The Greatest Happiness of the MINIMUM Number

I uploaded the manuscript titled "Equal Market Design I: Competitive Market Achieves the Greatest Happiness of the Minimum Number" at SSRN a month ago, which has been downloaded more than 500 times so far! I would like to say big thanks to all of those who kindly red it :)

The main purpose of the paper is to clarify the trade-off between efficiency and equality when redistribution by the third party is infeasible, which has not been documented (at least in a clear manner) in the literature. As a somewhat striking result, we show that the number of agents who engage in trades under market equilibrium must be minimum among all Pareto efficient and individually rational allocations. (provided that Pareto efficiency is modified from the standard definition in order to incorporate our presumption of no possible redistribution.)

While the market equilibrium (an intersection of supply and demand) maximizes total surplus, the sum of the agents' gains from trades measured in monetary value, it inevitably generates the agents who are left-behind from any trades. Moreover, the number of such left-behind agents are maximized through the competitive market. The intuition behind the result can be illustrated by the following figure.

In the figure, the number of successful trades or trade volume is 2. Consequently, 4 agents, 2 buyers and 2 sellers, are left-behind. As I explain in the paper, we can strictly increase the number of (mutually beneficial) trades if alternative buyer-seller matchings are possible. While such allocations reduce total surplus, equality is improved in the sense that more agents can enjoy surplus from trades and less agents become left-behind. That is, there exists a trade-off between efficiency (= total surplus) and equality (= trade volume).

The above finding is not restricted to a specific example; I formally show that there exists a strictly more equal allocation than the competitive equilibrium under very weak assumptions. Therefore, we could essentially say that equilibrium allocations under competitive markets are most unequal (if Pareto efficiency and individual rationality are considered to be least requirements for any sensible allocation). This finding may suggest a potential limitation of market economy even if no market failure is presupposed.

To check the further argument, please click here and download my paper! It is extremely short (the current version has only 8 pages), intuitive, and non-technical. In fact, it is by far the least technical among the papers that I have ever written, but I believe that its message and policy implication are the most significant.

For example, consider a labor market. My finding implies that employment is minimized if the market works competitively. By contrast, some frictions or social mechanisms that prevent the market from being competitive may help to create additional job opportunities. This insight is consistent with the findings in experimental economics in its early literature: decentralized markets typically result in excess quantity. See my blog article "Chamberlin vs. Smith in Roth (1995)" for the detail.

I am now trying to extend the model from a homogeneous good market to a general two-sided matching market with monetary transfers. New findings will soon be available!


Theory Seminar (Amir) at UTS

I attended a theory seminar at the University of Technology, Sydney (UTS) yesterday by Professor Rabah Amir from Univ. of Iowa, talking about the following theoretical IO paper:

Amir, R., Ericksonz, P., and Jin, J. "On the microeconomic foundations of linear demand for differentiated products"

This paper provides theoretical foundations of the multi-variate linear demand function for differentiated products that is widely used in industrial organization. While quasi-linear preferences with quadratic utility function (for goods) is known to yield linear demand (Singh and Vives, 1984), the authors extend the result. Namely, under quasi-linear environment, they show that the resulting demand system becomes linear and well defined if and only if the underling utility function (for goods) must be strictly concave and quadratic. The "only if" part implies that the use of linear demand that does not satisfy the law of demand ought to be regarded with some suspicion.

A theoretical punch line of the paper is that any (linear) demand system derived by consumer's utility maximization must satisfy the law of demand. I believe that we can establish a slightly more general result than this in the following way. It is known that (see, for example, Mas-Colell et.al. 1995) if a (Marshallian) demand function satisfies the weak axiom of revealed preferences and Walras' law , then its Hicksian demand satisfies the law of demand. If we further assume that there is no income effect, then the (Marshallian) demand function also satisfies the law of demand, since the two demand functions must have identical derivatives w.r.t. prices. The following is a summary the idea.

(1) weak axiom + (2) Walras's law => law of demand for Hicksian demand

(3) no income effect => Marshiallian demand = Hicksian demand

(1) weak axiom + (2) Walras's law + (3) no income effect => law of demand

The above argument does not presume any consumer's preferences or utility maximization behind the demand function. By contrast, the current paper associates consumer's demand with his/her utility function. Now, note that if a demand function is derived by utility maximization with an increasing utility function, then conditions (1) and (2) automatically hold. If utility function is quasi-linear and a consumer has enough income, then (3) is also guaranteed. Since these are exactly what the paper assumes, the sufficient conditions (1), (2), and (3) above are satisfied and thus the law of demand (for Marshallian demand) naturally follows.

Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic theory. New York: Oxford university press.
Singh, N., & Vives, X. (1984). Price and quantity competition in a differentiated duopoly. The RAND Journal of Economics, 546-554.


Chamberlin vs. Smith in Roth (1995)

I found very interesting description in The Handbook of Experimental Economics, Chapter 1: Introduction to Experimental Economics by Prof. Alvin Roth. He refers two pioneering papers on experimental economics, Chamberlin (1948) and Smith (1962), which investigate commodity markets under different trading rules and check whether competitive equilibrium could be established.
[T]he basic design of Chamberlin (1948) for inducing individual reservation prices and aggregate supply and demand curves has become one of the most widely used techniques in experimental economics.

Interestingly, Chamberlin (1948) reports systematic gap between his experimental results and the competitive equilibrium. The experimental design and its results are summarized as follows.
Chamberlin created an experimental market by informing each buyer and seller of his reservation price for a single unit of an indivisible commodity, and he reported the transactions that resulted when buyers and sellers were then free to negotiate with one another in a decentralized market.
The experiment he reported involved 46 markets, with slightly varying equilibrium prices. He observed that the number of units transacted was greater than the competitive volume in 42 of these markets and equal to the competitive volume in the remaining 4 markets, while the average price was below the competitive price in 39 of these markets and higher in the rest.

By contrast, (repeated) double auctions experimented by Smith (1962) result in convergence to the equilibrium.
One important form of market organization is the double auction market, first experimentally studied by Smith (1962), who observed rapid convergence to competitive equilibrium when the market was repeated several times with stationary parameters.

One might be puzzled why the number of transaction in ALL experiments in Chamberlin (1948) exceed or equal to the competitive volume: none falls below it. My recent article, Equal Market Design I: Competitive Market Achieves the Greatest Happiness of the Minimum Number, provides a theoretical account on this puzzle. This short paper (only 8 pages!) shows that the number of agents who engage in trades under any market equilibrium is MINIMUM among all Pareto efficient and individually rational allocations in the environment where redistribution by the third party is infeasible, i.e., no monetary transfers beyond buyer-seller pairs are prohibited. So, if buyers and sellers in Chamberlin's experiments somehow manage to reach the PE and IR allocations (it is likely although I have to check the original data if available), the number of transaction must be weakly larger than that of competitive equilibrium.

I would like to thank Prof. Morimitsu Kurino who pointed out the possible connection between experimental studies and my research, explicitly mentioning the above two papers, i.e., Chamberlin (1948) and Smith (1962).

Chamberlin, E. H. (1948). An experimental imperfect market. The Journal of Political Economy, 95-108.
Kagel, J. H. and Roth, A. E. (1995). The handbook of experimental economics. Princeton, NJ: Princeton university press.
Smith, V. L. (1962). An experimental study of competitive market behavior. The Journal of Political Economy, 111-137.
Yasuda, Y. (2016). Equal Market Design I: Competitive Market Achieves the Greatest Happiness of the Minimum Number, mimeo. SSRN#2755893