The estimated demand function for a given prospected service is given by: Q = 8 – p. In the market there is only one firm in the condition to deliver this service. The total cost function of this firm is given by the following expression: TC(Q) = 8 + 2Q + Q^2.  Gauge the following statements:This market should exist from a social welfare point of view and it is a natural monopolyThis market should not exist from a social welfare point of viewAll the three other options are incorrectThis market should exist from a social welfare point of view but it is not necessarily a natural monopoly

To gauge these statements, we first need to understand the concepts of social welfare and natural monopoly.

Social welfare is the overall well-being of the whole society. The existence of a market from a social welfare point of view depends on whether the total benefits to society (consumer surplus + producer surplus) are greater than the total costs to society.

A natural monopoly occurs when a single firm can serve the entire market demand at a lower cost than any combination of two or more firms.

1. "This market should exist from a social welfare point of view and it is a natural monopoly"

To verify this statement, we need to check if the total benefits are greater than the total costs and if the firm can serve the entire market at a lower cost than any combination of two or more firms.

The demand function is Q = 8 - p. This implies that p = 8 - Q. The revenue function of the firm is R(Q) = pQ = Q(8 - Q).

The total cost function is TC(Q) = 8 + 2Q + Q^2.

The profit function is π(Q) = R(Q) - TC(Q) = Q*(8 - Q) - (8 + 2Q + Q^2) = -Q^2 + 6Q - 8.

Setting the derivative of the profit function equal to zero gives us the profit-maximizing quantity Q* = 3. Substituting Q* = 3 into the demand function gives us the price p* = 5.

The total benefit to society is the area under the demand curve up to the quantity Q*, which is (1/2)Q^2 = 4.5. The total cost to society is the area under the total cost curve up to the quantity Q*, which is TC(Q*) = 8 + 2*3 + 3^2 = 17.

Since the total benefit (4.5) is less than the total cost (17), this market should not exist from a social welfare point of view.

Also, since the total cost function is quadratic (and therefore convex), the firm can serve the entire market at a lower cost than any combination of two or more firms, so it is a natural monopoly.

2. "This market should not exist from a social welfare point of view"

This statement is correct, as we have shown above.

3. "All the three other options are incorrect"

This statement is incorrect, as the second statement is correct.

4. "This market should exist from a social welfare point of view but it is not necessarily a natural monopoly"

This statement is incorrect, as we have shown that the market should not exist from a social welfare point of view, but it is a natural monopoly.