Question

The government is considering the construction of a network of sensors designed to detect the presence of aliens (a public good). Sigourney has demand for the public good given by Q=30-3P. Arnold has demand for the public good given by Q=36-6P. Suppose further that the marginal cost of installing the sensors is constant at $10 and the maximum number of sensors is 30.

               a) Find the equation for the social marginal benefit curve (collective demand curve), assuming Sigourney and Arnold are the only members of society.

               b) Solve for the optimal number of sensors mathematically

               c) When might the private market supply at least some sensors on their own? Explain.

Answer 1

The government is considering the construction of a network of sensors designed to detect the presence of aliens (a public good).

Sigourney has demand for the public good given by P = 10 - Q/3. Arnold has demand for the public good given by P = 6 - Q/6. Marginal cost is constant at $10 and the maximum number of sensors is 30.

a) Social marginal benefit curve has an equation given by P = 10 - Q/3 + 6 - Q/6 or P = 16 - 0.5Q.

b) Optimal number of sensors is found at

P = MC

16 - 0.5Q = 10

Q = 6/0.5 = 12 sensors

Sigourney will contribute 10 - 12/3 = $6 and Arnold will contribute P = 6 - 12/6 = $4

c) Private market will consider sensors as private good and so the market demand is Q = 30 - 3P + 36 - 6P or P = 66/9 - Q/9. Now maximum price consumers can afford is 66/9 = 7.33 Hence when marginal cost of sensors falls below $7.33, private market can supply some sensors.