1. a) Assume that a regulator knows a natural monopoly firm's total cost function (from which the regulator can derive the firm's AC and MC curves), and the market demand function is common knowledge for the firm and the regulator. Assume the firm sells its product at a uniform price. The regulator wants the firm to produce the socially optimal output level, Q*. The regulator has the power to grant a subsidy to the firm to enable it to break even when it produces Q*. Graphically depict the amount of the subsidy.

 b) Now suppose the market demand function is common knowledge for both the firm and the regulator, but the regulator is completely ignorant about the firm's cost functions. Assuming that the regulator has the authority to provide whatever subsidy would be needed to induce the firm to produce the socially optimal output Q* (although the regulator does not know the value of Q*), explain how the regulator could incentivize the firm to produce the socially optimal level of output.