Question

A monopolist barber shop sells its services to a continuum of consumers positioned on the [0, 1]interval. They are uniformly distributed on it. The barber shop is positioned at x̄ = 1/4. A consumer whose position is x belonging to [0, 1] values the barber service as 1 and incurs transportation cost |x − x̄| if he goes to the barber shop. The barber shop charges price p to all customers.

a. Find the optimal price p that the barber shop charges to all the customers.

b. Suppose now that the barber shop can issue n% discount cards and distribute them uniformly among half of consumers living to the right (that is, x ≥ x̄) – suppose that it is possible to distribute cards in such a way. Find optimal p and n.

c. Answer the previous question but assuming that the discounts are for those who live to the left of the barber shop (that is, x ≤ x̄).

Answer 1

a. The monopolist's profit maximizing level of output is found by equating its marginal revenue with its marginal cost, which is the same profit maximizing condition that a perfectly competitive firm uses to determine its equilibrium level of output.

Monopolies have much more power than firms normally would in competitive markets, but they still face limits determined by demand for a product. Higher prices (except under the most extreme conditions) mean lower sales. Therefore, monopolies must make a decision about where to set their price and the quantity of their supply to maximize profits. They can either choose their price, or they can choose the quantity that they will produce and allow market demand to set the price.

Since costs are a function of quantity, the formula for profit maximization is written in terms of quantity rather than in price. The monopoly's profits are given by the following equation:

π=p(q)q−c(q)

In this formula, p(q) is the price level at quantity q. The cost to the firm at quantity q is equal to c(q). Profits are represented by π. Since revenue is represented by pq and cost is c, profit is the difference between these two numbers.

b. As a result, the first-order condition for maximizing profits at quantity q is represented by:

0=∂q=p(q)+qp′(q)−c′(q)

The above first-order condition must always be true if the firm is maximizing its profit - that is, if p(q)+qp′(q)−c′(q) is not equal to zero, then the firm can change its price or quantity and make more profit.

Marginal revenue is calculated by p(q)+qp′(q), which is derived from the term for revenue, pq. The term c′(q) is marginal cost, which is the derivative of c(q). Monopolies will produce at quantity q where marginal revenue equals marginal cost. Then they will charge the maximum price p(q) that market demand will respond to at that quantity.

Consider the example of a monopoly firm that can produce widgets at a cost given by the following function:

c(q)=2+3q+q²

If the firm produces two widgets, for example, the total cost is 2+3(2)+2²=12. The price of widgets is determined by demand:

p(q)=24-2p

When the firm produces two widgets it can charge a price of 24-2(2)=20 for each widget. The firm's profit, as shown above, is equal to the difference between the quantity produces multiplied by the price, and the total cost of production: p(q)q−c(q). How can we maximize this function?

Using the first order condition, we know that when profit is maximized, 0=p(q)+qp′(q)−c′(q). In this case:

0=(24-2p)+q(-2)-(3+2q)=21-6q

Rearranging the equation shows that q=3.5. This is the profit maximizing quantity of production.

The key points are fivefold.

1. First, marginal revenue lies below the demand curve. This occurs because marginal revenue is the demand, p(q), plus a negative number.
2. Second, the monopoly quantity equates marginal revenue and marginal cost, but the monopoly price is higher than the marginal cost.
3. Third, there is a deadweight loss, for the same reason that taxes create a deadweight loss: The higher price of the monopoly prevents some units from being traded that are valued more highly than they cost.
4. Fourth, the monopoly profits from the increase in price, and the monopoly profit is illustrated.
5. Fifth, since—under competitive conditions—supply equals marginal cost, the intersection of marginal cost and demand corresponds to the competitive outcome.

We see that the monopoly restricts output and charges a higher price than would prevail under competition.

c. The Profit of the Monopolist when the buyers value is located in both unequal and one of the two interval scenario is high as Marginal revenue is higher than Marginal cost.The Value of Profit in second scenario is higher than first scenario.

3. Consumer surplus is when a consumer derives more benefit (in terms of monetary value) from a good or service than the price they pay to consume it. Imagine you are going to an Electronics store to buy a new flat panel TV. Before you go to the store, you decide to yourself that you are not going to pay more than $750 for a TV. This $750 is your maximum willingness to pay for the TV. After entering the store, you find a TV you really like for only $500! Since you were willing to pay $750 for the TV, and you only ended up paying $500 for it, you have saved $250. This $250 is called consumer surplus by economists, because it is the “extra” or “surplus” value you received from the good beyond the price you paid for it.

But figuring out an individual’s surplus isn’t good enough. We want to figure out the total amount of surplus for all consumers in the economy and derive the total consumer surplus. An easy way to visualize is shown to the right. In this mini economy we have 5 consumers, and we line them up left to right by their willingness to pay (consumer 1 is willing to pay more than consumer 2, etc.). You can see that each consumer pays the same price for the good, so their surplus is calculated as the difference between their willingness to pay, and the actual amount they have to pay.

For the first consumer, he is willing to pay $20, but only has to pay $5, so he gets a surplus of $15. The next consumer is willing to pay $16, but only has to pay $5, so he gets a surplus of $11. Using the same logic, the third, fourth, and fifth consumers have surplus values equal to $5, $3, and $0 (because their maximum willingness to pay is equal to the price, so consumer surplus is zero).

To get total surplus we add these values up, so $15+$11+$5+$3=$34. The total consumer surplus in this economy is $34.

. In economics, a deadweight loss (also known as excess burden or allocative inefficiency) is a loss of economic efficiency that can occur when equilibrium for a good or service is not achieved or is not achievable.

Suppose that the demand curve is represented by P = 10 - 2Q and MC = 2.

1. Find Qc

To find Qc we need to find the point where MC = the demand curve.

Therefore, we let 2 = 10 - 2Q. We solve for Q and find that Q = 4.

Therefore, Qc = 4.

2. Find Qm

This point corresponds to the point where Marginal revenue (MR) = Marginal Cost (MC)

Firstly, we need to know what the marginal revenue equation is. Well, if the demand curve is linear (a straight line) then it will always have a slope twice the size of the demand curve and the same intercept term. Since demand is: P = 10 - 2Q this means that MR = 10 - 4q.

Now we equate MR = MC such that 2 = 10 - 4Q and re-arranging we will find Q = 2.

Therefore, Qm = 2

3. Find price

To find the price, we get our function P = 10 - 2Q and we substitute in our value for Qm.

P = 10 - 2(2) = 6

We now have all the pieces of information that we need. If we plug them all into our DWL formula 1÷2 (P - MC) (Qc - Qm) we will get:

1÷2 (6 - 2) (4 - 2) = 4

therefore, our dead weight loss will be 4. And that's how we calculate the size of the dead weight loss!