State-Contingent Prices and the Value of Future Consumption
Summary
The document explains the meaning of state-contingent prices in a two-period asset-pricing setup. A security associated with a particular future state pays one unit if that state occurs and nothing in other states; its price today is the price of that contingent claim. Multiplying this price by a state-contingent quantity gives the current value of the payoff in that state.
When future consumption or income is represented as a vector across possible states, its total present value is the sum of each state’s price multiplied by the corresponding amount. The budget constraint therefore equates the value of current and future consumption to the value of current and future income. The explanation relies on the simplified setup in which tomorrow resolves into one of a known set of states. It clarifies the notation and valuation logic, but does not develop equilibrium pricing, uncertainty preferences, or how the state prices are determined.
Key ideas
- A state-contingent claim pays in one specified future state and has a price today.
- The price of a payoff amount in one state is the state price multiplied by that amount.
- The present value of a state-contingent income or consumption vector is the sum of state prices times state amounts.
- The budget constraint equates the total value of consumption with the total value of income.
- The explanation assumes a two-period setup with a known set of possible future states.
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Full text
# What is the meaning of multiplying price of contingent claim with e.g consumption level?
# What is the meaning of multiplying price of contingent claim with e.g consumption level?
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?
## Answer by Magic is in the chain (score 2, accepted)
https://quant.stackexchange.com/a/49396
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.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.