Editing Equity premium puzzle (section)

== Theory ==

The economy has a single representative household whose preferences over stochastic consumption paths are given by:

:<math>E_0 \left[\sum_{t=0}^\infty \beta^t U(c_t)\right]</math>

where <math display="inline">0<\beta<1</math> is the subjective discount factor, <math display="inline">c_t</math> is the per capita consumption at time <math display="inline">t</math>, U() is an increasing and concave utility function. In the Mehra and Prescott (1985) economy, the utility function belongs to the constant relative risk aversion class:

:<math>U(c, \alpha) = \frac{c^{(1-\alpha)}}{1-\alpha}</math>

where <math display="inline">0< \alpha < \infty </math> is the constant relative risk aversion parameter. Note that <math>\lim_{\alpha \to 1 } U(c,\alpha) = \ln(c)</math>. Weil (1989) replaced the constant relative risk aversion utility function with the Kreps-Porteus nonexpected utility preferences.

:<math>U_t = \left[c_t^{1-\rho}+\beta (E_t U_{t+1}^{1-\alpha})^{(1-\rho)/(1-\alpha)}\right]^{1/(1-\rho)}</math>

The Kreps-Porteus utility function has a constant intertemporal elasticity of substitution and a constant coefficient of relative risk aversion which are not required to be inversely related - a restriction imposed by the constant relative risk aversion utility function. Mehra and Prescott (1985) and Weil (1989) economies are a variations of Lucas (1978) pure exchange economy. In their economies the growth rate of the endowment process, <math display="inline">x_t</math>, follows an ergodic Markov Process.

:<math>P \left [x_{t+1} = \lambda_j | x_t = \lambda_i \right] = \phi_{i,j} </math>

where <math display="inline">x_t \in \{\lambda_1,...,\lambda_n\}</math>. This assumption is the key difference between Mehra and Prescott's economy and Lucas' economy where the level of the endowment process follows a Markov Process.

There is a single firm producing the perishable consumption good. At any given time <math display="inline">t</math>, the firm's output must be less than or equal to <math display="inline">y_{t}</math> which is stochastic and follows <math display="inline">y_{t+1}=x_{t+1} y_{t}</math>. There is only one equity share held by the representative household.

We work out the intertemporal choice problem. This leads to:

:<math>p_t U'(c_t) = \beta E_t[(p_{t+1} + y_{t+1}) U'(c_{t+1})]</math>

as the fundamental equation.

For computing stock returns

:<math>1 = \beta E_t\left[\frac{U'(c_{t+1})}{U'(c_t)} R_{e, t+1}\right]</math>

where

:<math>R_{e, t+1} = (p_{t+1} + y_{t+1}) / p_t</math>

gives the result.<ref>{{cite report |last1=Mehra |first1=Rajnish |title=The Equity Premium: Why is it a Puzzle? |date=February 2003 |doi=10.3386/w9512 |doi-access=free |website=National Bureau of Economic Research}}</ref>

The derivative of the [[Lagrange multiplier|Lagrangian]] with respect to the percentage of stock held must equal zero to satisfy necessary conditions for optimality under the assumptions of [[Rational pricing|no arbitrage]] and the [[law of one price]].