Financial conditions and supply decisions when firms are risk averse

Abstract

Extending earlier work by Greenwald and Stiglitz (Q J Econ 108:77–114, 1993) on the role of a firm’s equity position and bankruptcy costs in determining its production decision we show that, even if bankruptcy costs are ignored, a firm’s decision makers’ risk aversion, whether they are owner-entrepreneurs or hired managers, can give rise to the same results. What is more, we argue that, in the presence of risk aversion, increased variance of the output price affects a firm’s supply decision as the sum of an impact and an indirect effect. Under reasonable assumptions the impact effect prevails and then output decreases. We show this to hold for risk attitudes represented both by CARA and by CRRA utility functions. Finally, we explore the dynamics of the equity base. We provide examples in which the accumulation of net worth slows down as a consequence of an increase of risk.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    We don’t consider bankruptcy costs because, from an analytical point of view, we want to focus on risk aversion as a driving force for our results. This does not mean that we think that bankruptcy is costless. But showing the results without including in the objective function the Greenwald–Stiglitz bankruptcy cost enhances the relevance of the point we want to make for risk aversion.

  2. 2.

    As Greenwald and Stiglitz (1993, pp. 77–78) write: “...a general theory requires a portfolio approach to firm decision making”.

  3. 3.

    In the appendix of their 1988-Working Paper Greenwald and Stiglitz argue that, under the additional assumption (not made as a prerequisite in the formulation of the corresponding proposition) that “an increase in uncertainty increases the likelihood of bad events” (p. 37), the marginal cost of bankruptcy increases, and hence output decreases. Since it is not universally given that the likelihood of bad events is increased with an increase in uncertainty, the authors support their statement by hinting to the further possible assumption of an increase in the hazard rate \(f\left( {\overline{u}}\right) /\left( 1-F\left( {\overline{u}}\right) \right) \), where f is the density of the firm’s relative output price u, F the corresponding distribution and \({\overline{u}}\) the critical value of u below which the firm is bankrupt (Greenwald and Stiglitz 1988, p. 37, fn. 33). In distinction to this, in our model uncertainty will be captured by the price’s variance \(\sigma _{u}^{2}\) which will explicitly appear as a parameter in agents’ objective functions and render the reasoning simpler and more transparent.

  4. 4.

    To mention just a few representative contributions in this line of thought, see Stiglitz (2011, 2018) and Blanchard (2018).

  5. 5.

    When capital markets are affected by informational imperfections such as asymmetric information, a financing hierarchy (pecking order) can be envisaged. Internal finance is the most preferred source of finance. As to external sources, for the firm credit has a cost advantage over the issue of new equities (Fazzari et al. 1988). A different wording is used in the literature on the so-called “credit view”: bank loans are imperfect substitutes of the issue of new equities (credit is “special”). See, for instance, Bernanke and Blinder (1988).

  6. 6.

    More precisely, net worth is a pre-determined variable (the lagged level of a state variable). The law of motion of net worth will be analyzed in Sect. 6.

  7. 7.

    We follow Greenwald and Stiglitz closely in the definition of profit. The term uq is revenue (in real terms). Operating (i.e. non-bankruptcy) costs coincide with debt commitments, i.e. payments (interest and principal) due on outstanding debt \((1+r)b.\) Using the definiton of profit as the difference between revenue and costs and the definition of debt as the difference between the wage bill and internal finance, i.e. \(b=w\ell -a,\) it turns out that \(\pi =uq-(1+r)w\ell +(1+r)a.\) This definition of profit differs from the traditional one for the presence of the term \((1+r)a\), which is a negative component of cost.

  8. 8.

    Here is one of the two major differences with respect to Greenwald and Stiglitz (1993): while they subtract a bankruptcy cost \(c_{t}^{i}F\left( {\overline{u}}_{t+1}^{i}\right) \) from the firm’s profit, we don’t have that term (see Greenwald and Stiglitz 1993, equation (13), p. 88).

  9. 9.

    This is the second major difference with respect to Greenwald and Stiglitz (1993): we assume as objective function an expected utility function of profit rather than an expected profit function. Equivalently one could say that their von-Neumann Morgenstern utility function is the identity function \(V\left( \pi \right) =\pi \).

