Consider the following economy with two agents and two consumption goods, one of which is used...

 

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Consider the following economy with two agents and two consumption goods, one of which is used to produce the other. Let Household h's consumption of the two goods be denoted and respectively, where h = 1,2. Preferences: Household h has preferences given by: Endowments: U₁(₁,2)=2 nr +3 ln x h = 1,2. Both households are endowed with 2 units of good one (i.e. w). There is no endowment of good two. Technology: Let Y₂ denote the total amount of good two available to the economy. It is produced using good one, by means of the following technology: Y₂ = [4 - (x³ + x²)] where the term in brackets on the right-hand side of the equation above is the amount of good one that is used as an input to production of good two rather than consumed. Consider the problem of a "social planner" who maximizes an equally weighted sum of agents' utilities subject to the feasibility constraints. a. Write down the social welfare maximization problem that the planner solves. b. Use this problem to derive necessary conditions for an allocation to be Pareto efficient in this economy. c. Calculate the allocation that maximizes social welfare. d. Show how problems of this sort can be used to find all of the Pareto efficient allocations for this economy. Note: You do not have to find all of these allocations, just show how you could do so. Consider the following economy with two agents and two consumption goods, one of which is used to produce the other. Let Household h's consumption of the two goods be denoted and respectively, where h = 1,2. Preferences: Household h has preferences given by: Endowments: U₁(₁,2)=2 nr +3 ln x h = 1,2. Both households are endowed with 2 units of good one (i.e. w). There is no endowment of good two. Technology: Let Y₂ denote the total amount of good two available to the economy. It is produced using good one, by means of the following technology: Y₂ = [4 - (x³ + x²)] where the term in brackets on the right-hand side of the equation above is the amount of good one that is used as an input to production of good two rather than consumed. Consider the problem of a "social planner" who maximizes an equally weighted sum of agents' utilities subject to the feasibility constraints. a. Write down the social welfare maximization problem that the planner solves. b. Use this problem to derive necessary conditions for an allocation to be Pareto efficient in this economy. c. Calculate the allocation that maximizes social welfare. d. Show how problems of this sort can be used to find all of the Pareto efficient allocations for this economy. Note: You do not have to find all of these allocations, just show how you could do so.

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