Summary
The European Union is increasingly relying on direct payments to support farm incomes. Recent research has shown that a direct payment may increase production and investment by risk-averse farmers via a link between wealth, risk aversion and decision making. This paper shows that, even in the absence of risk aversion, a direct payment may stimulate farm investment. With lenders using a standard insolvency rule for determining bankruptcy, the direct payment raises the expected value of marginal investment because it reduces the risk of bankruptcy over the farmer's operating time horizon. The investment response to the direct payment is larger for a farmer with an intermediate versus low or high level of equity, and for a farmer with a long versus short-time horizon.
1. Introduction
Following the 2003 Luxembourg Agreement, most direct payments within the European Union's (EU) Common Agricultural Policy (CAP) are now combined in a direct single farm payment (Binfield et al., 2004; Roche and McQuinn, 2004; Sckokai and Anton, 2005). Similar policy shifts towards direct payments based on historical allocations rather than current enterprise decisions and crop yields have occurred in the US and other developed countries (Adams et al., 2001; Orden, 2005). The EU's single farm payment is decoupled from (i.e. not directly linked to) current production. This feature is important because a payment without any production impact qualifies for WTO ‘green box’ status, which means that the amount of payment is not subject to a WTO binding.
Even though a direct payment based on a historical reference is generally classified as decoupled from production, it is widely recognised that such a payment, especially if provided countercyclically to farm income, can have an impact on farm production and thus on trade flows. Most importantly, a direct payment raises wealth and possibly reduces risk, both of which will induce a risk-averse farmer to increase production (Hennessy, 1998; OECD, 2001; Anton and Le Mouel, 2004). If the direct payment is fixed over time (as opposed to countercyclically), the widely accepted assumption of decreasing absolute risk aversion is sufficient to ensure that the marginal impact of the payment on farm production is positive and a decreasing function of the farmer's wealth (Hennessy, 1998). Other links between a direct payment and farm production include a binding credit constraint and the anticipation by the farmer that future payments will eventually be based on the current level of production (OECD, 2001).
In this paper, a simple stochastic dynamic programming model is used to examine the link between a direct payment and farm investment. Understanding the investment response to a direct payment is important because an increase in investment typically results in higher farm production in both the short and long run. The analysis assumes that a profit-maximising farmer simultaneously receives a direct payment and makes a one-shot investment decision at date 0 with the aim of maximising the present value of expected year T farm equity. The farm is declared bankrupt if farm equity erodes to zero before date T, even though expected date T equity is positive when bankruptcy is declared. It is this risk of insolvency-driven bankruptcy that creates a link between the direct payment and farm investment. Since most real-world farmers face some risk that their assets will be seized and their farm operations will be terminated because farm equity has eroded to zero, it follows from this analysis that even in the absence of risk aversion, a direct payment and farm investment are typically linked.1
A second finding of this paper is that although the probability of bankruptcy is higher with a lower level of farm equity, it is not the case that the investment response to a direct payment is largest for a low-equity farmer. The analysis shows that the investment response to a direct payment is comparatively small for either a low or high-equity farmer and is comparatively large for a medium-equity farmer. This result is important given that a majority of real-world farmers are likely to fall into a medium-equity category. A third finding of this paper is that the investment response to a direct payment is stronger the longer the farmer's time horizon. Although the issue of time horizon for a farmer is complicated because farms are often owned by the same family for successive generations, this latter result does suggest that, all else equal, a younger farmer is likely to have a greater response to a direct payment than an older farmer. The theoretical finding that medium-equity farmers with comparatively long time horizons are expected to have the biggest investment response to a direct payment lends itself nicely to future empirical analysis.
In the following section, a brief review of the relevant literature on capital investment is provided. Section 3 introduces the assumptions of the model, a first-best investment benchmark is created and the stochastic dynamic programming problem is set up for the farmer when lenders employ a bankruptcy rule. Section 4 presents formal results for the two-period case, which lead to an explicit description of the linkage between the direct payment and farm investment. A generalisation of the results to more than two periods is presented in Section 5. Section 6 provides concluding comments.
