<p>Since your question has the tag “price discrimination” I’m going to assume the two demand functions are on the same good, on two different markets.</p>
<p>When you say “optimal bundle” I think you mean “equilibrium”.</p>
<p>The functional forms of your demand functions tell me that by <span class="math-container">$k$</span> in the <span class="math-container">$Q_1$</span>,<span class="math-container">$Q_2$</span> functions, you mean <span class="math-container">$p_1$</span>,<span class="math-container">$p_2$</span>, respectively. <span class="math-container">$p_1$</span>,<span class="math-container">$p_2$</span> shouldn’t be assumed equal in a question about price discrimination.</p>
<p>In a price discrimination situation, both markets are independent, so the monopolist wants to maximize profits on each of them.</p>
<p>The profit function in market <span class="math-container">$1$</span> is</p>
<p><span class="math-container">$\Pi_1 = q_1(36-q_1) - 12 q_1$</span></p>
<p><span class="math-container">$\Pi_1 = 36 q_1 - q_{1}^2 - 12 q_1$</span></p>
<p><span class="math-container">$\Pi_1 = 24 q_1 - q_{1}^2$</span></p>
<p>The optimality condition is set its derivative to <span class="math-container">$0$</span></p>
<p><span class="math-container">$\frac{d\Pi_1}{dq_1} = 24 - 2 q_1 =0 \implies q_{1}^\star = 12$</span></p>
<p>Substituting on inverse demand <span class="math-container">$1$</span>,</p>
<p><span class="math-container">$p_{1}^\star = 36 - q_1 = 36 - 12 = 24$</span></p>
<p>The profit function in market <span class="math-container">$2$</span> is</p>
<p><span class="math-container">$\Pi_2 = q_2 \cdot \frac{36-q_2}{2} - 12 q_2$</span></p>
<p><span class="math-container">$\Pi_2 = 18 q_2 - \frac{1}{2} q_{2}^2 - 12 q_2$</span></p>
<p><span class="math-container">$\Pi_2 = 6 q_2 - \frac{1}{2} q_{2}^2$</span></p>
<p>The optimality condition is set its derivative to 0</p>
<p><span class="math-container">$\frac{d\Pi_2}{dq_2} = 6 - q_2 = 0 \implies q_2^\star = 6$</span></p>
<p>Substituting on inverse demand <span class="math-container">$2$</span>,</p>
<p><span class="math-container">$p_{2}^\star = \frac{36-6}{2} = \frac{30}{2} = 15$</span></p>
<p>We now have the equilibrium quantities produced and prices. With this, we can calculate the monopolist’s profit on each market and its total profit.</p>
<p>The monopolist’s profits on each market are</p>
<p><span class="math-container">$\Pi_{1}^\star = p_1^\star q_1^\star - TC(q_1^\star) = 24 \cdot 12 - 12 \cdot 12 = 12 (24-12) = 12 \cdot 12 = 144$</span></p>
<p><span class="math-container">$\Pi_{2}^\star = p_2^\star q_2^\star - TC(q_2^\star) = 15 \cdot 6 - 12 \cdot 6 = 6(15-12) = 6 \cdot 3 = 18$</span></p>
<p>The monopolist’s total profit is</p>
<p><span class="math-container">$\Pi^\star = \Pi_{1}^\star + \Pi_{2}^\star = 144 + 18 = 162$</span></p>

CC BY-SA 4.0
2023-04-04T23:55:26.607Z
so if I needed to determine how many entries to bundle for each group and the price to charge for the bundles and compute the profits earned from implementing this strategy, would this be the answer
John Phillip
CC BY-SA 4.0
2023-04-05T03:09:32.683Z
@JohnPhillip Yes. You find the optimal quantity and price in each market, then add the profits from both markets.
Nicolas Torres
CC BY-SA 4.0
2023-04-04T14:33:14.640Z
2023-04-04T14:52:52.977Z
2023-04-04T14:52:52.977Z
Nicolas Torres
<p>If I am given two demand functions <span class="math-container">$Q1 = 36-k$</span> and <span class="math-container">$Q2 = 36-2k$</span> and <span class="math-container">$MC=12$</span>, to find the optimal bundle would I first solve for quantity by finding the inverse of both demand functions and solving for <span class="math-container">$Q$</span>, then using <span class="math-container">$Q$</span> to find the price e.g</p>
<p><span class="math-container">$k1=36-Q1$</span>,</p>
<p><span class="math-container">$36-2Q2=12$</span>,</p>
<p><span class="math-container">$24=2Q1$</span>,</p>
<p><span class="math-container">$12=q1$</span>,</p>
<p><span class="math-container">$p(12)=36-12 = 24$</span></p>
<p>with the optimal bundle just being the price and quantity found. Or would i solve it a different way.</p>

