Definition
It is the bridge between the microeconomic theory I have learned and the economic model that is practical (computable).
(1) Start from cost minimization problem (variable: z, parameter: w, q)
(2) Derive conditional demand z(w, q) and cost function c(w, q)
Note: they are already the solution of the cost minimization problem.
(3) Rewrite with unit cost (c(w,q)/q)
(4) Linearize conditional demand and cost function with proportional change (dx/x, x is the variable of interest)
This transformation is similar with logarithm transformation because of their relationship.
This percentage change offers a natural decomposition of changes in conditional demand to:
- expansion effect (change of q)
- substitution effect (change of relative price, unit cost / price of this input)
- biased technical change (change in the input-specific parameter)
- neutral technical change (change in the general technical parameter)
Rule of hat calculus
Upper case: level of variable
Lower case or hat: percentage change of variable
Greek letter: non-variable scalar (constant parameter)
Linearization of product with scalar
X = αY
x = y
Proof:
since α is constant, 
, so x = y
Linearization of product
X = Y × Z 
x = y + z
X= Y / Z x = y - z
(YZ)' = Y'Z + Z'Y
So we have x = y + z. But it only works for tiny change, and can be regarded as approximation for larger change
Linearization of sum
Also denoted as Jone's algebra:
Total derivative:
Where 
is the share of X term in original Z:
Note: note how do I derive the equation from minor change (dX,. dY, dZ) to percentage change (z, x, y, or minor change divided by original value)
Linearization of exponent
x = αy
Linearization of growth
e is natural logarithm base, t is time, g is growth rate
y = g
Note
Reference: Hat calculus: https://www.uvm.edu/~wgibson/CYU/CYU_hats.pdf