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Unpublished memorandum, 2013.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 c13-math-0001 on investment is

1 equation

where c13-math-0003 is a Wiener process. The process c13-math-0004 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 c13-math-0005. The dynamics of the state variable, which can be interpreted as representing the state of the production technology, is given by

2 equation

where c13-math-0007 is a Wiener process independent of c13-math-0008. The parameters c13-math-0009, and ϕ are functions of c13-math-0010 and t. The stochastic part of c13-math-0011 will be called the production risk, and the stochastic part of c13-math-0012 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, c13-math-0013. 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, c13-math-0014. Then the short rate of interest in the economy in equilibrium is

3 equation

This is a consequence of the equation for λ in Example 1 of Vasicek (2005) with c13-math-0016. Eq. (3) holds for any specification of the processes c13-math-0017, 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 c13-math-0018 be the state price density process (in Vasicek (2005), results are stated in terms of the so-called numeraire portfolio c13-math-0019. The equilibrium value of the short rate is given by

4 equation

where

5 equation

is the average coefficient of risk tolerance, weighted by the end-of-period wealth levels.

To derive Eq. (4), note that

6 equation

where N0 = W(0)/A(0) and

7 equation

The state price density process given by

8 equation

is a function of c13-math-0025, and t. Put

9 equation

and write c13-math-0027. The process c13-math-0028 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 c13-math-0030. 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 c13-math-0031. Consequently, c13-math-0032 and therefore also c13-math-0033 are martingales and

11 equation

At T,

The individual end-of-period wealths are given by

13 equation

and Eq. (12) can be written in the form

From the coefficients of c13-math-0038 in the expansion of c13-math-0039 (or from Eq. (48) of Vasicek (2005) with c13-math-0040), it follows that

15 equation

and therefore

Eq. (4) follows from the relationship c13-math-0043 and from Eqs. (16), (14).

In terms of the underlying economic variables, Eq. (4) can be written as

17 equation

References

  1. Vasicek, O. A. (2005). “Economics of Interest Rates.” Journal of Financial Economics, 76, 293–307.
  2. Vasicek, O. A. (2013). “General Equilibrium with Heterogeneous Participants and Discrete Consumption Times.” Journal of Financial Economics, 108, 608–614.
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