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How to build a numerical model II: Balancing the world

This video provides main insights into the basic principles of equilibrium models. Watch Hannes Weigt explain more.
This time we want to get familiar with the concept of equilibrium models, a flexible and widely applied approach of economic modeling. Now, what exactly is an equilibrium? Roughly speaking, an equilibrium defines a state 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 modeling, 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 case of flow conservation constraints in networks.
Another famous economic equilibrium concept is the Nash equilibrium, when strategic actors enter the market frame. Basically, also the solution of an optimisation problem defines an equilibrium– the point in which the defined objective is maximised or minimised.
To set up an equilibrium model, one needs to know what conditions need to hold in order for a state to be in equilibrium. Those conditions can then be transferred into equations that can be solved. Keeping things simple, one can distinguish two main forms of those so-called complementarity conditions. First, we have to understand why a firm produces something or consumers are buying something. Those are activities of the different market actors, and they correspond to the zero-profit logic. If the costs of an activity are higher than the resulting revenue, or 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 second type of complementarity constraints are market clearing conditions that correspond to prices. If the supply on a 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. 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.
Let’s again use a little test example to better understand how to set up an equilibrium model. This time, we assume a company has a single power plant and wants to decide if it should sell its energy given a known price p. We need to define conditions which have to hold in the resulting equilibrium. First, the company will produce and sell its output if the market price is at least as high as its production cost c. And consequently not produce if the market price is lower than its cost. But the firm needs to take into account that its power plant has a limited capacity. This can be captured by a market clearing constraint.
The supply on this market is the available plant capacity q max, and the demand is the actual output of the firm. The corresponding price can be interpreted as the shadow price on capacity, p cap. This capacity price is actually a component on the cost side of the firm. So we need to further adjust the activity condition by incorporating p cap into the zero-profit constraint. Now we have set up a simple equilibrium formulation of our problem. We have two variables– output Q and the capacity price p cap– and two equations. In a similar fashion, the market representation can be extended to capture for example investment decisions, making the plant capacity q max a variable, or by introducing price-sensitive consumers.

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 top-down modeling; termed computable general equilibrium (CGE), while the former is used for bottom-up models. 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 mathematical functionalities of equilibrium models or the interplay with optimization models you may want to focus 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, WP-EM-38 (Journal Version: Abrell, J., & Weigt, H. (2012). Combining energy networks. Networks and Spatial Economics, 12(3), 377-401)

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