# Consider a monopolist that faces the following equations: Market demand for monopolist's product:...

## Question:

Consider a monopolist that faces the following equations: Market demand for monopolist's product: Q = 100 - P TC for monopolist: TC = 20Q + (3/10) Q2 a. Write the demand equation in terms of Q.

b. Given the above information, what is the profit maximizing price and quantity for the monopolist?

c. Calculate the monopolist's total profits.

d. Suppose the monopolist is regulated to produce where price equals average total cost (average cost pricing). Calculate the quantity the monopolist will produce and the price that will be charged after regulation.

e. Calculate the level of profits for the regulated monopoly.

f. What do you think happens to the consumer surplus under regulation compared to a scenario with no regulation? What do you think happens to deadweight loss when we move from an unregulated to a regulated monopolist?

## Regulated and Unregulated Monopoly:

Monopoly is the sole trader of a good or service in the whole economy therefore, he's referred to as the price maker. Due to lack of competitors, monopolist would be tempted to charge high price level to maximize profit. Since government is concerned about the welfare of its citizens, it regulates these monopolies to ensure that they don't exploit the consumers.

## Answer and Explanation:

**a).**

Calculate the inverse demand equation using direct demand equation by making price *P* the subject of the formula:

{eq}Q=100-P\\P=100-Q {/eq}

**b).**

The monopolist produce where marginal revenue is equal to marginal cost. Use 'twice-as-steep' rule to get marginal revenue from inverse demand function:

{eq}P=100-Q\\MR=100-(2\times 1)Q\\MR=100-2Q {/eq}

Take the first partial derivative of total cost function with respect to output to get the marginal cost:

{eq}\displaystyle MC=\frac{\partial TC}{\partial Q}: 20+\frac{3}{5}Q {/eq}

Equate marginal revenue and marginal cost and solve for monopolist output:

{eq}\displaystyle 100-2Q=20+\frac{3}{5}Q\\500-10Q=100+3Q\\13Q=400\\Q=30.77\,\text{units} {/eq}

Plug in the monopolist output in the inverse demand function and solve for price:

{eq}P=100-(30.77)\\P=$69.23 {/eq}

**c).**

Profit is the difference between total revenue and total cost:

{eq}\prod=TR-TC\\TR=P\cdot Q\\=$69.23\times 30.77\\=$2130.21\\TC=20(30.77)+\frac{3}{10}(30.77)\\=$899.44\\\therefore \prod =$2130.21-$899.44\\=$1230.77 {/eq}

**d).**

Divide the total cost function with output *Q* to get the average total cost:

{eq}\displaystyle ATC=\frac{TC}{Q}\\\displaystyle =\frac{20Q+\frac{3}{10}Q^2}{Q}\\=\displaystyle 20+\frac{3}{10}Q {/eq}

Equate the average total cost with inverse demand function and solve for output:

{eq}P=ATC\\\displaystyle 20+\frac{3}{10}Q=100-Q\\200+3Q=1000-10Q\\13Q=800\\Q=61.54\,\text{units} {/eq}

Plug in the output back into the inverse demand function and solve for price:

{eq}P=100-(61.54)\\P=$38.46 {/eq}

**e).**

Calculate the profit when monopoly is regulated:

{eq}TR=$38.46\times 61.54\\=$2366.83\\TC=\displaystyle 20(61.54)+\frac{3}{10}(61.54^2)\\=$2366.95\\\therefore \prod=$2366-$2366\\=0 {/eq}

The regulated monopoly will make zero economic profit.

**f).**

- Consumer surplus:

Consumer surplus is less in unregulated monopoly than in regulated monopoly. This is because charging price above the marginal cost (unregulated monopoly) converts some of the consumer surplus to be monopolist additional revenues. This make consumer surplus to be less.

- Deadweight loss:

Deadweight loss is a total welfare loss that neither goes to consumer nor to the producer. It result from charging price above the marginal cost which make some buyers who are willing and able to pay price equal to marginal cost not to be served. Unregulated monopoly courses a lot of deadweight loss than regulated monopoly.

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Chapter 3 / Lesson 13