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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:

(ππ,π§π§)=(ππ)1/2(24 β π§π§)1/2

where π is the amount of coconuts he consumes and π§π§ is the time he expends working in a day (there 24 hours available per day). The production function of coconuts is =(π§π§)1/2 , where ππ is the amount of coconuts produced and π΄π΄ is a measure of productivity.

- a) Compute the all the feasible allocations.
- b) Compute the excess demand functions of coconuts and labor and verify Walrasβs Law. In order to do so, denote by ππ the price of coconuts and by π€π€ the wage rate.
- c) Compute the Walrasian Equilibrium of this economy. DO NOT FORGET to compute equilibrium quantities.
- d) Compute the Pareto efficient allocations.
- e) Show that the Walrasian Equilibrium allocation is Pareto efficient.
- f) Comparative Statics: Compute the effect of an increase in π΄π΄ on the Walrasian Equilibrium of this economy.

Problem 3: 2x2x2 Economy

Consider an economy with two citizens (Β andΒ π΅π΅), two goods (ππΒ andΒ ππ) and two factors of production (πΎπΎ and πΏπΏ). We will use the following notation:

πΎπΎππ is the amount of capital employed in the production of ππ,

πΎπΎππ is the amount of capital employed in the production of ππ,

πΏπΏππ is the amount of labor employed in the production of ππ,

πΏπΏππ is the amount of labor employed in the production of ππ,

π₯π₯ is the quantity of good ππ produced,

π¦π¦ is the quantity of good ππ produced,

π₯π₯π΄π΄ is the quantity of good ππ consumed by citizen π΄π΄,

π¦π¦π΄π΄ is the quantity of good ππ consumed by citizen π΄π΄,

π₯π₯π΅π΅ is the quantity of good ππ consumed by citizen π΅π΅,

π¦π¦π΅π΅ is the quantity of good ππ consumed by citizen π΅π΅.

Problem 4: (1 point)

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