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Accounting relationship of GTAP one region model


Accounting relationships

Accounting relationship are equations that always hold no matter how do we shock the model.


They are basically:


For each accounting relationship, it is possible to relax it, in order to change exogenous variables. For example, for market clear condition, price are set to be endogenous. If we need to model an exogenous change on price, then the original market clear condition may not hold, so we need to add a slack variable to make sure the accounting equation still hold.


Here are five major category of accounting equations and their associated slack variables:



These accounting equations will be discussed below.
Note: refers to sets in GTAP model to check the set of commodities in equations below.


Tradable market clear condition

Description
total quantity of tradable goods supply (produced) equals to its demand by different sources. It described the demand of goods (products or factor input)


It determines the endogenous variable: market price PM. If PM is set to be exogenous (or change exogenously), we need to add slack variable to make sure the equation still hold.


Representation in quantity term

Level form



Where:


Note in baseline data, TRADSLACK = 0. It represents the excess supply or the difference between supply and demand


Linearized (percentage change) form
convert the equation from level form to percentage form, for the convenience of solving model numerically
Total derivative of the equation above:

Recall that percentage change of variable X is defined as : x = (dX / X) * 100
So dX = Xx / 100


However, for the percentage change of TRADSLACK, it is defined as

The reason is that if the denominator is TRADSLACK, it is possible to cause "divided by zero" problem.


Then we have the linearized representation of this condition:


Representation in value term

Level form
Multiply market price PM on both side to the level equation above, to convert quantity to value:


Linearized form
Similarly, convert linearized form equation in quantity to value by multiplying PM on both side of

And have:

If market price PM is endogenous, TRADSLACK remains zero before and after model simulation


Share form
From the linearized form with value, we can further divide both sides with VOM, and have



Where SHRFM, SHRPM and SHRGM are share of initial demands (firm, private, government) in the initial supply.


This equation is called "MKTCLTRD" in GTAPO.TAB


For detailed procedure, see the note of linearization of market clearing condition


Mobile endowment market clear condition

Mobile endowment market clear condition. It describes the demand of mobile factor inputs (only used by producers)



Representation in quantity term

Level form

where


Linearized form


Representation in value term

Level form

Note: in GTAP notation, we use "M" for value related variable, in order to mark the value is measured in which price.


Linearized form
(equation MKTCLENDWM)

Note that in the original model, the endowment are fixed supply, so the LHS of the equation is always zero.


Sluggish endowment market clear condition

For sluggish endowment, the market price is not determined by market clear condition. Instead, we have:




Where:


Here QFE is determined by CES assumption of firm technology.


Supply price PMS(i,j) is determined by the CET assumption on the imperfect mobility of the endowment.


PMS adjusts the imbalance between QOES and QFE.


Zero pure profit condition

Zero profit equation for a firm in sector j is given by the total output value equals to total input value (endowment input and intermediate input)


It is for the supply side (comparing with the market clear condition for demand side)


Level form


Or with slack variable:

Note: it is like the column in IO table.


Note: to understand the value transmission, consider the formula below (for understanding)



Where PTAX refers to tax and is positive
VOA and left: column of IO table
VOM and right: row of IO table.


Zero profit condition is always written in value terms and evaluated at price faced by the firm (agent's price "A")


A firm with exogenous input and output price uses a given technology to choose output level. Zero profit equation determines the level of output.


As long as the general equilibrium nature of the model is reserved, we have PROFITSLACK = 0 after shock.


Note: here we do not consider the specific production function or production technology. So it is not from optimizing a specific production, but just a condition from the economic intuition.


We can further write it as price-quantity notation:

Where:

When we measure the value of intermediate input from demand side, we also use the quantity as QF, but with price PM. That is to say, intermediate input's value measured on demand side is VFM and has price PM, while on supply side it is VFA with price PF.


Linearized form
Using "with constant return to scale and cost minimization, price-weighted output quantity change equals to sum of price weighted input quantity change", we have the following equation, where percentage change of PROFITSLACK is evaluated related to VOA:


Linearized form by taking total derivative with respect to quantity and price of the equation above.



This is equation ZEROPROFITS in GTAPO.tab.


For detailed procedure, see the note of linearization of zero profit condition


Income - expenditure balance

Income- expenditure balance conditions include:


For expenditure side, we have:

private expenditure equals to expenditure on all tradable commodities it purchases:

Note: VPA is the expenditure. The income from selling inputs is VFA (or VOA) for inputs in endw_comm set.


Government expenditure equals to expenditure on all tradable commodities it purchases:

Total expenditure in the economy:

Where:


Note that in the market clear condition of tradable goods, value of private and government are represented as VPM and VGM respectively. So I think here we use "A" because we consider private and government as agents, and consider specific price modifier (tax or subsidy) applied on them.


On income side, we have:

Formation of income:

Where:


Finally, we have the income-expenditure (household budget) balance:
INCOME = EXPENDITURE = PRIVEXP + GOVEXP + SAVE


Note: when we say "household budget" here, the "household" refers to the regional household or the whole economy, which takes all income and separate to private household (the "household" in common saying), government, and saving.


Saving-investment balance

We have saving-investment balance:
NETINV = SAVE
or
VOM("cgds") - VDEP = SAVE


SAVE: the demand of cgds commodity


NETINV = VOM("cgds") - VDEP = GROSSINV - VDEP: supply of cgds commodity


We also have:
SAVE = PSAVE × QSAVE
NETINV = GROSSINV - VDEP

VOA("cgds") = PS("cgds") × QO("cgds") = PCGDS × QCGDS


Note: cgds is in the set of PROD_COMM and CGDS_COMM (see sets in GTAP model) but not in TRAD_COMM. So:


To reflect the excess of investment supply over saving, we add a slack variable:
NETINV = SAVE + WALRASLACK


Similarly, we define to avoid "dividing by zero" problem.


Total differentiate the equation above, we have it in the linearized form
NETINV × walras_sup = SAVE × walras_dem + NETINV × walraslack


Where:


We can write it as
walras_sup = walras_dem + walraslack


Where


Procedure to get the second equation:
NETINV = GROSSINV - VDEP
total differentiate it:
dNETINV = dGROSSINV - dVDEP


NETINV walras_sup = GROSSINV qcgds - VDEP kb