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Q2. The two goods in a single-consumer competitive economy have the demand functions 𝑞1 = 𝑝1−2 and 𝑞2 = 𝑝2−3, where 𝑝 and 𝑞 denote price and quantity, respectively. The (constant) marginal cost of producing good 1 is 2 and the constant marginal cost of producing good 2 is 3.
- Calculate the elasticity of demand in the markets for good 1 and good 2. [8%]
Suppose that the government imposes specific unit commodity taxes of 𝑡1 and 𝑡2 on the two produced goods.
- If the government wishes to raise revenue 𝑅 from market 1, satisfying 𝑅 = 0.05 , what level must 𝑡1 be set at? [8%]
- Show that, if 𝑡1 and 𝑡2.are set optimally, they satisfy the relationship
𝑡1𝑡2 = 6(𝑡1 − 𝑡2) [8%]
- Suppose that the tax rate that you have found in (b) to raise 0.05 from market 1 was set optimally. What will the tax rate in market 2 be in an optimal commodity tax structure and how much tax revenue will the government raise from market 2? [8%]
- What, if any, lessons do you draw from the comparison of the tax yields from markets 1 and
2 in this case? [8%]
Each of the 2 identical consumers in an economy with a single private good and a single pure public good has a utility function given by
(𝑥𝑖, 𝐺) = 𝑥𝑖 + 𝐺1/2
Here, 𝑥𝑖 is 𝑖′𝑠 consumption of the private good and 𝐺 is the aggregate provision of the sole public good. Each consumer has an endowment 𝑤. This can be used to purchase the private good or the public good, each of which has a price of unity.
- Calculate the Nash equilibrium level of 𝐺. [7%]
- Suppose there were, instead, 𝑛 > 2 identical consumers. What would be the aggregate spending on the public good? [7%]
- How do you explain your finding in (b)? [6%]