The pricing kernel

In our previous discussion we noted that the state prices depend on the likelihood of the states, and on the attitude of the agents towards those states. The next step is to try and decompose the state prices in the following way

$$q_{k}=\pi _{k}m_{k}$$.

$\pi _{k}$ is the probability of $\omega _{1}=k$, while $m_{k}$ is the price that you would pay to enjoy £1 tomorrow, knowing that tomorrow's state is going to be $\omega _{1}=k$. It is very important to note that although the agent defines $m_{k}$ as if she knew $\omega _{1}$, it will not be equal to the price of the bond. This is because different states imply different effects on her utility. We call the collection of $%% \left\{ m_{k}\right\}$ the pricing kernel. We can now rewrite the bond and asset price equations as

$$b_{0} = \sum_{k}\pi _{k}m_{k}=\mathbf{E}\left[ m\right]$$

$$c_{0} = \sum_{k}\pi _{k}m_{k}c_{k}=\mathbf{E}\left[ mc\right]$$

The above two equations have very strong connections with the general equilibrium asset pricing theory. We might as well recall that

$$b_{0} = \mathbf{E}\left[ \mathcal{M}_{1}\right]$$

$$c_{0} = \mathbf{E}\left[ \mathcal{M}_{1}c\right]$$

where $\mathcal{M}_{1}=\frac{d\mathcal{U}_{1}/dC}{d\mathcal{U}_{0}/dC}$ is the marginal rate of substitution. Therefore we observe that the pricing kernel is proportional to the marginal rate of substitution. It is then clear how the preferences define the pricing kernel, translating the state changes into changes in the consumption of the representative agent.