<p>To illustrate what Tirole has done, let's consider a simpler environment.</p>

<p>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</p>

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

<p>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>

<p>$$ p_x x +p_y y = m $$</p>

<p>We can rearrange the above equality to</p>

<p>$$ y = \frac{m - p_x x}{p_y} $$</p>

<p>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</p>

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

<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2017-02-03T20:08:37.847Z
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
CC BY-SA 3.0
2017-02-03T18:56:35.700Z
2017-02-03T18:56:35.700Z
Theoretical Economist
<p>I am dealing with a constrained optimization problem found in Tirole's <em>Theory of corporate finance</em>. 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.</p>

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

<p>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.
$$\int_0^\infty[R-w(R)-[1-y(R)]K]p(R)dR\geq I-A$$</p>

<p>My question relates to the following.
The author argues that, since the second constraint binds at equilibrium, <em>it can be added to the objective function</em>. The problem is then equivalent to minimizing the expected audit costs:
$$K\left[\int_0^\infty[1-y(R)]p(R)dR\right]$$
<em>subject to the two above constraints</em>.
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!</p>

CC BY-SA 3.0
2017-02-03T17:25:33.327Z
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
CC BY-SA 3.0
2017-02-03T20:00:49.433Z
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
<microeconomics><optimization><corporate-finance><contract-theory>
Adding a non-binding constraint to the objective function
jlol
# Adding a non-binding constraint to the objective function

(Asked by: username_0 on 2017-02-03)

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!

**Tags:** microeconomics, optimization, corporate-finance, contract-theory
> **Comments:**
> **username_1**: 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"?  
> **username_0**: 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.  

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## 1 answers

### Answer by username_2 on 2017-02-03. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: 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!  

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