2. Relevant literature on capital investment
Much of the literature on capital investment is not specific to agriculture. An older literature shows that with convex capital adjustment costs and a sequence of investment decisions, greater price uncertainty induces more investment by a risk-neutral firm (e.g. Hartman, 1972; Abel, 1983). Zeira (1990) showed that this linkage may be reversed if the firm is risk averse. More recently, it has been shown that because of a real option, investment irreversibility will induce a firm to delay the investment decision, and the delay is longer (although under some conditions it is shorter) with a higher level of price variability (e.g. McDonald and Siegel, 1986; Dixit and Pindyck, 1994). These models of capital investment would be useful for analysing the investment response to a direct payment if the payment is known to change price variability.
With reference to bankruptcy, Mahul (2000) used a two-stage model to show how the risk of asset liquidation due to financial insolvency induces a risk-neutral firm to under-invest in a productive activity. Milne and Robertson (1996) and Holt (2003) showed that the threat of bankruptcy induces a profit-maximising firm to hold a relatively high level of cash reserves, which results in a smaller stream of dividend payments and flow of funds into investment. Finally, Vercammen (2000) showed that an implicit aversion towards bankruptcy creates a real option, and this option induces a profit-maximising firm to delay its investment decision. These studies, which demonstrate how bankruptcy risk distorts a risk-neutral firm's decision making, provide a starting point for analysing the linkage between a direct payment and farm investment.
More relevant for the current analysis is the literature that deals with finance, investment and the cost of capital (see Fama and Miller, 1972, Chapter 4, and Nickell, 1978, Chapter 8 for a good overview). These models, which endogenise the financial structure of the investing firm (i.e. optimal debt-to-equity ratio), are grounded in Modigliani–Miller's well-known theorem (see Fama and Miller, 1972: 150–180, for a description), which states that with efficient markets and no transaction costs, the value of the firm and the incentive to invest are not dependent on the firm's financial structure. When limited liability and bankruptcy are considered, the independence between the firm's choice of investment and financial structure breaks down. Firms with lower debt face a lower effective cost of capital, and as a result invest relatively more aggressively.2
The linkage between a direct payment and farm investment has been subjected to very little empirical verification. Building on the risk aversion models of direct payments and production, Sckokai and Moro (2005) theoretically established a positive link between farm investment and a direct payment. They went on to use specialised arable crop data from the Italian Farm Accounting Data Network to provide empirical support of their conjecture. The results of other empirical analyses indirectly support the direct payment–farm investment linkage by establishing that, all else equal, higher debt farmers tend to be less productive (Lee and Chambers, 1986; Weersink et al., 1990; Whittaker and Morehart, 1991). If more productive farmers invest more aggressively, then the direct payment will increase farm investment by raising farm productivity. In somewhat related research, Roche and McQuinn (2004) used portfolio theory to show that the risk-reducing properties of a direct payment will induce a farmer to shift to a riskier crop mix. Lagerkvist (2005) used a stochastic dynamic programming model to examine how policy reform uncertainty affects farmers' land investment decisions and the price of farmland.
3. Model set-up
3.1. Basic assumptions
A representative risk-neutral farmer will operate for T years, unless bankruptcy forces the farmer to stop operating earlier. The farmer's objective is to maximise expected terminal equity, which is discounted farm equity at the end of year T.3 At the end of year 0, the farmer receives a one-off direct payment of size S, which is assumed to be independent of all farm attributes (e.g. acreage, commodity mix, input use, etc.). This payment is used to reduce outstanding farm debt from D0 to D0 − S. The farmer also has a one-off, irreversible investment opportunity at the end of year 0, the full amount of which must be financed with new debt. Specifically, the farmer must choose the level of investment, I, such that the farm's capital stock increases from k0 to k0 + I at the end of year 0 and then remains at level k0 + I until year T or bankruptcy occurs. The unit price of capital is assumed constant over time and is equal to P. In addition to the direct cost of investment, IP, the farmer incurs an external capital adjustment cost c(I) where c′(I) > 0 and c″(I) > 0.4 If the investment in question is farmland, then c(I) may reflect the farmer's upfront cost of upgrading machinery, buildings and human capital.
Let WT−i (DT−i, I) denote expected terminal equity at the end of year T − i for a farmer who is operating with DT−i units of debt and k0 + I units of capital. Expected terminal equity at the time of the investment decision and receipt of the direct payment (i.e. year 0) is given by W0 (D0, I), where D0 = D0 − S + IP + c(I) is post-investment debt at the end of year 0. The farmer chooses I to maximise W0 (D0, I). Of particular interest in this analysis is how the size of the direct payment, S, affects the farmer's optimal choice of I.
