   |
|
||||
| |
|
|
|
|
|
A Stochastic Optimal Control Formulation of the Consumption/Debt Decision Octavio Ramirez, Charles B. Moss, and William G. Boggess Octavio Ramirez is a professor and head of the Economics and Sociology of Production and Conservation Area at CATIE; Charles Moss is an associate professor in the Food and Resource Economics Department at the University of Florida; and William Boggess is a professor and head of the Department of Agricultural and Resource Economics at Oregon State University. The authors would like to thank the two anonymous journal referees for their helpful comments, and Kim Box for her editorial assistance. This is Florida Agricultural Experiment Station journal series no. R-05975. Abstract The risk-balancing optimal debt model proposed by Collins has been the basis of several studies into the effects of policy on debt choice in agriculture; however, the formulation does not explicitly model the consumption/investment tradeoff facing the farm firm. This study reformulates the optimal debt model using stochastic optimal control to directly address the consumption/investment tradeoff. Key words: debt, investment/consumption decisions. Article <top> Collins’ 1985 formulation of the risk-balancing debt choice framework has led other researchers to conduct studies into factors affecting agricultural debt levels. Featherstone, Moss, Baker, and Preckel examined the implications of traditional commodity programs on optimal debt, and Moss, Ford, and Boggess derived the implications of changes in tax code on debt levels. Empirically, verification of the original Collins formulation was provided at the aggregate level by Moss, Shonkwiler, and Ford. Firm-level evidence supporting Collins’ formulation was reported by Ahrendsen, Collender, and Dixon. Given the above studies based on Collins’ formulation, we conclude that the risk-balancing model of debt has been useful in the analysis of a variety of policy questions faced by agriculture during the last decade. The model’s appeal stems from its simplicity and the intuitive reasonableness of its results; however, much of the appeal and simplicity of Collins’ model results from the collapsing of the decision-making process into a single period. This simplification has two important limitations. First, the model’s results are myopic in the sense that the optimal leverage decision is not conditioned on the future possible time paths of the exogenous variables (for example, cost of capital and rate of return on assets). Second, since Collins’ framework subsumes the consumption/investment decisions into the objective function, the static model provides little insight into the dynamic nature of the consumption/investment tradeoff. The consumption/investment decision in the dynamic model is central to the decision facing the farm firm. A stochastic optimal control formulation, which explicitly models the consumption/ debt tradeoff faced by decision makers, is developed in this study. The following section records the development of a dynamic theoretical framework based on the assumption that decision makers maximize a discounted stream of expected utility by simultaneously making consumption/investment and leverage decisions in each time period in response to exogenous changes in the cost of capital and the moments of the rate of return on assets. A numerical example of the optimal consumption/debt formulation is then presented. A Theoretical Model of Farm Debt and Consumption Decisions <top> As noted above, one of the most powerful aspects of Collins’ model is its simplicity and the intuitive appeal of its results. In extending the model to more fully capture the tradeoffs implied by debt decisions, it is necessary to sacrifice some of the basic simplicity of the model; however, the intuition of the more complicated model should agree with the implications of the simple risk-balancing model. The development of a stochastic control formulation of the optimal debt problem based on a power utility function is recorded below. Given this basic relationship, we then present an empirical model that is used to estimate the movement of debt, equity, and consumption over time. As with Collins’ model, the derivation begins with the rate of return on equity, RE, which is defined as
where rp is the net returns to agricultural assets, A is total agricultural assets, i is the capital gains rate, K is the cost of debt capital, and δ is the leverage rate. The second equality in equation (1) reformulates the general equality, collapsing the operating returns to agricultural assets and the capital gains into a single term, RA, which denotes the total returns to agricultural assets. Given the rate of return to equity defined in equation (1), the change in equity at time t can be defined as
where E(t) is farm equity and C(t) is the level of farm consumption. Recognizing the stochastic nature of RA(t) and K(t), equation (2) can be rewritten as
where μA(t) is the anticipated or expected rate of return to assets, σA(t) is the standard deviation of RA(t), and dz(t) is the increment of a stochastic process that obeys Brownian motion. Equation (3) can be represented as
