We will analyze a representative agent economy and see how shocks and intertemporal discounted tradeoff drives asset prices.
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.
In other words for any contingent payoff profile across states, there exists a portfolio of traded securities that replicates it exactly. This imposes no arbitrage.
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 |
| $Q_t(s^{t+1}\mid s^t)$ | Price of an Arrow security paying 1usd in $s^{t+1}$, traded in $s^t$ | Equilibrium object |
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 $M_{t,t+k} = M_{t,t+1} \cdot M_{t+1,t+k}$.
Define return $R_{t + 1}$ of a general asset with price $P_t$ and defined divident stream $\{D_{\tau}\}$ as:
$$ R_{t + 1} = \frac{P_{t + 1} + D_{t + 1}}{P_t} $$where $P_{t + 1}, P_{t}$ are prices at time $t, t + 1$ resp and $D_{t + 1}$ is period $t + 1$ dividend.
Corollary
$$ > 1 = \mathbb E_t[M_{t, t + 1}R_{t + 1}]. > $$
Some Common Asset’s 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.