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Consider the Arrow-Debreu economy withNstates. Suppose that there are two agents in the economy,j= 1, 2. Let the incomes of the two agents be as

Consider the Arrow-Debreu economy withNstates. Suppose that there are two agents in the economy,j= 1, 2. Let the incomes of the two agents be as follows:

y(1)= (z11+h1 ,...,zN1+hN)

y(2)= (z11-h1 ,...,zN1-hN)

where bothzandhare variables that may vary across states. Assume that zi>0 for alli, butE[h] = 0. Furthermore, suppose that both agents' utility function isu(c) =ln(c).

  1. For any given Arrow-Debreu pricesp1,...,pN, solve for the two agents's optimal consumption in each state.
  2. What is the correlation between the two agent's consumptions?
  3. Use the market-clearing condition to solve for the pricepias a function ofziandi.
  4. Under what condition will the individuals' consumptions be constant across states? In that case, what are the risk-adjusted probabilities (i.e.,qi=pi/(p1+...+pN) , i= 1,...,N )?

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