At the end of year t ∈ {1,2, … , T}, the constant-returns-to-scale farmer earns stochastic net revenue Rt ∈ ( − ∞, + ∞) per unit of farm capital and then uses the full amount of this revenue to pay down debt. Each year, net unit revenue is independently drawn from a probability density function f(R), which has mean R̄ and a corresponding distribution function F(R). In addition to having the standard properties, it is assumed that f′(R) > 0 for R ∈ ( − ∞, R̄), and f (R)/F(R) is a decreasing function for R ∈ ( − ∞, R̄).5 We assume that R̄ > r0P, where r0 is the lender's opportunity cost of capital. The R̄ > r0P restriction implies that the expected present value of the investment and the expected surplus earned on existing farm capital are positive with the riskless rate r0.
The lending sector is competitive, and the equilibrium rate of interest on farm debt is set such that a lender expects to earn zero profits when a loan is provided to a farmer. A lending contract, which is assumed to be renegotiated on a year-to-year basis and which is secured with the farmer's capital, does not specify a maximum payment but does specify a minimum payment equal to the accrued interest on outstanding debt. If the farmer is unable to make the minimum payment because of low farm revenue, then new debt is issued to cover the payment shortfall.6 Following Stiglitz (1972), if the issue of new debt will result in negative equity for the farmer, then because of the short-term nature of the lending contract and limited liability for the farmer, the lender must write off the excess debt (hereafter referred to as ‘bad debt’). After writing off bad debt, the lender must choose one of two options. First, the lender can allow the farmer to continue operating the following year as a zero-equity client. Second, the lender can force the farm into bankruptcy by costlessly seizing and selling the farm's capital assets. Independent of which option is chosen, at the beginning of each year the lender sets the interest rate on outstanding debt, rt, at some level above the competitive rate, r0, to compensate for the possibility of writing off bad debt.
In the appendix, it is shown that when DT−1 ≤ (k0 + I)P, then the R̄ > r0P assumption ensures that a solution for equation (3) exists. This condition makes sense because the lender sets rt−1 to earn zero expected surplus, and R̄ > r0P implies positive expected surplus for farm capital. Two comparative static results for r*t−1 are also derived and presented in appendix A.1. Specifically, it is shown that r*t−1 is an increasing function of farm debt, Dt−1, and the level of investment, I. Both these findings are expected because an increase in both Dt−1 and I increase the probability of bankruptcy.
3.2. First-best outcome
Here, we create a first-best benchmark by assuming that a lender extends a zero-equity loan to a zero-equity farmer rather than forcing that farmer into bankruptcy. If the lender operates in a competitive environment with full information and no transaction costs, then this no-bankruptcy strategy is an expected equilibrium outcome, and it will lead to the first-best market outcome. Indeed, competition among lenders will ensure that bankruptcy is never declared because, given the assumption that R̄ > r0P, a lender can always find a sufficiently large value for rt such that a non-negative return is expected from a zero-equity farmer [this result follows from the proof in appendix A.1 that equation (3) has a solution even when Dt = (k0 + I)P].
Equation (6) can be used to establish the following result (the proof of all formal results are contained in the appendix).
Result 1
In a first-best environment where lenders never force a zero-equity farmer into bankruptcy, the optimal level of farm investment is independent of farm debt and the direct payment.
Result 1 is consistent with the Modigliani–Miller theorem. In an efficient capital market with perfect information and no taxes or transaction costs, the value of the farm and the farmer's incentive to invest are independent of the farm's debt level and thus of the probability that bad debt will be written off by the lender. The value of the farm and the optimal level of investment in this first-best world are the same as in a riskless world where each year the farmer earns net unit revenue, R̄, with certainty. In this first-best scenario, a direct payment cannot have an impact on the farmer's investment decision because the payment lowers farm debt but does not change the expected surplus of the investment.10
3.3. Bankruptcy considerations
In the previous section, it was assumed that a lender always provides a loan to a zero-equity farmer because even with previous bad debt and current zero equity, the rate of interest can be raised to a sufficiently high level to allow the lender to break even on the loan. In reality, farmers with zero equity and a history of bad debt are often forced into bankruptcy, even if the farm capital is expected to generate a positive surplus.11 Indeed, Miller (1962) demonstrated how bankruptcy costs can result in credit rationing by forward-looking lenders. Similarly, Kim (1978) showed that the optimal debt capacity set by a lender is typically less than 100 per cent of the value of the borrower's assets. Costly bankruptcy generally implies that lenders may deny a loan to a zero-equity, bad-debt farmer, even if continuation of the farm operation is socially efficient (see Stiglitz, 1972, and Bulow and Shoven, 1978, for more discussion).
