An Arrow-Debreu model concerns the interaction of consumers and producers in markets. Lars Mathiesen  proposed a representation of this class of models in which two types of equations define an equilibrium: zero profit and market clearance. The corresponding variables defining an equilibrium are activity levels (for constant-returns-to-scale firms) and commodity prices.5
Commodity markets merge primary endowments of households with producer outputs. In equilibrium the aggregate supply of each good must be at least as great as total intermediate and final demand. Initial endowments are exogenous. Producer supplies and demands are defined by producer activity levels and relative prices. Final demands are determined by market prices.
Economists who have worked with conventional textbook equilibrium models can find Mathiesen's framework to be somewhat opaque because many quantity variables are not explicitly specified in the model. Variables such as final demand by consumers, factor demands by producers and commodity supplies by producers, are defined implicitly in Mathiesen's model. For example, given equilibrium prices for primary factors, consumer incomes can be computed, and given income and goods prices, consumers' demands can then be determined. The consumer demand functions are written down in order to define an equilibrium, but quantities demanded need not appear in the model as separate variables. The same is true of inputs or outputs from the production process: relative prices determine conditional demand, and conditional demand times the activity level represents market demand. Omitting decisions variables and suppressing definitional equations corresponding to intermediate and final demand provides significant computational advantages at the cost of a somewhat more complex model statement.
For concreteness I now turn to specific features of the GTAP model. The core model described here is a static, multi-regional model which tracks the production and distribution of goods in the global economy. In GTAP the world is divided into regions, and each region's final demand structure is portrayed by a representative agent who allocates expenditure across goods so as to maximize welfare, with fixed levels of investment and public output. Production incorporates intermediate inputs, and primary factors include skilled and unskilled labor, land, resources and physical capital. The dataset includes a full set of bilateral trade flows with associated transport costs, export taxes and tariffs.
In the following section, before writing down the equilibrium conditions per se, I describe production technology and producer choices. I then outline the structure of private and public final demand. Finally, I write down the zero profit and market clearance equations.