed that labor-saving technology was one of the factors that led to labor migration. The influx of these new immigrants changed the structure of the urban labor market and, ultimately, affected the development model of the entire society.

3. Materials, Methods, and Results

3.1. Baseline Model

Consider a dairy farm that operates with conventional milking technology (CMT). The amount of labor devoted to production is denoted by L. For simplicity, our model does not consider farmers’ decision on other inputs’ use, but the model can be expanded to consider such inputs following the formulation of [68]) (adding the decisions on variable-input use will not alter the main conclusions of this paper.) The farm household has $\overline{L}$ units of labor available. Under CMT, production function is given by $f_0\left(L\right)$. Output, milk, is denoted by x, which is assumed to be a numeraire good.

With the new technology, Automatic Milking Systems (AMSs), the farm also makes the decision on whether to adopt them and change the labor input use accordingly. Under AMS technology, production function is given by $f_1\left(L\right)$. For both production technologies ($i=0,1$), we assume a diminishing return for the marginal product of labor, i.e., $f_i^{\prime }\left(L\right)>0$ and $f_i^{{}^{\prime \prime }}\left(L\right)<0$. To capture the productivity improvement from the new technology, add the assumption that for all labor-use level L, the output under the new technology is always higher: $f_1\left(L\right)>f_0\left(L\right),\forall L$. There is a fixed cost of adopting an AMS system. Following [69,70], assume that the investment can be measured by the same unit as the numeraire good. $x_0$ denotes the fixed cost of investment in the AMS system.

Under the general-equilibrium framework, we consider a utility-maximizing dairy-farm household, with utility that comes from the consumption of the output and leisure (that is, labor hours not devoted to production activities). The utility function of the household is $U(x,L)$. Utility is increasing in the consumption good x. From the standard labor-supply literature, assume utility is increasing in leisure, which is $\overline{L}-L$. Thus, utility is decreasing in the labor hours L devoted to producing the good. That is: $U_x>0,U_L<0$.

In sum, the farm household’s problem can be written as:

(1)$\underset{i,x,L}{max}U(x,L),s.t.f_i\left(L\right)-i·x_0=x,L\le \overline{L}.$

The Lagrangian of the problem is:

(2)$\mathcal{L}=U(x,L)+{\lambda }_1(x-f_i\left(L\right)-i·x_0)+{\lambda }_2(\overline{L}-L),$

where ${\lambda }_1$ and ${\lambda }_2$ are the Lagrange multipliers for the production constraint and labor-availability constraint.

Following [71], this optimization problem is solved through two stages: in the first stage, the farm household makes the technology decision. That is, the discrete choice between $i=0$ and $i=1$. In the second stage, the farm household decides optimal labor use under the chosen technology.

Using backward induction, first determine the optimality conditions for the second-stage problem. For both $i=0,1$, the first-order conditions are given by:

(3)$U_x+{\lambda }_1=0,$

and

(4)$U_L-{\lambda }_1{f}_i^{\prime }\left(L\right)-{\lambda }_2=0,$

with complementary slackness condition:

(5)${\lambda }_2\ge 0,\overline{L}-L\ge 0,{\lambda }_2(\overline{L}-L)=0,x=f_i\left(L\right)-i·x_0.$

From the optimality conditions, we can see that, when the labor-availability constraint is not binding, the first-order conditions can be rewritten as:

(6)$\frac{U_x}{U_L}=-\frac{1}{f_i^{\prime }\left(L\right)}.$

This is the standard marginal rate of substitution equal to the marginal rate of the transformation condition. When the labor-availability constraint is binding, the condition has to be adjusted by the relative shadow costs of the two constraints:

(7)$\frac{U_x}{U_L}=-\frac{1}{f_i^{\prime }\left(L\right)}+\frac{{\lambda }_1}{{\lambda }_2}.$

Let $V^0,V^1$ denote the indirect utility (or maximized utility) under the old technology and the new technology, respectively. In the first stage of the problem, the farm household chooses the technology that brings about higher utility:

(8)$U=max\{V^0,V^1\}.$

Definition 1.

Threshold-fixed cost of the new technology, denoted by $x^s$, is such that the household’s utility is the same under the old and new technologies:$V^0=V^1$.

Note that from the prior definitions, the adoption threshold is different from the definitions in the technology-adoption literature, such as [8,71]. In [8], the technology-adoption threshold is where a farm’s profit is the same under the old and new technologies. In this article, since the general-equilibrium framework emphasizes both the production and consumption side of the farm household’s decision-making, the adoption threshold is defined by the utility function instead of by profit.