  10. 10.

    As a referee has pointed out, occasionally the price of some agricultural products (coffee) or mining products (crude oil, copper) drops by more than 50%, though this is not the case for a representative firm.

  11. 11.

    See e.g. distortion risk measures like the GlueVaR, as in Belles-Sampera et al. (2014).

References

  1. Belles-Sampera J, Guillén M, Santolino M (2014) Beyond value-at-risk: GlueVaR distortion risk measures. Risk Anal 34(1):121–134

    Article  Google Scholar 

  2. Bernanke B, Blinder A (1988) Credit, money, and aggregate demand. Am Econ Rev Pap Proc 78:435–439

    Google Scholar 

  3. Bernanke B, Gertler M (1989) Agency costs, net worth and business fluctuations. Am Econ Rev 79:14–31

    Google Scholar 

  4. Bernanke B, Gertler M (1990) Financial fragility and economic performance. Q J Econ 105:87–114

    Article  Google Scholar 

  5. Blanchard O (2018) On the future of macroeconomic models. Oxford Rev Econ Policy 34(1–2):43–54

    Article  Google Scholar 

  6. Fazzari S, Hubbard G, Petersen B (1988) Financing constraints and corporate investment. Brook Pap Econ Act 1:141–206

    Article  Google Scholar 

  7. Gale Douglas, Hellwig M (1985) Incentive-compatible debt contracts: the one-period problem. Rev Econ Stud 52:647–663

    Article  Google Scholar 

  8. Greenwald B, Stiglitz J (1988) Financial market imperfections and business cycles. NBER working paper no. 2494

  9. Greenwald B, Stiglitz J (1989) Toward a theory of rigidities. Am Econ Rev Pap Proc 79(2):364–369

    Google Scholar 

  10. Greenwald B, Stiglitz J (1993) Financial market imperfections and business cycles. Q J Econ 108:77–114

    Article  Google Scholar 

  11. Myers S, Majluf N (1984) Corporate financing and investment decisions when firms have information that investors do not have. J Financ Econ 13:187–221

    Article  Google Scholar 

  12. Nielsen CK (2015) The loan contract with costly state verification and subjective beliefs. Math Soc Sci 78:89–105

    Article  Google Scholar 

  13. Stiglitz J (2011) Rethinking macroeconomics: what failed, and how to repair it. J Eur Econ Assoc 9(4):591–645

    Article  Google Scholar 

  14. Stiglitz J (2018) Where modern macroeconomics went wrong. Oxford Rev Econ Policy 34(1–2):70–106

    Google Scholar 

  15. Townsend R (1979) Optimal contracts and competitive markets with costly state verification. J Econ Theory 21:265–293

    Article  Google Scholar 

  16. Tulli V, Weinrich G (2015) Using value-at-risk to reconcile limited liability and the moral-hazard problem. Decis Econ Finance 38:93–118

    Article  Google Scholar 

Download references

Acknowledgements

This paper has greatly benefitted from valuable and helpful comments and suggestions by Domenico Delli Gatti and two anonymous referees. The responsability for remaining shortcomings is the authors’ only.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vanda Tulli.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

A)Derivation of condition (13) We have \(\pi \ge 0\) if and only if

$$\begin{aligned} u\ge \frac{\left( 1+r\right) w\phi \left( q\right) }{q}-\frac{\left( 1+r\right) a}{q} \end{aligned}$$
(26)

and we know that, for any w and r, the firm always chooses \(q\le {\hat{q}} =\left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{(1+r)w}\right) \). Moreover, by de l’Hospital \(\lim _{q\rightarrow 0}\frac{\left( 1+r\right) w\phi \left( q\right) }{q} = \lim _{q\rightarrow 0}\left( 1+r\right) w\phi ^{\prime }\left( q\right) = 0\), while

$$\begin{aligned} \frac{\partial }{\partial q}\left( \frac{\left( 1+r\right) w\phi \left( q\right) }{q}\right) =\frac{q\left( 1+r\right) w\phi ^{\prime }\left( q\right) -\left( 1+r\right) w\phi \left( q\right) }{q^{2}} \end{aligned}$$

