Two individuals A and B have utility functions defined over two goods, a private good 'x') and a public good, 'F. The utility function of an agent 'i' is defined by ui = 2 log xi + log F where F =FA + FB. Each agent has 200 units of private goods x, as his endowment and 1 unit of private good can be transformed into 1 public good F. Answer the following (a) Find the Nash equilibrium values of FA and FB? (b) What is the Pareto optimal level of F? (c) In what condition the Pareto optimal level of F will not depend on the number of private goods? (d) What would be the consumption of x by each agent if they contribute equally towards the public good (F)? (e) Would it change if an agent had a larger endowment of x to begin with?
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- Which of the follwoing theoretical results is concerned with the relationship of factor endowments and the distribution of payoffs in an economy? Rybczynski Theorem Ricardian Equivalence Heckscher-Ohlin Theorem Stolper-Samuelson TheoremChris and Dana live in an exchange economy with two goods: good Q and good R. Chris starts off with an endowment of 6 units of Q and 10 units of R. Dana starts off with an endowment of 8 units of Q and 8 units of R. Suppose that the price of good R is pR=1 and the price of good Q is pQ=2. a )At these prices, does the market clear? Yes or no? Explain your answer. b) What relationship must hold between the consumption of each agent and the price of the two goods at the market clearing equilibrium? Write the equationA fixed amount of a good has to be allocated between two individuals, h = 1; 2 with utility functions Where xh is the amount of the good allocated to consumer h. a. How should be allocated to maximize a utilitarian SWF? Illustrate the answer graphically. How do the optimal values of x1 and x2 change among the cases a1 2, a1 = a2, and a1 > a2? b. What is the allocation maximizing the Bernoulli-Nash SWF ? Illustrate graphically. How do the optimal values of x1 and x2 change with the preference parameters a1 and a2? c. What is the allocation maximizing the maxim in-Rawlins SWF? Illustrate graphically. How does the allocation change with preference parameters a1 and a2?
- Bluth’s preferences for paper and houses can be expressed as Ub(p, h) = 2pb + hb, while Scott’s preferences can be expressed as Us(p, h) = ps + 2bs. Bluth begins with no paper and 10 houses, whereas Scott begins with 10 units of paper and no houses. 1. Is the starting endowment Pareto efficient? Justify your answer using an Edgeworth box? Determine whether each of the following price pairs is consistent with a competitive equilibrium. If yes, determine the resulting allocation of goods, sketching that equi- librium in your Edgeworth box. If not, explain why not (for what good is there a shortage, for what good is there a surplus?) pp =$3 and ph =$1 along with pp =$1 and ph =$1 Assume that the price of houses is $1. Given that price, determine the highest price pp that is consistent with a competitive equilibrium.Heterogeneity in an endowment economy Suppose we have two typesof households: A and B. The utility maximization problem for a consumerof type i is max InCi,t+β ln Ci,t+1 Ct,Ct+1subject to Ci,t +Ci,t+1/ 1 + rt = Yi,t + Yi,t+1 / 1 + rt Note that the A and B households have the same discount rate and the same utility function. The only thing that is possibly different is their endowments. 1, Write down the Euler equation for households A and B.Consider an economy with a private good and public good. The economy consists of two consumers whose utility functions are: u1(x1,y)=1/2 ln x1 + 1/2 ln y and u2(x2,y)=1/3 ln x2 + 2/3 ln y (better written equations attached) The endowment of consumer 1 is w1=2, and the endowment of consumer 2 is w2=3, both in units of private goods. The production of public good uses the linear Technology y=z. Find Lindahl equilibrium.
- Consider an Edgeworth box economy, where preferences and endowments are given by. ( attached below ) In the following, use the normalization p2 = 1.(a) Find all the Pareto optimal allocations (use x1 to parameterize thePO allocations). (b) Suppose the government wants individual 1 to have the bundle x1 = (5, 9). Show whether the government can induce this outcome an equilib- rium with transfers. Find the supporting prices and transfers if possible. (c) State the Second Fundamental Theorem of Welfare and explain how the result in part (b) relates to that theorem. (d) Suppose the government does not care what bundle individual 1 gets and only wants to ensure that the individual 1 receives utility level equal to 45. Show whether this can be achieved as an equilibrium with transfers. Find the supporting prices and transfers if possible.Consider a community of two families A and B, who contribute EA and EB dollars to the community’s events budget. Community events are a public good and the total quality of events provided E = EA + EB. Assume each family has an annual income of $60 which it splits between the events budget and all other consumption spending X. Family utility functions are given by UA = XA^1/2 E^1/2 and UB = XB^1/2E^1/2. (a) What event budget would maximize the social welfare function W = UA + UB? (Study tip: You can either set up and solve the social planner’s problem, or recall the necessary condition for optimal public good provision. I recommend the latter.) (b) If households privately decide how much to contribute to the event budget, what will the total budget be? (Study tip: Begin by finding each household’s best response to the other, then solve. You can use the fact that the problem is symmetric to save time in this step.)Suppose that 2 roommates, Andy and Bob, are trying to pick an apartment in Chicago. Locations can be chosen from set of alternatives A={ x: x exists [0,1]}. Andy and Bob both want to minimize their daily commute but they work at different locations: Andy at xA=0.3, while Bob at xB=0.6. Specifically, their utility functions are: ui(x)= -(x-xi)2. Question: What is the set of all Pareto Efficient outcomes in A, assuming no money can be exchanged.
- Consider an economy with a private good and public good. The economyconsists of two consumers whose utility functions are:u1(x1, y) = 1/2 ln x1 +1/2 ln y and u2(x2, y) = 1/3 ln x2 +2/3 ln y.The endowment of consumer 1 is w1 = 2, and endowment of consumer 2 is w2 = 3, both in units of private goods. The production of public good uses the linear technology y = z. 1. Find Lindahl equilibrium.Consider agent A with (inverse) demand curve for the public good and agent B with inverse demand , where prices are measured in £ per unit. The marginal cost of producing the public good is £10 per unit. What is the Pareto efficient level of the public good? Explain. Illustrate in a graph. [Hint: Compute the marginal social benefit of the public good by adding up the demand curves vertically, over the p’s - image attached that has the inverse demand functionConsider the model of a rational consumer that cares about consumption of private goods and consumption of broadcast public television (a public good). Suppose that the total level of broadcast public television provided through voluntary contributions is 10 hours of programming. Then the government decides to raise money through a tax and provide 10 hours of programming to the public. What would we predict about crowd-out of voluntary contributions to broadcast public television when government does this? How would the answer change if consumers get warm glow utility from donating to broadcast public television in addition to utility from the public television itself? (Be specific.)