where ΔE is the change in equity, μ is the expected change in equity over time, Δt is the time increment, σ is the standard deviation of the change in equity, and Δz is a Wiener process. A Brownian motion process has three important characteristics. First, the current value of the process contains all the relevant information necessary to predict future values; this is referred to as the Markov property. Second, the Brownian motion increments are independent across time. And third, the variance of a Brownian motion process is a linear function of the time increment. Of these properties, equation (3) satisfies the Markov property by construction. Similarly, the independence of Δz can be imposed by assumption. Thus, the only remaining point is the demonstration of the linear relationship between variance and the time increment Δt. Following Dixit and Pindyck, the Wiener increment is defined as
where εt is defined as a standard normal random variable. The Wiener process then has an expected value of zero and a variance of Δt. The expectation and variance of equation (4), given the results of equation (5), yield:
and (6)
where Exp[×] denotes the expectation operator. The expected change in equity over time is then the mean rate of return on agricultural equity net of consumption, or
while the variance of the change in equity over time is a linear function of increment in time,
Given the movement of equity over time, the optimal debt and consumption paths can be derived under the assumption of constant relative risk aversion. The entrepreneur is assumed to maximize the expected present value of the utility of consumption discounted at rate r,
subject to the equation of motion (3) and an initial level of equity E(0) = E0, where b is a parameter that incorporates the Arrow-Pratt relative risk-aversion coefficient for the power utility function.1 1The use of this formulation of the utility function is based on previous work by Merton within the stochastic control formulation. In general, the power utility function is written as
where r is the Arrow-Pratt relative risk-aversion coefficient. After this reformulation, we see that the Arrow-Pratt relative risk-aversion coefficient for the power utility function formulation in equation (9) is -b +1. The stochastic optimal control problem defined in equations (3) and (9) is composed of three types of variables: (a) the exogenous path variable, (b) the control variable, and (c) the state variable. The exogenous path variable cannot be controlled by the decision maker. In the current analysis there are two exogenous path variables. First, the cost of capital varies over time but along a known path (that is, the decision maker knows the cost of capital at each point in time). The second exogenous path variable, the rate of return on assets, varies over time according to a known distribution. Specifically, the decision maker does not know the exact realization of the rate of return on assets, but knows how the mean and the variance of the rate of return on assets change over time. The control variable, the second type of variable used to define stochastic optimal control problems, can be varied by the farmer at each instant in time. In this analysis the control variables are the leverage position (i.e., the debt-to-asset ratio) and the level of farm consumption.2 2The process of specifying the debt-to-asset position as a control variable has its pitfalls. In reality, changing the debt position of a farm may be costly or impossible due to fixities within the credit system. First, a farmer who wishes to expand the use of debt may be required to pay financing fees. Second, lending restrictions (such as maximum debt-to-asset ratios) may be used by the lending institutions to limit risk to the financial institutions. In addition, the investments themselves may tend to be lumpy. For example, farmland in many states is typically purchased by the quarter-section. The final class of variable is the state variable, which cannot be directly controlled but can be influenced by the level of the control variables. The choice of these variables is subject to the realizations of the exogenous path variables. In this formulation the endogenous state variables are the equity and asset levels. Solving the optimal control problem as specified by equations (3) and (9), using traditional techniques described in Kamien and Schwartz, yields the following optimal paths for the control variables (refer to the Appendix for a detailed presentation of the specific derivation process):
and (10)
where
D is the fraction of equity used for consumption in each period. The optimal leverage result in equation (10) is consistent with the optimal leverage conditions derived by Collins. Specifically, optimal leverage increases with an increase in expected return on assets over the cost of capital, decreases with an increase in the variance of agricultural returns, and declines with increased risk aversion. The solution
in equations (10) and (11) also provides insights into changes in consumption.
