Lapan Econ 603

Spring 1999

Midterm Exam

Answer any two questions; answer all parts to each question.

1. Answer all parts.

a) Prove that a competitive equilibrium is Pareto efficient. State the critical assumptions used in the proof and show where the proof would fail if there were a consumption tax on one of the outputs.

b) Consider an exchange economy with two goods (C,F) and two people (A,B). The total endowment of goods for this economy is given by: ; however, (through trade or production) the alternative aggregate endowment vector of can be obtained.

i. Which endowment point would "society" prefer if preferences are given by:


where is the consumption vector of household h.

ii. Which endowment point would "society" prefer if, instead, preferences are given by:


where is the consumption vector of household h.

Carefully explain your answer to parts i and ii.

c) Consider a model with L goods and F firms. Let denote the feasible production set for firm f, and let denote a feasible netput vector for the firm, where .

i. Briefly describe how the aggregate production set is derived and discuss the relationship between profit maximization and efficiency and the role (if any) played by convexity.

Next, assume there are three firms and three goods; goods 1 and 2 can be consumed by households, while good 3 is a pure intermediate good (like steel). The endowment vector is given by: . Each firm’s production technology is given by:



where is firm i’s netput of good j. Thus, firm 1 uses goods 2 and 3 to produce good 1, firm 2 uses good 2 to produce good 3, and firm 3 uses good 2 to produce good 1.

ii. Derive the production possibility frontier for this economy in (good 1, good 2) space (as good 3 is not consumed, the only restriction on it is that its net output be non-negative). Can every efficient point be supported as a profit maximizing allocation? What production inefficiency, if any, would result from a tax on good 3? Explain.

2. Answer all parts

a) Briefly discuss the issues that arise in proving the existence of a competitive equilibrium. Next, consider the following model of a two person (A,B), two good (C,F) exchange economy. Preferences for each person are given by:



is the consumption vector of household h. Each person's endowment vector is given by: where: (i.e., total endowments are: (6,2)).

i. Will a competitive equilibrium exist for this economy? If so, find the equilibrium price as a function of the individual endowment vector; if not, explain why not. (NOTE - your answer should depend on person A's share of the endowment of each good).

ii. If an equilibrium exists, will it be Pareto efficient? Can all Pareto efficient equilibria be supported as a competitive equilibrium with transfers? Can some of the Pareto efficient allocations be supported as a competitive equilibrium with transfers? Explain carefully.

b) Consider a model with three goods, plus pollution (z). Let good one be the numeraire, and assume it can be consumed by households or used by firms to produce goods two and three. There are N identical households, with preferences given by:


where is the household’s consumption of private goods, and z denotes the pollution, which harms households. There are three firms in the economy; two firms can produce good two, and one firm can produce good three; production of good two produces pollution, while production of good three does not. The production technology for each firm is as follows:





where is firm i's netput of good j, and total pollution is given by: .
Note that each firm uses only good one as an input to produce its product; also note that firms one and two have different production technologies for both the good and the bad (pollution).

Finally, each household is endowed with units of good one. Assume that in equilibrium (or in an efficient allocation) all households receive the same consumption vector. The resource constraint for good one is then given by:

i. Find the competitive equilibrium for this economy and discuss why it is inefficient.

ii. Find the (symmetric) Pareto efficient equilibrium for this economy and compare to the competitive equilibrium. In describing the Pareto efficient equilibrium, carefully specify which firm(s) produce(s) good two; show how this equilibrium changes as the size (N) of the economy grows. (symmetric equilibrium means each household receives the same consumption vector).

iii. Briefly discuss what policy is required to make the competitive equilibrium Pareto efficient. Then, assuming the only feasible policy is to tax or subsidize good three, discuss what the optimal policy would be and relate your answer to the sign of (in answering the last part concerning the appropriate tax/subsidy on good three, you may assume that only firm one can produce good two; also, you do not need to explicitly solve for the tax/subsidy on good three, but you should justify whether a tax or subsidy is the appropriate policy).

3. Answer all parts and subparts.

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 are the input prices in sector i, for labor and capital, respectively. Assume that labor used in sector x is subject to an ad valorem tax at rate , but no other factor is taxed (thus: ), where (W,R) are the net returns each factor receives.

i. Given the total supply of K and L, find the general equilibrium supply curves. Also, discuss which efficiency conditions, if any, are violated due to this tax.

ii. Suppose demand for final goods is given by: , where I denotes total income. Find the equilibrium output price and input prices as functions of the tax rate. (let good y be the numeraire).

iii. Assuming the tax rate on labor in sector x cannot be changed (for political reasons), could a tax (or subsidy) to output of good x improve efficiency? Could it restore full efficiency? Explain. (an intuitive argument will suffice here; you do not need to provide the mathematical analysis).

b) Consider a model with two households (A, B) and two goods (1, 2). Households have identical preferences given by:

; h= A, B; where is household h's consumption of good i.

The endowment vector for each household is: (i.e., household A is endowed with 10 units of good one, and household B with 26 units of good one). Good two can be produced using inputs of good one as follows:

where denotes output of good 2, and denotes input of good 1 (which can be supplied by either household).

i. Given the total endowment vector , find the set of Pareto efficient allocations for this economy. Show how each can be supported as a competitive equilibrium with transfers.

Next, assume that lump-sum transfers are not feasible and that the only feasible policy for redistributing income is to tax purchases of good 2, and redistribute the proceeds in a non-discriminatory fashion to each household as follows:

; where:

is the net transfer to household i (=A,B), with the government budget constraint implying: ; is a positive lump-sum transfer that is the same for all households; and is the tax rate on the household's purchase of good 2 . (Note that the government cannot tax the household's consumption of good one).

ii. Given this tax/transfer scheme, can every Pareto efficient allocation be supported as a competitive equilibrium? If not, show the set of feasible utility allocations that can be achieved with this tax scheme and find the optimal scheme if the government's objective is to maximize the utility of the poorer person (household A).