# Adding a non-binding constraint to the objective function

I am dealing with a constrained optimization problem found in Tirole's Theory of corporate finance. My question is not related to the details of this model, but just to provide some context, we are solving for the optimal contract to secure outside financing in a setting with costly state verification. $y(\hat{R})$ is the probability of no audit when income $\hat{R}$ is reported. Audits cost $K$ to the lender. $w_0(\hat{R},R)$ is the reward for the borrower in case of no audit, and $w_1(\hat{R},R)$ in case of audit.

The objective function is: $$\max\limits_{y(\cdot),w_0(\cdot,\cdot),w_1(\cdot,\cdot)}\left\{\int_0^\infty w(R)p(R)dR\right\}$$

It is subject to the incentive constraint for the borrower to report the true income $R$: $$w(R)=\max\limits_{\hat{R}}\{y(\hat{R})w_0(\hat{R},R)+(1-y(\hat{R}))w_1(\hat{R},R)\}$$ and to the break-even constraint for lenders who provide $I-A$ to the entrapreneur. $$\int_0^\infty[R-w(R)-[1-y(R)]K]p(R)dR\geq I-A$$

My question relates to the following. The author argues that, since the second constraint binds at equilibrium, it can be added to the objective function. The problem is then equivalent to minimizing the expected audit costs: $$K\left[\int_0^\infty[1-y(R)]p(R)dR\right]$$ subject to the two above constraints. I don't understand why we could add the binding constraint to the objective function and find an equivalent problem subject to the same constraints. Is there a general optimization principle that I missed along the way? Thanks!

• By the verb "added" is it meant literally the mathematical operation of addition, or just "incorporated", which means nothing more than "if the apparently _in_equality constraint, will hold as an equality, then we treat it as an equality to begin with, and insert it into the objective function"? – Alecos Papadopoulos Feb 3 '17 at 17:25
• I thought Tirole meant it as an addition, possibly due to his dubious phrasing and my not being the sharpest tool in the shed. I was wrong. – jlol Feb 3 '17 at 20:00

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} &\quad f(x) + y \\ \text{s.t.} &\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.

• Wow. I did not even try a substitution. I thought Tirole had to be using some optimization result I was not aware of. Clearly, you are right. Thank you! – jlol Feb 3 '17 at 20:08