(25 points) Consider a public good model with two consumers and one producer. The cost function of the producer is given by u_(1)(m,x)=m+10x-x^2 u_(2)(m,x)=m+14x-2x^2 defined on the consumption set X=Rtimes [0,bar (q)] where bar (q)=3.5 Consumers have some initial endowment of the numeraire m and a zero endowment of the public good z. (a) (5 points) Find the socially optimal level of public good. (b) (5 points) Find the equilibrium level of public good under the private provision. (c) (5 points) For this part only, suppose that the government can use a production subsidy (i.e., a subsidy per unit produced). What level of production subsidy rate is required to achieve the socially optimal level of the public good under private provision. Provide intruition. Calculate the amount of financing needed to run this subsidy program. (d) (5 points) Now suppose that consumers receive a cashback bonus of ap from the gov- ernment for each unit of public good they purchase at price p.Find the cashback bonus rate that restores the socially optimal level of public good. Calculate the amount of financing needed to run this cashback program. (e) (5 points) Finally.suppose that only consumer 2 can receive a cashback bonus of op from the government for each unit of public good she purchase at price p (i.e., consumer 1 is not eligible for this program, but he still can buy good x at price p). Find the cashback bonus rate that restores the socially optimal level of public good. Calculate the amount of financing needed to run this cashback program. A Public Good Model

(a) To find the socially optimal level of the public good, we need to maximize the sum of the consumers' utilities subject to the cost constraint of the producer. The social welfare function is given by: The cost constraint of the producer is given by: To maximize the social welfare function subject to the cost constraint, we take the derivative of the social welfare function with respect to x and set it equal to zero: Solving for x, we get: Therefore, the socially optimal level of the public good is x = 4.(b) To find the equilibrium level of the public good under private provision, we need to find the level of x that maximizes the sum of the consumers' utilities, given their individual cost constraints. The individual cost constraints are given by: To find the equilibrium level of x, we take the derivative of each consumer's utility function with respect to x and set it equal to zero: Solving for x, we get: Since the consumers have different cost constraints, they will choose different levels of x. The equilibrium level of the public good is the average of the two levels: (c) To achieve the socially optimal level of the public good under private provision, the government can use a production subsidy. The production subsidy per unit produced is denoted by s. The new cost constraint of the producer is given by: To find the level of s that achieves the socially optimal level of x = 4, we set the derivative of the social welfare function with respect to x equal to zero and solve for x: Solving for x, we get: Substituting x = 4 into the new cost constraint, we get: Since the cost constraint of the producer must be satisfied, we have: Solving for s, we get: Therefore, a production subsidy rate of s = 8 is required to achieve the socially optimal level of the public good under private provision. The amount of financing needed to run this subsidy program is equal to the total subsidy paid to the producer, which is given by: (d) To restore the socially optimal level of the public good, consumers can receive a cashback bonus of ap for each unit of public good they purchase at price p. The new utility functions of the consumers are given by: To find the level of ap that restores the socially optimal level of x = 4, we set the derivative of the social welfare function with respect to x equal to zero and solve for x: Solving for x, we get: Substituting x = 4 into the new utility functions, we get: $u_{2}(m,4) = m + 14(4) - 2