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In the textbook Asset Pricing by John Cochrane, on p. 57, a budget constraint of a Lagrange optimization is: $c + \Sigma_s pc(s) c(s) = y + \Sigma_s pc(s) y(s) $

$pc(s)$ is "price today of contingent claim" (p54) (I am not sure whether it is "today" in this context). $y(s)$ and $c(s)$ are respectively the state-contingent income and state-contingent consumption.

What is the meaning of $pc(s) c(s)$ and $pc(s) y(s)$? Isn't $pc(s)$ about security? Why can it be multiplied with income and consumption?

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You can assume two periods economy: calling them today and tomorrow is a convenient representation that is easy to relate to. Today is certain, tomorrow is not- the number of states is known, and the economy will be in one of these states tomorrow. A generic state is represented by s. $pc(s)$ is the today price of a security that will pay one unit if state s occurs tomorrow and zero in all other states. The price of a security that pays x if state s occurs is $pc(s) x$, and if you have a security that pays x(s) in state s, so think of x as vector now, then the price would be sum across the states $\sum_s{pc(s) x(s)}$.

The constraints you have copied is just stating that the total value of consumption must be equal to total income. Their today flows are c and y, and their tomorrow state contingent flows are multiplied by the respective state contingent prices.

Hope this helps!

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  • $\begingroup$ Thank you for your answer. Would you mind if you clarify what is the interpretation of "tomorrow state contingent flows multiplied by the respective state contingent prices"? (I understand the meaning behind $\sum_s{pc(s) x(s)}$, but why multiply $pc(s)$ (which is about security) with income and consumption?) $\endgroup$
    – Aqqqq
    Commented Oct 28, 2019 at 7:36
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    $\begingroup$ Your income and consumption are expressed in terms of ‘money’. So you can compute their present value as you would for any payoff. If you have higher income in states which are more desirable, then surely that is worth more. Multiplying by state prices would reflect that. $\endgroup$
    – Magic is in the chain
    Commented Oct 28, 2019 at 7:52

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