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The classical bargaining model for organized labor.

What is known today as "classical economics" spanned a period from roughly the 1750s to the 1870s. We date its beginning with Richard Cantillon's Essai sur lan Nature du Commerce en General in 1755 (1964), and its ending with J. E. Caimes's Some Leading Principles of Political Economy Newly Expounded, published in 1874. (1) In reading the primary texts of the Classical school, we notice divergent views on many issues: production, exchange, money, technology, etc. However, one common thread runs through the corpus of knowledge known as classical economics. Classical economists believe that economic value, which is the basis for all discussions pertaining to markets and prices, was determined by the costs of the factors needed to produce the good in question. Thus, classical economics is characterized by a "cost of production" theory of value. The distribution of the gains from production was initially tied to this theory of value but was eventually treated as its own theory of distribution. Once the theories of production and distribution became disentangled, economists were able to envision ways to influence distributional outcomes that could alleviate the suffering of the working classes.

Classical economists treated all economic production as a function of three categories of factors: land, labor, and capital. The price paid by a consumer for a finished product was dependent upon the total value of the good's production as derived from the value of the land, labor, and capital necessary for the good's completion. Consequently, the value of a good is measured in terms of the returns to the factors of production. This point is important in understanding the dynamics of the classical economic model. Prices reflect the payments made to the factors of production, so the returns to capital (i.e., profit) and to labor are inversely related when prices are stable. Thus, increasing profits always occur at the expense of the other factors of production. This, of course, includes the payments to labor. As capitalist profits rise in this model, wages fall, and vice versa.

In as much as the capitalist or entrepreneur is required to combine the factors of production, it is easy to see how the classical economists could arrive at the conclusion that the capitalists and the laborers would eventually become antagonistic. The classical doctrine regarding wages is pessimistic. Classical economists developed the notion of an "iron law of wages," that wages paid to labor would trend toward a level of bare subsistence as capitalists endeavored to increase their profits in the face of diminishing returns to production. This long-run trend would result in the "stationary state," where society is characterized by static equilibrium of bare subsistence wages for the vast majority of people. The classical economists as a group tended to be egalitarian in their thinking, writing, and action. In word and deed, they consistently championed the rights of the working classes. (2) This social philosophy, coupled with the belief in a miserable stationary state, led the classical economists to develop an emphasis on the importance of labor in the computation of factor costs.

In this paper, we explore the classical labor theory of value and the implications it produces for a theory of distribution. In particular, we focus on the work of John Stuart Mill, who produced the most comprehensive and synthesized treatment of classical economics. Mill's extension of the classical labor theory of value provides for a separate theory of distribution. This theory in turn offers an interesting economic rationale for the existence of labor unions as a means of improving the desirability of the stationary state. We extend the classical model by presenting one of John Stuart Mill's unique contributions to classical value theory, which allowed him to posit that society is governed by different laws for production and distribution. In contrast to earlier classical writers, Mill believed that the recognition that production and distribution are separate occurrences gave reason to be optimistic about the long run. To Mill, the perfection of society, seen in a prosperous stationary state, was in part achieved by labor unions securing hitherto unrealized advantages to labor. Moreover, Mill, through his economic argument in favor of organized labor, actually foresaw the modern literature on uncertainty and information. We illustrate this contribution by way of an example that captures the distributional gains that workers enjoy from repeated negotiations between unions and employers.

Classical Wage Theory

The relationship of wages to profits in the classical model can be seen using Ricardo's construct. (3) Ricardo's distributional model is based on the assumption that wages and profits are a function of agricultural production and manufacturing and, from these, "natural prices" of food and manufactures could be derived. To begin, Ricardo assumes that in the long run manufacturing output per year will equal the additions to money supply per year, which he considers to be the quantity of gold manufactured. The "natural price" of manufactures would therefore be the ratio of the additions to the gold supply divided by annual output of manufactures. We can see from this relationship that, if the output of gold remains constant and we have increasing returns in manufacturing, the natural price of manufacturers will decrease. In the case of agriculture, the classical economists were aware that land quality and proximity had an impact on agricultural production. The output of food per worker was expected to diminish over time as marginal land, farther away from population centers, is brought into cultivation. The natural price of food can also be thought of as a ratio of the additions to the money supply to annual increases in agricultural output. In contrast to the natural price of manufacturers, Ricardo expected the natural price of food to rise over time. This would take place because of the diminishing returns experienced in agriculture.

