I have this problem in asset pricing that I don't know how to solve. Here it is: Consider an economy with a complete set of Securities and $N$ states of the world Tomorrow. Assume that there are two alternative set of assets: either a) there is a complete set of Arrow-Debreu Securities with price $q_{\theta}$ for the one with nonzero payoff in state $\theta$ or b) there is a complete set of complex Securities (and no A-D security) denoted by $j=1,...,N$ with price $q_j$.

Consider the equilibrium of the economy with the two different asset structures (and assume it is unique). $(...,\hat{q}_{\theta},...)$ is the equilibrium of the economy with a complete set of A-D Securities; $(...,\bar{q}_{j},...)$ is the equilibrium in the economy with a set of complete Securities, $j=1,...,N$.

Show that, for each asset $j$, $$\bar{q}_{j}=\frac{1}{1+r_f}\sum_{\theta}\hat{q}_{\theta} R_{j \theta}$$

My approach:

Since markets are complete, $R$ denotes the $N\times N$ full-ranked matrix of payoffs for the securities, with typical column $$ R_j = [R_{j1},..., R_{jN}] $$

Let $\bar{q_j} = [q_{j1},...,q_{jN}]$ be the vector of asset prices. Then, it is possible to determine a vector $\psi$ such that: $$ \bar{q_j} = \psi_{\theta} R_{\theta} $$ Where $\psi_{\theta}$ is the price of the portfolio of complex securities delivering a unit of A-D security. Therefore, $\bar{q}_j$ can be explicited as a summation, and discounting at period one with the risk free rate, yields $$ \bar{q}_{j}=\frac{1}{1+r_f}\sum_{\theta}\hat{q}_{\theta} R_{j \theta} $$

Now, I don't know if my solution is correct, because to me it seems too much naive... I hope that someone can help!



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