B1 A small village is populated by two individuals, A and B .   Both villagers use wood fired stoves for cooking and heating. An individual i  = A, B who burns an amount xi  of wood generates pollution.

Aggregate pollution is c(xA + xB )2 . Gathering xi units of wood from a nearby forest causes individual i a utility cost of kxi(2) .  Individuals derive utility from burning wood and suffer disutility from pollution and wood gathering:

uA = xA - c(xA + xB )2 - kxA(2)  and uB  = xB - c(xA + xB )2 - kx B(2) .

(a) Derive the Nash equilibrium (xA(N), xB(N)) of the wood burning game       between A and B .  How does the households’ equilibrium be-       haviour react to k and to c?                                                     [10]

(b) Derive the utilitarian optimum (xA(*), xB(*)), maximising the sum of utilities.  Is it a Pareto improvement over the Nash equilibrium outcome? Compare the pollution in both outcomes. Explain.      [10]

(c) Propose a policy solution that the villagers can achieve without outside help and that would ensure that the Nash equilibrium under your policy would coincide with the utilitarian optimum. Explain why.                [10]

(d) Identify an assumption of the model that is not reflecting the       real world and would affect optimal policy. Explain how it would       affect the model and the optimal policy.             [10]

B2 In an economy a representative household has income w and derives

utility from consuming avocadoes xa  and beer xb  and has the utility function

u = ^xa +^xb .

The price of avocadoes is normalised to 1, the price of beer is p. Both avocadoes and beer are subject to a per unit tax, at rates ta and tb  respectively.

(a) Set up the optimal consumption problem of the household and derive the optimal consumption ratio xa(*)/xb(*), depending on tax rates ta  and tb . Explain.                                                           [10]

(b) Suppose the government needs to fund expenditure R > 0 through      the commodity tax, which raises revenue taxa(*) + tbxb(*) . Given this       constraint, will the household’s utility be maximised if ta  = tb ?       Explain or compute.            [10]

(c) Suppose the government is planning to tax the consumption of       beer.   Would a per unit or an ad valorem tax be preferable?       Propose and explain two arguments (based on the model or the       real world).                  [10]

(d) Suppose now that households are heterogenous in their prefer-       ences and their abilities to generate income, i.e. their productiv-       ities.  Discuss two issues that now affect the design of optimal       commodity taxation.                      [10]

B3 A small economy is populated by three voters 1, 2 and 3, and con- templates whether to accept an offer by a foreign country for a new, important trade agreement.  The alternatives are to accept (A), to reject (R), or to postpone and renegotiate (P). The voters’ prefer- ences over these alternatives are given by this payoff matrix:

Voting is between any two alternatives, the alternative with a majority of votes wins.

(a) What is the utilitarian optimal choice? Is any alternative a Con-       dorcet winner?                 [10]

(b) Suppose a vote between A and R is called.  Derive the voters’       best responses and a Nash equilibrium of the voting game.  Is       voting in this Nash equilibrium truthful? Is it Pareto optimal?     [10]

(c) Suppose now that first a vote is called whether to have the vote between R and A or whether to postpone the decision (i.e. choos- ing alternative P). Derive the Nash equilibrium of the two stage game, supposing that voters vote truthfully in the second stage, if a second vote is called.                                                          [10]

(d) Two different proposals for the wording of the vote between A       and R are discussed: ”Sign the trade agreement, yes or no?” and       ”Forego the trade agreement, yes or no?” . Using concepts from       economic theory, argue whether actual behaviour of voters can       be expected to depend on which wording is used.                       [10]