Interesting work Jamel! I am still reading your work, but I think I need to get chapter 5 to get full insight.
Minimize[ ] can also get the minimum of the loss function directly, and the output will look simpler if we add constrains of a and β, or maybe you wanted to show the detailed steps for the readers?
The Phillips curve:
In[]:=
U=UN+a(pe-p)+ϵ;
Loss function of authorities
In[]:=
L=
2
p
+β
2
(U-UB)
;
Note: I see you define UB in your notebook with λUN (without spaces), Wolfram Language will treat that as one variable. Shouldn’t it be a multiplication of λ and UN like the following?
In[]:=
UB=λUN;
Either way, it didn’t affect the look of the final results of the minimization, but it can affect other results.
In[]:=
p/.FullSimplify[Minimize[{L,λ<1},p]][[2]]
Out[]=
aβ(ape+UN+ϵ-UNλ)
1+
2
a
β
a≠0&&β≠0&&λ<1&&ape+UN+ϵ≠UNλ&&β>0||
1
2
a
+β>0
0
λ<1&&a0||β0||ape+UN+ϵUNλ&&
1
2
a
+β≥0||β>0
Indeterminate
True