Chapter 13
Independence of Production and Technology Risks
Unpublished memorandum, 2013.
The economies investigated in Vasicek (2005, 2013) (Chapters 11 and 12 of this volume) contain a production process whose rate of return 
on investment is
where 
is a Wiener process. The process 
represents a constant return-to-scale production opportunity. The amount of investment in production is determined endogenously.
The parameters of the production process can themselves be stochastic, reflecting the fact that production technology evolves in an unpredictable manner. It is assumed that their behavior is driven by a Markov state variable 
. The dynamics of the state variable, which can be interpreted as representing the state of the production technology, is given by
where 
is a Wiener process independent of 
. The parameters 
, and ϕ are functions of 
and t. The stochastic part of 
will be called the production risk, and the stochastic part of 
will be called the technology risk.
In many applications, it is realistic to assume that the technology risk is independent of the production risk—that is, 
. For instance, if the production is farming, progress in development of new agricultural methods, hybrids, fertilizers, and so on is independent of weather. When the two risks are independent, some special cases attain.
Case 1. Suppose all investors have the same degree of risk tolerance, 
. Then the short rate of interest in the economy in equilibrium is
This is a consequence of the equation for λ in Example 1 of Vasicek (2005) with 
. Eq. (3) holds for any specification of the processes 
, and for any investors' consumption time preferences.
Case 2. Suppose that the time preference functions of all participants are concentrated at the point T. In other words, each participant maximizes the expected utility of end-of-period wealth. Let 
be the state price density process (in Vasicek (2005), results are stated in terms of the so-called numeraire portfolio 
. The equilibrium value of the short rate is given by
where
is the average coefficient of risk tolerance, weighted by the end-of-period wealth levels.
To derive Eq. (4), note that
where N0 = W(0)/A(0) and
The state price density process given by
is a function of 
, and t. Put
and write 
. The process 
is a martingale,
Expand the left-hand side of (10) by Ito's lemma to obtain a partial differential equation for R in the variables 
. Since the coefficients of that equation are all independent of L, differentiating both sides with respect to L produces the same equation for the derivative 
. Consequently, 
and therefore also 
are martingales and
At T,
The individual end-of-period wealths are given by
and Eq. (12) can be written in the form
From the coefficients of 
in the expansion of 
(or from Eq. (48) of Vasicek (2005) with 
), it follows that
and therefore
Eq. (4) follows from the relationship 
and from Eqs. (16), (14).
In terms of the underlying economic variables, Eq. (4) can be written as
References
- Vasicek, O. A. (2005). “Economics of Interest Rates.” Journal of Financial Economics, 76, 293–307.
- Vasicek, O. A. (2013). “General Equilibrium with Heterogeneous Participants and Discrete Consumption Times.” Journal of Financial Economics, 108, 608–614.