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 ,...,z_N¹+h_N)

y⁽²⁾= (z₁¹-h₁ ,...,z_N¹-h_N)

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

1. For any given Arrow-Debreu pricesp₁,...,p_N, 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 pricep_ias a function ofz_iand_i.
4. Under what condition will the individuals' consumptions be constant across states? In that case, what are the risk-adjusted probabilities (i.e.,q_i=p_i/(p₁+...+p_N) ,i= 1,...,N )?