A monopoly firm faces a demand curve given by the following equation: P = $500 − 10Q, where Q equals quantity sold per day. Its marginal cost curve is MC = $100 per day. Assume that the firm faces no fixed cost. You may wish to arrive at the answers mathematically, or by using a graph (the graph is not required to be presented), either way, please provide a brief description of how you arrived at your results.
a) How much will the firm produce?
b) How much will it charge?
c) Can you determine its profit per day? (Hint: you can; state how much it is.)
d) Suppose a tax of $1,000 per day is imposed on the firm. How will this affect its price?
e) How would the $1,000 per day tax its output per day?
f) How would the $1,000 per day tax affect its profit per day?
g) Now suppose a tax of $100 per unit is imposed. How will this affect the firm’s price?
h) How would a $100 per unit tax affect the firm’s profit maximizing output per day?
i) How would the $100 per unit tax affect the firms profit per day?
@Samuel
"There's no AC so I can't locate it."
=> Since the marginal cost is stable (although that is very unlikely to happen in the real world it is the case here) and there is no fixed cost, the AC is 100.
a) To determine how much the firm will produce, we need to find the quantity at which marginal cost equals marginal revenue. Marginal revenue (MR) is the change in total revenue resulting from selling one additional unit, and it is equal to the price for a monopoly firm. So, in this case, MR = P = $500 - 10Q.
As the marginal cost (MC) is constant at $100 per day, we can set MR equal to MC and solve for Q:
$500 - 10Q = $100
-10Q = $100 - $500
-10Q = -$400
Q = 40
So, the firm will produce 40 units per day.
b) To determine how much the firm will charge, we need to substitute the quantity into the demand equation:
P = $500 - 10Q
P = $500 - 10(40)
P = $100
So, the firm will charge $100 per unit.
c) To determine the firm's profit per day, we need to calculate total revenue (TR) and total cost (TC).
TR = P * Q
TR = $100 * 40
TR = $4,000
TC = MC * Q
TC = $100 * 40
TC = $4,000
Profit (π) = TR - TC
Profit = $4,000 - $4,000
Profit = $0
Therefore, the firm's profit per day is $0.
d) When a tax of $1,000 per day is imposed, it will affect the firm's price. The tax will increase the firm's costs, which will have to be passed on to the consumers in the form of higher prices. The new price (P') can be calculated by adding the tax to the original price:
P' = P + Tax
P' = $100 + $1,000
P' = $1,100
So, the price will increase to $1,100 per unit.
e) The $1,000 per day tax will influence the firm's output per day. To determine the new quantity (Q'), we need to subtract the tax from the original price and solve for Q:
P' = $500 - 10Q'
$1,100 = $500 - 10Q'
10Q' = $500 - $1,100
10Q' = -$600
Q' = -60
However, quantity cannot be negative, so the firm will not produce any output with this tax imposed.
f) With no output being produced due to the $1,000 per day tax, the firm will have zero revenue, and its profit per day will be equal to the negative amount of the tax. Therefore, the firm will have a profit per day of -$1,000.
g) If a tax of $100 per unit is imposed, it will affect the firm's price. The tax will increase the marginal cost of producing each unit, leading to a higher price. The new price (P') can be calculated by adding the tax to the original price:
P' = P + Tax
P' = $100 + $100
P' = $200
So, the price will increase to $200 per unit.
h) To determine the firm's profit-maximizing output per day with the $100 per unit tax, we need to calculate the quantity at which marginal cost equals marginal revenue. The new marginal revenue (MR') equation will be:
MR' = P' = $200 - 10Q'
Setting MR' equal to MC, we get:
$200 - 10Q' = $100
-10Q' = $100 - $200
-10Q' = -$100
Q' = 10
So, the firm's profit-maximizing output per day will be 10 units.
i) With the $100 per unit tax, the firm's profit per day can be calculated using the new revenue and cost. The new total revenue (TR') can be calculated by multiplying the new price by the quantity:
TR' = P' * Q'
TR' = $200 * 10
TR' = $2,000
The new total cost (TC') can be calculated by multiplying the new marginal cost by the quantity:
TC' = MC * Q'
TC' = $100 * 10
TC' = $1,000
Profit (π') = TR' - TC'
Profit = $2,000 - $1,000
Profit = $1,000
Therefore, the firm's profit per day with the $100 per unit tax is $1,000.
To answer these questions, we need to apply basic principles of microeconomics. Let's break down each question step by step:
a) To determine the firm's profit-maximizing output level, we need to find the quantity at which marginal cost (MC) equals marginal revenue (MR). In a monopoly, the marginal revenue is given by the equation MR = P(1-1/η), where η is the price elasticity of demand.
In this case, the demand equation is P = $500 - 10Q. To find η, we differentiate the demand equation with respect to Q:
dP/dQ = -10
Then substitute this value into the expression for η:
η = -Q(P/dQ) / P = -Q(-10) / (500-10Q) = 10Q / (500-10Q)
Now, set MR equal to MC and solve for Q:
MC = MR
$100 = P(1-1/η)
$100 = (500-10Q)(1-1/(10Q/(500-10Q)))
$100 = (500-10Q)(1-500/(10Q))
$100 = (500-10Q)(1-50/Q)
$100 = (500-10Q)(Q-50)/Q
$100Q = (500-10Q)(Q-50)
Solving this equation will give us the profit-maximizing quantity.
b) To find the price charged by the firm, we substitute the quantity found in part a) into the demand equation: P = $500 - 10Q.
c) To determine the profit per day, we need to calculate the total revenue and subtract the total cost. Total revenue is simply the quantity sold multiplied by the price charged, while total cost is given as the product of quantity and marginal cost.
d) When a tax of $1,000 per day is imposed on the firm, the price charged by the firm will increase. To determine the new price, subtract the tax amount from the original price.
e) The tax will affect the firm's output per day. To find the new quantity, subtract the change in quantity caused by the tax from the initial quantity.
f) The tax will affect the firm's profit per day. To find the new profit, calculate the difference between the new total revenue and the new total cost.
g) When a tax of $100 per unit is imposed, the price charged by the firm will increase. To determine the new price, add the tax amount to the original price.
h) The tax per unit will affect the firm's profit-maximizing output per day. To find the new quantity, subtract the change in quantity caused by the tax from the initial profit-maximizing quantity.
i) The tax per unit will affect the firm's profit per day. To find the new profit, calculate the difference between the new total revenue and the new total cost.
Use the demand curve to derive MR.
NOTE: MR is always two times steeper than the demand curve if the equation is linear.
Hence, MR=$500-20Q.
Use MC=MR
100=500-20Q
20Q=400
Q=20
P=500-10(20)=400.
Profit per day= There's no AC so I can't locate it.
In terms of tax. If it is a lump sum tax and proportional tax it won't affect the current profit maximizing price and qty.
However, tax per unit will affect profit maximizing.