An economy consists of two individuals, A and B and two goods, a private good, which is called good 1, and a public good, which is called good 2. Each individual i=A,B is endowed with 24 units of time, which can be used to produce an amount x1i of a private good at a labour cost of x1i units of time, and an amount x2i of a public good at a labour cost of 2x2i units of time.
Individual i ‘s utility function is given by ui(x1i,x2)=(x1i)12(g)12, for both i=A,B and where g denotes the aggregate provision of the public good, i.e. g=x2A+x2B.
(a) Derive the allocation that maximises utilitarian welfare (i.e. the sum of both individual’s utility functions).
(b) Suppose both individuals decide on provision of feasible bundles (x1A,x2A) and (x1B,x2B) simultaneously and have full information about the other individual’s utility function. Derive the Nash equilibrium of the noncooperative game.
(c) Explain concisely (max three sentences) why there is usually under provision of the public good (you can refer to (i) and (ii),but you can also explain this more generally).
(d) Suggest policies that ensure the optimal provision of the public good (max three sentences).
Solution
(a) To maximize utilitarian welfare, we need to maximize the sum of both individuals’ utility functions. This can be done by setting the marginal rate of substitution equal to the ratio of the costs of production for both goods. The optimal allocation is achieved when the following condition is met:
(x1A)−12(2x2A)−12=12
and
(x1B)−12(2x2B)−12=12
(b) In a noncooperative game, each individual will choose their provision of goods to maximize their own utility, taking the other individual’s provision as given. The Nash equilibrium is achieved when no individual can increase their utility by unilaterally changing their provision. This occurs when the following conditions are met:
(x1A)−12(2x2A)−12=(x1B)−12(2x2B)−12
(c) There is usually under-provision of the public good because individuals do not take into account the positive externalities their provision of the public good has on others. In a noncooperative game, each individual only considers their own utility, leading to a lower provision of the public good than what would maximize total welfare.
(d) Policies that ensure the optimal provision of the public good could include subsidies for the production of the public good, regulations requiring a certain level of provision, or the creation of a public agency to provide the good. These policies can help align individual incentives with social welfare, leading to a higher provision of the public good.
