In the problem below, u is a utility function; $\beta$ is a discount factor; pc(s) is the price for a contingent claim for state s. c is initial consumption and and y is initial wealth. s represents a specific state.
$$max \ u(c) + \Sigma_s\beta\pi(s)u[c(s)]$$ $$ s.t. \ c + \Sigma_s pc(s) c(s) = y + \Sigma_s pc(s)y(s)$$

Can anyone help me understand the budget constraint?


So in an Arrow-Debreu market, there is a price for every asset/claim for every state of the world. The states $s$ refer to this.

For your utility maximization problem, you are maximizing a combination of what you consume, $u(c)$, and the expected profit from your portfolio of claims, which can then be put into future consumption.

$\pi(s)$ describes the profit in state $s$, and that is multiplied by the utility of consumption in that state $s$. In this case, I guess the maximizer does not seem to worry about variance, so maximizing sum of those possible outcomes in all states of the world suffices. Finally, $\beta$ represents the fact that you prefer to consume immediately instead of investing and waiting to see the state of the world.

So the budget constraint can be roughly described as such: whatever you decide to consume now plus the cost of buying claims that will give you money to consume later must equal whatever money you start with plus whatever money you get from buying those cool claims.

  • $\begingroup$ So i guess the right handside of the budget constraint means "money you start with plus whatever money you get from buying those cool claims", but isnt the payoff of contingent claims guaranteed to be one? What does y(s) represent? $\endgroup$ – user1559897 May 15 '16 at 13:10
  • $\begingroup$ y(s) I think means future wealth given state s $\endgroup$ – Kitsune Cavalry May 15 '16 at 16:37
  • $\begingroup$ What does mean to multiple y(s)with PC(s) then? $\endgroup$ – user1559897 May 15 '16 at 16:39
  • $\begingroup$ $pc(s)$ could be the price you sell your claims back for and/or payoff of the claims. I'm actually not sure why you would multiply it by $y(s)$. Does the problem give you any other information specifically about the form of $y(s)$? What does the vector describe exactly? $\endgroup$ – Kitsune Cavalry May 15 '16 at 23:19
  • 2
    $\begingroup$ @KitsuneCavalry it is your contingent wealth. That is you will have $y(s)$ units of wealth tomorrow in state $s$. This you can sell to someone today at the price $pc(s)$, hence the total value of your future wealth in $y(s)$ is the product of the two. $\endgroup$ – Giskard Jun 14 '16 at 8:49

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