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Consider the following economy with production.  There is a worker endowed with 1 unit of time, which can be allocated either to leisure or labor.  There is a firm with a technology that turns a worker’s labor (the complement of his leisure) into bread.  That technology satisfies: b(l)=l^(1/2)

Where l∈[0,1]  denotes the worker’s labor and b(l) is the amount of bread produced as a function of labor input. The  worker’s  utility  function  from  labor  supplied  and bread consumed is: u(l,b) =αln(b)−l,0< α <2

1. What is the set of Pareto optimal allocations in this economy? 

2. Provide prices and an allocation that, together, constitute a competitive equilibrium in this economy. 

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Joshua StredderLv10
9 Apr 2021
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Related questions

Let us have an economy that lives only for 2 periods. The representative household has a utility function that values consumption goods for any period as follows

u(c) = ln(c).

The valuation throughout both periods can be represented as a total utility function that discounts future utility with a β discount parameter as follows,

U(c1, c2) = u(c1) + βu(c2) = ln(c1) + β ln(c2),

where ct represent consumption good in period t for every period t = 1, 2. The budget constraint for this person in every period is expressed as

(1+τtc)ct + xt =(1−τtI)(wtnt +rtkt), t=1,2.

where xt represent investment in period t, wt the wages earned by hours worked, rt the return of capital in period t, nt the hours worked supplied by the household in every period t, kt the stock of capital that the household has in period t, τtc the tax rate imposed by the government in consumption goods in every period t, and τtI the tax rate imposed by the government to total household income in every period t. In every period, the total amount of hours that the representative household is of H ̄ = 1. Investment in every period can be defined by the following law of motion of capital

kt+1 = xt +(1−δ)kt, t=1,2.

Acknowledge that k3 = 0 because there is no period 3 in this economy. The household starts with a given amount of capital k ̄1 > 0. The representative firm produces good by the following production function in every period t,

Yt =ztKtαNt1−α, t=1,2.

There is a government that covers government expenditures in period 1 and period 2 by taxing capital rent and labor. Hence, their budget can be expressed as

τtcct +τtI(wtnt +rtkt)=gt, t=1,2.

a) Before writing any competitive equilibrium definition or the Social Planner Problem, can we modify the budget constraint in 
order to get rid a variable in every period? Explain.

b) Define the Social Planner Problem.

c) Solve the Social Planner Problem. This is, find the Euler equation in terms of exogenous variables and the only endogenous variable k2.

d) Define the tax distorted competitive equilibrium of this economy.

e) Solve the representative household problem.

f) Solve the representative firm problem in period 1 and 2.

g) Find the optimal Euler equation in terms of exogenous variables and the only endogenous variable k2.

h) Imagine that the government can only implement income taxes. This is τtc = 0 for every period t = 1, 2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem?

i) Imagine that the government can only implement consumption taxes. This is τtI = 0 for every period t = 1,2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem? 


PART A

1. In which of the following situations is the Coase theorem likely to apply? Explain.

a. The neighbor who lives below you in the apartment building loves to play old John Denver records at loud volume early in the morning. In fact, his favorite song seems to be “Thank God I’m a Country Boy.” You hate old John Denver songs and this one in particular.

b. You have just moved to Beijing and quickly discover that you are allergic to smog. It causes you great pain in breathing and limits your ability to work. Furthermore, you have medical expenses related to treatment for its health effects on you.

c. Medical research in the 1990s has revealed a specific cause of birth defects related to a certain drug prescribed for pregnant women in the 1960s (thalidomide).

d. The water in the huge underground Ogalalla aquifer in the Great Plains region of the United States is being nitrate contaminated due to agricultural fertilizer use throughout the region. As a homeowner in this region, you find that your water must be treated in order to be safe for drinking.

2. One of the most striking aspects of cities in the former Soviet Union is the poor condition of public places. Streets, parks, libraries, museums, concert halls, and public gathering places of every description are all in very poor condition, revealing years of neglect and lack of maintenance.

Explain this phenomenon in light of the definition of property rights and the transition from socialism to a market economy.

3. Suppose that the marginal benefit associated with corn production is MB = 2.5. The marginal private cost of production is MPC = 2 + 0.1Q, where Q measures bushels of corn produced in thousands. The marginal damages caused by pollution in corn production are given by MD = 0.05Q.

a. Compute the privately optimal output and the socially optimal output.

b. Compute new solutions for an increase in MB to MB = 3. Explain what has happened and why. c. Demonstrate the effects of an increase in MD to MD = 0.075 compared to the solutions in part a.

4. Policy analysts in the Department of Natural Resources have estimated that the marginal benefits from water pollution abatement are given by the function MB ¼ 0.90 À 0.03A, where A is a measure of the abatement intensity. Industry experts have estimated that the marginal cost of abatement activity is MC = 0.30 + 0.09A.

a. Determine the optimal level of abatement activity.

b. Explain what happens when new technology reduces the marginal cost of abatement to MC = 0.25 + 0.06A. 5. Consider the pollution reduction mechanism known as emissions trading. a. What is the economic rationale for permitting emissions trading?

b. If the objective is to minimize the cost of emissions reductions, explain what economic principle should be followed in allocating emissions reductions between two countries.

c. Suppose the marginal cost curves for emissions reductions in countries A and B are MCA = 10 + 0.04qA and MCB = 5 + 0.02qB. What is the most efficient allocation of an emission reduction of q0 between the two countries? (Assume that the emissions reductions of the two countries must add up to the total emissions reduction: q0 = qA = qB.)

PART B

1. Which of the following do you consider pure public goods? Private goods? Why?

a. Wilderness areas

b. Satellite television

c. Medical school education

d. Public television programs

e. Automated teller machine (ATM)

2. Indicate whether each of the following statements is true, false, or uncertain, and justify your answer.

a. Efficient provision of a public good occurs at the level at which each member of society places the same value on the last unit.

b. If a good is nonrival and excludable, it will never be produced by the private sector.

c. A road is nonrival because one person’s use of it does not reduce another person’s use of it.

d. Larger communities tend to consume greater quantities of a nonrival good than smaller communities.

3. Tarzan and Jane live alone in the jungle and have trained Cheetah both to patrol the perimeter of their clearing and to harvest tropical fruits. Cheetah can collect 3 pounds of fruit an hour and currently spends 6 hours patrolling, 8 hours picking, and 10 hours sleeping.

a. What are the public and private goods in this example?

b. If Tarzan and Jane are each currently willing to give up one hour of patrol for 2 pounds of fruit, is the current allocation of Cheetah’s time Pareto efficient? Should he patrol more or less?

4. The aircraft company Airbus receives much of its funding from European governments. Airbus recently decided to build a new 550-seat mega-jetliner, with duty-free shopping courts and restaurants on board. The project has experienced production delays as well as cost overruns, and it now appears that there will be very few buyers. An industry expert says the idea from the start was “nonsense”

Is public sector production of aircrafts ever justified?

Explain why it could lead to the apparently ill-advised decision to build the mega-jetliner.

5. Suppose that there are only two fishermen, Zach and Jacob, who fish along a certain coast. They would each benefit if lighthouses were built along the coast where they fish. The marginal cost of building each additional lighthouse is $100. The marginal benefit to Zach of each additional lighthouse is 90 − Q, and the marginal benefit to Jacob is 40 − Q, where Q equals the number of lighthouses.

a. Explain why we might not expect to find the efficient number of lighthouses along this coast.

b. What is the efficient number of lighthouses? What would be the net benefits to Zach and Jacob if the efficient number were provided?

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