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Equilibrium models

This video provides main insights into the basic principles of equilibrium models. Watch Hannes Weigt explain more.
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In this lecture, we want to get familiar with the second main type of economic models next to optimizations namely, equilibrium models. Now, before we have a look at the modelling details, what exactly is an equilibrium? Roughly speaking, an equilibrium defines a stage which is in balance, and nobody has an incentive to deviate from it. So you could say it’s a stable point. There are different reasons why something can be in equilibrium. In economic modelling, the most common form of equilibrium is the competitive market equilibrium, namely, that supply and demand are in balance. But the same logic of balancing some input and output also has meaning beyond economic concepts for example, in cases of flow conservation constraints in networks.
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What gets in also has to get out. Another famous economic equilibrium concept is the Nash equilibrium, when strategic actors enter the market framework. Basically, also the solution of an optimisation problem defines an equilibrium the point in which the defined objective is maximised or minimised. Let’s again start with a simple example to get an idea what an equilibrium model is. Assume you’re a firm deciding how much energy to supply with your power plant. You know your cost function, linking your output decision, q, with the resulting cost c. You also know the market price p. Now, your decision will be to produce as much energy with your plant such that the resulting marginal cost level is equal to the market price.
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If you would produce more, you would make a loss, as the costs of those additional units would be higher than the market price you can obtain. If you would produce less, you actually forgo some possible profit, as your costs are lower than the market price, and you could still sell some more. The point where your output results in a cost level that matches the market price marks the equilibrium the perfect competitive equilibrium, to be more precise. In a more general setting, equilibrium models can be structured into two elements representing market interactions, termed complementarity conditions. First, we have to understand why a firm produces something or consumers are buying something.
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Those are activities of the different market actors, and they correspond to the so-called zero-profit logic. If the costs of an activity are higher than the resulting revenue basically, the price one can get, the activity will not be carried out, as it would make a loss. Correspondingly, if the activity is carried out meaning it has a positive value costs and revenue have to be equal. The activity makes zero profit. The same logic applies to the demand side of a market. Consumers are only willing to buy a product if their buying costs, defined by the market price, are lower or equal to the benefit they obtain from buying the product.
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The second type of complementarity constraints are market clearing conditions that correspond to prices. They bring the supply and demand side together. If the supply on the market exceeds demand, we don’t face any scarcity, and prices should be zero. In turn, if prices are positive, supply and demand have to be in balance in other words, to produce a supply exactly as much as the consumers are willing to buy for the resulting price. If consumers are willing to pay more, producers will increase supply until the new market equilibrium is reached, and vice versa. Please note that both the costs and what can be understood as market is rather general and does not need to refer to pure production costs or observable markets.

Contrary to an optimization problem, an equilibrium model has no objective but consists of the respective conditions that need to hold when the market or system is in equilibrium, meaning in a stable state. As explained in the tutorial, there are two general equilibrium conditions we need to account for: zero-profit conditions and market clearing conditions. Both conditions follow a simple logic: what economic incentives drive the behaviour of the market participants?

In case of the zero-profit condition the underlying question relates to the activity of the involved market actors: what incentivizes an actor to become active (ie produce or consume a good)? It’s the obtainable benefit! A firm will start to produce if the price it can get for its product is at least as high as its production costs. The firm will stop increasing its production when the price equals its production costs (at least in a perfect competitive market) → the condition is in equilibrium.

In case of the market clearing conditions the underlying question relates to the basic economic principle of supply and demand. If a market is undersupplied (supply < demand) the resulting high price will give producers an incentive to increase production (via the zero-profit condition) up to the point where the market is in equilibrium (supply = demand). If the market is oversupplied (supply > demand) the product has no value, as there are excess supplies available. The market at hand does not need to be a ‘real’ market; ie also production constraints represent a supply (available production capacity) and demand (requested output) logic.

Equilibrium models are flexible designs suited for a large variety of model applications. Generally two types of equilibrium models are distinguished: partial equilibrium models that capture only one or a subset of markets/sectors while prices and quantities on other markets are taken as given; and general equilibrium models that capture the whole economy with many interacting markets/sectors. The latter is the main approach in large scale numerical top-down modeling; termed Computable General Equilibrium (CGE), while the former is also used for numerical bottom-up models and for conceptual evaluations. In general, equilibrium models are the first choice when deviating from perfect competition or monopolistic competition in a market setting.

In the literature recommendation below, you find a combined natural gas and electricity market model (Abrell and Weigt, 2010). As natural gas is a fuel input in electricity generation the two markets present a classical example of upstream and downstream markets. In addition both markets rely on networks introducing spatial components into the market design. The paper at hand provides a comprehensive design of both markets. As we did not yet cover the detailed mathematical functionalities of equilibrium models you may want to focus your reading on the above described decision conditions for zero-profit constraints and market clearing.

Abrell, J. & Weigt, H. (2010). Combining Energy Networks. Dresden University of Technology. (Journal Version: Abrell, J. & Weigt, H. (2012). Combining energy networks. Networks and Spatial Economics, 12(3), pp. 377-401.)

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