<price-discrimination>
How do i find the optimal bundle from two demand functions
John Phillip
# How do i find the optimal bundle from two demand functions

(Asked by: username_0 on 2023-04-04)

If I am given two demand functions $Q1 = 36-k$ and $Q2 = 36-2k$ and $MC=12$, to find the optimal bundle would I first solve for quantity by finding the inverse of both demand functions and solving for $Q$, then using $Q$ to find the price e.g

$k1=36-Q1$,

$36-2Q2=12$,

$24=2Q1$,

$12=q1$,

$p(12)=36-12 = 24$

with the optimal bundle just being the price and quantity found. Or would i solve it a different way.

**Tags:** price-discrimination

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## 1 answers

### Answer by username_1 on 2023-04-04. Score: 2

Since your question has the tag “price discrimination” I’m going to assume the two demand functions are on the same good, on two different markets.

When you say “optimal bundle” I think you mean “equilibrium”.

The functional forms of your demand functions tell me that by $k$ in the $Q\_1$,$Q\_2$ functions, you mean $p\_1$,$p\_2$, respectively. $p\_1$,$p\_2$ shouldn’t be assumed equal in a question about price discrimination.

In a price discrimination situation, both markets are independent, so the monopolist wants to maximize profits on each of them.

The profit function in market $1$ is

$\Pi\_1 = q\_1(36-q\_1) - 12 q\_1$

$\Pi\_1 = 36 q\_1 - q\_{1}^2 - 12 q\_1$

$\Pi\_1 = 24 q\_1 - q\_{1}^2$

The optimality condition is set its derivative to $0$

$\frac{d\Pi\_1}{dq\_1} = 24 - 2 q\_1 =0 \implies q\_{1}^\star = 12$

Substituting on inverse demand $1$,

$p\_{1}^\star = 36 - q\_1 = 36 - 12 = 24$

The profit function in market $2$ is

$\Pi\_2 = q\_2 \cdot \frac{36-q\_2}{2} - 12 q\_2$

$\Pi\_2 = 18 q\_2 - \frac{1}{2} q\_{2}^2 - 12 q\_2$

$\Pi\_2 = 6 q\_2 - \frac{1}{2} q\_{2}^2$

The optimality condition is set its derivative to 0

$\frac{d\Pi\_2}{dq\_2} = 6 - q\_2 = 0 \implies q\_2^\star = 6$

Substituting on inverse demand $2$,

$p\_{2}^\star = \frac{36-6}{2} = \frac{30}{2} = 15$

We now have the equilibrium quantities produced and prices. With this, we can calculate the monopolist’s profit on each market and its total profit.

The monopolist’s profits on each market are

$\Pi\_{1}^\star = p\_1^\star q\_1^\star - TC(q\_1^\star) = 24 \cdot 12 - 12 \cdot 12 = 12 (24-12) = 12 \cdot 12 = 144$

$\Pi\_{2}^\star = p\_2^\star q\_2^\star - TC(q\_2^\star) = 15 \cdot 6 - 12 \cdot 6 = 6(15-12) = 6 \cdot 3 = 18$

The monopolist’s total profit is

$\Pi^\star = \Pi\_{1}^\star + \Pi\_{2}^\star = 144 + 18 = 162$

> **Comments:**
> **username_0**: so if I needed to determine how many entries to bundle for each group and the price to charge for the bundles and compute the profits earned from implementing this strategy, would this be the answer  
> **username_1**: @JohnPhillip Yes. You find the optimal quantity and price in each market, then add the profits from both markets.  

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