We assume here that a lender will maintain the lending relationship with the farmer if current equity is positive (i.e. bad debt was not written off the previous year), and will force the farmer into bankruptcy if current equity is zero because bad debt was written off the previous year. The lending constraint given by equation (3), which defines the rate of interest for a particular level of debt and capital stock, remains the same as in the first-best case, except that the constraint is valid now only when the farm in question has positive equity. The farmer's value function also remains the same as in the first-best case except that the last term in equation (2), which is expected terminal equity when beginning from a zero-equity position in year t + 1, is replaced by 0 to reflect the bankruptcy rule employed by the lender.
4. Theoretical results for the two-period case
In appendix A.4, several important properties of the R*T−1 function given by equation (10) are established, including: (i) R*T−1 < R̄, (ii) dR*T−1 /dI > 0 for (k0 + I)P > D0 and dR*T−1/dI → 0 for (k0 + I)P → D0 and (iii) dR*T−1/dD0 > 0 for (k0 + I)P > D0. In words, the unit net revenue that triggers bankruptcy is less than R̄, is an increasing function of both I and D0 with positive initial equity and is independent of I with zero initial equity. The intuition behind these results is that higher values of I and D0 imply higher leverage, and higher leverage generally raises minimum revenue requirements to avoid bankruptcy. The one exception is when the farmer has no equity, in which case additional investment does not change the farmer's equity position and R*T−1 no longer changes with I.
In equation (11), c′(I) is the marginal external cost of the investment and the remaining terms sum to the expected marginal increase in terminal equity due to the investment, accounting for the probability of bankruptcy after one year of operation.
The following result can now be established.
Result 2
Regardless of the level of initial equity, Γ(D0, S) > 0, which implies that a farmer invests less than the first-best level if the lender uses a bankruptcy rule.
Result 2 can be explained as follows. Equation (12) shows that the bankruptcy rule reduces the expected marginal value of the investment for two reasons. First, with probability F(R*T−1), the surplus from marginal investment, δ2 (R̄ − r0P), may be lost due to bankruptcy. Second, the added investment raises the probability of bankruptcy by an amount dF(R*T−1)/dI, and when bankruptcy does occur, the surplus from all the farm's capital, (k0 + I)δ2 (R̄ − r0P), is lost. This twofold reduction in the expected marginal value of the investment due to the bankruptcy rule, as measured by Γ(D0, S), induces the farmer to invest at a level that is less than first-best.
The next result establishes how the marginal impact of investment on the probability of bankruptcy depends on the farmer's initial equity.
Result 3
The marginal impact of investment on the probability of bankruptcy, dF(R*T−1)/dI, is positive but it approaches zero as initial farm equity, (k0 + I)P − D0, approaches either zero or an arbitrarily high level.
Result 3 is used in the following analysis to help characterise the linkage between the direct payment and farm investment. Result 3 establishes the intuitive result that a high-equity farmer has a negligible probability of bankruptcy, and so additional investment has a negligible impact on the probability of bankruptcy. In addition, a farmer with initial equity close to zero has a relatively high probability of bankruptcy, but additional investment has a negligible impact on his probability of bankruptcy. This latter result emerges because, with a negligible level of equity, the minimum level of net unit revenue required to avoid bankruptcy, R*T−1, changes very little as investment increase, since the equity position remains approximately equal to zero. The most important implication of Result 3 is that additional investment has the greatest impact on the probability of bankruptcy for a farmer with a ‘medium’ equity level.
The main result of this paper can now be established.
Result 4
The marginal impact of the direct payment on farm investment is positive, with maximum impact typically occurring for a farmer with a ‘medium’ level of equity.
