# From General Equilibrium to Mechanism Design

When starting to do research in the computer science side of general equilibrium theory, I often struggled to connect general equilibrium models to the more popular mechanism design models in computer science, e.g., multi-item auctions. Aside from that, in many expositions about general equilibrium theory (at least in the computer science side of things), there seems to be no mention of mechanism design (Cheung et al., Zhang). Similarly, when one is taught mechanism design little, if any, mention to general equilibrium is made. The goal of this post is to put together the pieces that lead to the development of general equilibrium theory and mechanism design and explain how general equilibrium theory is essentially a subfield of mechanism design.

In order to understand how general equilibrium theory and mechanism design relate to each other, one has to look at how both subjects come to being. Following the second world war, European and American governments had to preside over the reconstruction of devastated economies. This required an improved understanding of the role of public expenditure in economic activity since the war led to an unprecendented increase in the role of government in the economy, a change that was unsustainable: While in 1944, the American government's spending at all levels accounted for 55 percent of gross domestic product (GDP), by 1947, government spending had dropped 75 percent in real terms, or from 55 percent of GDP to just over 16 percent of GDP. (Bureau of Economic Anal- ysis)

Fortunately, when the Second World War (WW2) broke out, many European academics escaping the war moved to the United States helping the development of rigrous mathematical models of markets which would be key to help manage the role of public expenditure in economic growth. These academics re-emerged primarily at the University of Chicago, where the Cowles Commission was founded in 1939^{1}, with the maxim, “Science is Measurement” (Mitra-Kahn). The Cowles commission aimed to link mathematics and economics (Mitra-Kahn), and played a crucial role in the development of mathematical microeconomic models that have become the foundation of modern economics.

These efforts that were initiated by the Cowles Commision culminated to the seminal work of Keneth Arrow and Gérard Debreu which proved the existence of general equilibria in a very general market setting (Arrow and Debreu). Arrow and Debreu's proof meant that in perfectly competitive economies, which are defined by a series of rigrourous assumptions in their paper (Arrow and Debreu), a general equilibrium^{2} which consists of consumptions, productions and supporting prices exist, i.e., consumptions, productions, and prices for each good which are consistent with each other and that 1) maximize the utility of buyers, 2) maximize the profit of firms, and 3) guarantee that the demands in each market in the economy is equal to their supply. Perhaps more importantly, in seperate papers, Arrow and Debreu independently but simultaneously proved the *first and second welfare theorems of economics* which stated that 1) general equilibrium consumptions and productions are pareto-efficient and 2) any pareto-optimal tuple of consumptions and productions can be made into a geneal equilibrium (Debreu, Arrow).

Arrow and Debreu's results implied that free^{3} markets are then an efficient way to allocate resources since they result in a pareto-optimal distribution of resources---an inference which is contingent on the global stability of general equilibria, i.e., free markets actually settling into a general equilibrium, which still remains unknown (Cheung et al.). Unfortunately, Arrow and Debreu's results provided ambiguous conclusions on how the transition away from a war economy with high public expenditure. On one hand, Arrow and Debreu's results suggested that a post-war economy with no governmental intervention and no public expenditure was optimal as free markets are a pareto-efficient mechanism of ressource allocation. On the other hand, real world markets rarely would ever truly be free and it seemed like decreasing public expenditure significantly could be a disastrous economic choice as Paul Samuelson, 1970 Nobel Prize laureate, wrote in 1943: “some ten million men will be thrown on the labor market.” (Mitra-Kahn) warning that it would be “the greatest period of unemployment and industrial dislocation which any economy has ever faced.” (Mitra-Kahn)

The issue of public expenditure in a post-WW2 would become a central theme in Samuelson's research who would provide the first rigorous definition of public goods. Based on his definition Samuelson would then derive what came to be known as the *Samuelson condition*: the first order optimality condition associated with the optimal provision of public goods in terms of the demand and supply for public goods^{4} (Samuelson, “The Pure Theory of Public Expenditure,” Samuelson (“Diagrammatic Exposition of a Theory of Public Expenditure”)). Samuelson's work was in some sense ground breaking as it moved the problem of allocating public spending from the realm of political theory to the realm of general equilibrium theory. Samuelson's analysis would be studied more rigorously in subsequent general equilibrium models culminating eventually in a generalization of Arrow-Debreu's model of a competitive economy called the *private ownership economy with public goods*, a model which differentiates between public and private goods (Mitra-Kahn).

