Consider a country with 1 million identical consumers. Each consumer utility function is given by U(q, m) = ln(q) + m - 2E, where q denotes the amount of GPUs that she consumes, m denotes the amount of $ consumed, and E denotes the total level of an environmental pollutant in the country.

GPUs are produced by competitive firms with a constant marginal cost of 1$/unit, and no fix or semi-fixed costs.

Producing GPUS generates pollution: each GPU produced produces two additional units of the pollutant .

1. At the Pareto optimal allocation, what is the level of GPU consumption for each consumer?

In the rest of the problem, assume that consumers taken the total level of pollution E as fixed when making decisions.

2. At the market equilibrium, what is the level of GPU consumption by each consumer?

This was found to be 1, by maximizing ln(q) - pq with respect to q for consumers where p = 1 as the firm has constant marginal cost of 1$/unit.

3. What is the deadweight loss at the market equilibrium? (you can approximate to the nearest trillion)

4. What is the magnitude of the optimal Pigouvian tax (in $/unit)? Assume that the tax is paid by the consumer.

This is found to be 4000000 by using the fact that marginal damage is equal to the magnitude of optimal Pigouvian tax, and total marginal damage here is 2q*100000*2, as we let E = 2q, and there are 1000000 consumers.

I need help with part 1 and 3.