Equation (12) can be used to explain Result 4. Consider the first bracketed term, F(R*T−1). It was previously established that the minimum net unit revenue that triggers bankruptcy, R*T−1, and thus the probability of bankruptcy, F(R*T−1), are increasing functions of initial farm debt, D0. Moreover, F(R*T−1) takes on a value near zero for very low levels of farm debt (i.e. high equity). Now consider the second bracketed term, (k0 + I)(dF(R*T−1)/dI), which can be interpreted as the marginal impact of investment on the probability of bankruptcy. Result 3 established that this function begins near zero, grows to a positive value and then shrinks to near zero as farm debt changes from a low to a high level or, equivalently, farm equity changes from a high to a low level. These two individual components of equation (12), together with their sum which represents the net effect of D0 on Γ(D0, S), are illustrated in Figure 1. The particular shape of the Γ(D0, S) function (top schedule) is central to this analysis.
Result 4 is stated in terms of the marginal impact of the direct payment, S, on the farmer's incentive to invest. Initial debt is D0 = D0 − S + I + c(I), and Γ(D0, S), which is given by equation (12), is inversely related to the farmer's incentive to invest (see proof of Result 2). It therefore follows that the slope of the Γ(D0, S) function in Figure 1 can be interpreted as the marginal impact of the direct payment on the farmer's incentive to invest. In Figure 1, using this relationship, it is shown that the marginal impact of the direct payment on investment is initially weak, strengthens and then once again weakens as farm equity decreases from a high level to zero. Consequently, the impact of the direct payment on investment is largest for a farmer with a medium level of equity. The direct payment effect weakens as equity declines from a medium to a low level because the marginal impact of the investment on the probability of bankruptcy weakens as equity declines, and so the direct payment has a diminishing stimulating effect.
It is important to note that underlying Result 4 is the assumption that the model's parameter values are unaffected by changes in the direct payment.13 In reality, one would expect a direct payment to be partially capitalised into the price of the asset. Capitalisation implies that P will increase with S, in which case the linkage between the direct payment and investment will be overstated in the current analysis. Similarly, the direct payment may be provided as compensation for the removal of other programmes, which raise expected net unit revenue, R̄, and reduce revenue variability. In this model, a simultaneous increase in S and decrease in R̄ will have an ambiguous impact on optimal investment because a reduction in R̄ implies a smaller incentive to invest for the farmer. Similarly, a simultaneous increase in both S and revenue variability will have an ambiguous impact on optimal investment because rising revenue variability increases the probability of bankruptcy, which in turn reduces the farmer's incentive to invest.
5. Longer time horizon and numerical example
5.1. Longer time horizon
Given the structural equivalence of the general T-year model and the specific two-year model, the results of the two-year model easily generalise to the T-year case. This is because all relevant impacts of a longer time horizon can be analysed by comparing the probability of bankruptcy variables, F(R*T−2) and Fi*(R*T−i). The following result can therefore be established:
Result 5
For a given level of farm equity, the marginal impact of the direct payment on the farmer's choice of investment is larger when the farmer has a longer time horizon.
Result 5 is expected because a longer time horizon implies a higher cumulative probability of bankruptcy, and it is the probability of bankruptcy that creates the linkage between the direct payment and investment. For a high-equity farmer, the cumulative probability of bankruptcy and the marginal impact of investment on the probability of bankruptcy are both relatively small and thus relatively unresponsive to an increasing length of time horizon. For a low-equity farmer, the cumulative probability of bankruptcy is relatively high, even with a short time horizon, and it does not increase substantially as the time horizon lengthens. The investment response to a direct payment therefore increases most quickly with a lengthening time horizon if the farmer has a medium level of equity. This result is not formally established, but it appears to follow from the analysis mentioned earlier.
5.2. Numerical example
We assume that R is a normally distributed random variable with mean R̄ = 1.1 and standard deviation σ = 2.14 Also suppose that P = 10 and r0 = 0.1, which implies that R̄ − r0P = 0.1 (the expected net rate of surplus from the investment, R̄/P − r0, is therefore 1 per cent). Finally, assume that k0 = 1 and T ∈ {2,3}. Using equation (9) and equation (A7) from the appendix, the date 0 expected marginal value of the investment can be calculated for the T = 2 and T = 3 bankruptcy case by subtracting expected terminal equity with I = 0 from expected terminal expected equity with I = 0.1. Now repeat this procedure with equation (4) to obtain a measure of the date 0 expected marginal value of investment for the T = 2 and T = 3 first-best case. The difference between the date 0 expected marginal value of investment with and without the bankruptcy rule provides an estimate of Γ(D0, S).
