Definition
Where:
- i and j belong to TRAD_COMM
- EP: own price elasticity for i = j, cross price elasticity for i not equals j (ordinary, uncompensated, marshallian)
- EY: income (per capita) elasticity (expenditure / Engel)
- yp: % change of total revenue
- pop: % change of population
- pp: % change of sector's price
- qp: % change of sector's output demand
For the system of N sectors, the total number of elasticities needed is: N*N (EP) + N (EY)
It is called "generic demand equation" because this function is invariant to the choice of functional form.
Development of generic demand equation
First, we solve the utility maximization problem of consumer:
Utility function: 
Budget constraint at YH.
Advantages of utility maximizing approach are:
- This approach gives the welfare interpretation of the results
- This approach restricts the number of independent elasticities need to be provided.
From the solution of utility maximization problem, we have the unrestricted demand equation (level form, single household,
Take total derivative of the demand equation, we have the linearized form:
Demand for Individual and aggregated households
It takes population growth rate explicitly (as POP below)
Demand function for single household
Demand function for aggregated household
Relation between individual and aggregated demand and income
Where:
- YH: income of single household
- YP: income of aggregate household
- Qi: demand of output i for single household
- QPi: demand of output i for aggregated household
- POP: population
Linearized form:
yh = yp - pop
Then unrestricted demand equation (linearized, % change) for single household and aggregated household are:
Alternative form: Hicksian and substitution elasticities
Hicksian (compensated) demand function: from expenditure minimization problem:
Level form: 
Linearized form: 
CP: the compensated (Hicksian) elasticities
CP can be converted to Marshallian elasticities EP via Slutsky decomposition:
Recall that CONSHR means the share of consumption by each sector over total expenditure of consumption (see the example here)
CP(i,j) is further related with the Allen elasticity of substitution 
:
General restrictions of demand system
The generic demand equation has several important features:
- Homogeneity in price (neutrality of money, Cournot aggregation):
When prices and income increase by the same percentage, the demand quantity is unchanged, or:
- Engel aggregation (budget constraint):
When just income goes up by 1%, the total (summed) value of demand goes up by 1%.
- Symmetry of APEs:
These features also serve as general restrictions of demand system, they reduce the number of independent elasticities by about half.
If we specify the functional form of utility, the number of elasticities can be further reduced, and elasticities become functions of budget shared and a limited number of parameters.