Differentiating A Numerical Example <top> To demonstrate the intuition behind the model and how the optimal choices change in response to changes in the model’s parameters, we offer a numerical example. The parameterization of the model is based on risk-aversion parameter b, which is determined by the individual’s preference and three exogenous values (the expected return on agricultural assets, the variance of those returns, and the riskless discount rate). Ramirez provides an initial estimate of the relative risk-aversion coefficient using aggregate data. Specifically, he estimates the risk-aversion parameter by applying the generalized method of moments (GMM) to the debt-to-asset and consumption representation in equation (10). Hansen proposed the GMM estimator for cases in which some of the explanatory variables are either unobservable or measured with error. The full information, multiple-equation estimation yields an estimated b for domestic agriculture of -1.57. The asymptotic standard error of the estimate for b under the GMM estimation procedure is 0.44.
For comparison, an implicit estimate of b can be obtained by substituting the average rate of return on agricultural assets, standard deviation of return on agricultural assets, cost of capital, and the average debt-to-asset ratio observed for aggregate domestic agriculture. The average rate of return on agricultural assets, including capital gains, for 197294 was 11.88%, with a standard deviation of 9.01%.3 The average cost of capital, derived by dividing total interest paid by the total level of debt, was 8.87%. The average debt-to-asset ratio during this time period was calculated to be 17.19%, which would imply a b coefficient in equation (9) of -2.0704. This coefficient is well within the .95 level of confidence estimated by Ramirez. 3The average rate of return and standard deviation on the rate of return on agricultural assets and the cost of debt for the 197294 period were computed from U.S. Department of Agriculture data. The rate of return on agricultural assets was computed as net farm income, adding back interest and net rent paid to landlords divided by agricultural assets excluding the farm household. The capital gains rate was approximated by the change in real estate values divided by the average investment in real estate. One disadvantage of using aggregate data to parameterize the debt model is that aggregation tends to reduce the volatility of return. Intuitively, a national volatility tends to remove regional volatility from the data. Hence, the risk-aversion coefficient that is imputed using aggregate data will be biased toward increased risk aversion. Using Ramirez’s estimated risk-aversion coefficient, the elasticities of the debt-to-asset ratio, level of consumption, and probability of equity loss are presented in Table 1. The results show that an increase in the expected rate of return on agricultural assets implies an increase in the optimum debt-to-asset position. In addition, an increase in the variability of the rate of return on agricultural assets is associated with a decline in the optimum debt-to-asset position. As reported in Table 1, the debt-to-asset ratio is fairly elastic with respect to changes in either the expected rate of return on agricultural assets or the variance of that rate of return. However, this elasticity must be viewed within the stylized context of the model and the absence of financial fixities encountered by farm firms. The magnitude of either elasticity appears to be fairly sensitive to the relative risk-aversion coefficient. For example, the elasticity of the optimum debt-to-asset ratio with respect to a change in the variance amplifies from -2.2533 to -4.8173 when the risk-aversion coefficient is changed from Ramirez’s estimate of -1.57 to the sample estimate of -2.07. To parameterize consumption and the elasticity of consumption, the average expected value of inflation over the sample period from the Livingston survey was used to discount consumption. Specifically, the Federal Reserve Bank of Philadelphia surveys public and private economists on their expectations of the CPI one year into the future. This measure of inflation was used by Alston to estimate the effect of anticipated inflation on farmland values. The average expectation of inflation over the sample period was 2.62%. Given this approximation, the optimum rate of consumption of farm equity is 7.76%, assuming Ramirez’s risk-aversion coefficient, and 8.06% at the sample implied estimate. The estimated elasticities indicate that an increase in the expected rate of return on agricultural assets increases the rate of consumption, while an increase in the volatility of asset returns causes the rate of consumption to decline. Further, consumption appears to be elastic with respect to changes in expected return and inelastic with respect to changes in the variance of expected returns. In