Consider a Cournot duopoly with the following inverse demand function: P = 50 - 0.2Q1 - 0.2Q2. The firms' marginal cost are identical and given by MCi(Qi) = 2. Based on this information firm 1 and 2's reaction functions are

A) r1(Q2) = 4.8 - 0.5Q1 and r2(Q1) = 4.8 - 0.5Q2.

B) r1(Q2) = 4.8 - 0.5Q2 and r1(Q2) = 4.8 - 0.5Q1.

C) Q1 = 25 - 0.75Q2 and Q2 = 25 - 0.75Q1.

D) Q1 = 120 - 0.5Q2 and Q2 = 120 - 0.5Q1.

B) r1(Q2) = 4.8 - 0.5Q2 and r1(Q2) = 4.8 - 0.5Q1.

To find the reaction functions of firm 1 and firm 2 in a Cournot duopoly, we need to maximize their individual profits with respect to their respective quantities produced.

Step 1: Determine the profit function for each firm.
The profit function for firm i is given by πi(Qi, Qj) = (P - MCi) * Qi, where P is the price, MCi is the marginal cost of firm i, and Qi and Qj are the quantities produced by firm i and firm j, respectively.

In this case, since the firms have identical marginal costs (MC1 = MC2 = 2), the profit function for both firms is:
πi(Qi, Qj) = (P - 2) * Qi.

Step 2: Determine the best response of firm 1 (r1) and firm 2 (r2) to the quantity chosen by the other firm.
The best response of a firm is the quantity that maximizes its profit, given the quantity chosen by the other firm.

To find r1(Q2), we maximize the profit function of firm 1 with respect to Q1, holding Q2 constant:
Maximize π1(Q1, Q2) = (P - 2) * Q1, subject to the inverse demand function P = 50 - 0.2Q1 - 0.2Q2.

To find r2(Q1), we maximize the profit function of firm 2 with respect to Q2, holding Q1 constant:
Maximize π2(Q2, Q1) = (P - 2) * Q2, subject to the inverse demand function P = 50 - 0.2Q1 - 0.2Q2.

Step 3: Calculate the reaction functions.

To maximize the profit functions, we substitute the inverse demand function into the profit functions and differentiate with respect to the respective variables, setting the derivatives equal to zero.

For firm 1:
Maximize π1(Q1, Q2) = (50 - 0.2Q1 - 0.2Q2 - 2) * Q1
= (48 - 0.2Q1 - 0.2Q2) * Q1.
Taking the derivative with respect to Q1:
dπ1/dQ1 = 48 - 0.4Q1 - 0.2Q2 - 2,
Setting dπ1/dQ1 = 0,
48 - 0.4Q1 - 0.2Q2 - 2 = 0,
46 - 0.4Q1 - 0.2Q2 = 0,
0.4Q1 + 0.2Q2 = 46.

For firm 2:
Maximize π2(Q2, Q1) = (50 - 0.2Q1 - 0.2Q2 - 2) * Q2
= (48 - 0.2Q1 - 0.2Q2) * Q2.
Taking the derivative with respect to Q2:
dπ2/dQ2 = 48 - 0.2Q1 - 0.4Q2 - 2,
Setting dπ2/dQ2 = 0,
48 - 0.2Q1 - 0.4Q2 - 2 = 0,
46 - 0.2Q1 - 0.4Q2 = 0,
0.2Q1 + 0.4Q2 = 46.

Therefore, the correct answer is option C:
Q1 = 25 - 0.75Q2 and Q2 = 25 - 0.75Q1.

To find the reaction functions for the firms in a Cournot duopoly, we need to maximize each firm's profit with respect to their own output, taking into account the other firm's output as a parameter.

First, let's determine the profit function for firm 1. The profit function is given by the difference between total revenue and total cost:

Profit1 = TR1 - TC1

Total revenue (TR1) is the product of price (P) and quantity (Q1):

TR1 = P * Q1

Total cost (TC1) is the product of marginal cost (MC1) and quantity (Q1):

TC1 = MC1 * Q1

Substituting the given inverse demand function (P = 50 - 0.2Q1 - 0.2Q2) and marginal cost (MC1 = MC2 = 2):

TR1 = (50 - 0.2Q1 - 0.2Q2) * Q1
TC1 = 2 * Q1

Now, we can express the profit function for firm 1:

Profit1 = (50 - 0.2Q1 - 0.2Q2) * Q1 - 2 * Q1

To maximize this profit function, we can take the derivative with respect to Q1, set it equal to zero, and solve for Q1:

dProfit1/dQ1 = (50 - 0.2Q1 - 0.2Q2) - 0.4Q1 - 2 = 0
50 - 0.2Q1 - 0.2Q2 - 0.4Q1 - 2 = 0
-0.6Q1 - 0.2Q2 + 48 = 0
-0.6Q1 = 0.2Q2 - 48
Q1 = (0.2Q2 - 48) / -0.6
Q1 = (48 - 0.2Q2) / 0.6

This gives us the reaction function for firm 1, denoted as r1(Q2).

Similarly, we can repeat the same steps to find the reaction function for firm 2, denoted as r2(Q1). By following the same procedure, we find:

Q2 = (48 - 0.2Q1) / 0.6

Therefore, the reaction functions for firm 1 and firm 2 are:

r1(Q2) = (48 - 0.2Q2) / 0.6
r2(Q1) = (48 - 0.2Q1) / 0.6

Comparing these equations to the answer choices:

A) r1(Q2) = 4.8 - 0.5Q1 and r2(Q1) = 4.8 - 0.5Q2
B) r1(Q2) = 4.8 - 0.5Q2 and r1(Q2) = 4.8 - 0.5Q1
C) Q1 = 25 - 0.75Q2 and Q2 = 25 - 0.75Q1
D) Q1 = 120 - 0.5Q2 and Q2 = 120 - 0.5Q1

We can see that the correct answer is option A) r1(Q2) = 4.8 - 0.5Q1 and r2(Q1) = 4.8 - 0.5Q2.