An important result that emerged from this line of work started by Samuelson and which was proven by Foley is that an equilibrium called the *Lindahl-Foley equilibrium* exists and the first and second welfare theorems apply to the Lindahl-Foley equilibrium in the private ownership economy with public goods (Duncan K Foley, Duncan Karl Foley).^{5} Foley's results confirmed the role that government have been attributed in achieving Pareto optimality via optimal redistribution policies (Mitra-Kahn). Although a gross simplification of the conclusion, this meant that governments could direct their public expenditures using the first order optimality conditions determined by Samuelson in a post-WW2 world to provide optimal levels of public goods, ensuring their economies settle into a pareto-optimal outcome^{6}!

Unfortuntaly, Samuelson only provides the (first order) conditions for a provision of public goods to be optimal, i.e., conditions for the market for public goods to be in equilibrium, (Samuelson, “The Pure Theory of Public Expenditure”), but he does not tell us how to *compute* an optimal provision of public goods. It turns out that the computation of the optimal provision of public goods, is not so straightforward since it requires the policy maker to know the *true* preferences of the consumers! However, in such models consumers have an incentive to lie about their preference over public goods in order to increase their utility at equilibrium. This means that if the policy maker elicits the preference of consumers over public goods the consumers will lie (as they are rational), which will then lead the policy maker to compute a non-optimal provision of public goods.

The issue of *incentive-compatibility*, i.e., the consumers reporting their preferences truthfully, in the provision of public goods is exactly what led to the development of mechanism design. In order to ensure that economies settled in pareto-optimal outcomes, governments had to decide their equilibrium level of public expenditure, yet to compute their equilibrium level expenditures, governments had to be able to elicit the true preference of consumers over public goods. This required the development of the mechanism design literature which would provide a new formalization of social and economic interactions which accounted for incentives. The consistent objective of the mechanism design literature which was introduced in seminal papers by Hurwicz in the 60s and 70s (Hurwicz, “Optimality and Informational Efficiency in Resource Allocation Processes,” Hurwicz (“On Informationally Decentralized Systems”), Hurwicz (“Outcome Functions Yielding Walrasian and Lindahl Allocations at Nash Equilibrium Points”)), was to provide such a framework that *also* encompassed general equilibrium theory. This means that mechanism design is a generalization of general equilibrium theory. More precisely, the general equilibrium or the competitive equilibrium is a particular outcome of a game just like the VCG outcome is, and the Walrasian mechanism, i.e., the function which takes as input the preferences of consumers, computes and outputs a competitive equilibrium, is an instance of a mechanism just like the VCG mechanism is.

More importantly, however, history tells us that without general equilibrium theory, there would be no mechanism design. Perhaps an analogy is fitting here: *General equilibrium is to mechanism design, as the normal distribution is to probability theory*. Just like one cannot imagine a theory of probability without a solid understanding of the statistical and algorithmic properties of the normal distribution, one cannot expect a proper understanding of mechanism design with a proper understanding of the economic and algorithmic properties of general equilibria. History is the bigest testament to this statement, without general equilibrium theory's incentive issue, Hurwicz and many other would have not gotten inspired to introduce mechanism design. It is for that reason that if we would like to push the boundaries of our understanding of mechanism design further, we have to push our understanding of general equilbrium theory further!

# Works Cited

Arrow, Kenneth J. “An Extension of the Basic Theorems of Classical Welfare Economics.” *Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability*, University of California Press, 1951, pp. 507–32.