addition to changes in leverage and consumption, this formulation also can be used to examine the effect of changes in the expected rate of return on agricultural assets and variance of the rate of return on the probability of equity loss. Following the procedure discussed in Moss, Ford, and Boggess, the probability of equity loss under Ramirez’s risk-aversion coefficient is .1548. The elasticity of this figure indicates that the probability of equity loss increases with an increase in the expected return on agricultural assets and declines with an increase in the variability of agricultural returns.4 For the stated risk-aversion coefficient, this result implies that the increased leverage resulting from the increased rate of return more than offsets the increases in profitability. 4Moss, Ford, and Boggess develop the probability of equity loss as a measurement of the riskiness of a particular debt-to-asset position. To analyze factors that cause this riskiness to change, they note that the distribution of the rate of return to equity can be transformed into a standard normal distribution as
The probability
of equity loss is then proportional to E[Φ]. They then differentiate
E[Φ] with respect to tax code parameters. In this study,
we prefer to leave the probability of equity loss unnormalized:
where Departing from the single estimate of risk aversion based on aggregate data, Table 2 depicts the effect of risk aversion on optimal leverage, consumption, and the probability of equity loss. As expected, reductions in the relative risk-aversion coefficient are associated with increased use of debt. In addition, consistent with the results of Moss, Ford, and Boggess, these increased levels of leverage imply increased probabilities of equity loss. Table 2 also shows that consumption is an increasing function of risk aversion. The latter result is not obtainable from the static formulation. The increasing rate of consumption suggests that the decision maker consumes a higher fraction of equity in each period, implying that a smaller percentage of income is being reinvested. To further develop this aspect of the model, we divide the expectation of equation (3) by the level of equity at time t, and substitute the optimal debt and consumption relationships from equation (10), yielding:
where D is defined in equation (11). This change in equity is the rate at which income is reinvested.
Using the original risk-aversion coefficient from Ramirez, we find that the expected return on equity is .125, implying a net reinvestment rate of .047, or that 37.9% of the operating income is reinvested. The last three rows of Table 2 present the expected return on equity, the net reinvestment rate, and the percentage of income reinvestment. The findings suggest that the net reinvestment increases as risk aversion declines; however, this increase has two effects. First, the rate of consumption declines as risk aversion increases. Second, the expected return on equity increases as risk aversion declines. Conclusions and Implications <top> Using stochastic optimal control, a formulation of the consumption/ investment decision of the farm firm is developed in this study. The formulation has two advantages over the traditional static leverage formulation proposed by Collins. First, the stochastic optimal control formulation allows for an explicit consumption/investment tradeoff model. This tradeoff is subsumed into the objective function in Collins’ risk-balancing formulation. Second, this formulation is not myopic in the sense that it considers the entire path of the exogenous variables. These additional implications, with regard to consumption/investment decisions at the firm level, may be particularly important in determining the response of farm firms to changes in agricultural policies. A numerical example—based on expected returns on agricultural assets, the variance of returns on agricultural assets, the cost of capital, and the implied risk-aversion coefficient—is provided. The results indicate that the debt-to-asset ratio is elastic with respect to changes in the expected rate of return on assets and the variance of the rate of return on assets; however, both consumption and the probability of equity loss are elastic with respect to changes in the expected rate of returns to agricultural assets, but inelastic with respect to changes in the variance of returns to agricultural assets. These results suggest that optimal debt is more responsive to changes in volatility than to consumption or the probability of equity loss. In addition, the results indicate that the rate of reinvestment declines as risk aversion increases. This rate decline can be attributed to two factors. First, as the risk aversion declines, the return on equity declines, and the debt-to-asset ratio falls. Second, as risk aversion increases, the relative rate of consumption increases. These two effects are compounded