Once natural prices of food and manufactures are obtained, the "natural wage" becomes the sum of the natural prices of manufactures and food required to sustain laborers and their dependents. As Eltis (2000) illustrates, Ricardo believed that in an expanding economy, it would be necessary for workers to receive rising money wages to offset the impact of diminishing returns in agriculture. The money wage could fall over time if changes in technology created greater labor productivity in agriculture and industry. Ricardo, however, believed that in the long run the loss in agricultural productivity brought about by diminishing returns would outweigh any productivity gains experienced by improved technology. As a result, the natural wage would tend to rise.

We are now able to see the classical relationship between wages and profits. According to Ricardo, annual agricultural and manufacturing output would, in the aggregate, be valued at the annual addition to the money (i.e., gold) supply. This would correspond to the capitalists' revenue for selling manufactured goods and agricultural products. If, as assumed by Ricardo, capital to labor ratios are the same in agriculture and manufacturing, and if workers are paid the natural wage, profits are then measured as the gold produced minus the natural wage. From this expression we can see the classical notion that profits and wages become antagonistic. Again, Ricardo and other classical economists did not overlook the possibility that technology changes could increase productivity in agriculture and manufacturing. Ricardo just believed that over time, lower land fertility would have greater impact on the natural price of food. Because of declining land fertility in an expanding economy, capitalists would be required to pay higher wages. These higher wages would reduce profits per worker. Declining profits per worker would be a contributing factor to leading the economy to the stationary state.

Classical Long Run Conclusions: Mill's Reinterpretation

The most advanced statement of classical economic theory occurs with John Stuart Mill's book Principles of Political Economy, published in 1848. In many ways, Mill's book was a synthesis of economic thought, from Adam Smith to the present. Mill did, however, advance the classical theory in new directions. One of the unique elements of Mill's theory was his belief that society was governed by different sets of economic laws: one, laws of production, and two, laws of distribution. On the one hand, the laws of production are fixed. These laws are physical laws dictated by the production technology. An example would be the law of diminishing returns. The results of diminishing returns were uncontested because these outcomes could be easily verified. On the other hand, Mill viewed the laws of distribution as variable, with deliberate changes affecting desired social outcomes. Examples given by Mill include progressive income taxes, death taxes, land reform, emigration, and population control. What is interesting to note here is that, according to Mill, the separation of these economic laws allows for the possibility of society being able to alleviate the misery implied by the classical notion of the stationary state (King and Yanochik 2011).

In this way, Mill differed from the other classical economists. Whereas previous classical writers believed the stationary state was an outcome of stagnation and poverty, Mill believed that the stationary state was a desirable social outcome. Although stagnation would result from these laws of distribution, incentives to work would not be impaired. The stationary state would take place at the most efficient level of production and all individuals would enjoy a relatively prosperous existence. In addition, Mill believed that in the stationary state, people would be relieved of the tedious and arduous jobs that were common in the earlier classical conception. The stationary state was to Mill an outcome to be strived for by society.

According to Mill, the mentality of society would have to accept these ideas which limited fortunes and population. When this change of mindset was complete, the laws of distribution could be constructed to bring about the prosperous stationary state. Laws that pertained to organized labor were one example of these distributional laws. Mill understood that wages were the result of bargaining but that, in most cases, the worker was at a disadvantage as compared to the employer in knowing the market value of his work. As a result, Mill believed that organized labor could serve a valuable market function by providing information to workers about their market value. The organization of workers would put them in a better position than individual workers to bargain for wages. It is interesting to note that Mill considered labor unions, because of these functions, to be an important component of market-based, competitive economic systems. (4) We can also state that Mill, in his recognition of the importance of information in labor negotiations, anticipated the modern literature on labor market information and uncertainty. (5)

