Description of the Alvarez-Lucas algorithm
It is used to solve for the changes in the equilibrium wages and prices.
1) Start with the equation for the change in prices (equation 8 in the paper).
Express the price equation in logs. pf and wf express the change in prices and wages. We can put all the price equations
together as ln(pf)=h(ln(pf),ln(wf)). If we fixed ln(wf) then h(.,ln(wf)) maps g(ln(wf)) into (Tg)(ln(wf))=h(g(ln(wf)),ln(wf)),
with a fixed point ln(pf)(ln(wf))=(Tln(pf))(ln(wf)).This mapping T is a contraction.
Thus, to find the vector of normalized price changes, given a vector of
relative wage changes, simply iterate on the mapping T .
2)Having solved for ln(pf)(ln(wf)),solve for the counterfactual
expenditure share matrix as a function of ln(wf).
3)Defines the excess of demand equation zw.We have that zw=0 evaluated at the equilibrium relative wage changes.
For some v in the interval (0,1] define the mapping tw.
A fixed point of T satisfies the equilibrium condition of
zero excess demand. It maps a change in wages vector into a new change in
wages vector.
4) Having solved for the equilibrium change in wages, we can now solve for
the equilibrium change in prices and the counterfactual expenditure share
matrix.