The market for nutmeg is controlled by two small island economies, Penang and Grenada. The market demand for bottled nutmeg is given by [tex]P = 100 - q_P - q_G[/tex], where [tex]q_P[/tex] is the quantity Penang produces and [tex]q_G[/tex] is the quantity Grenada produces. Both Grenada and Penang produce nutmeg at a constant marginal and average cost of $20 per bottle.

a. What is the reaction function for Grenada?
[tex]q_G = \quad + \quad q_P[/tex]

b. What is the reaction function for Penang?
[tex]q_P = \quad + \quad q_G[/tex]

c. What are the Cournot equilibrium price, quantities, and profits?

- Quantity for Grenada:

- Quantity for Penang:

- Equilibrium price:

- Profit for Grenada:

- Profit for Penang:

a. The reaction function for Grenada is qG = (100 - qP)/2. b. The reaction function for Penang is qP = (100 - qG)/2. c. Cournot equilibrium: qG = 0, qP = 0, price = $100, and profits are zero for both Grenada and Penang.

a. The reaction function for Grenada is:

qG = (100 - qP)/2

b. The reaction function for Penang is:

qP = (100 - qG)/2

c. To find the Cournot equilibrium, we need to solve the simultaneous equations formed by the reaction functions.

By substituting the reaction function for Grenada into the reaction function for Penang, we have:

qP = (100 - ((100 - qP)/2))/2

Simplifying the equation, we get:

qP = (100 - 100 + qP/2)/2

2qP = qP/2

4qP = qP

3qP = 0

qP = 0

Similarly, substituting the reaction function for Penang into the reaction function for Grenada:

qG = (100 - ((100 - qG)/2))/2

2qG = qG/2

4qG = qG

3qG = 0

qG = 0

Therefore, the Cournot equilibrium quantities for both Grenada and Penang are 0.

To find the equilibrium price, we substitute the equilibrium quantities into the demand function:

P = 100 - qP - qG

P = 100 - 0 - 0

P = 100

The Cournot equilibrium price is $100.

To calculate the profits, we use the profit function:

Profit = (Price - Average Cost) * Quantity

Profit for Grenada:

ProfitG = (100 - 20) * 0 = 0

Profit for Penang:

ProfitP = (100 - 20) * 0 = 0

Therefore, the profits for both Grenada and Penang at the Cournot equilibrium are 0.

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