A Bargaining Model Based on Mill's Assertion

Mill's insight suggests a game theoretic model of bargaining from modern economics. One effect of organizing labor is that it changes the "higgling" (6) between the two sides from a static bargaining process to a dynamic game wherein the employer deals not with the individual worker in a single period but with the union in multiple periods. Consider the following simplified model. Workers have a value to an employer, [pi], that is distributed uniformly on support [0,[[pi].sub.H]], The lone employer is informed about the realized value of [pi] while the workers know only the distribution. We assume both sides to be risk neutral and allow the worker or union to make wage demands, which the employer may then accept or reject. This setup allows us to maintain the simplicity of the model while highlighting the role of information in the outcome. We first consider the case of a single non-union employee who makes an offer to an employer that the employer accepts or rejects and then we compare this to a situation in which the union can make offers on the employee's behalf. Because a union will interact with the firm repeatedly, we allow the union to make a second offer in the case that the first offer is declined. As suggested by Mill, this repeated interaction with the employer provides the union valuable information that allows for an updating of beliefs and a more precise estimate of the employee's value to the firm.

In the case of an employee making an offer without the assistance of a union, the firm will accept an offer of w if the firm's privately realized value of [pi] lies in the range [w, [[pi].sub.H]], which happens with probability ([[pi].sub.H] - w)/[[pi].sub.H]. The worker therefore chooses a wage offer, w, to maximize w([[pi].sub.H] - w)/[[pi].sub.H]. The worker's optimal wage offer in this case is

[w.sup.*]([[pi].sub.h]) = [[pi].sub.h]/2. (1)

This offer will be accepted by all firms for which [pi] > [[pi].sub.H]/2, leaving the worker with an expected payoff of

[[pi].sub.H]/4. (2)

In this case there is only a fifty percent chance of a worker and firm striking a deal. A more involved model could include heterogeneous prior beliefs on the part of a stream of workers that would lead to a variety of wage demands and thus the reasonable belief on the part of the firm that some future worker will make an acceptable offer. One could likewise expand the model to include the worker's future prospects. Although these extensions are beyond the purpose of the current example, they too would serve to highlight the role and value of information in forming the "laws of distribution."

On the other hand, if we allow labor to "stand out for terms" with "organized concert" as Mill suggested by forming a union that engages repeatedly with the firm, then the workers will update their beliefs (i.e., learn) regarding the true nature of [pi]. The following simplified analysis is adopted from Gibbons (1992). Consider a two period model in which the union makes a wage offer, [w.sub.1], in the first period which the firm accepts or rejects. The payoffs if the firm accepts the offer are [w.sub.1] for the worker and ([pi] - [w.sub.1]) for the firm. If the firm rejects the first period offer, then the union makes another offer, [w.sub.2], in the second period. The payoffs if the union accepts the offer in the second round are [delta][w.sub.2] for the worker and [delta]([pi] - [w.sub.2]) for the firm. If the firm rejects the second period offer then payoffs are zero for both the union and the firm. A perfect Bayesian equilibrium in this model consists of wage offers, [w.sub.1] and [w.sub.2]([w.sub.1]), for the union, acceptance rules for the firm, and union beliefs in each round regarding the value of a worker to the firm such that the wage offers maximize the union's payoff given its beliefs and the firm's subsequent strategy, the firm's acceptance rules maximize the firm's expected payoff, and the union's beliefs are determined via Bayes' rule. We ensure a sub-game perfect outcome by examining the second period first.

Having offered a wage, [w.sub.1], resulting in a rejection in the first round, the union updates its beliefs about the firm's type. The firm's decision to reject the first period offer, [w.sub.1], while expecting a second period offer of [w.sub.2] indicates that the firm prefers either rejecting both offers or accepting [w.sub.2] rather than accepting wq. The payoff from rejecting both wage offers is zero and the payoff from accepting [w.sub.2] in the second period is [delta]([pi] - [w.sub.2]). Therefore, the union updates its beliefs to incorporate that the firm rejected wq either because [pi] - [w.sub.1] <0 or because [pi]-[w.sub.1] < [delta]([pi] - [w.sub.2]). The highest type that would have rejected the first period offer of [w.sub.1] is therefore [pi]=max{[w.sub.1], ([w.sub.1] - [delta][w.sub.2])/(1-[delta])}. The union's belief is now that the firm's type is uniformly distributed on [0, [[pi].sub.1]]. Given this belief, the optimal offer in the second period is [w.sub.2] = [[pi].sub.1]/2, which yields the following implicit function for [[pi].sub.1], the marginal type from the first round: [pi]=max{[w.sub.1], ([w.sub.1] - [delta][[pi].sub.1]/2)/(1-[delta])}. Notice that [w.sub.1] [greater than or equal to]([w.sub.1] - [delta][[pi].sub.1]/2)/(1-[delta]) leads to a contradiction since it implies both that [[pi].sub.1] = [w.sub.1] and that [w.sub.1] [less than or equal to][[pi].sub.1]/2. Therefore, it must be the case that [w.sub.1] < ([w.sub.1] - [detla][[pi].sub.1]/2)/(1-[delta]) and [[pi].sub.1] = ([w.sub.1] - [delta][[pi].sub.1]]/2)/(1-[delta]). Also, note that the optimal offer in the second round is lower than that from the one-shot game and a hire is therefore more likely to occur in the presence of a union. Solving for the highest type that would have rejected the first offer (who does so because he prefers accepting the second period offer to accepting the first period offer), we obtain [[pi].sub.1]([w.sub.1])=[2w.sub.1]/(2 - [delta]) and therefore