A natural question is whether the farm will use more or less labor under the new technology. Proposition 1 is aimed at answering this question.

Proposition 1.

At adoption threshold $x^s$ , farms adopting the new technology will use more labor and produce more output than non-adopting farms.

Proof. 

See Appendix A. □

Figure 2 illustrates Proposition 1, when the labor availability constraint is non-binding. The gray curves are indifference curves. Since utility is decreasing in labor, the indifference curves are upward sloping. Under the old technology $f_0$, the marginal rate of substitution equal to the marginal rate of the transformation condition dictates that the indifference curve is tangent to the production function. Consequently, optimal labor use is at $L^0$, and the farm household achieves utility of $U^0$. At adoption threshold $x^s$, the farm is indifferent between the old and new technology. This implies that the production function under the new technology, $f_1$ subtracting the investment $x^s$, must be tangent to the same indifference curve $U^0$. As predicted in Proposition 1 and shown in Figure 2, the farm household will use more labor ($L^1>L^0$), and the consumer will use more output, under the new technology at the adoption threshold.

The intuition behind Proposition 1 is that labor-saving technology makes labor input relatively less valuable in the general-equilibrium framework. At the adoption threshold, the two production functions have to be tangent to the same indifference curve. Thus, the marginal rate of substitution must be different under the two production technologies. This, in turn, implies that the marginal rate of transformation under the two production technologies must be different at the adoption threshold. Moreover, since the output under the new technology is always higher, the only possible scenario is that the marginal rate of transformation is higher under the new production technology than the old technology. Therefore, we must confirm that the farms adopting the new technology will use more labor.

Proposition 1 reconciles the first puzzling, stylized fact in AMS adoption: that labor-saving technology may not necessarily save labor. From the survey of dairy farms in Idaho [7], a majority of the farms that are interested in adopting an AMS system report that they do not expect fewer labor hours to be used in production. This is consistent with the finding from [72], which found no significant savings in labor compared to conventional milking systems. Overall, Proposition 1 predicts that when adoption is at early stage, adopters are more likely to be close to the adoption threshold, which means that they are more likely to expand production to compensate for the fixed cost of investment. Moreover, when the adoption threshold reduces, it will be more likely to observe significant savings in labor.

3.2. Heterogeneity

In this subsection, we explore the role of heterogeneity in the adoption of an AMS and aim to provide an explanation to why small farms could be more likely to adopt an AMS than large farms. Again, for simplicity, we assume that farms are heterogeneous in one dimension: size of farm. Farm size is measured by an index $\alpha$, $\alpha$, which is ranging from 0 to 1, where a higher value of the index indicates a larger farm. Furthermore, we assume that a larger farm and a smaller farm differ in two important ways. First, we modify the baseline model as follows: following [71], let $h_i\left(\alpha \right)·L$ be the efficiently used labor under any production technology i, so the production of the output becomes a function of the efficiently used labor. We assume that a larger farm achieves a higher labor efficiency than smaller farms. This could be due to the fact that a larger farm possesses more managerial capital than a smaller farm. Formally, the assumption can be written as:

(9)$f_i(h_i\left({\alpha }_1\right)·L)>f_i(h_i\left({\alpha }_2\right)·L),\forall i=0,1,{\alpha }_1>{\alpha }_2.$

Second, we assume that different-sized farms value their leisure time differently (or their marginal utility of labor). To reflect this point, we modify the baseline model and assume that $U=U(x,v(\alpha \left)L\right)$, where

(10)$U(x,v\left({\alpha }_1\right)·L)>U(x,v\left({\alpha }_2\right)·L),\forall {\alpha }_1>{\alpha }_2.$

In this way, we are effectively assuming that larger farms value their labor more (or their leisure time less) than smaller farms. For simplicity, in this subsection, we focus on the interior solution for the farm and assume a non-binding labor-availability constraint. Then, for a farm with size of $\alpha$, the utility-maximization problem can be rewritten as:

(11)$\underset{i,x,L}{max}U(x,v\left(\alpha \right)L),s.t.f_i\left(h_i\left(\alpha \right)L\right)-i·x_0=x.$

In addition, the optimality conditions become:

