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I am reading following article: https://www.tse-fr.eu/sites/default/files/medias/doc/by/nguyen/1august13_jmerev_gln.pdf and trying to code the optimization problem given in the page 6 (eq P-6). The Lagrangian and the associated first order conditions are given in the page 14 & 15 (eq 12-18). As of now I am thinking discount rate is zero. Suppose first we are trying to solve this over finite time horizon instead of infinity

  1. How do I get optimal paths of control and state variables?

Here is the Mathematical formula given in the paper. Problem The Lagrangian and the associated conditions are given here: Lagrangian eqs Here is my effort but not sure how to proceed.I tried NDSolve with some initial conditions but it didn't work.

<< VariationalMethods`
Clear["Derivative"]
ClearAll["Global`*"]
u[c_] := Log[c]; (*Utility Function*)
\[Psi] = 0.36;
f[k_, l_] := ((k)^\[Psi])*((1 - 
      l)^(1 - \[Psi])); (*Physical Production function*)
Subscript[\[Phi], 1] = 2;
Subscript[\[Phi], 2] = 0.1;
Subscript[\[Phi], 3] = 1;
g[m_] := Subscript[\[Phi], 3]*(m + Subscript[\[Phi], 1])^
   Subscript[\[Phi], 2] - 
  Subscript[\[Phi], 3]*Subscript[\[Phi], 1]^Subscript[\[Phi], 
   2]; (*health production function*)
Subscript[\[Alpha], 1] = 0.023;
Subscript[\[Alpha], 2] = 0.023;
Subscript[\[Alpha], 3] = 1;
Subscript[\[Gamma], 1] = 1.01;
Subscript[\[Gamma], 2] = 1;
Subscript[\[Gamma], 3] = 1;
\[Alpha][h_] := 
 Subscript[\[Alpha], 1] + 
  Subscript[\[Alpha], 2]*
   Exp[-Subscript[\[Alpha], 3]*h]; (*transmission rate*)
\[Gamma][h_] := 
 Subscript[\[Gamma], 1] - 
  Subscript[\[Gamma], 2]*
   Exp[-Subscript[\[Gamma], 3]*h]; (*recovery rate*)
Subscript[\[Delta], K] = 0.05; (*Physical Capital Depreciation rate*)
\

Subscript[\[Delta], H] = 0.05; (*Health Capital Depreciation rate*)
d = 0.005; (*Death rate*)
b = 0.037; (*Birth rate*)
L[c_, k_, l_, m_, 
  h_, \[Lambda]1_, \[Lambda]2_, \[Lambda]3_, \[Mu]1_, \[Mu]2_] := 
 u[c] + (\[Lambda]1*(f[k, l] - c - 
      m - (Subscript[\[Delta], K] + b - d)*k)) + (\[Lambda]2*(g[
       m] - (Subscript[\[Delta], H] + b - \[Mu])*
       h)) + (\[Lambda]3*((1 - 
        l)*(b + \[Gamma][h] - \[Alpha][h]*l))) + \[Mu]1*(1 - 
     l) + \[Mu]2*m;
eqs = EulerEquations[
  L[c[t], k[t], l[t], m[t], 
   h[t], \[Lambda]1[t], \[Lambda]2[t], \[Lambda]3[t], \[Mu]1[
    t], \[Mu]2[t]], {c[t], k[t], l[t], m[t], 
   h[t], \[Lambda]1[t], \[Lambda]2[t], \[Lambda]3[t], \[Mu]1[
    t], \[Mu]2[t]}, t]
eqs // TableForm
{c[t], k[t], l[t], m[t], h[t]} /. 
  FullSimplify[
   Solve[eqs, {c[t], k[t], l[t], m[t], 
     h[t], \[Lambda]1[t], \[Lambda]2[t], \[Lambda]3[t], \[Mu]1[
      t], \[Mu]2[t]}]] // TableForm
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