[w.sub.2]([w.sub.1]) = [w.sub.1]/(2-[delta]). (3)

The problem for the union is now to choose [w.sub.1] to maximize [w.sub.1] ([[pi].sub.H] - [[pi].sub.1] ([w.sub.1]))/[[pi].sub.H] + [delta][w.sub.2]([w.sub.1])([[pi].sub.1]([w.sub.1]) - [w.sub.2] ([w.sub.1]))/[[pi].sub.1] ([w.sub.1]). The solution to this maximization problem is given by

[w.sub.1.sup.*] = [[pi].sub.H]/2 - [delta][[pi].sub.H]/8. (4)

Note that the first round offer, [w.sub.1.sup.*], is less than the equilibrium offer in the one-shot game given in Eq. (1) and is equal to that offer in the event that second round payoffs are completely discounted (i.e., for [delta]=0). The reason for lowering the first period offer relative to that from the non-union case is that the firm will now reject offers that would have been acceptable in expectation of a better second round offer. The marginal benefit of increasing the wage demand due to the existence of the second round is simply outweighed by the marginal cost in terms of an increased likelihood of rejection. Plugging the optimal first round offer into Eq. (3), yields

[W.sub.2.sup.*] = ([[pi].sub.H]/2 - [delta][[pi].sub.H]/8)/(2 - [delta]). (5)

To fully characterize the perfect Bayesian equilibrium, we combine these strategies for the union with the union's beliefs that the true value of [pi] is uniformly distributed on [0, [[pi].sub.H]] in the first round and, in the case that [w.sub.1.sup.*] is rejected in the first round, that [pi] is uniformly distributed on [0, [[pi].sub.1] ([w.sub.1.sup.*])]. The equilibrium strategy for the firm is to accept the first round offer if [pi] [greater than or equal to] [[pi].sub.1] ([w.sub.1.sup.*]), accept the second round offer if [[pi].sub.1]([w.sub.1.sup.*])> [pi] > [w.sub.2.sup.*], and not to hire otherwise. Using Eqs. (4) and (5), the union or worker's expected payoff in this game is given by:

[[pi].sub.H]/4 + [[delta].sup.2][[pi].sub.H]/[32(2-[delta])], (6)

which is strictly larger than the expected payoff given by Eq. (2). While this analysis is greatly simplified, it serves as a modern equivalent of Mill's assertion that organizing can give workers information about the wage that the demand for their labor would justify and shows that such organization of labor will lead to a higher expected worker payoff than they would receive by acting alone. At the same time, the updating of beliefs means that the firm is not only more likely to be successful in hiring a worker but also does so at a lower expected wage. The firm therefore benefits more from unionization than the worker does in our model, although the importance of this result should not be overstated given the simplified nature of the model.

Conclusion

For all the advances made by members of the classical school, they were not able to arrive at a theory of distribution that described mutually beneficial outcomes for both workers and capitalists. Classical theory was based on a cost of production theory of value, and as such, was unable to explain wages as a function of marginal productivity, and therefore how profits and wages could increase at the same time. Because of diminishing productivity in agricultural production, economic expansion would require capitalists to pay higher wages to workers. As wages rose, capitalist profits would necessarily fall. According to Ricardo, over time this process would lead society to the stagnant stationary state.