(12)$U_x+{\lambda }_1=0,$

and

(13)$U_{L}v\left(\alpha \right)-{\lambda }_1{f}_i^{\prime }\left(L\right)h_i\left(\alpha \right)=0,$

which can be further rewritten as:

(14)$\frac{U_x}{U_L}=-\frac{1}{f_i^{\prime }\left(L\right)}\frac{1}{v\left(\alpha \right)h_i\left(\alpha \right)}.$

Under such a formulation, the adoption threshold $x_s$ is no longer a constant number but a function of $\alpha$. Notably, when $x_s\left(\alpha \right)$ is decreasing in $\alpha$, we will have a smaller farm that faces a smaller adoption threshold than a large farm. Proposition 2 explores such conditions.

Proposition 2.

$x_s\left(\alpha \right)$ is decreasing in α, if $\frac{h_1\left(\alpha \right)}{h_0\left(\alpha \right)}$ is decreasing in α.

Proof. 

See Appendix B. □

The intuition behind Proposition 2 is that, in the same spirit as [71], when $\frac{h_1\left(\alpha \right)}{h_0\left(\alpha \right)}$ is decreasing in $\alpha$, smaller farms experience higher labor-efficiency gains under a new technology than large farms. Thus, smaller farms require a lower adoption threshold to justify the investment. In addition, the general-equilibrium framework allows for another explanation: smaller farms value their leisure time more than large farms (that is, $v\left(\alpha \right)$ is decreasing in $\alpha$).

Proposition 2 reconciles the second puzzling, stylized fact in AMS adoption: that the smaller farms are more likely to adopt an AMS than large farms. Proposition 2 suggests that smaller farms may face a lower adoption threshold than large farms. This is consistent with [73], which compared the profitability of AMS dairies and found that smaller-sized dairies could be more profitable using an AMS. Moreover, the value of leisure time is documented in [3], which suggests that an important benefit of an AMS for small farms is the increased flexibility among its operators.

It should be also noted that when labor availability constraint is introduced to the model, a third explanation could also be introduced: value of off-farm income. Consider a farm that adopts an AMS, where the labor availability constraint is binding. Then, the shadow price of the labor-availability constraint, ${\lambda }_2$, can be interpreted as not only how much this farm household values its leisure time, it also represents the value of this farm household’s off-farm income.

4. Discussion

In general, our model predictions are consistent with prior findings in the literature. [74] conducted a survey in 2013 of dairy producers in Maryland and Pennsylvania, investigating factors affecting their decision-making in transitioning to an AMS. Results indicate the most influential aspects for AMS adoption were the ability to improve herd management and increase availability of family time. The greatest concerns were in the return on investment and the challenges of changes in management style. Another finding was the positive correlation between the (higher) levels of education of a dairy farmer and their willingness to adopt an AMS. Similarly, [3] conducted an empirical study of AMS adoption in Norway and found that the main factors affecting AMS-technology adoption included a reduced and increased flexibility of the workload. Finally, [73] compared the profitability of AMS dairies with 120, 240, and 1500 cow-herd size versus parlor systems, finding higher profitability for the smaller-sized dairies. These empirical findings confirm the importance of $v\left(\alpha \right)$ and $\frac{h_1\left(\alpha \right)}{h_0\left(\alpha \right)}$ in the decision-making for various-sized farms. Although it is not the main focus of this paper to discuss market equilibrium, our framework allows for the discussion on this topic. This framework is especially useful to analyze the impact of labor-market fluctuations on the adoption of labor-saving technologies. In this section, we briefly outline such a framework, which incorporates the joint labor–market equilibrium and machinery–market equilibrium. Following [75], a market-equilibrium condition in our model is a joint market clearing of the output market, labor market, and machinery market.

Let X be the set of farmers. The set of adopters A is all the farmers adopting the AMS system: $A=\{\alpha \in X|x_0<x^s\left(\alpha \right)\}$. Then, the set of non-adopters is the complement of A, denoted by $A^c$. Next, the aggregate supply of output, $Q^s$, is given by the total production from both the adopters and non-adopters:

(15)$Q^s=\underset{x\in A}{\int }{f}_1\left(h_1\left(\alpha \right)L^1\right)dx+\underset{x\in A^c}{\int }{f}_0\left(h_0\left(\alpha \right)L^0\right)dx.$

Similarly, aggregate labor demand ($L^d$) can be written as the labor demand from both the adopters and non-adopters:

(16)$L^d=\underset{x\in A}{\int }{L}^{1}dx+\underset{x\in A^c}{\int }{L}^{0}dx.$

Finally, machinery demand ($M^d$) is the total number of AMS systems demanded from adopters:

(17)$M^d=\underset{x\in A}{\int }{𝟙}_{x\in A}dx$

We can now define the joint equilibrium for the three markets.

