Assume that there are 1000 identical firms each with the marginal cost function of MC=10 + 2Q and the market demand function for the product is also given as P=1900-0.001Q, find the equilibrium of a firm in the market.
To find the equilibrium of a firm in the market, we need to use the following steps:
1. Set the marginal cost equal to the market price to get the optimal quantity for the firm:
MC = P
10 + 2Q = 1900 - 0.001Q
Solving for Q:
2.001Q = 1890
Q = 944.03
2. Plug the optimal quantity in the demand function to get the market equilibrium price:
P = 1900 - 0.001Q
P = 1900 - 0.001(944.03)
P = 1891.06
3. To check if this is a competitive equilibrium, we need to verify that each firm is making a profit at this price and quantity.
Total revenue for each firm:
TR = P x Q
TR = 1891.06 x 944.03
TR = 1,785,601.98
Total cost for each firm:
TC = MC x Q
TC = (10 + 2Q) x Q
TC = (10 x Q) + (2 x Q^2)
TC = (10 x 944.03) + (2 x 944.03^2)
TC = 1,889,310.56
Profit for each firm:
π = TR - TC
π = 1,785,601.98 - 1,889,310.56
π = -103,708.58
Since the profit is negative, each firm is not making a profit at this price and quantity. Therefore, this is not a competitive equilibrium and the market needs to adjust to a lower price or less output to reach a stable equilibrium.
To find the equilibrium of a firm in the market, we need to set marginal cost (MC) equal to the price (P) and solve for the quantity (Q) at that point.
Given:
MC = 10 + 2Q
P = 1900 - 0.001Q
Setting MC equal to P:
10 + 2Q = 1900 - 0.001Q
Rearranging the equation:
2Q + 0.001Q = 1900 - 10
2.001Q = 1890
Q = 1890 / 2.001
Calculating Q:
Q ≈ 944.53
Now, we can substitute this value of Q back into the demand equation to find the equilibrium price (P):
P = 1900 - 0.001Q
P = 1900 - 0.001 * 944.53
Calculating P:
P ≈ $951.63
Therefore, the equilibrium for a firm in the market is a quantity of approximately 944.53 and a price of approximately $951.63.
To find the equilibrium of a firm in the market, we need to determine the quantity and price at which the firm maximizes its profit. This occurs when the firm's marginal cost (MC) is equal to the market price (P).
First, let's find the equilibrium quantity (Q) by setting MC equal to P and solving for Q:
MC = P
10 + 2Q = 1900 - 0.001Q
Rearranging the equation, we get:
2Q + 0.001Q = 1900 - 10
Combining like terms, we get:
2.001Q = 1890
Dividing both sides by 2.001, we find:
Q ≈ 945.53
So the equilibrium quantity for the firm is approximately 945.53 units.
Next, let's find the equilibrium price (P) by substituting the equilibrium quantity into the market demand function:
P = 1900 - 0.001Q
P = 1900 - 0.001(945.53)
P ≈ 1900 - 0.946
P ≈ 1899.054
So the equilibrium price for the firm is approximately $1899.054.
Therefore, the equilibrium for the firm in the market is an output quantity of approximately 945.53 units and a price of approximately $1899.054.