The early classical economists were led to a belief that a stationary state would involve a miserable subsistence level wage for the working class. John Stuart Mill, one of the last of the classical economists, had a different idea about the outcome of society in the stationary state. In this paper, we have presented Mill's theory that production and distribution were inherently different concepts. According to Mill, while the laws of production were fixed by technology, distributional outcomes were dependent upon the interaction between labor and the owners of capital. From this conception, Mill was able to envision a potentially optimal stationary state which economic policy could and, according to Mill, should be constructed to bring about.

As mentioned, the classical economists, including Mill, predated the marginalist revolution, and therefore did not have a correct understanding of wage determination. Classical economists did, however, understand the problems of wage negotiations between employers and employees. Mill gave the most accurate and precise account of the root of this problem. An individual worker, acting independently, does not have the ability to accurately assess his value in the marketplace. Without this assessment, workers would be left with a diminishing share of the value of production. Labor organizations have the ability to get the data needed in order to determine labor's market value. From Mill's theoretical dichotomy, we see an economic theory that justifies the potential market clearing function of organized labor. We have presented an example illustrating how an alteration to the laws of distribution as simple as allowing labor to organize can help alleviate the conditions of the working class in the classical notion of the stationary state. Mill's thoughts on labor not only gave classical economists a different perspective on the stationary state but foreshadowed the economic literature on imperfect information.

DOI 10.1007/s11293-015-9463-5

References

Akerlof, G. A. (1970). The Market for "Lemons": quality uncertainty and the market mechanism, Quarterly Journal of Economics, 84(3), 488-500.

Bai, C., & Wang, Y. (2003). Uncertainty in labor productivity and specific human capital investment. Journal of Labor Economics, 21(3), 651-675.

Bhaskar, V, Manning, A., & To, T. (2002). Oligopsony and monopsonistic competition in labor markets. Journal of Economic Perspectives, 16(2), 155-174.

Cairnes, J. E. (1874). Some leading principles of political economy newly expounded. New York: Harper & Brothers.

Cantillon, R. (1964). Essai sur Ian Nature du Commerce en General. New York: Augustus M. Kelley Publishers.

Dixit, A. K., & Pinkyck, R. S. (1994). Investment under uncertainty. Princeton: Princeton University Press.

Eltis, W. (2000). The classical theory of economic growth (2nd ed.). Hampshire: Palgrave.

Gibbons, R. (1992). Game theory for applied economists. Princeton: Princeton University Press.

Jevons, W. S. (1881). Richard Cantillon and the Nationality of Political Economy. Contemporary Review.

King, J. T., & Yanochik, M. A. (2011). John Stuart Mill and the economic rationale for organized labor. The American Economist, 56(2), 28-34.

Mill, J. S. (1987). Principles of political economy. Fairfield: Augustus M. Kelley Publishers.

Prendergast, C. (2002). Uncertainty and Incentives. Journal of Labor Economics, 20(2), S115-S137.

Sowell, T. (1994). Classical economics reconsidered. Princeton: Princeton University Press.

Stigler, G. (1962). Information in the Labor Market. The Journal of Political Economy, 70(5), 94-105.

Mark A. Yanochik [1] * John T. King [1]

Published online: 26 June 2015

([mail]) Mark A. Yanochik

myanochi@georgiasouthem.edu

[1] Department of Finance and Economics, Georgia Southern University, Statesboro, GA 30460-8151, USA

(1) According to W. S. Jevons (1881), Cantillon wrote the first modem, systematic treatise on economics. Cairnes's book signaled the end of classical economics because he wrote it as a defense of the new paradigm in economics that emerged in 1871, the "marginalist revolution."

(2) Sowell (1994), p. 29

(3) This section is based on Eltis (2000), pp. 186-198.

(4) Mill (1987), Book V, Chapter X, pp. 936-937.

(5) The modern literature on this topic begins with Stigler's seminal paper (1962). Also see (for example) Bhaskar, et. al. (2002), Prendergast (2002), and Bai and Wang (2003). Mill's analysis of labor negotiations bears some resemblance to the "lemons problem" (Akerlof 1970), where labor may be subjected to a "loss in resale value" due to asymmetric information about product quality (Dixit and Pindyck 1994, p. 249). The key difference between Mill and Akerlof is that in labor negotiations we have an uninformed seller making an offer to an informed buyer rather than an informed seller making an offer to an uninformed buyer.

(6) This was Mill's term for "haggling."
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