There are two goods markets, and the government considers introducing a per-unit tax on each. For simplicity, the marginal costs of firms in both markets are constant, and those of the first and second goods are given by and , respectively. Therefore, given that t and t denote the amount of tax per unit of the first and second goods, the equilibrium prices are given by P = + t and p = + t, respectively. The government knows the utility function of the consumer in the market of the first goods, which is given as follows: 3 x+1-Px where x and 1 denotes consumption and income. On the other hand, the government does not know the utility function of the consumer in the market of the second goods and thinks that one of the following two utility functions U2 and U takes place with probability 0.5: U(x) = = 1. 1 U (x) = 2 x + 1 - Px U (x)=2x2 + 1 - P2x2 where I > 12. The government needs at least R tax revenue: i.e., R = tx + tx. Moreover, since the government does not know the utility function of the consumer in the market of the second goods, it wants to maximize, instead of social welfare, the expectations of social welfare, W(U, U2, U2): 2. 3. 1 W(U, U2, U) = 0.5 W (U, U) + 0.5 W (U,U) where W (U, U) is the social welfare function and is given by W (U,U?) = log U + log U?. Find consumer consumption (t) of the consumption (t) of utility function U2, and second goods. Find the function from policy variables (t, t) to expected social welfare. Formalize the government's optimization problem. In addition, find the Lagrange function L(t, t, 2). (Note: You need not to solve the problem.) first goods in equilibrium, consumer (t) of function U in the market of the