by the percentage of reinvestment dropping from 73.0% to 34.2% as the relative risk aversion increases from .25 to 2.25. While formulation offers direct advantages at the firm level through explicit incorporation of the consumption/ investment decision, the theoretical model—which is built upon Collins’ original framework—provides an empirical approximation for the examination of aggregate debt response. For example, an empirical counterpart to the optimal debt and consumption model could be used to extend the results of Moss, Shonkwiler, and Ford; however, two caveats involving the effect of risk on financial decision must be recognized. First, aggregation of farm-level returns to aggregate returns implies a reduction in variance. This reduction suggests a downward bias in the estimation of the relative risk-aversion coefficient. (In aggregate, the debt-to-asset level represents the average leverage position of all firms in the sector while the aggregate risk reduces by aggregation.) The second problem involves the aggregation of individual attitudes. Further research into the potential bias of these two distortions is warranted. The empirical applications of this framework, beginning with the reexamination of previously published results, are numerous. For example, Featherstone et al.’s examination of the effect of government policy on optimal debt could be expanded to also examine changes in the firm’s reinvestment rate. Another possibility would be to apply the stochastic control formulation to changes in taxation as Moss, Ford, and Boggess did. Statistical applications of the new formulation may be challenged by the lack of data on consumption, but integration of the consumption/investment decision should improve the empirical results. References <top>
Appendix: Derivation of Optimal Debt and Consumption Paths <top> The derivation of the optimal debt and consumption path directly follows the discussion of stochastic optimal control in Kamien and Schwartz. In addition, the development of the optimal debt and consumption paths in this study can be viewed as a reformulation of Merton’s two-asset version of the lifetime portfolio problem. Starting with equations (3) and (4) from the text, we define the value of the optimum consumption path as described in Kamien and Schwartz:
s.t.:
The dynamic programming approximation to this problem is then written as:
The second term in the expectation can be approximated with a second-order Taylor series expansion as:
Applying Ito’s lemma and substituting the stochastic process yields:
Subtracting
Dividing
through by Δt, and moving
Finally,
replacing
To simplify
the solution of equation (A7), we substitute
The respective
optimum solutions for
and (A9)
Substituting these solutions into (A8), rearranging terms, and simplifying yields the following nonlinear second-order differential equation:
Following
Merton, one possible solution to the general differential form is
Since B
proves not to be a function of
where D
is the fraction of equity consumed. The optimal paths presented in our
study can be obtained by substituting the appropriate derivatives into
|
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
AEM Home 
Site Map
Contact Us Cornell
© 2002
Cornell University |
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
(1)
(2)
(3)
(4)
(5)
(7)
(8)
(9)
(11)
defined
in equation (10) with respect to the mean return on assets in period
t indicates that the effect of μA(t)
on consumption is dependent on the relative risk-aversion coefficient.
If 0 < b < 1, then DD/DμA(t)
< 0, assuming the expected rate of return to agricultural assets
is greater than the cost of capital. In this scenario, consumption declines
as the mean rate of return on agricultural assets increases. Conversely,
if b < 0, then DD/DμA(t)
> 0, implying that the rate of consumption increases as the mean
return on assets increases. Similarly, if 0 < b < 1, then
consumption increases with an increase in the variance, and if b
is less than zero, then the consumption declines with an increase in
variance.
is the change in probability, or the probability distribution function
of
We define the elasticity
of the probability of equity loss by differentiating 
with respect to the expected rate of return on agricultural assets and
the variance of the rate of return on agricultural assets, and dividing
each respective differential by
(12)
(A1)
(A2)
(A3)
(A4)
from both sides of the
expectation then yields:
(A5)
to the right-hand side of the equality yields:
(A7)
yielding:
and 
are then:
(A10)
where B is some positive function of the problem’s parameters.
Computing
and 
substituting the
results in the differential equation, and simplifying yields:
(A11)
is indeed a solution for the nonlinear differential equation given above.
While this derivation is consistent with Merton, we reformulate the
results in (A11) to focus on the consumption decision— specifically,
(A12)