<p>Consider a firm in a two period world. The firm operates as a monopolist in the first period. In the second period, another firm will enter and the two firms will act as Cournot duopolists (i.e., in the second period the two firms compete in quantities as they will simultaneously decide on their production levels and then price will be determined from the demand unction and market clearing). The market demand is identical in each period and is described by the function <span class="math-container">$Q=100-P$</span> where <span class="math-container">$0\le P\le 100 $</span> in dollar per unit and <span class="math-container">$Q$</span> is the number of units demanded. In the first period, the firm has a constant marginal cost of 20 TL per unit. The firm learns from its first period production and thus has a lower cost in the second period. In particular, its second period marginal cost is a constant <span class="math-container">$\\\$$$(20-(Q_1/5))$</span> per unit where <span class="math-container">$Q_1$</span> is the number of units that the firm sold in the first period. So, the more it produces in the first period, the lower its second period cost. Unfortunately, the firm is full of corporate spies and its competitor in the second period will also learn and have the same marginal cost.</p>
<blockquote>
<p>I want to I identify a subgame perfect equilibrium of this two stage game, assuming that the firm present in both periods maximizes the (undiscounted) sum of its profits from both periods.</p>
</blockquote>
<p>My solution does not make sense. Therefore, I don’t write down my attempt. I will appreciate if you help me to deal with this question. Thank you.</p>
<hr />
<p>My Solution:</p>
<p>I apply the backward induction.</p>
<p><strong>Second period:</strong> For qiven <span class="math-container">$Q_1$</span> solve for the Cournot problem:</p>
<p>For firm 1,</p>
<p><span class="math-container">$$\pi_1^c = (100 - (q_1 +q_2))q_1 - (20 - \frac{Q_1}{5})q_1$$</span></p>
<p><span class="math-container">$$\frac{\partial \pi_1^c}{\partial q_1}= 100-2q_1-q_2 +\frac{Q_1}{5}=0$$</span></p>
<p><span class="math-container">$$q_1 = 40 - \frac{q_2}{2} +\frac{Q_1}{10}$$</span></p>
<p>Similarly, for firm 2</p>
<p><span class="math-container">$$\pi_2^c = (100 - (q_1 +q_2))q_2 - (20 - \frac{Q_1}{5})q_2$$</span></p>
<p><span class="math-container">$$\frac{\partial \pi_2^c}{\partial q_2}= 100-2q_2-q_1 +\frac{Q_1}{5}=0$$</span></p>
<p><span class="math-container">$$q_2 = 40 - \frac{q_1}{2} +\frac{Q_1}{10}$$</span></p>
<p>When I solve together</p>
<p><span class="math-container">$$q_2 = 40 +\frac{Q_1}{10} + \frac{1}{2}(40 - \frac{q_2}{2} +\frac{Q_1}{10})$$</span></p>
<p><span class="math-container">$$q_1 = q_2 = \frac{80}{3}+\frac{Q_1}{15}$$</span></p>
<p><strong>Second period:</strong> Firm 1 is a monopolist</p>
<p><span class="math-container">$$\pi_1 =(100-Q_1)Q_1- 20 Q_1$$</span></p>
<p><span class="math-container">$$\frac{\partial \pi_1}{\partial Q_1}= 100 - 2Q_1 - 20 =0$$</span></p>
<p><span class="math-container">$$Q_1=40$$</span></p>
<p>So, SPNE = <span class="math-container">$\{88/3, 88/3\}$</span></p>

CC BY-SA 4.0
2022-08-16T11:17:59.520Z
For given first-period $Q_1$, 1. calculate the firm's first-period profit, 2. calculate the firm's second-period profit in Cournot equilibrium, 3. add those two. Then maximize the sum via choice of first-period $Q_1$.
VARulle
CC BY-SA 4.0
2022-08-18T13:56:54.690Z
Dear @VARulle I tried to do your hint many times. But I could not find a proper answer. I am confused too much. Please can you show the solution. I will apperciate you if you help me.
studentp
CC BY-SA 4.0
2022-08-18T14:08:01.380Z
There's a policy here against posting (and answering) something resembling a homework question, and against posting questions that show no own effort in answering them. So the least you have to do is post your own (failed) attempts at following these steps.
VARulle
CC BY-SA 4.0
2022-08-18T15:20:17.880Z
Dear @VARulle you are so right. I am so sorry for disturbing you for this issue. I added my solution attempt. I corrected my all mistakes which I see. But, can you please check my answer as well ? Thank you so much:)
studentp
CC BY-SA 4.0
2022-08-19T16:20:35.947Z
What happens to the term $-20q_1$ when you differentiate the profit function of firm 1? (Same for firm 2.)
VARulle
<microeconomics><game-theory><self-study><nash-equilibrium>
Find SPNE in the two stage game
studentp
# Find SPNE in the two stage game

