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A crash course in general equilibrium theory

#### December 6, 2022

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#### Written by Ronaldakven

I have always had a weakness for history lessons. Although very different from the standard optimal control problems and differential equations I am solving in my studies, I find that the lessons we can learn from great previous minds are both insightful and fascinating. Combining both my favorite economics concepts and a history lesson, today I want to give you a small insight into the development of general equilibrium (GE) theory.

### What is general equilibrium theory?

In 1874, French neoclassical economist Léon Walras came up with a model to determine the appropriate prices in an economy, in his work ‘Elements of Pure Economics’. This was the foundation of general equilibrium theory, a branch of economic theory via which economists wanted to better understand the broad economy and which mechanisms pin down prices of goods. A standard general equilibrium model contains several goods markets that jointly determine all the prices of the goods. If the price of one good increases, there will be, ceteris paribus, effects on the prices of other goods via demand and supply effects.

A subset of models in general equilibrium, the so-called partial equilibrium models, considers a more specific version of models by simplifying assumptions that leave out certain markets. As an example, a general equilibrium model could for instance include both households and firms whose decisions jointly determine wages, interest rates and so forth. On the other hand, a partial equilibrium model would for instance only consider the household’s side and take wages and interest rates as exogenously given.

### What did the early economists actually mean by equilibrium?

Among other concepts such as Walras’s law (where loosely speaking, if of the goods markets clear, all markets clear), Walras introduced the Walrasian equilibrium. In this model with a variety of goods markets, prices of capital goods are the same, regardless whether the good is an input or output. Walras was a very theoretical economist (as all of the mathematical economists are) and was mostly interested in the stability and uniqueness of equilibria. Besides that, he was working on the proofs of existence of equilibria via so-called tatonnement processes (but I am not going to bore you with that). In the spirit of Walras, a market equilibrium will only hold when all markets simultaneously clear.

In the 1920s, Alfred Marshall introduced his theory of supply and demand in equilibrium models. In this partial equilibrium, prices are determined by only considering the price of one good and keeping all other prices constant. This only works sufficiently when shifts in demand curves do not drive shifts in supply curves, which was part the critique of economists such as Piero Sraffa. Saffra argued that in reality, an increase in demand for a certain good drives shifts in supply curves of substitutes for that specific goods and subsequently also pushes the supply curve of the good for which the demand increased. The framework of Marshall thus only can be used as a method of analysis (which was also more of Marshall’s idea as he wanted to use economic thinking as a more practical tool, useful for social reforms).

After Saffra, economists Arrow, Debreu and McKenzie introduced the modern concept of GE. In his paper in 1959, Debreu laid out the framework which was heavily discussed and used by other scholars. Later, Arrow and Debreu came up with a more detailed definition of an equilibrium in a goods market; goods are characterized by four aspects, namely the time of delivery, the place of delivery, under which circumstances the goods were to be delivered and the nature of the goods. In the model with complete markets Arrow and Debreu are proposing, it comes as no surprise that there is a continuum of prices that characterize every possible good and delivery combination. After Arrow and Debreu, economists were also drawn towards the concept of incomplete markets, where in the state space of possible outcomes we cannot span all possible outcomes to fully utilize and optimize resource usage over space and time.

### General equilibrium and welfare theory

Next, I will explain some of the properties of the competitive general equilibrium proposed by Arrow and Debreu. The Fundamental Theorems of Welfare Economics were introduced by Adam Smith back in the 1700s and have been a key point in the characterization of the modern notion of equilibrium theory.

First, under a few assumptions regarding the preferences of consumers, the presence of complete markets and with perfect information, the market equilibrium will be Pareto efficient. This is the First Fundamental Theorem of Welfare Economics. The theorem implies that the market will allocate the goods in such a way that there will be no reallocation possible so that one consumer would be better off without making another consumer worse off. By construction, the aforementioned assumptions will be Pareto efficient. When externalities are present, market failures will arise and thus there will be a loss of welfare.

Second, not every allocation of resources that is efficient can constitute an equilibrium. A Pareto efficient allocation should be supported by an appropriate set of prices. With redistribution of initial endowments of the agents in the economy, the market itself will do the rest and set prices in such a way that the markets will clear in a Pareto efficient way. Contrary to the first Fundamental Theorem, the Second Fundamental Theorem of Welfare Economics also needs an assumption regarding preferences of consumers and production sets. However, the two theorems above do not say anything about stability, uniqueness and existence. Existence can for instance be proved by the Brouwer Fixed-Point theorem (introduced in the course IME), but that is left as homework for the reader.

### Macro meets Micro: Dynamic Stochastic General Equilibrium modelling

Modern day economists are less focussed on the nitty gritty details of existence or uniqueness of equilibria that were of great interest to the mathematical economists in the 19th and 20th century. However, more practical problems such as smoothing of business cycles and effective monetary policy that scholars try to answer at central banks are relying on models that use the basic concepts by Walras and others. For one, a modern class of models called Dynamic Stochastic General Equilibrium (DSGE) are very prominent in the literature of monetary theory. These models usually consist of households that decide on how much to consume today and how much to save for a rainy day tomorrow (Dynamic). They are also Stochastic as the economy is exposed to exogenous shocks (such an increase in productivity); agents thus have to form expectations regarding the condition of the economy tomorrow. Finally, the prices in the model adjust in such a way so that all markets clear (General Equilibrium). With these models, economists can make an estimate of the changes in endogenous variables (such as output, employment, consumption etcetera) when the central bank raises the interest rates. Although some people might say that the works of the early economists are outdated, I find it insightful and inspiring to see that contemporary economic models are still drawing inspiration from them. I hope you have learnt something from this short history lesson!