Motivation
We want a unified framework to answer two questions in a stochastic economy:
- How do agents price any asset (bonds, stocks, firms)?
- What objective function should a firm maximize?
The key insight: under complete markets, Arrow-Debreu prices pin down a unique stochastic discount factor (SDF), which prices every asset via a single formula. In a representative-agent economy, completeness imposes no additional restrictions on allocations (zero net supply clears trivially), but it gives us clean pricing machinery.
Stochastic Environment
Primitives. Time is discrete, $t = 0, 1, 2, \ldots$ A random variable $s_t$ (the “state”) summarizes all exogenous shocks at date $t$. The initial state $s_0$ is known.
Histories. A history of shocks through date $t$ is the tuple
$$s^t = (s_0, s_1, \ldots, s_t).$$We write $s^{t+1} \geq s^t$ to mean that $s^t$ is a prefix of $s^{t+1}$ (i.e., the first $t+1$ entries of $s^{t+1}$ coincide with $s^t$).
Probabilities. Let $\Pr(s^t)$ denote the unconditional probability of history $s^t$. Conditional probabilities are
$$\Pr(s^{t+1} | s^t) = \frac{\Pr(s^{t+1})}{\Pr(s^t)} \quad \text{if } s^{t+1} \geq s^t, \qquad \Pr(s^{t+1} | s^t) = 0 \quad \text{otherwise}.$$Complete Markets Assumption
Assumption (Complete Markets). There exist enough securities (in zero net supply) to span every possible realization of shocks.
This means: for any contingent payoff profile across states, there exists a portfolio of traded securities that replicates it exactly.
Notations:
Write as $e_i(s_t)$ the endowment of consumer $i$ when current shock is $s_t$, which is exogenous (aka fixed given $s_t$). Write as $c_{i,t}(s^t)$ the consumption of $i$ in history $s^t$ — though it’s technically a choice variable, it is pinned down given $s^t$ and price. Further:
| Symbol | Description | Determined by |
|---|---|---|
| $e_i(s_t)$ | Endowment of consumer $i$ when current shock is $s_t$ | Fixed (exogenous) |
| $c_{i,t}(s^t)$ | Consumption of agent $i$ in history $s^t$ | Choice variable |
| $B_i^{AD}(s^t)$ | Agent $i$’s holding of Arrow-Debreu security for history $s^t$ | Choice variable |
| $B_{i,t}(s^{t+1})$ | Agent $i$’s holding of Arrow security for $s^{t+1}$, purchased in $s^t$ | Choice variable |
| $Q^{AD}(s^t)$ | Price of an Arrow-Debreu security paying $1 in $s^t$ | Equilibrium object |
| $Q_t(s^{t+1} | s^t)$ | Price of an Arrow security paying $1 in $s^{t+1}$, traded in $s^t$ |
Two Market Structures
We consider two equivalent formulations of complete markets. They differ in when securities are traded, not in the set of achievable allocations.
Arrow-Debreu (A-D) Securities
Definition. An Arrow-Debreu security $B^{AD}(s^t)$ is a claim that pays $1 if and only if history $s^t$ is realized, and $0 otherwise. One such security exists for every possible history $s^t$, $t \geq 1$.
Trading protocol: All A-D securities are traded once, at date 0, at prices $Q^{AD}(s^t)$. After the portfolio is chosen, no further trading occurs; consumption is determined by endowments plus security payoffs.
Consumer $i$’s problem (A-D):
$$ \begin{aligned} & \max_{\{c_{i,t}(s^t)\}} \quad \sum_{t=0}^{\infty} \sum_{s^t} \beta^t \Pr(s^t)\, U_i\!\bigl(c_{i,t}(s^t)\bigr)\\ & \text{s.t.} \quad c_{i, t}(s^t) \le B^{AD}_i(s^t) + e_i(s_t), \forall t\ge 0, s^t.\\ & \qquad \sum_{t=1}^{\infty} \sum_{s^t} Q^{AD}(s^t)B^{AD}_i(s^t) \leq 0. \end{aligned} $$We would normalize $Q^{AD}(s^0) = 1$.
Market clearing (zero net supply):
$$ \sum_{i=1}^{I} B_{i,t}^{AD}(s^t) = 0 \quad \text{for all } s^t. $$Arrow Securities
Definition. An Arrow security $B_t(s^{t+1})$ is a claim, traded in history $s^t$, that pays $1 if and only if tomorrow’s history is $s^{t+1} \geq s^t$, and $0 otherwise. One such security exists for every successor history.
