Market demand is given as QD = 200 – 3P. Market supply is given as QS = 2P + 100. Each identical firm has MC = 0.5Q and ATC = 0.25Q. What is each firm’s profit?
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To find each firm's profit, we need to determine the equilibrium price and quantity in the market first.
Market equilibrium occurs when the quantity demanded (QD) equals the quantity supplied (QS). Therefore, we can set QD equal to QS and solve for the equilibrium price.
QD = QS
200 - 3P = 2P + 100
Next, we can solve this equation to find the equilibrium price:
200 - 100 = 2P + 3P
100 = 5P
P = 100 / 5
P = 20
So, the equilibrium price in the market is P = $20.
To find the equilibrium quantity, we can substitute this value of P back into either QD or QS:
QD = 200 - 3P
QD = 200 - 3(20)
QD = 200 - 60
QD = 140
Therefore, the equilibrium quantity in the market is Q = 140.
Now, we can determine the individual firm's quantity produced by equating the market demand and supply to their cost and finding the intersection point:
QD = 200 - 3P = Q = MC
200 - 3P = 0.5Q
Substituting Q = MC, we get:
200 - 3P = 0.5(0.5Q)
200 - 3P = 0.25Q
Replacing Q with 140 (equilibrium quantity), we have:
200 - 3P = 0.25(140)
200 - 3P = 35
3P = 200 -35
3P = 165
P = 165 / 3
P = 55
Thus, each firm will produce Q = MC = 0.5Q = 0.5(140) = 70 units at a price of P = $55.
Now, let's calculate each firm's profit. The profit for each firm can be calculated using the following formula:
Profit = Total Revenue - Total Costs
Total Revenue is calculated by multiplying the quantity produced (Q) by the price (P). In this case:
Total Revenue = Q x P
Total Revenue = 70 units x $55 per unit
Total Revenue = $3,850
Total Costs include both variable and fixed costs. Variable costs can be calculated by multiplying the quantity produced (Q) by the Marginal Cost (MC). Fixed costs are not provided in this question, but we'll assume they are zero for simplicity.
Variable Costs = Q x MC
Variable Costs = 70 units x 0.5Q
Variable Costs = 0.5(70)^2
Variable Costs = $1,225
Given that the Average Total Cost (ATC) is 0.25Q, we can calculate the total costs (Variable Costs + Fixed Costs) by multiplying the Average Total Cost by the quantity produced (Q):
Total Costs = Q x ATC
Total Costs = 70 units x 0.25Q
Total Costs = 0.25(70)^2
Total Costs = $612.50
Finally, we can calculate the profit for each firm:
Profit = Total Revenue - Total Costs
Profit = $3,850 - $612.50
Profit = $3,237.50
Therefore, each firm's profit is $3,237.50.