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 C 1 and C 2 , respectively Therefore, given that t 1 and t 2 denote the amount of tax per unit of the first and second goods, the equilibrium prices are given by p 1 C 1 t 1 and p 2 C 2 t 2 , respectively The government knows the utility function of the consumer in the market of the first goods, which is given as follows U 1 (x 1 ) (3 2)x 1 2 3 I 1 p 1 x 1 where x 1 and I 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 U L 2 and U H 2 takes place with probability 0 5 U L 2 (x 2 ) 2x 2 1 2 I L 2 p 2 x 2 U H 2 (x 2 ) 2x 2 1 2 I H 2 p 2 x 2 where I H 2 I L 2 The government needs at least R tax revenue t 1 x 1 t 2 x 2 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 1 , U L 2 , U H 2 ) W(U 1 , U L 2 , U H 2 ) 0 5W(U 1 , U L 2 ) 0 5W(U 1 , U H 2 ) where W(U 1 , U i 2 ) is the social welfare function and is given by W(U 1 , U i 2 ) log U 1 log U i 2 Find consumer consumption x 1 (t 1 ) of the first goods in equilibrium, consumer consumption x L 2 (t 1 ) of utility function U L 2 , and x H 2 (t 1 ) of function U H 2 in the market of the second goods Find the function from policy variables (t 1 , t 2 ) to expected social welfare Formalize the government's optimization problem( maxW(U(x(t)))) In addition, find the Lagrange function (t 1 , t 2, ) (Note You need not to solve the problem )

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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

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 C₁ and C₂, respectively. Therefore, given that t₁ and t₂ denote the amount of tax per unit of the first and second goods, the equilibrium prices are given by p₁= C₁+ t₁ and p₂= C₂+ t₂, respectively. The government knows the utility function of the consumer in the market of the first goods, which is given as follows:

U¹(x₁) = (3/2)x₁^2/3 + I¹ - p₁x₁

where x₁ and I¹ 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 U_L²and U_H² takes place with probability 0.5:

U_L²(x₂) = 2x₂^1/2 + I_L² - p₂x₂

U_H²(x₂) = 2x₂^1/2 + I_H² - p₂x₂

where I_H² > I_L². The government needs at least R tax revenue t₁x₁ + t₂x₂. 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(U¹, U_L², U_H²):

W(U¹, U_L², U_H²) = 0.5W(U¹, U_L²) + 0.5W(U¹, U_H²)

where W(U¹, U_i²) is the social welfare function and is given by W(U¹, U_i²) = log U¹ + log U_i².

Find consumer consumption x¹(t₁) of the first goods in equilibrium, consumer consumption x_L²(t₁) of utility function U_L², and x_H²(t₁) of function U_H² in the market of the second goods.
Find the function from policy variables (t₁, t₂) to expected social welfare.
Formalize the government's optimization problem(= maxW(U(x(t)))). In addition, find the Lagrange function (t₁, t_2,). (Note: You need not to solve the problem.)