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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 − 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 𝑞𝑞 is the amount of coconuts produced and 𝐴𝐴 is a measure of productivity.
- a) Compute the all the feasible allocations.
- 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 𝑤𝑤 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 𝐴𝐴 on the Walrasian Equilibrium of this economy.
Problem 3: 2x2x2 Economy
Consider an economy with two citizens ( and 𝐵𝐵), two goods (𝑋𝑋 and 𝑌𝑌) 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,
𝐾𝐾𝑌𝑌 is the amount of capital employed in the production of 𝑌𝑌,
𝐿𝐿𝑋𝑋 is the amount of labor employed in the production of 𝑋𝑋,
𝐿𝐿𝑌𝑌 is the amount of labor employed in the production of 𝑌𝑌,
𝑥𝑥 is the quantity of good 𝑋𝑋 produced,
𝑦𝑦 is the quantity of good 𝑌𝑌 produced,
𝑥𝑥𝐴𝐴 is the quantity of good 𝑋𝑋 consumed by citizen 𝐴𝐴,
𝑦𝑦𝐴𝐴 is the quantity of good 𝑌𝑌 consumed by citizen 𝐴𝐴,
𝑥𝑥𝐵𝐵 is the quantity of good 𝑋𝑋 consumed by citizen 𝐵𝐵,
𝑦𝑦𝐵𝐵 is the quantity of good 𝑌𝑌 consumed by citizen 𝐵𝐵.
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.