Lapan                                                                                                                        Econ 603

Spring 2001

Midterm Exam

a)      State the second welfare theorem, indicate its importance and give a sketch of how to prove it (a formal proof is not needed; just indicate the crucial steps in the proof).  In particular, identify the crucial assumptions in this proof, including the role of the convexity assumption.  Discuss the relationship between the first and second welfare theorems, and indicate why the assumptions required to prove the two theorems differs.

b)      Next, consider a two person (I, II), two good (M, F) exchange economy.    Suppose preferences are given by:
Person I:                 Person II:
Given the aggregate endowments,  such that:  :

i.         Find the set of Pareto efficient allocations for this economy.  Which allocations are supportable as competitive equilibrium?

ii.       Given the aggregate endowments of the two goods, find the set of endowments to each individual that result in a competitive equilibrium .  Compare your answer to part (i).

c)      Finally, consider the same preferences as in part (b), but now assume there is no initial endowments of goods (M, F).  Instead, there is a single input with fixed supply  that can produce output according to the following production technology:

i.         Find the set of Pareto efficient allocations for this economy.  Which allocations are supportable as competitive equilibria?  Relate your answer to your answer to part (a).

ii.       For what set of initial endowments of L across the two individuals does a competitive equilibrium exist?

a)      Prove that a competitive equilibrium is Pareto efficient.  Show what step(s), if any, in your proof would fail to hold if there is a tax on a final product consumed by households.

b)      Consider a two period (t=1,2), two person (A,B) exchange economy in which two goods (C,M) are consumed in each period (hence, a total of four goods).  Individuals have identical preferences given by:

where is person i’s (=A,B) consumption vector in period t (=1,2), and  denotes the “discount” rate for future consumption.  The individuals’ endowment vectors are:

where, for example,  is person A’s first period endowment of good C, and  is that person’s second period endowment of good M.  {HINT: It may help simplify your work if you note preferences are identical and homothetic and that the total endowment of each good is the same}.

i.         Find equilibrium prices, individual consumption vectors and individual utility assuming that borrowing or lending is not possible  (hence, the only possible trades are within period trades, such as M for C in period 1, or M for C in period 2).  Is this allocation Pareto efficient?  Explain.

ii.       Find the equilibrium prices, consumption and utility assuming that borrowing or lending is possible (hence, all possible trades are feasible).  Compare the resulting utility level for each household.  (for computational simplicity, assume  when comparing utilities).

c)      Consider a two good economy with three firms.  The firms have the following technology:

Firm 1:             Firm 2:
Firm 3:
where  is a constant and  denotes firm j’s netput of good i.

i.         Derive the aggregate production set (technology) for this economy.  Assuming A=1 and the initial endowment of good 2 is 30, state the production possibility frontier.

ii.       Suppose that .  Will profit maximization lead to efficient production in this case?  Explain, and if you answer no indicate what action (or policy) is required to insure that profit maximization leads to efficient production.

a)      Consider a two good (X,Y), two factor (K,L) general equilibrium model where production exhibits constant returns to scale.  Production functions (and dual cost curves) are given by:

where  is the price paid for capital and labor, respectively, in sector i.  If factors are freely mobile between sectors and there is no differential taxation, then  and .  Finally, let  denote the factor endowments and assume:

i.         Given output prices, derive the general equilibrium supply curves.  Since, at the firm level, marginal cost is constant (i.e., independent of output level), does that imply these general equilibrium supply curves are infinitely elastic?  Explain your answer.

ii.       Given output prices, show how a tax on labor used in sector X will affect equilibrium output levels and factor prices (for wages, distinguish between the wages paid by firms in both sectors).  How do your results for the impact of this tax differ from what you would expect to find in a partial equilibrium model?  Explain.

b)      Using the same production structure as in part (a), suppose all individuals in the economy have identical preferences given by:  .

i.         Assume, as in part (a), that labor used in sector X is subject to a percent tax , so that .  Find the competitive equilibrium factor prices, output prices and outputs as a function of the tax rate, and show how each changes with the tax rate.

ii.       What inefficiency, if any, does this tax cause?  Assuming this tax on labor in sector X cannot be eliminated, could a consumption tax (or subsidy) on good X  improve welfare?  Explain, and indicate which policy is appropriate  (you do not have to prove your answer, but you should give a clear explanation of the reasoning behind your answer).

c)      Consider the following simplified general equilibrium model.  Each of H individuals have preferences given by:

;

where  is the consumption vector chosen by household h,  and  stands for air quality, which is the same for all households.  Let good M be the numeraire good.  Households are endowed with units of good M, and with fractional ownership in each of the  J firms.   The cost function for each firm is: ,  where:   denotes the output vector of the firm, and  denotes the amount of good M required to produce that output vector.  Assume good M and goods {1,…L} are rival (private) goods, so the usual feasibility conditions hold for those goods.

i.         Assuming air quality (Z) is constant, state the mathematical conditions required for Pareto efficiency and interpret them economically.

ii.       Next, assume that air quality is a decreasing function of  (the aggregate output of good one).  Restate the efficiency conditions for this case and show how they differ from part (i).  State what policy, if any, is required to insure that a competitive equilibrium is Pareto efficient.