Definition
Closure: the classification of variables (exogenous or endogenous) in the model.
- endogenous variables: values solved in the model
- exogenous variables: determined outside of the model. In analysis, they are given as shocks
Criteria of a valid closure
- The number of endogenous variables equals to the number of equations.
- Equations are independent
- Closures make economic sense
GE closure
A standard GTAP closure (GE closure) is based on the following assumptions:
- all market in equilibrium (market clear)
- all firms have zero profits
- regional household on budget constraint
Recall that in the regular requirement of GE model, we have:
- budget constraint condition from consumer's problem
- zero profit condition from producer's problem
- all market clear is required by the model
This closure is defined by:
Exogenous
pop - population
psave - numeraire
profitslack incomeslack endwslack - slack variables
cgdslack tradslack - slack variables
ao af afe ava - technical change
dpfpriv dpfgov dpfsave - preferences
to tfe tf tg tp - policy variables
qo(ENDW_COMM); - endowments
Rest endogenous;
Walras' s law in GE closure
Walras's law tells that when the assumptions of GE closures are met, when the global investment must equal global savings.
In the model, we set the price of capital goods, psave, to be the numeraire, and all other prices are relative price of psave. In that case, psave is set to be exogenous since its value does not change in the GE model.
The linearized form of Walras's law is given by:
walras_sup = walras_dem +walraslack
where:
- walras_sup is the linearized form of NETINV (net investment)
- walras_dem is the linearized form of SAVE, which equals GLOBINV (global investment)
- walraslack: an endogenous slack variable. walraslack = 0 indicates that Walras' law holds and SAVE = NETINV
Note: this relationship is given by the save in GTAP
Slack variables in GE closure
Similar with walraslack, we also include multiple slack variables on the right-hand side of equations that represents GE assumptions. In GE closure, these slack variables are exogenous are does not change, in order to guarantee those GE assumptions and associate equilibrium conditions always hold.
Market clearing conditions
- tradable commodities - tradslack
- mobile endowment commodities - endwslack
- sluggish endowment commodities - endwslack
- Zero pure profit condition - profitslack
- Income-expenditure balance - incomeslack
- Saving-investment balance - walraslack
PE closure
When one or more equilibrium conditions are not satisfied, the closure becomes PE closure. Sometimes, we need to use a PE closure to isolate certain partial equilibrium effects.
For example, let us consider the market clearing condition of commodity i (recall the market clearing condition in the accounting relationship).
In GE closure, the model solves the price of commodity i, pm(i) endogenously so the market clearing condition holds and tradslack(i) is an exogenous variable that tradslack(i) = 0.
If we would like to fix or manually change pm(i), then it is not guaranteed anymore that pm(i) can satisfy the market clearing condition, and the model becomes a PE model.
In that case, we need to swap the endogeneity / exogeneity between pm(i) and the slack variable tradslack(i), in order to shock pm(i) as exogenous variable, and solve tradslack(i) as an endogenous variable to represent that the market clearing condition does not hold anymore. Also, we need to swap between walraslack (initially endogenous, now exogenous) and psave (initially exogenous, now endogenous) because Walras's law does not hold in PE closure.
Here we summarize the swaps needed to convert a GE closure to PE closure, given the partial equilibrium effects one would like to research:
- Swap tradslack(i) with pm(i) to fix market price, PM(i)
- Swap endwslack(i) with pm(i) to fix market price, PM(i)
- Swap profitslack(i) with qo(i) to fix output level, QO(i)
- Swap incomeslack(i) with y to fix regional income
- Swap walraslack with psave if swapping any of the above