(Asked by: username_0 on 2022-08-15)

Consider a firm in a two period world. The firm operates as a monopolist in the first period. In the second period, another firm will enter and the two firms will act as Cournot duopolists (i.e., in the second period the two firms compete in quantities as they will simultaneously decide on their production levels and then price will be determined from the demand unction and market clearing). The market demand is identical in each period and is described by the function $Q=100-P$ where $0\le P\le 100 $ in dollar per unit and $Q$ is the number of units demanded. In the first period, the firm has a constant marginal cost of 20 TL per unit. The firm learns from its first period production and thus has a lower cost in the second period. In particular, its second period marginal cost is a constant $\\\$$$(20-(Q\_1/5))$ per unit where $Q\_1$ is the number of units that the firm sold in the first period. So, the more it produces in the first period, the lower its second period cost. Unfortunately, the firm is full of corporate spies and its competitor in the second period will also learn and have the same marginal cost.

> I want to I identify a subgame perfect equilibrium of this two stage game, assuming that the firm present in both periods maximizes the (undiscounted) sum of its profits from both periods.

My solution does not make sense. Therefore, I don’t write down my attempt. I will appreciate if you help me to deal with this question. Thank you.

---

My Solution:

I apply the backward induction.

**Second period:** For qiven $Q\_1$ solve for the Cournot problem:

For firm 1,

$$\pi\_1^c = (100 - (q\_1 +q\_2))q\_1 - (20 - \frac{Q\_1}{5})q\_1$$

$$\frac{\partial \pi\_1^c}{\partial q\_1}= 100-2q\_1-q\_2 +\frac{Q\_1}{5}=0$$

$$q\_1 = 40 - \frac{q\_2}{2} +\frac{Q\_1}{10}$$

Similarly, for firm 2

$$\pi\_2^c = (100 - (q\_1 +q\_2))q\_2 - (20 - \frac{Q\_1}{5})q\_2$$

$$\frac{\partial \pi\_2^c}{\partial q\_2}= 100-2q\_2-q\_1 +\frac{Q\_1}{5}=0$$

$$q\_2 = 40 - \frac{q\_1}{2} +\frac{Q\_1}{10}$$

When I solve together

$$q\_2 = 40 +\frac{Q\_1}{10} + \frac{1}{2}(40 - \frac{q\_2}{2} +\frac{Q\_1}{10})$$

$$q\_1 = q\_2 = \frac{80}{3}+\frac{Q\_1}{15}$$

**Second period:** Firm 1 is a monopolist

$$\pi\_1 =(100-Q\_1)Q\_1- 20 Q\_1$$

$$\frac{\partial \pi\_1}{\partial Q\_1}= 100 - 2Q\_1 - 20 =0$$

$$Q\_1=40$$

So, SPNE = $\{88/3, 88/3\}$

**Tags:** microeconomics, game-theory, self-study, nash-equilibrium

> **Comments:**
> **username_1**: For given first-period $Q_1$, 1. calculate the firm's first-period profit, 2. calculate the firm's second-period profit in Cournot equilibrium, 3. add those two. Then maximize the sum via choice of first-period $Q_1$.  
> **username_0**: Dear @username_1 I tried to do your hint many times. But I could not find a proper answer. I am confused too much. Please can you show the solution. I will apperciate you if you help me.  
> **username_1**: There's a policy here against posting (and answering) something resembling a homework question, and against posting questions that show no own effort in answering them. So the least you have to do is post your own (failed) attempts at following these steps.  
> **username_0**: Dear @username_1 you are so right. I am so sorry for disturbing you for this issue. I added my solution attempt. I corrected my all mistakes which I see. But, can you please check my answer as well ? Thank you so much:)  
> **username_1**: What happens to the term $-20q_1$ when you differentiate the profit function of firm 1? (Same for firm 2.)  

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## 0 answers