Definition 2.

The joint supply-chain-market clearing condition is determined by the following set of conditions:

• Clearing of the output market:$Q^s=Q^d$.

• Clearing of the labor market:$L^s=L^d$.

• Clearing of the machinery market:$M^d=M^s$.

Here, the demand for output, labor supply, and machinery supply are given exogeneously. This allows comparative-static analysis of a shock in one market on the equilibrium in another market.

5. Conclusions

In this article, we provide a general-equilibrium model to explain two stylized facts in the adoption of AMS. We show that, under certain conditions, farms adopting a labor-saving technology may, in fact, use more labor to compensate for the loss in the value of labor; and second, when smaller farms experience more labor efficiency gains or value their leisure time more (or for off-farm income), they are more likely to adopt the new technology than larger farms.

These findings enrich the large body of the technology-adoption literature. Ref. [76] indicates how the diffusion of technology adoption has generally been approached according to two venues: (i) A disequilibrium process, where the introduction of an innovative technology creates (market) uncertainty regarding its operation conditions and performance. Our paper strengthens this argument by providing a new angle on the heterogeneity of performance of early adopters. (ii) A decision-theoretic nature, viewing diffusion as an equilibrium process. This equilibrium process affects adoption in accordance to the varying levels of benefits received from different (potential) technology adopters. Ref. [76] argues that the disequilibrium process is relevant to study technology that is completely functional when used in any pertinent operation. Conversely, some technology will be effective depending on the size and characteristics of the operation, thus applying a decision-theoretic nature to study the process is appropriate for this latter case. Ref. [77] studies different farm-level characteristics to identify factors that affect the role of awareness of (precision agriculture) technology, and its impact in the diffusion process of technology adoption. Features such as farm size, operator education, level of time dedicated to farming (full-time over part time) had a substantial positive effect, while the age of the operating farmer had a negative effect on adoption. Our paper expands this line of research by providing theoretical foundations of the general-equilibrium effects of technology adoption.

This paper has three main limitations. First, our conceptual model captures one input, labor, without including other important inputs such as land and capital use. The simplification allows of tractable solution. However, excluding other inputs limits the applicability of the model. For example, it is difficult to explore capital-labor substitution within our conceptual framework. Second, we explore the heterogeneity of adopters in one dimension: farm size. In reality, the heterogeneity could be coming from multiple sources, which deserves further investigation. Third, another important limitation of the paper is we cannot explicitly add financial and resource constraint to the model. This is particularly important for illustrating barriers to entry (see discussion in [11]) and the heterogeneity of resource constraints ([78]).

For future research, an important element is to explore factors beyond the size of labor. As discussed in the literature, herd size and labor costs are not the sole relevant drivers of AMS adoption, as interest in technology and openness to collaboration are also main factors that lead to adopting an AMS. Northern Europeans tend to be early adopters of technology, and returns are not the only priority in their decision-making process [3]. Ref. [79] studied technology diffusion, determining that geography plays a major role for two reasons: proximity to product and interaction with adopters. Both of these aspects are pertinent characteristics of European dairy farmers, possibly serving as additional catalysts in their adoption of an AMS. Another important element is to further investigate the role of big data in technology adoption ([80,81]).

Author Contributions

Conceptualization, X.D.; methodology, X.D.; formal analysis, X.D. and Z.Y.; investigation, H.T.; resources, X.D. and H.T.; supervision, L.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. Number of dairy farms in the states of Idaho and Washington, in 2002 and 2017. Source: USDA Agricultural Census.

Figure 2. Illustration for Proposition 1.

Table 1

Literature Classification.