Figure 2 shows the estimated loss in the expected marginal value of the investment due to the bankruptcy rule, Γ(D0, S), as a function of the farmer's pre-investment debt-to-asset ratio for both T = 2 and T = 3.15 As expected, because of the increased risk of bankruptcy, Γ(D0, S) is larger and thus the optimal level of investment is lower with a higher level of initial debt. Moreover, Γ(D0, S) is larger for the longer time horizon. Notice that the marginal impact of a direct payment on the farmer's choice of investment, as measured by the slope of the schedules in Figure 2, is largest when the farmer has an intermediate level of equity and when the time horizon is longer rather than shorter.
6. Conclusions
The purpose of this paper was to construct a theoretical model of farm investment in order to examine the impact of a direct payment on the farmer's choice of investment. The three main results of this analysis can be described as follows. First, if lenders use a standard bankruptcy rule, a direct payment is expected to induce higher investment by a farmer even if the farmer is assumed to be risk neutral rather than risk averse. Second, the investment response to a direct payment is comparatively small for a farmer with either a high or low level of equity and is comparatively large for a farmer with a medium level of equity. Third, all else equal, the investment response is larger the longer the farmer's time horizon.
The first result emerges because the risk of bankruptcy reduces the farmer's expected value of marginal investment by implicitly raising the farmer's expected cost of capital. A direct payment lowers the risk of bankruptcy, and this reduction in risk induces the farmer to invest more aggressively. The bankruptcy rule used in this analysis is the standard ‘seize assets if equity erodes to zero’ rule that is used by most lending institutions. This rule is motivated by costs of bankruptcy, which have not been explicitly included in the model. The bankruptcy rule reduces the farmer's expected value of marginal investment, and hence creates a link between the direct payment and investment, because in the absence of bankruptcy, farm equity is expected to grow over time. It is interesting to note that in the current model, a direct payment raises investment towards the first-best level, unlike a traditional price support, which may raise investment above the first-best level.
The analysis presented here is general in many respects, but it is also based on a number of simplifying assumptions, several of which have already been discussed. In most cases, relaxing a particular assumption will result in a largely predictable weakening or strengthening of the direct payment impact on investment. In general, the fundamental linkage between the risk of bankruptcy, a direct payment and farm investment appears to be robust and it has strong intuitive appeal. Finally, this paper should prove useful as a guide to empirical analysis. Specifically, empirical models should be designed to allow for a maximum direct payment impact when income variability is high, the farmer's debt-to-asset ratio takes on an intermediate value and the farmer's time horizon is relatively long.
Appendices
A.1. Proof that a solution for equation (3) exists, and comparative static analysis for rt−1*
By differentiating with respect to rt−1, it is easy to verify that the left-hand side of equation (A1) is a strictly decreasing, positive-valued function of rt−1, ranging from a high of ∞ as rt−1 → − ∞ to a low of 0 as rt−1 → ∞. Because the right-hand side of equation (A1) is positive by assumption, it follows from the mean value theorem that a value for rt−1 exists which solves equation (3).
Equations (A2) and (A3) can be signed positive after substituting equation (3) into the respective numerators. (Call this result Lemma A1).
A.2. Derivation of the WT−i FB(DT−i, l) function given by equation (5)
A.3. Derivation of WT−i (DT−i, I) function given by equation (8)
A.4. Properties of the R *T−1 function given by equation (10)
Proof of Result 2
Set equation (6) with T = 2 equal to equation (11) and then cancel terms to obtain c′ (IFB) = c′ (I*) + Γ (D0, S). Noting that R̄ > r0P, the c″ (I*) > 0 assumption implies that I* < IFBif Γ (D0, S) > 0 and that the gap between I*and IFB increases with a larger value for Γ (D0, S). From equation (12), Γ(D0, S) > 0 with (k0 + I)P > D0 because R̄ > r0P by assumption, F(R*T−1) > 0 and dF (R*T−1)/dI = f (R*T−1)(dR*T−1/dI) > 0. Noting that f′ (R) > 0 for R < R̄, these latter inequalities follow from Lemmas A2 and A3. Q.E.D.