Arrow, Kenneth, and Gerard Debreu. “Existence of an Equilibrium for a Competitive Economy.” *Econometrica: Journal of the Econometric Society*, JSTOR, 1954, pp. 265–90.

Bureau of Economic Anal- ysis, U.S. Department of Commerce. *National Income and Product Accounts Tables, 1940‒1947” (Gdp Accounts in Billions of Chained 1937 Dollars)*. http://bea.gov/iTable/iTable.
cfm?ReqID=9&step=1. Accessed 28 Dec. 2021.

Cheung, Yun Kuen, et al. “Tatonnement Beyond Gross Substitutes? Gradient Descent to the Rescue.” *Proceedings of the Forty-Fifth Annual Acm Symposium on Theory of Computing*, Association for Computing Machinery, 2013, pp. 191–200, https://doi.org/10.1145/2488608.2488633.

Debreu, Gerard. “The Coefficient of Resource Utilization.” *Econometrica: Journal of the Econometric Society*, JSTOR, 1951, pp. 273–92.

Foley, Duncan K. “Lindahl’s Solution and the Core of an Economy with Public Goods.” *Econometrica: Journal of the Econometric Society*, JSTOR, 1970, pp. 66–72.

Foley, Duncan Karl. *Resource Allocation and the Public Sector*. Yale University, 1966.

Hurwicz, Leonid. “On Informationally Decentralized Systems.” *Decision and Organization: A Volume in Honor of J. Marschak*, North-Holland, 1972.

---. “Optimality and Informational Efficiency in Resource Allocation Processes.” *Mathematical Methods in the Social Sciences*, Stanford University Press, 1960.

---. “Outcome Functions Yielding Walrasian and Lindahl Allocations at Nash Equilibrium Points.” *The Review of Economic Studies*, vol. 46, no. 2, JSTOR, 1979, pp. 217–25.

Mitra-Kahn, Benjamin H. “General Equilibrium Theory, Its History and Its Relation (If Any) to the Market Economy.” *City University, London the New School for Social Research, New York*, 2005.

Samuelson, Paul A. “Diagrammatic Exposition of a Theory of Public Expenditure.” *The Review of Economics and Statistics*, JSTOR, 1955, pp. 350–56.

---. “The Pure Theory of Public Expenditure.” *The Review of Economics and Statistics*, vol. 36, no. 4, JSTOR, 1954, pp. 387–89.

Zhang, Li. “Proportional Response Dynamics in the Fisher Market.” *Theor. Comput. Sci.*, vol. 412, no. 24, Elsevier Science Publishers Ltd., May 2011, pp. 2691–98, https://doi.org/10.1016/j.tcs.2010.06.021.

The Cowles Commission later relocated to Yale in 1955, and was re-named as the Cowles Foundation. Some prominent economists associated with the foundation are: Arrow, Debreu, Stiglitz, and Tobin (Mitra-Kahn)↩

An alternative name for general equilibrium is competitive or Walrasian equilibrium. Note, however, that general equilibrium refers to a situtation in which all markets within an economy are in equilibrium, while a competitive or Walrasian equilibrium's definition restricts itself to a single market. In that sense, a general equilibrium is more general. The distinction is often blurred in computer science works as mathematically in most models of interest the economy can often be seen as one big market.↩

A free market in this sense is one that resembles Arrow-Debreu's model of a competitive economy.↩

Note that Samuelson's analysis makes no mention to equilibrium concepts. The lack of equilibrium analysis is not surprising as Arrow-Debreu's work defining a general equilibrium was published at the same time as Samuelson's paper.↩

Lindahl can be seen as a figure similar to Walras in public economics, having first pondered on the issue of equilibrium public expenditure, hence the inclusion of his name in the equilibrium concept.↩

Notice that if the market for public goods is out of equilibrium, i.e., the policy maker has not produced an optimal level, i.e., market clearing level, of public goods, by Walras' law there must exist another market out of equilibrium, which is why understanding optimal levels of government spending is key to ensuring that the economy settles into a pareto-optimal outcome.↩