To determine how much the firm will produce?

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 unit. 

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?

Let's analyze each part of the question step by step:

a) To determine how much the firm will produce, we need to find the profit-maximizing quantity, where marginal cost (MC) equals marginal revenue (MR). The demand curve is given as P = $500 - 10Q, which can be rearranged to find MR:

MR = d(P)/dQ = d($500 - 10Q)/dQ = $500 - 20Q

Setting MR equal to MC, we get:

$500 - 20Q = $100

Solving for Q:

20Q = $500 - $100 20Q = $400 Q = $400 / 20 Q = 20 units

So, the firm will produce 20 units per day.

b) To determine how much the firm will charge, we can use the demand curve:

P = $500 - 10Q P = $500 - 10(20) P = $500 - $200 P = $300

The firm will charge $300 per unit.

c) Profit per day can be calculated as:

Profit = (P - MC) * Q Profit = ($300 - $100) * 20 Profit = $200 * 20 Profit = $4,000 per day

So, the firm's profit per day is $4,000.

d) If a tax of $1,000 per day is imposed on the firm, its price will be affected. To find the new price, we need to account for the tax in the cost function:

New MC = MC + Tax New MC = $100 + $1,000 New MC = $1,100

To determine the new price, set MR equal to the new MC:

$500 - 20Q = $1,100

Solving for Q:

20Q = $500 - $1,100 20Q = -$600 Q = -$600 / 20 Q = -30 units

Since we can't have a negative quantity, the firm will not produce any units, and the price becomes irrelevant.

e) With a tax of $1,000 per day, the firm will reduce its output to zero per day.

f) If the firm doesn't produce any units due to the tax, its profit per day will be zero because there are no sales.

g) Now suppose a tax of $100 per unit is imposed. This tax will affect the firm's price by increasing its marginal cost (MC). The new MC becomes:

New MC = MC + Tax per Unit New MC = $100 + $100 New MC = $200 per unit

To determine the new price, set MR equal to the new MC:

$500 - 20Q = $200

Solving for Q:

20Q = $500 - $200 20Q = $300 Q = $300 / 20 Q = 15 units

Now, let's calculate the new price:

P = $500 - 10Q P = $500 - 10(15) P = $500 - $150 P = $350 per unit

So, the firm will charge $350 per unit with a $100 per unit tax.

h) With the $100 per unit tax, the firm's profit-maximizing output per day can be found by setting MR equal to the new MC:

$500 - 20Q = $200

Solving for Q:

20Q = $500 - $200 20Q = $300 Q = $300 / 20 Q = 15 units

The firm will produce 15 units per day.

i) To calculate the firm's profit per day with the $100 per unit tax, use the profit formula:

Profit = (P - New MC) * Q Profit = ($350 - $200) * 15 Profit = $150 * 15 Profit = $2,250 per day

So, with the $100 per unit tax, the firm's profit per day is $2,250.