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This a long homework assignment. For this reason you can get extra points. General Equilibrium and Welfare Economics are two important topics in Economics. The fundamental theorems of welfare economics are particularly interesting because we can use them to think in a systematic way about the role of markets and governments.

 

Problem 1: Pure Exchange Economy

 

 

Problem 2: One Consumer and One Producer Economy

 

Robinson Crusoe is the only survivor in his island. He produces coconuts employing his labor as input. His utility function is given by:

 

(𝑐𝑐,𝑧𝑧)=(𝑐𝑐)1/2(24 βˆ’ 𝑧𝑧)1/2

where 𝑐 is the amount of coconuts he consumes and 𝑧𝑧 is the time he expends working in a day (there 24 hours available per day). The production function of coconuts is =(𝑧𝑧)1/2 , where π‘žπ‘ž is the amount of coconuts produced and 𝐴𝐴 is a measure of productivity.

 

  1. a) Compute the all the feasible allocations.
  2. b) Compute the excess demand functions of coconuts and labor and verify Walras’s Law. In order to do so, denote by 𝑝𝑝 the price of coconuts and by 𝑀𝑀 the wage rate.
  3. c) Compute the Walrasian Equilibrium of this economy. DO NOT FORGET to compute equilibrium quantities.
  4. d) Compute the Pareto efficient allocations.
  5. e) Show that the Walrasian Equilibrium allocation is Pareto efficient.
  6. f) Comparative Statics: Compute the effect of an increase in 𝐴𝐴 on the Walrasian Equilibrium of this economy.

 

Problem 3: 2x2x2 Economy

 

Consider an economy with two citizens (Β  andΒ  𝐡𝐡), two goods (𝑋𝑋  andΒ  π‘Œπ‘Œ) and two factors of production (𝐾𝐾 and 𝐿𝐿). We will use the following notation:

 

𝐾𝐾𝑋𝑋 is the amount of capital employed in the production of 𝑋𝑋,

πΎπΎπ‘Œπ‘Œ is the amount of capital employed in the production of π‘Œπ‘Œ,

𝐿𝐿𝑋𝑋 is the amount of labor employed in the production of 𝑋𝑋,

πΏπΏπ‘Œπ‘Œ is the amount of labor employed in the production of π‘Œπ‘Œ,

π‘₯π‘₯ is the quantity of good 𝑋𝑋 produced,

𝑦𝑦 is the quantity of good π‘Œπ‘Œ produced,

π‘₯π‘₯𝐴𝐴 is the quantity of good 𝑋𝑋 consumed by citizen 𝐴𝐴,

𝑦𝑦𝐴𝐴 is the quantity of good π‘Œπ‘Œ consumed by citizen 𝐴𝐴,

π‘₯π‘₯𝐡𝐡 is the quantity of good 𝑋𝑋 consumed by citizen 𝐡𝐡,

𝑦𝑦𝐡𝐡 is the quantity of good π‘Œπ‘Œ consumed by citizen 𝐡𝐡.

 

Problem 4: (1 point)

 

  1. a) (0.50 points) Define a Pareto efficient allocation.
  2. b) (0.25 points) Briefly explain the first fundamental theorem of welfare economics.
  3. c) (0.25 points) Briefly explain the second fundamental theorem of welfare economics.

 

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