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Let the market demand be

Q=120-2 p

, where

\mathrm{p}

is the market price an...

Jun 10, 2024

Solution by Steps

step 1

The market demand function is given by

Q = 120 - 2p

step 2

The total market output

Q

is the sum of the outputs of the two firms:

Q = q_1 + q_2

step 3

Substitute

Q = q_1 + q_2

into the demand function:

q_1 + q_2 = 120 - 2p

step 4

Solve for

p

p = \frac{120 - (q_1 + q_2)}{2}

step 5

The revenue for firm 1 is

R_1 = p \cdot q_1 = \left(\frac{120 - (q_1 + q_2)}{2}\right) q_1

step 6

The cost for firm 1 is

C_1 = 12q_1

step 7

The profit for firm 1 is

\pi_1 = R_1 - C_1 = \left(\frac{120 - (q_1 + q_2)}{2}\right) q_1 - 12q_1

step 8

Differentiate

\pi_1

with respect to

q_1

and set the derivative to zero to find the reaction function:

\frac{d\pi_1}{dq_1} = 60 - q_1 - \frac{q_2}{2} - 12 = 0

step 9

Solve for

q_1

q_1 = 48 - \frac{q_2}{2}

step 10

Similarly, for firm 2, the profit function is

\pi_2 = \left(\frac{120 - (q_1 + q_2)}{2}\right) q_2 - 12q_2

step 11

Differentiate

\pi_2

with respect to

q_2

and set the derivative to zero:

\frac{d\pi_2}{dq_2} = 60 - q_2 - \frac{q_1}{2} - 12 = 0

step 12

Solve for

q_2

q_2 = 48 - \frac{q_1}{2}

Answer

The Cournot reaction functions are

q_1 = 48 - \frac{q_2}{2}

and

q_2 = 48 - \frac{q_1}{2}

Part (b): Draw the reaction functions for both firms.

step 1

Plot the reaction function

q_1 = 48 - \frac{q_2}{2}

on a graph with

q_1

on the y-axis and

q_2

on the x-axis

step 2

Plot the reaction function

q_2 = 48 - \frac{q_1}{2}

on the same graph with

q_2

on the y-axis and

q_1

on the x-axis

step 3

Label the axes and the points where the reaction functions intersect the axes

step 4

The reaction functions intersect the axes at

(q_2 = 0, q_1 = 48)

and

(q_1 = 0, q_2 = 48)

Answer

The reaction functions intersect the axes at

(q_2 = 0, q_1 = 48)

and

(q_1 = 0, q_2 = 48)

Part (c): Find the Cournot (Nash) equilibrium firm outputs and market price p.

step 1

Set the reaction functions equal to each other:

q_1 = 48 - \frac{q_2}{2}

and

q_2 = 48 - \frac{q_1}{2}

step 2

Substitute

q_2 = 48 - \frac{q_1}{2}

into

q_1 = 48 - \frac{q_2}{2}

q_1 = 48 - \frac{48 - \frac{q_1}{2}}{2}

step 3

Simplify and solve for

q_1

q_1 = 32

step 4

Substitute

q_1 = 32

back into

q_2 = 48 - \frac{q_1}{2}

q_2 = 32

step 5

The market price

p

is found by substituting

q_1 + q_2 = 64

into the demand function:

p = \frac{120 - 64}{2} = 28

Answer

The Cournot (Nash) equilibrium firm outputs are

q_1 = 32

and

q_2 = 32

, and the market price is

p = 28

Part (d): Suppose the two firms decide to collude and form a cartel. How much will the cartel produce?

step 1

The total output of the cartel maximizes joint profit. The joint revenue is

R = p \cdot Q = \left(\frac{120 - Q}{2}\right) Q

step 2

The joint cost is

C = 12q_1 + 12q_2 = 12Q

step 3

The joint profit is

\pi = R - C = \left(\frac{120 - Q}{2}\right) Q - 12Q

step 4

Differentiate

\pi

with respect to

Q

and set the derivative to zero:

\frac{d\pi}{dQ} = 60 - Q - 12 = 0

step 5

Solve for

Q

Q = 48

Answer

The cartel will produce

Q = 48

Part (e): The two firms agree to split the cartel output and profit equally. If firm 1 cheats on the cartel agreement, how much will it produce?

step 1

If firm 1 cheats, it will produce more than its agreed share. Let

q_1

be the output of firm 1 and

q_2 = 24

(firm 2's agreed share)

step 2

Firm 1 maximizes its profit:

\pi_1 = \left(\frac{120 - (q_1 + 24)}{2}\right) q_1 - 12q_1

step 3

Differentiate

\pi_1

with respect to

q_1

and set the derivative to zero:

\frac{d\pi_1}{dq_1} = 60 - q_1 - 12 = 0

step 4

Solve for

q_1

q_1 = 48

Answer

If firm 1 cheats on the cartel agreement, it will produce

q_1 = 48

Key Concept

Cournot reaction functions and equilibrium

Explanation

The Cournot reaction functions describe how each firm's output depends on the other firm's output. The equilibrium is found where these reaction functions intersect.