Entry and output Due by 5:00pm on Thursday 26 May, in your tutor's box. You don't need to type up your answers; legible handwriting is fine. The inverse demand for electricity is p = a - q. Electricity is provided by a single clean generator, M, and by n identical dirty generators in a competitive fringe. Total electricity generated is q = qm + nqf . The government, G, deals with the negative externality from dirty generation by requiring each dirty generator to surrender an allowance for each KwH generated.

The steps of the game are:

1. G chooses how many allowances for dirty energy to provide, ` ∈ R+

2. M observes ` and chooses how much (clean) energy to generate, qm∈ R+

3. Each member of the competitive fringe chooses how much (dirty) energy to generate, qf∈ R+, which determines how many allowances it must purchase.

4. Each consumer observes the price of electricity, and chooses how much to use.

a) In step 4, each consumer chooses how much electricity, e, and how much of a num´eraire good, y, to consume. Her utility is u(e) + y which is made up of her benefit from electricity, u(e) = ae - e 2/2, and her benefit from consuming the num´eraire. Her budget constraint is pe + y = I. 1 Solve her utility maximisation problem to identify her inverse demand for electricity. It should have the parameter a in it somewhere.

b) In step 3, each generator in the competitive fringe observes the price of electricity, p, and the price of a dirty generation allowance, φ. It chooses qf to maximise its profits, pqf - φqf - c(qf ), where its generation costs are c(qf ) = mq2 f /2. Characterise profit maximisation by taking a first-order condition wrtqf . Solve your first-order condition for qf to obtain an expression for the supply curve for electricity by a single member of the competitive fringe, qf = Sf (p - φ). 1Note that we are not allowing for two-part tariffs or block pricing.

c) The market for allowances clears when the demand for allowances (equal to the supply of dirty electricity) matches the supply of allowances (fixed at the quantity that G chose in step 1): nSf (p-φ) = `. Write down this market clearing condition, using the expression for the supply by a dirty generator, Sf (p - φ), that you identified in question (b). Solve your market clearing condition to find the equilibrium price of an allowance and the quantity of dirty electricity generated, both as functions of `.

d) In step 2, M maximises its profits which are pqm - κqm, facing the inverse demand for electricity of p = a - qm - nqf . M anticipates that the competitive fringe will generate nqf = ` KwHs. Set up M's profit maximisation problem under these assumptions. Take a FOC to characterise M's choice of qm.

e) How much electricity will be generated in equilibrium, qm +nqf ? Your answer should be a function of ` rather than a number.

f) In step 1, G chooses `. It wishes to maximise welfare, which is the benefit from consumption minus the sum of generation costs and the harm from emissions:2 W(`) = u(qm + nqf ) + I - nc(qf ) - κqm - nqfH, where H is the marginal harm (negative externality) from dirty generation. Can you derive an expression for G's choice of how many allowances to provide for dirty generation, `?

g) Can you provide any intuition for your result in (f)? For example, should the government provide more allowances when the harm from dirty generation is high or when it is low? Why? Should there be more or fewer allowances when generation costs are high? Why? 2The function u and income I were introduced in question (a). Effectively consumer benefit is a function of electricity consumed, e, but as this is equal to electricity generated, qm + nqf , the latter can be substituted in to the expression for utility.