Adding a non-binding constraint to the objective function
To illustrate what Tirole has done, let's consider a simpler environment.

Consider a utility maximisation problem over two goods, $x$ and $y$. The consumer has utility function $u(x,y) = f(x) + y$, where $f$ is strictly increasing and strictly concave. The consumer's problem is thus

$$ \begin{align}
\max_{x,y} &amp;\quad f(x) + y \\
\text{s.t.} &amp;\quad p_x x + p_y y \le m
\end{align} $$

Given the conditions on $f$, we know that the budget constraint must bind. That is to say that in any solution to the above maximisation problem, it must be the case that

$$ p_x x +p_y y = m $$

We can rearrange the above equality to

$$ y = \frac{m - p_x x}{p_y} $$

We can thus incorporate this fact into our objective function and use the expression derived above for $y$ to replace it. Thus, our optimisation problem now becomes

$$ \max_x \quad f(x) +  \frac{m - p_x x}{p_y}$$

which, in a way, is a simpler problem, given that I now no longer have to worry about the Kuhn-Tucker conditions. The second-order conditions are also now much easier to verify.

Tirole does something similar to this. He uses the fact that a constraint is binding to simplify the objective function by expressing the objective in terms of the binding constraint.