please help to finish the problem set

1 General equilibrium with two-person exchange Consider an economy populated by two consumers indexed by 𝑖 = {1,2}. Let 𝜔𝑖 = (𝑥1 𝑖 , 𝑥2 𝑖 ) denote person 𝑖’s initial allocation (endowments) of goods 𝑥1 and 𝑥2. Assume 𝜔1 = (13,1) and 𝜔2 = (1,13). Each individual has an identical utility function given by 𝒰𝑖(𝑥1 𝑖 , 𝑥2 𝑖 ) = 2 ln 𝑥1 𝑖 + 2 ln 𝑥2 𝑖

(a) Draw an Edgeworth box for this economy. Mark a point that corresponds to the initial allocation.

(b) Provide a brief verbal definition of Pareto efficiency. Argue that this definition is formally equivalent to the equality of the agents’ marginal rates of substitution.

(c) Determine whether the initial allocation is Pareto efficient.

(d) Provide a brief definition of the contract curve. Without any formal derivation, illustrate how the contract curve would be derived in the context of an Edgeworth box.

(e) What are the implications with regard to economic efficiency of any competitive equilibria that might be reached in the context of this economy?

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