Suppose that individual demand for a product is given by
QD
= 1000 − 5P. Marginal revenue is MR = 200 − 0.4Q, and marginal cost is constant at $20. There are no fixed costs.
Well, well, well, looks like we're diving into the world of demand, revenue, and costs! Hold on tight, it's going to be a bumpy, yet entertaining, ride!
First, let's start with the demand equation: QD = 1000 − 5P. The QD represents the quantity demanded by individuals at a given price. In this case, the higher the price (P), the lower the quantity demanded. Keep this in mind as we move forward!
Now, let's move on to marginal revenue (MR), which is all about how much additional revenue you get from selling one more unit of your product. The MR equation given is MR = 200 − 0.4Q. According to this equation, the marginal revenue decreases as the quantity (Q) increases. It's like when you're at a buffet and the more plates you pile up, the less excited you are about getting another plate. Trust me, I've been there!
And finally, we have our constant marginal cost of $20. That means, no matter how many units you produce, it'll cost you a simple $20 for each one. It's like having an annoying monthly subscription fee that never changes. Thank you, capitalism!
Now, since there are no fixed costs, we can focus purely on marginal revenue and marginal cost. To maximize profit, we want to produce the quantity where MR equals MC (marginal cost), or in plain terms, where the money we bring in from selling an extra unit equals the money it costs us to make that unit. It's a delicate balancing act, just like juggling while riding a unicycle!
So, set MR equal to MC and solve for Q! In this case, 200 - 0.4Q equals 20. Once you find that magical Q value, just substitute it back into the demand equation (QD = 1000 − 5P) to find the corresponding price (P).
And just like that, voila! You've unlocked the secrets of demand, revenue, and costs. Who says economics can't be a barrel of laughs? Well, besides my numerous economic professors who never cracked a single joke...
To find the profit-maximizing quantity and price for this product, we need to use the marginal revenue (MR) and marginal cost (MC) approach.
1. Find the profit-maximizing quantity:
Since the marginal cost is constant at $20, we can set it equal to the marginal revenue and solve for the quantity:
MC = MR
$20 = 200 - 0.4Q
0.4Q = 200 - $20
0.4Q = $180
Q = $180 / 0.4
Q = 450
Therefore, the profit-maximizing quantity for this product is 450.
2. Find the corresponding price:
To find the price, we can substitute the quantity we found into the demand equation and solve for P:
QD = 1000 - 5P
450 = 1000 - 5P
5P = 1000 - 450
5P = 550
P = 550 / 5
P = $110
Therefore, the profit-maximizing price for this product is $110.
3. Calculate the total profit:
To calculate the total profit, we need to subtract the total cost from the total revenue at the profit-maximizing quantity:
Total Revenue = Price * Quantity
Total Revenue = $110 * 450 = $49,500
Total Cost = Marginal Cost * Quantity
Total Cost = $20 * 450 = $9,000
Total Profit = Total Revenue - Total Cost
Total Profit = $49,500 - $9,000 = $40,500
Therefore, the total profit at the profit-maximizing quantity and price for this product is $40,500.
To find the equilibrium price and quantity of the product, we need to set the quantity demanded equal to the quantity supplied and solve for the price.
Step 1: Set Quantity Demanded (QD) equal to Quantity Supplied (QS)
QD = QS
Step 2: Write the equation for Quantity Demanded (QD)
QD = 1000 - 5P
Step 3: Write the equation for Quantity Supplied (QS)
Since there are no fixed costs and the marginal cost is constant at $20, the equation for Quantity Supplied (QS) is:
QS = MC
Step 4: Substitute the value of Marginal Cost (MC) into the equation for Quantity Supplied (QS)
QS = 20
Step 5: Set the Quantity Demanded (QD) equal to the Quantity Supplied (QS)
1000 - 5P = 20
Step 6: Solve for the price (P)
Subtract 1000 from both sides to isolate -5P on one side:
- 5P = 20 - 1000
Combine like terms on the right side:
-5P = -980
Divide both sides by -5 to solve for P:
P = -980 / -5
P = 196
Therefore, the equilibrium price (P) is $196.
Step 7: Substitute the equilibrium price into the equation for Quantity Demanded (QD) to find the equilibrium quantity (Q)
QD = 1000 - 5P
Substitute P = 196 into the equation:
QD = 1000 - 5(196)
QD = 1000 - 980
QD = 20
Therefore, the equilibrium quantity (Q) is 20.