<p>If <span class="math-container">$IR_L$</span> is not binding, you can increase both <span class="math-container">$T_L$</span> <em>and <span class="math-container">$T_H$</span></em> by the same amount: Such increases do not violate either of the following:</p>
<ol>
<li><span class="math-container">$IR_L$</span> is not violated as it's initially not binding;</li>
<li>Neither <span class="math-container">$IC_H$</span> nor <span class="math-container">$IC_L$</span> is affected because both sides increase by the same amount; and</li>
<li><span class="math-container">$IR_H$</span> is not affected because it's implied by <span class="math-container">$IR_L$</span> and <span class="math-container">$IC_H$</span> (and the fact that <span class="math-container">$CS_H(p)&gt;CS_L(p)$</span> for all <span class="math-container">$p$</span>).</li>
</ol>

CC BY-SA 4.0
2022-07-08T08:47:55.383Z
Thanks! what about showing that $IC_H$ is binding which seems to be the more difficult proof?
John Pine
CC BY-SA 4.0
2022-07-09T02:22:37.233Z
@JohnPine: If $IC_H$ doesn't bind, then increase $T_H$, which makes $IC_L$ easier to satisfy and doesn't affect $IR_H$ because of point 3 above.
Herr K.
CC BY-SA 4.0
2022-07-08T01:16:27.143Z
2022-07-08T01:16:27.143Z
Herr K.
<p>Two types of customers, equal proportions:</p>
<p><span class="math-container">$q_H(p_H)=20-0.5p_H$</span>,<br />
<span class="math-container">$q_L(p_L)=20-p_L$</span></p>
<p>The firm cant differentiate between them.
It wishes to create a menu of two possible taarifs, <span class="math-container">$(p_H,T_H),(p_L,T_L)$</span></p>
<p><span class="math-container">$T_i$</span> a fixed cost on customer <span class="math-container">$i$</span> and <span class="math-container">$p_i$</span> is the price per good <span class="math-container">$q$</span>.</p>
<ol>
<li>Consumer surplus is <span class="math-container">$CS_H=q_H^2$</span>,<span class="math-container">$CS_L=q_L^2/2$</span></li>
<li>The target function</li>
</ol>
<p><span class="math-container">$$
\max{\pi} = p_H(q_H)q_H+p_L(q_L)q_L+T_H+T_L-4q_H-4q_L
$$</span>
<span class="math-container">$$
=36q_H-2q_H^2+16q_L-q_L^2+T_H+T_L
$$</span>
<span class="math-container">$$st.$$</span>
<span class="math-container">$$
IR_H: CS_H(p_H)-T_H\geq0
$$</span>
<span class="math-container">$$
IR_L:CS_L(p_L)-T_L\geq0
$$</span>
<span class="math-container">$$
IC_H:CS_H(p_H)-T_H\geq CS_H(p_L)-T_L
$$</span>
<span class="math-container">$$
IC_L: CS_L(p_L)-T_L \geq CS_2(p_H)-T_H
$$</span>
Now, I know that <span class="math-container">$IR_L$</span> and <span class="math-container">$IC_H$</span> are binding constraints because there is no distortion at the top and the low demand consumer will have a 0 net surplus, but cant seem the formalize a mathematical proof for it.</p>
<p>I've tried first supposing <span class="math-container">$IR_L$</span> is not binding, thus you could raise <span class="math-container">$T_L$</span> and consumer L will still purchase, but got stuck showing it won't hinder <span class="math-container">$IC_L$</span>. Proving <span class="math-container">$IC_H$</span> is binding seems to be more difficult.</p>
<p>I would greatly appreciate your assistance! thank you.</p>
<p>BTW</p>
<p>it's not homework, but a marathon for a final exam in four days :)</p>

<microeconomics><proof>
Formal proof that IR_L IC_H are binding constraints
John Pine
# Formal proof that IR_L IC_H are binding constraints

(Asked by: username_0 on 2022-07-07)

Two types of customers, equal proportions:

$q\_H(p\_H)=20-0.5p\_H$,  
$q\_L(p\_L)=20-p\_L$

The firm cant differentiate between them.
It wishes to create a menu of two possible taarifs, $(p\_H,T\_H),(p\_L,T\_L)$

$T\_i$ a fixed cost on customer $i$ and $p\_i$ is the price per good $q$.

1. Consumer surplus is $CS\_H=q\_H^2$,$CS\_L=q\_L^2/2$
2. The target function

$$
\max{\pi} = p\_H(q\_H)q\_H+p\_L(q\_L)q\_L+T\_H+T\_L-4q\_H-4q\_L
$$
$$
=36q\_H-2q\_H^2+16q\_L-q\_L^2+T\_H+T\_L
$$
$$st.$$
$$
IR\_H: CS\_H(p\_H)-T\_H\geq0
$$
$$
IR\_L:CS\_L(p\_L)-T\_L\geq0
$$
$$
IC\_H:CS\_H(p\_H)-T\_H\geq CS\_H(p\_L)-T\_L
$$
$$
IC\_L: CS\_L(p\_L)-T\_L \geq CS\_2(p\_H)-T\_H
$$
Now, I know that $IR\_L$ and $IC\_H$ are binding constraints because there is no distortion at the top and the low demand consumer will have a 0 net surplus, but cant seem the formalize a mathematical proof for it.

I've tried first supposing $IR\_L$ is not binding, thus you could raise $T\_L$ and consumer L will still purchase, but got stuck showing it won't hinder $IC\_L$. Proving $IC\_H$ is binding seems to be more difficult.

I would greatly appreciate your assistance! thank you.

BTW

it's not homework, but a marathon for a final exam in four days :)

**Tags:** microeconomics, proof


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## 1 answers

### Answer by username_1 on 2022-07-08. Score: 1

If $IR\_L$ is not binding, you can increase both $T\_L$ *and $T\_H$* by the same amount: Such increases do not violate either of the following:

1. $IR\_L$ is not violated as it's initially not binding;
2. Neither $IC\_H$ nor $IC\_L$ is affected because both sides increase by the same amount; and
3. $IR\_H$ is not affected because it's implied by $IR\_L$ and $IC\_H$ (and the fact that $CS\_H(p)>CS\_L(p)$ for all $p$).

> **Comments:**
> **username_0**: Thanks! what about showing that $IC_H$ is binding which seems to be the more difficult proof?  
> **username_1**: @username_0: If $IC_H$ doesn't bind, then increase $T_H$, which makes $IC_L$ easier to satisfy and doesn't affect $IR_H$ because of point 3 above.  

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