There are two goods markets, and the government considers introducing a per-unit tax on each. For...

 

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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 are constant, and those of the first and second goods are given by ₁ and ₂, 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 = ₁ + t₁ and p₂ = ₂ + t₂, respectively. The government knows the utility function of the consumer in the market of the first goods, which is given as follows: 3² x₁³+1¹-P₁x₁ where x₁ and 1¹ 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: U¹(x₁) = = 1. 1 U² (x₂) = 2 x² + 1² - P₂x₂ U (x₂)=2x2 + 1 - P2x2 where I > 12. The government needs at least R tax revenue: i.e., R = 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, U2, U2): 2. 3. 1 W(U¹, U2, U³) = 0.5 W (U¹, U²) + 0.5 W (U¹,U²) where W (U¹, U²) is the social welfare function and is given by W (U¹,U?) = log U¹ + log U?. Find consumer consumption ¹(t₁) of the consumption (t₂) of utility function U2, and second goods. Find the function from policy variables (t₁, t₂) to expected social welfare. Formalize the government's optimization problem. In addition, find the Lagrange function L(t₁, t₂, 2). (Note: You need not to solve the problem.) first goods in equilibrium, consumer (t₂) of function U in the market of the 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 ₁ and ₂, 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 = ₁ + t₁ and p₂ = ₂ + t₂, respectively. The government knows the utility function of the consumer in the market of the first goods, which is given as follows: 3² x₁³+1¹-P₁x₁ where x₁ and 1¹ 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: U¹(x₁) = = 1. 1 U² (x₂) = 2 x² + 1² - P₂x₂ U (x₂)=2x2 + 1 - P2x2 where I > 12. The government needs at least R tax revenue: i.e., R = 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, U2, U2): 2. 3. 1 W(U¹, U2, U³) = 0.5 W (U¹, U²) + 0.5 W (U¹,U²) where W (U¹, U²) is the social welfare function and is given by W (U¹,U?) = log U¹ + log U?. Find consumer consumption ¹(t₁) of the consumption (t₂) of utility function U2, and second goods. Find the function from policy variables (t₁, t₂) to expected social welfare. Formalize the government's optimization problem. In addition, find the Lagrange function L(t₁, t₂, 2). (Note: You need not to solve the problem.) first goods in equilibrium, consumer (t₂) of function U in the market of the

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1In equilibrium the consumer will consume x of the first good and y of the second good The consumers ... View the full answer