What is the difference between the term "competitive equilibrium" and the term "general equilibrium"? Here in econ.SE, they are two different tags, so there is probably a difference between them, but I don't understand what exactly.
A competitive equilibrium ("Walrasian Equilibrium")'s defining characteristic is that it's competitive. It's about an equilibrium in which market forces (say, consumers, firms)' supply and demand responds to prices, and prices respond to supply and demand, and no Pareto-improving trade possibility remains in the end.
To be technical, consider a market with $I$ agents, $J$ goods, and initial endowments $E$ of shape $I\times J$. Also, each agent has rational preferences over the of goods.
A CE consists of a ($J\times 1$) vector of prices $P$ and a ($I\times j$) vector of allocations $X_i$ with two characteristics:
- Feasibility: $\sum_i J_i \geq \sum_i X_i \, \forall j\in J$
- Rationality: $X_i \succsim_i \, H \forall H: HP \leq E_iP, \forall i$
The general equilibrium's defining characteristic is that it is an equilibrium on more than one market; as opposed to the partial equilibrium in which we hold at least one price fixed and analyze the response of other markets/prices only.
So, what's the difference?
It's about the emphasis. Any GE is a CE, but not any CE is necessarily a GE. We could look at a CE of consumer goods, but not analyze labor supply (i.e., not even model labor). Would this then really be a general equilibrium? Mostly, it is about the field.
In macroeconomics, it is a standard procedure to shut off several general equilibrium response mechanisms (for example, fix labor supply) and first look at a toy model in which only one margin is allowed to vary, and understand the mechanism - the partial equilibrium. Then, after that was understood, understand to which extent the mechanism is valid and relevant once one allows other margins to adjust. Therefore, here, it is important to be able to emphasize the difference - i.e. partial versus general equilibrium.
In other fields, one may not care as much about "completely different" markets. The distinction is for example not as important in many IO topics, and hence many there use the notion of the competitive equilibrium instead.