Now if I eliminate all the unused code and ask it to evaluate only the expression that you wish to maximize

In[1]:= Phi[x_] = CDF[NormalDistribution[], x];
phi[x_] = PDF[NormalDistribution[], x];
Pi0sL[\[Mu]_, x0m_, x0s_] = p x0m - (sg + (c + m0) tg) x0s + Phi[(x0m - \[Mu])/\[Sigma]] (-p x0m + (eg + sg+
   (c + m0) tg) x0s + (-eg + p - sg - c tg - m0 tg) \[Mu]) + Phi[(x0s - \[Mu])/\[Sigma]] ((-eg - sg - (c +m0)
   tg) x0s + p \[Mu] + (eg - p + sg + c tg + m0 tg) \[Mu]) + phi[(x0m - \[Mu])/\[Sigma]] (eg - p + sg + c tg+
   m0 tg) \[Sigma] + phi[(x0s - \[Mu])/\[Sigma]] (-p \[Sigma] + (-eg + p - sg - c tg - m0 tg) \[Sigma]);
Pi0sR[\[Mu]_, x0m_, x0s_] = -(c + m0 + (1 + tg) r0) (x0s - x0m) - (tg (c + m0) + sg) x0s + p (-(x0s - \[Mu])
   Phi[(x0s - \[Mu])/\[Sigma]] - \[Sigma] phi[(x0s - \[Mu])/\[Sigma]] + x0s);
Pi0s[\[Mu]_, x0m_] := Max[FindMaxValue[{Pi0sL[\[Mu], x0m, x0s], 0 <= x0s <= x0m}, {x0s, x0m}],
   FindMaxValue[{Pi0sR[\[Mu], x0m, x0s], 18 >= x0s >= x0m}, {x0s, x0m}]];
Pi0mLR[d_, \[Gamma]_, x0m_] := -(c + m0) x0m + \[Gamma] Pi0s[L + d, x0m] + (1 - \[Gamma]) Pi0s[L, x0m];
Pi0m[d_, \[Gamma]_] := FindMaxValue[{Pi0mLR[d, \[Gamma], x0m], 0 <= x0m <= 18}, {x0m, L}];
L = 4; \[Sigma] = 1; p = 20; c = 1; m0 = 3; m1 = 6; sc = 0; sg = 2; eg = ec = 2; r0 = r1 = 4; tg = tc = 0;
Pi0mLR[2, .5, x]

then I get

Out[9]= 88.4042 - 4 x

and with 0<=x the maximum of that is the 88.4042 that you are seeing.

So FindMaximum returning an incorrect value does not appear to be your problem.

If you change your

Plot[Pi0mLR[2, .5, x], {x, 0, 10}]

to

Plot[Evaluate[Pi0mLR[2, .5, x]], {x, 0, 10}]

then you will see

Why that happens is a whole different subject that I will leave to someone else to clearly explain.

If you change

FindMaximum[{Pi0mLR[2, .5, x], x >= 0}, x]

to

Trace[FindMaximum[{Pi0mLR[2, .5, x], x >= 0}, x]]

and evaluate the notebook then you will see multiple screens of the details of finding the maximum. The final steps of that are to maximize 88.4042 - 4 x such that x >= 0 and that results in x being zero and the maximum being 88.4042. The 88.4042 results from 53.2021 + 35.2021. The 35 is half of 70.4042, the 70.4042 is the max of 70.4042 and 68.49...

If you work your way carefully back through that step by step you should be able to find exactly where in your screenful of long equations every step that FindMaximum is taking to give you the result. This is much better than some other corners of Mathematica where there is no information available on exactly how it came to the conclusion.

A hint may be that if I give a starting estimate for x

FindMaximum[{Pi0mLR[2, .5, x], x >= 0}, {x, 1}]

it tells me

{88.4042, {x -> 0.}}

no matter what starting estimate I give it.

well that's my kernel name. In Evaluation => Notebook's Kernel you can specify your kernel.

Evaluate notebook gives me a stream of errors about not being a Real number when trying to evaluate Pi0m[2, .5]

Can you reproduce this when you download a fresh copy of your attachment and evaluate that?

Maximize returns the same solution, BTW. I'm attaching this haunted file.

Attachments: