Question

The inverse demand for sandwiches over the lunch hour is given by P = 8 – 0.01Q, where P is the price per sandwich and Q is the total number of sandwiches brought to market. There are two sandwich shops operating in the local market. Firm 1’s cost function is C₁ = 0.0002q₁², where q₁ is the number of sandwiches brought to market by Firm 1. Firm 2’s cost function is C₂ = 0.0002q₂², where q₂ is the number of sandwiches brought to market by Firm 2. The two firms compete by setting output (Cournot

Report the equilibrium output level of Firm 1

Report the equilibrium output level of Firm 2

Report the equilibrium price for sandwiches

Report the profit of Firm 1

Assume that there are two mineral springs each with zero costs of production. The market demand for the mineral water is given by Q = 1 – P.

Assume that each firm maximizes profit assuming that the other’s output will not change (Cournot behavior). Report the equilibrium price.

Assume that each firm maximizes profit assuming that the other’s output will not change (Cournot behavior). Report the profit of Firm 1.

Now suppose the two firms form a cartel. Report the equilibrium price, assuming the two firms split the profits equally.

Now suppose the two firms form a cartel. Report the profit of each firm, assuming the two firms split the profits equally.

Under these cost and demand conditions, what would be the socially optimal price?

Answer 1

Demand Function:

where P is the price of sandwich and Q is the total number brought to the market.

There are two firms who compete by setting output.

Firm 1's cost function :

Firm 2's cost function :

and

.

Firm 1's Profit function:

firm 2's profit function:

Since both the firm compete in quantities, each firm's objective is to optimise his profit for the given choice of his opponent.

therefore, for the given q₂, firm 1 chooses q₁ to maximize his profit

FOC:

Solve for q₁ for the given value of q_2:

The above expression represents the best response of firm 1 for the any quantity choice q₂ made by firm 2.

since both the firms have identical costs and hence identical profit maximization problem. we can directly write the best response function of firm 2 analogous to firm 1's.

from best response functions of firm 1 and firm 2, Solving for q₁ and q₂:

is the equilibrium output level of firm 1.

Put the value of into best response of firm 2:

is the equilibrium output level of firm 2.

is the total number of sandwiches brought into the market.

put the value of Q into demand function:

is the equilibrium price of the sandwiches.

Total revenue of firm 1:

Total cost of firm 1:

Profit of firm 1:

is the profit level of firm 1.