Consumer $i$’s problem (Arrow):
$$ \begin{aligned} & \max_{\{c_{i,t}(s^t),\, B_{i,t}(s^{t+1})\}} \quad \sum_{t=0}^{\infty} \sum_{s^t} \beta^t \Pr(s^t)\, U_i\!\bigl(c_{i,t}(s^t)\bigr)\\ & \text{s.t.} \quad c_{i,t}(s^t) + \sum_{s^{t+1} \geq s^t} Q_t(s^{t+1}|s^t)\, B_{i,t}(s^{t+1}) \leq e_i(s_t) + B_{i,t-1}(s^t) \quad \forall s^t. \end{aligned} $$Market clearing:
$$\sum_{i=1}^{I} B_{i,t-1}(s^t) = 0 \quad \text{for all } s^t.$$Equilibrium Prices
The FOC of the A-D problem (with normalization $Q^{AD}(s^0)=1$) gives, for any consumer $i$:
$$Q^{AD}(s^t) = \beta^t \Pr(s^t) \frac{U_{i,c}(s^t)}{U_{i,c}(s_0)}.$$Note here $U_{i,c}(s^t) \equiv U_i'\!\bigl(c_{i,t}(s^t)\bigr)$. We assume $c_{i, t}(s^t)$ is fixed conditional on $s^t$ realized, which can be obtained from the FOC.
Arrow Prices
The Euler equation from the sequential Arrow problem gives:
$$Q_t(s^{t+1}|s^t) = \beta\, \Pr(s^{t+1}|s^t) \frac{U_{c,i}(s^{t+1})}{U_{c,i}(s^t)} \quad \text{for all } i.$$Equivalence
The A-D price of any history decomposes as the product of one-step Arrow prices along that history:
$$Q^{AD}(s^t) = Q_0(s^1|s^0) \times Q_1(s^2|s^1) \times \cdots \times Q_{t-1}(s^t|s^{t-1}).$$The Stochastic Discount Factor (SDF)
Definition. The stochastic discount factor between dates $t$ and $t+k$ is
$$ > M_{t,t+k}(s^{t + l}) := \frac{\beta^k\, U_c(c_{t+k}(s^{t + k}))}{U_c(c_t(s^t))}. > $$
Note it satisfies the recursion $M_{t,t+k} = M_{t,t+1} \cdot M_{t+1,t+k}$.
Asset Pricing
The central pricing principle under the representative agent complete market assumption: any asset’s price equals the expected discounted value of its payoffs under the SDF. Below, all expectations $\mathbb{E}_t[\cdot]$ are conditional on history $s^t$.
One-Period Risk-Free Bond
A bond purchased in $s^t$ that pays $1 in every $s^{t+1} \geq s^t$. Its price equals the sum of Arrow prices over all continuations:
$$Q_t^{rf} = \sum_{s^{t+1}\geq s^t} Q_t(s^{t+1}|s^t) = \mathbb{E}_t\!\left[\beta \frac{U_{c,t+1}}{U_{c,t}}\right] = \mathbb{E}_t[M_{t,t+1}].$$The one-period risk-free gross interest rate is $R_t^{rf} = 1 / Q_t^{rf}$.
Two-Period Risk-Free Bond
A bond purchased in $s^t$ that pays $1 in every $s^{t+2} \geq s^t$. Replicate by backward induction using Arrow securities:
$$Q_t^{rf,2} = \sum_{s^{t+2}\geq s^t} \Pr(s^{t+2}|s^t) \frac{\beta^2 U_c(s^{t+2})}{U_c(s^t)} = \mathbb{E}_t[M_{t,t+2}] = \mathbb{E}_t\!\left[\frac{\beta^2 U_{c,t+2}}{U_{c,t}}\right].$$Stock (Equity Claim on a Dividend Stream)
A firm pays stochastic dividends $\{D_t(s^t)\}$. Its ex-dividend price at $s^t$ is:
$$P_t = \mathbb{E}_t\!\left[\sum_{k=1}^{\infty} M_{t,t+k}\, D_{t+k}\right].$$Equivalently, in recursive form:
$$P_t = \mathbb{E}_t\!\bigl[M_{t,t+1}(P_{t+1} + D_{t+1})\bigr].$$Defining the gross return $R_{t+1} := (P_{t+1} + D_{t+1})/P_t$, this becomes the fundamental asset pricing equation, which must hold for any traded asset under no-arbitrage:
$$1 = \mathbb{E}_t[M_{t,t+1}\, R_{t+1}].$$Value of the Firm
Using a no-arbitrage argument, one can show that under complete markets, because all consumers value dividend streams identically using the same A-D prices, the firm’s objective is unambiguous:
$$ \begin{aligned} & \max_{\{D_t(s^t)\}} \quad \sum_{t=0}^{\infty} \sum_{s^t} Q^{AD}(s^t)\, D_t(s^t)\\ & \text{s.t. technological/resource constraints on } D. \end{aligned} $$firms maximize the present discounted value of dividends, using Arrow-Debreu prices as discount factors.