Context	Subject	Author(s)	Conclusion/Impact
Agricultural	LaborMarket	S.K.Jayasuriya& R.T.Shand (1986) [12]	Contributes to increased productivity,but lead to net reductions of agricultural labor use
		Gardner & Rausser (2001) [13]	Decrease in demand for farm labor
		Charlton (2019) [14]	Put downward pressure on wages in agricultural production
	Production	Murali & Balakrishnan (2012) [15];Taghinezhad et al. (2014) [16];Alexander et al. (2018) [17];Feng et al. (2013) [18];Charlton (2019) [14]	Reduce production costs and improve production efficiency
	EconomicStructure	Mueller et al. (2019) [19]	Played a fundamental role historically in the structural transformationof agrarian economies; diversify their farming activities and save time
Manufactory	LaborMarket	Berman et al. (1994) [20]	Drives the shift in labor demand away from unskilled and towardskilled labor in U. S. manufacturing over the 1980s.
Banking	LaborMarket	Fung (2006) [21]	Labor-saving technologies are associated with higherfirm-level employment
		Fung (2008) [22]	Increase the efficiency of human resources
High-tech industries	Labor Market	Coad & Rao (2011) [23]	Optimizing the labor demand-supply structure

Appendix A

The proof proceeds by calculating the constraint-optimization problem and then proof by contradiction. First, farmers will face the utility-maximization problem as follows:(A1)$$\underset{i,x,L}{max}U(x,L^i),s.t.f_i\left(L^i\right)-i·x_s=x.$$

Denote that i is the indicator for labor-saving-technology adoption or not (i =1 represents the labor-saving-technology adoption, otherwise i = 0). Next, set up the Lagrangian equation as:(A2)$$\mathcal{L}=U(x,L^i)+{\lambda }_1(x-f_i\left(L^i\right)+i·x_s).$$

By taking the first-order derivatives and acquiring the optimal conditions:(A3)$$U_x+{\lambda }_1=0$$ and (A4)$$U_L-{\lambda }_1{f}_i^{\prime }\left(L^i\right)=0,$$ then, we can conduct the marginal rate of substitution ($MR{S}_i$) and get following condition:(A5)$$\frac{U_x}{U_L}=-\frac{1}{f_i^{\prime }\left(L^i\right)}.$$

At the threshold, which makes farmers keep the same utility level ($U^0$) with a different MRS, we will have:(A6)$$f_1\left(L^1\right)-f_0\left(L^0\right)=x^s.$$

Assume that $F\left(L\right)$=$f_1\left(L\right)$−$f_0\left(L\right)$. By taking the derivatives w.r.t L, we could know that $F^{\prime }\left(L\right)$ = $f_1^{\prime }\left(L\right)$−$f_0^{\prime }\left(L\right)$. From our assumption that the marginal production after using labor-saving technology would be higher than farmers that do not use it, $f_1^{\prime }\left(L\right)$ > $f_0^{\prime }\left(L\right)$, we conclude that $f_1^{\prime }\left(L\right)$−$f_0^{\prime }\left(L\right)$> 0 and $F^{\prime }\left(L\right)$> 0, thus $F\left(L\right)$> 0, such that $f_1\left(L\right)$>$f_0\left(L\right)$.

Second, suppose that $L^0$>$L^1$. Then, from $f_i^{\prime }$> 0 and $f_i^{″}$< 0, we would have the following conditions:(A7)$$f_1\left(L^0\right)>f_1\left(L^1\right)>f_0\left(L^1\right)$$ and (A8)$$f_1^{\prime }\left(L^0\right)>f_0^{\prime }\left(L^1\right).$$

From the conditions above, we could know that $MR{S}_1$>$MR{S}_0$. Combining the diminishing return to the marginal increase in the labor inputs for $f_0\left(L\right)$ and $f_1\left(L\right)$, we can derive for a certain amount of labor $\overline{L}$ ($\overline{L}$>$L^0$), at which the two production curves will intersect and result in:(A9)$$f_1\left(\overline{L}\right)=f_0\left(\overline{L}\right).$$

However, we have known that $x^1=x^0+x^s$ at the threshold, which means $f_1\left(\overline{L}\right)$>$f_0\left(\overline{L}\right)$. So, $L^0$>$L^1$ violates our assumption and could not exist. Therefore, $L^1$>$L^0$.

Appendix B

This follows directly from (A1), except that the relevant threshold function $F\left(L\right)$ becomes:(A10)$$F\left(L\right)=f_1\left(h_1\left(L\right)L\right)-f_0\left(h_0\left(L\right)L\right)$$ and (A11)$$F^{\prime }\left(L\right)=f_1^{\prime }\left(h_1\left(L\right)L\right)h_1^{\prime }\left(L\right)-f_0^{\prime }\left(h_0\left(L\right)L\right)h_0^{\prime }\left(L\right).$$

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