Arrow–Debreu State Price

In one sentence

An Arrow–Debreu state price $q(s)$ is the today-price of receiving one unit of consumption (the numeraire) in a particular future state $s$.

How state prices work

Suppose there is uncertainty about tomorrow, with states $s \in S$. In a complete-markets Arrow–Debreu world, you can trade claims that pay off in each state.

An Arrow security for state $s$ pays 1 unit if and only if state $s$ occurs.
The state price $q(s)$ is the competitive price of that Arrow security (for the chosen numeraire).

Intuition: $q(s)$ is high when the state is likely and/or when an extra unit of consumption in that state is very valuable (high marginal utility).

Pricing any payoff with state prices

If an asset pays $x(s)$ units of the numeraire in state $s$ tomorrow, its price today is:

\[ P ;=; \sum_{s \in S} q(s), x(s). \]

This is the discrete-state version of “price equals discounted, risk-adjusted expected payoff”.

    flowchart TD
	  S["States tomorrow<br/>s ∈ S"] --> Q["State prices<br/>q(s)"]
	  X["Asset payoff by state<br/>x(s)"] --> PX["Price today<br/>P = Σ q(s) x(s)"]
	  Q --> PX

Link to probabilities and discounting (high level)

In many textbook setups, you can write $q(s)$ as “probability × discount × marginal-utility adjustment.” One common representation is:

\[ q(s) = \pi(s), m(s), \]

where $\pi(s)$ is the probability of state $s$ and $m(s)$ is a (state-dependent) stochastic discount factor that is larger in “bad times.”

Why economists use state prices

State prices are useful because they:

connect general equilibrium and finance (Arrow securities, contingent claims),
make welfare statements under uncertainty more transparent,
provide a clean benchmark for what “complete markets” would imply.

Arrow Security: A contingent claim that pays 1 unit in exactly one state and 0 otherwise.
State-Contingent Commodity: A good indexed by time and state (e.g., “one apple delivered tomorrow if state $s$ happens”).
Stochastic Discount Factor (SDF): A random variable $m$ such that $P = E[m x]$ for payoffs $x$.
Complete Markets: A setting where all relevant risks can be traded/insured via state-contingent claims.
General Equilibrium: A model where all markets clear simultaneously via prices.

Quiz

### What is a state price $q(s)$? - [x] The price today of receiving 1 unit of the numeraire if state $s$ occurs - [ ] The probability that state $s$ occurs - [ ] The interest rate in state $s$ - [ ] The price of a good today with certainty > **Explanation:** It is a *contingent* price: “pay 1 only in state $s$”. ### In discrete states, the price of an asset with payoff $x(s)$ can be written as: - [x] $P = \sum_s q(s)\,x(s)$ - [ ] $P = \sum_s \pi(s)$ - [ ] $P = x(s)$ for the most likely $s$ - [ ] $P = \max_s x(s)$ > **Explanation:** State prices weight each state’s payoff. ### State prices are most directly linked to which complete-markets instrument? - [x] Arrow securities (state-contingent claims) - [ ] Futures margin requirements - [ ] A central bank policy rate - [ ] Payroll taxes > **Explanation:** Arrow securities are the canonical state-contingent claims. ### Holding everything else fixed, a state that is “worse” (higher marginal utility of consumption) tends to have: - [x] A higher state price - [ ] A lower state price - [ ] The same state price - [ ] No state price by definition > **Explanation:** Payoffs in bad states are more valuable, so they command higher prices. ### True or False: State prices exist for every state in an Arrow–Debreu *complete markets* benchmark. - [x] True - [ ] False > **Explanation:** Complete markets mean every relevant contingency can be priced and traded. ### If an asset pays 1 unit in every state (a risk-free payoff of 1), its price is: - [x] $\sum_s q(s)$ - [ ] $\max_s q(s)$ - [ ] $\sum_s \pi(s)$ - [ ] Always 1 by definition > **Explanation:** Set $x(s)=1$ for all $s$ in $P=\sum_s q(s)x(s)$. ### A high state price is most consistent with a state that is: - [x] Valuable to insure (high marginal utility) and/or relatively likely - [ ] Always impossible to occur - [ ] Guaranteed to be “good times” for everyone - [ ] Irrelevant for welfare and trade > **Explanation:** “Bad states” with high marginal utility tend to have higher state prices. ### State prices are closely related to which object in modern asset pricing? - [x] The stochastic discount factor (pricing kernel) - [ ] The Consumer Price Index (CPI) - [ ] The unemployment rate - [ ] The money multiplier > **Explanation:** State prices can be represented as probability-weighted components of the SDF. ### True or False: Knowing all state prices allows you to price any payoff $x(s)$ by linearity. - [x] True - [ ] False > **Explanation:** In complete markets, any payoff can be decomposed into Arrow securities. ### Which statement is closest to the economic interpretation of $q(s)$? - [x] Today’s exchange rate between “a unit of consumption today” and “a unit delivered in state $s$ tomorrow” - [ ] Tomorrow’s realized inflation rate in state $s$ - [ ] The legal probability of state $s$ - [ ] A tax rate set by the government > **Explanation:** State prices are intertemporal/state-contingent relative prices.