P=15-Q/1000. Suppose there are two firms in this market. Compute equilibrium quantities and profits for each firm, and the equilibrium market price. Hint: Start with thinking about the number of loaves a firm will sell in a month — this quantity must be where its marginal revenue is equal to its marginal cost. But notice that firm 1’s marginal revenue will depend on firm 2’s quantity choice, and vice versa. To construct a spreadsheet, start by constructing a formula for firm 1’s marginal revenue as a function of its quantity choice and firm 2’s quantity choice. If firm 1 is making 1000 loaves and firm 2 is also making 1000 loaves, then firm 1’s marginal revenue is how its revenue changes if it increases quantity by one, assuming firm 2 keeps its quantity at 1000. Do the same firm firm 2, and then use Solver to find quantities where MR = MC for both firms.

Question

P=15-Q/1000. Suppose there are two firms in this market. Compute equilibrium quantities and profits for each firm, and the equilibrium market price. Hint: Start with thinking about the number of loaves a firm will sell in a month — this quantity must be where its marginal revenue is equal to its marginal cost. But notice that firm 1’s marginal revenue will depend on firm 2’s quantity choice, and vice versa. To construct a spreadsheet, start by constructing a formula for firm 1’s marginal revenue as a function of its quantity choice and firm 2’s quantity choice. If firm 1 is making 1000 loaves and firm 2 is also making 1000 loaves, then firm 1’s marginal revenue is how its revenue changes if it increases quantity by one, assuming firm 2 keeps its quantity at 1000. Do the same firm firm 2, and then use Solver to find quantities where MR = MC for both firms.

Answer 1

To find the equilibrium quantities and profits for each firm, and the equilibrium market price, we will follow these steps:

Step 1: Determine the marginal revenue for each firm.
Step 2: Set up the equations for each firm's marginal revenue equal to marginal cost.
Step 3: Use Solver to find the equilibrium quantities for both firms.
Step 4: Calculate the profits for each firm at the equilibrium quantities.
Step 5: Determine the equilibrium market price.

Let's go through each step in more detail:

Step 1: Determine the marginal revenue for each firm.
The marginal revenue (MR) is the change in total revenue resulting from a one-unit increase in quantity. In this case, for each firm, the marginal revenue will depend on the quantity chosen by both firms.

For Firm 1:
MR1 = d(R1)/dQ1 = 15 - Q1/1000 - (Q2/1000) * dQ1/dQ2

For Firm 2:
MR2 = d(R2)/dQ2 = 15 - Q2/1000 - (Q1/1000) * dQ2/dQ1

Note that dQ1/dQ2 and dQ2/dQ1 represent the partial derivatives of the quantities Q1 and Q2 with respect to each other.

Step 2: Set up the equations for each firm's marginal revenue equal to marginal cost.
Since the marginal revenue should be equal to the marginal cost (MC) for each firm at the equilibrium, we can set up the following equations:

For Firm 1:
MR1 = MC1

For Firm 2:
MR2 = MC2

Step 3: Use Solver to find the equilibrium quantities for both firms.
By solving the equations above simultaneously, we can find the equilibrium quantities for both firms.

Step 4: Calculate the profits for each firm at the equilibrium quantities.
Using the equilibrium quantities found in step 3, we can calculate the profits for each firm. The profit equation for each firm is:

Profit = (P - MC) * Quantity

Step 5: Determine the equilibrium market price.
The equilibrium market price can be found by substituting the equilibrium quantities into the demand equation:

P = 15 - (Q1 + Q2) / 1000

Using these steps, you can construct a spreadsheet and use Solver to find the equilibrium quantities and profits for each firm, as well as the equilibrium market price.

Answer 2

To compute the equilibrium quantities and profits for each firm, as well as the equilibrium market price, we need to follow a few steps:

Step 1: Construct the formulas for each firm's marginal revenue.
Firm 1's marginal revenue formula depends on firm 2's quantity choice and can be calculated as follows:
MR1 = P + (dP/dQ2) * Q2, where P is the price, Q2 is firm 2's quantity, and (dP/dQ2) is the derivative of the market price with respect to firm 2's quantity.

Similarly, firm 2's marginal revenue formula depends on firm 1's quantity choice and can be calculated as:
MR2 = P + (dP/dQ1) * Q1, where Q1 is firm 1's quantity, and (dP/dQ1) is the derivative of the market price with respect to firm 1's quantity.

Step 2: Set up the profit functions for each firm.
The profit function for each firm is given by:
Profit1 = (P - MC1) * Q1
Profit2 = (P - MC2) * Q2
where MC1 and MC2 represent the marginal costs for firms 1 and 2, respectively.

Step 3: Use Solver to find quantities where MR = MC for both firms.
Using a spreadsheet, you can input the given formula for MR1 and MR2, as well as the profit functions for each firm. Set up a solver tool to find the quantities of loaves where MR1 = MC1 and MR2 = MC2 simultaneously.

Step 4: Calculate the equilibrium market price.
Once you obtain the quantities for each firm, plug them back into the demand equation to calculate the equilibrium market price. The demand equation given is P = 15 - Q/1000.

Step 5: Calculate the profits for each firm using the equilibrium quantities and market price.
Substitute the equilibrium quantities and market price into the profit functions for each firm to calculate their respective profits.

By following these steps, you can compute the equilibrium quantities and profits for each firm, as well as the equilibrium market price.