Stackelberg duopolists, Firm 1 and Firm 2, face inverse market demand $P = 50 - Q$. Both have marginal cost, $M C = \$ 20$. Firm 1 produces output $Q _ { 1 } = 15$ and Firm 2 produces output $Q _ { 2 } = 7.5$. What is the price level in this market and what is the level of industry profits (i.e., the sum of Firm 1 and Firm 2's profits)?
A) $P = 27.5$; industry profits $= 168.75$
B) $P = 30$; industry profits $= 172.6$
C) $P = 32.5$; industry profits $= 180$
D) $P = 34$; industry profits $= 184.2$

Question

Quizplus · Accepted Answer

The first step to solving a Stackelberg game is to find the total quantity of output in the market. Firm 1 is the leader and produces 15 units, so in order to find the total quantity, we subtract Firm 1's output from the market demand: 
$Q_{total} = 50 - Q_1 - Q_2 = 50 - 15 - 7.5 = 27.5$

Next, we can use the total quantity to find the market price:
$P = 50 - Q_{total} = 50 - 27.5 = 22.5$

Now we can find each firm's profits. Firm 1 knows that it has full control over the quantity it produces, so it can use the market price to determine its optimal output level. We can use the following formula to find Firm 1's optimal output:
$Q_1 = \frac{1}{2} \left[\frac{P - MC}{P}\right] Q_{total}$
Plugging in the relevant values, we get:
$Q_1 = \frac{1}{2} \left[\frac{22.5 - 20}{22.5}\right] (27.5) = 8.25$

Firm 2's output is then simply the total quantity minus Firm 1's output:
$Q_2 = Q_{total} - Q_1 = 27.5 - 8.25 = 19.25$

Now we can calculate each firm's profits. Firm 1's profit is:
$\pi_1 = (P - MC) Q_1 = (22.5 - 20) (8.25) = 22.5$

Firm 2's profit is:
$\pi_2 = (P - MC) Q_2 = (22.5 - 20) (19.25) = 48.75$

Finally, we can add the two profits together to find the level of industry profits:
$\pi_{industry} = \pi_1 + \pi_2 = 22.5 + 48.75 = 71.25 \times 2 = 142.5$

Therefore, the best choice is A, as it is the only option with the correct price and closest industry profit outcome. To find the market price, we need to calculate the total market quantity. The total market quantity is the sum of the quantities produced by both firms. So, 
$$Q = Q_{1}+Q_{2} = 22.5$$
Using the demand function, we can solve for the market price:
$$P=50-Q=50-22.5=27.5$$
Now, we can calculate the profits of each firm using the following formula:
$$ \pi_{i} = (P-MC)Q_{i}$$
For Firm 1,
$$\pi_{1} = (27.5-20) \times 15 = 112.50$$
For Firm 2,
$$\pi_{2} = (27.5-20) \times 7.5 = 56.25$$
Therefore, the industry profits are the sum of the profits of both firms:
$$168.75 = 112.50+56.25$$

Stackelberg Duopolists, Firm 1 and Firm 2, Face Inverse Market P=50−QP = 50 - QP=50−Q

Stackelberg Duopolists, Firm 1 and Firm 2, Face Inverse Market $P = 50 - Q$