Consider the two-nation two-good model. Each nation has the same population and specializes in production of a particular good: nation-1 labour specializes in good-1, and nation-2 specializes in good-2. It takes \lambda 1 hours from nation-1 labour to produce each unit of good-1 and \lambda 2 hours from nation-2 labour to produce each unit of good-2. The total utility to a nation-1 household from consuming c1 good-1, c12 good-2, and supplying h1 hours is given by
u1(c1) + u2(c12) − h1 = \sigma 1(c1)\theta /\theta + \sigma 2(c12)\theta /\theta − h1,
and the total utility to a nation-2 household from consuming c21 good-1, c2 good-2, and supplying
h2 hours is given by
u1(c21) + u2(c2) − h2 = \sigma 1(c21)\theta /\theta + \sigma 2(c2)\theta /\theta − h2.
Now we consider the case where
\theta =0.7, \sigma 1 =3=\sigma 2,\lambda 1 =1.5=\lambda 2. (3)
a) Analyze the social planner problem, with weights \alpha 1 = 0.5 = \alpha 2, and solve for the optimal allocation.
b) The First Fundamental Theorem of Welfare states that the market equilibrium outcome corresponds to the solution of the social planner’s problem with some weights \alpha 1′ and \alpha 2′ . What do you think the weights are for the economy with parameters given by (3)? Can you obtain the equilibrium price and equilibrium outcomes directly from the solution to the social planner’s problem? If so, compute them.
c) One challenge from climate change is that consumption may bring negative externality in that more energy consumption or more driving could lead to disasters such as wild fire. Suppose that, for simplicity, such externality only occurs for certain type of goods but not others. More precisely, we assume that only the products of nation-1 will lead to global warming, and if the average consumption is C1, then it leads to a cost of vC1. Thus, the social planner’s problem is to choose [(c1, c12, h1), (c21, c2, h2)] to maximize
\alpha 1[u1(c1) + u2(c12) − h1] + \alpha 2[u1(c21) + u2(c2) − h2] − v(c1 + c21)/2.
Solve the social planner’s problem with parameters given by (3), v = 3, and weights \alpha 1 = 0.5 =
\alpha 2.