Proof of Result 3
With (k0 + I)P > D0, dF(R*T−1)/ dI = f (R*T−1)(dR*T−1/dI) > 0 from the proof for Result 2. Result 3 holds if f(R*T−1) → 0 as (k0 + I)P − D0 → ∞ and if dR*T−1/dI → 0 as (k0 + I)P − D0 → 0. From equation (10), (k0 + I)P − D0 → ∞ implies R*T−1 → − ∞, which in turn implies f(R*T−1) → 0 because f′ (R) > 0 for R < R̄. Lemma A4 established that dR*T−1/dI → 0 as (k0 + I)P → D0 Q.E.D.
Proof of Result 4
From dr*T−2/dI > 0 and dr*T−2/dS = − dr*T−2/dD0 < 0 (Lemma A1), it follows that equation (A11) takes on a negative value, which in turn implies dΓ/dS < 0.
The proof of the second part of equation (4) is available from the author upon request.
Acknowledgements
The author would like to thank the editor and three anonymous referees for many helpful comments and suggestions. Additional helpful comments were obtained from participants at an OECD workshop in Paris in May 2003. The OECD supported this project financially.
The effects of risk aversion and bankruptcy on the farmer's investment response to a direct payment are likely to be larger if these two features exist simultaneously rather than separately. The additive effects of risk aversion and bankruptcy are not considered in this paper.
The model developed in this paper is quite different from the general finance models of Fama and Miller (1972) and Nickell (1978). Indeed, in this model, new investment is fully financed with debt (since the farmer holds all the equity), whereas in the general models, the firm issues both equity and debt to finance new investment. In addition, an explicit solution to the T-period stochastic dynamic programming problem and comparative static results are formally derived in this paper but not in the general papers.
A more general model would allow the farmer to maximise the expected discounted utility of the consumption stream between year 1 and year T, plus expected terminal equity. Allowing for adjustments in consumption would reduce the farmer's susceptibility to bankruptcy, and thus would be likely to weaken the proposed linkage between the direct payment and investment.
The farmer operates with constant returns to scale (see what follows) and so the c″(I) > 0 assumption is necessary to ensure a finite optimal level of investment in the absence of bankruptcy risk. A capital adjustment cost function is a common feature of investment models (Nickell, 1978).
The normal distribution satisfies both these properties. Many of the results can be established with much weaker restrictions on f(R).
The model allows for R < 0, in which case the farmer's outstanding loan balance is increased, first, to make up the net revenue shortfall (i.e. cover non-interest farm expense) and second, to cover the outstanding interest payment.
The higher order moments of F(R), which do not explicitly appear in the analysis, primarily affect investment incentives through F(Rt *). For example, a higher variance for R will typically imply a higher value for F(Rt *) when F(Rt *) < R̄, as can be seen for the case where R is normally distributed, i.e. ∂F(Rt *)/∂σ > 0 when F(Rt *) = (2πσ2)−1/2 ∫ −∞Rt\,*e−0.5((R−R̄)/σ)2dR.
Equation (3), together with the R̄ > r0P assumption, ensures that Wt+1FB((k0 + I)P,I) > 0.
For notational convenience, salvage value is the sum of the year T value of the farm's assets and net farm revenue in year T.
The farmer receives an implicit benefit if bad debt is written off by the lender. This benefit is exactly offset by the higher borrowing cost, which results from the lender raising the lending rate above r0 to account for the expected cost of writing off bad debt. Hence, the value of the farm is independent of the level of debt carried by the farmer.
Real-world farms often escape bankruptcy but must undergo significant downsizing and financial restructuring when insolvency is reached. The results presented in this paper are likely to remain valid in a more general model that allows for financial restructuring rather than full-blown bankruptcy, provided that the restructuring generates an uncompensated loss for the farmer.
The DT−j argument within the RT−i*(DT−j, RT−i−1, RT−i−2, … , RT−j+1) function is suppressed for notational convenience.
The credit goes to an anonymous referee who pointed out this limitation of the analysis.
This example exploits the following property of the normal distribution (Johnson and Kotz, 1970): ∫ −∞R *Rf(R)dR = R̄Φ(R *) − σϕ(R *), where Φ(·) and ϕ(·) are, respectively, the cumulative distribution function and the probability density function for a standard normal random variable.
The T = 3 schedule fluctuates because of small approximation errors in the numerical integration method and the use of discrete versus continuous values for D0.