is positive if \(\phi ^{\prime }\left( q\right) >\phi \left( q\right) /q\) which is true due to \(\phi \left( 0\right) =0\) and strict convexity of \(\phi \). Thus \(\left( 1+r\right) w\phi \left( q\right) /q\) is strictly increasing and on \(\left[ 0,{\hat{q}}\right] \) reaches its maximum at \({\hat{q}}.\) Therefore, and since \(\left( 1+r\right) a/q\ge 0\),

$$\begin{aligned} supp\left( f\right) \sqsubseteq \left[ \frac{\left( 1+r\right) w\phi \left( \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\left( 1+r\right) w} \right) \right) }{\left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\left( 1+r\right) w}\right) },\infty \right) , \end{aligned}$$

i.e. (13), is a sufficient condition for (26) to hold for all \(u\in supp\left( f\right) \). Finally, to show

$$\begin{aligned} \frac{\left( 1+r\right) w\phi \left( \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\left( 1+r\right) w}\right) \right) }{\left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\left( 1+r\right) w}\right) }<1 \end{aligned}$$
(27)

let \(f_{1}\left( q\right) =\left( 1+r\right) w\phi \left( q\right) \) and \( f_{2}\left( q\right) =q\). Then \(f_{1}\left( 0\right) =0=f_{2}\left( 0\right) \), \(f_{1}^{\prime }\left( 0\right) =0<1=f_{2}^{\prime }\left( 0\right) \), \( f_{1}^{\prime }\left( {\hat{q}}\right) =1=f_{2}^{\prime }\left( {\hat{q}}\right) \) and \(f_{1}^{\prime }\left( q\right) <f_{2}^{\prime }\left( q\right) \) for all q s.t. \(0\le q<{\hat{q}}\). But this implies

$$\begin{aligned} f_{1}\left( {\hat{q}}\right) =\int _{0}^{{\hat{q}}}f_{1}^{\prime }\left( q\right) dq<\int _{0}^{{\hat{q}}}f_{2}^{\prime }\left( q\right) dq=f_{2}\left( {\hat{q}} \right) \end{aligned}$$

which is equivalent to (27). \(\square \)

B)\(\textit{Derivation}\, \textit{of}\, sgn\left( \partial q^{\star }/\partial \sigma _{u}^{2}\right) \textit{in}\, \textit{case}\, \textit{CRRA}\) We can calculate \(\partial q^{\star }/\partial \sigma _{u}^{2}\) as A/B, with A given by (16) and

$$\begin{aligned} B=\mu ^{\prime \prime }-\frac{1-\gamma }{2}\left\{ \phantom {\left. -\left( 2-\gamma \right) \frac{q^{\star }}{\mu }\frac{ \mu \left( \mu ^{\prime }+q^{*}\mu ^{\prime \prime }\right) -q^{*}\left( \mu ^{\prime }\right) ^{2}}{\mu ^{2}}\right\} \sigma _{u}^{2}.} \frac{\mu -q^{\star }\mu ^{\prime }}{\mu ^{2}}\left[ 2-\left( 2-\gamma \right) \frac{q^{*}}{\mu } \mu ^{\prime }\right] \right. \nonumber \\ \left. -\left( 2-\gamma \right) \frac{q^{\star }}{\mu }\frac{ \mu \left( \mu ^{\prime }+q^{*}\mu ^{\prime \prime }\right) -q^{*}\left( \mu ^{\prime }\right) ^{2}}{\mu ^{2}}\right\} \sigma _{u}^{2}. \end{aligned}$$

As already seen, A is positive for small \(\sigma _{u}^{2}\). Thus the sign of \(\partial q^{\star }/\partial \sigma _{u}^{2}\) is the same as that of B . To evaluate the latter, we write it, setting \(\mu ^{\prime }/\mu =z^{\prime }\) and \(\mu ^{\prime \prime }/\mu =z^{\prime \prime },\) as

$$\begin{aligned} B= & {} \mu ^{\prime \prime }-\frac{1-\gamma }{2}\left\{ \frac{1-q^{*}z^{\prime }}{\mu }\left[ 2-\left( 2-\gamma \right) q^{*}z^{\prime } \right] \right. \nonumber \\&\left. -\left( 2-\gamma \right) \frac{q^{*}}{\mu }\left[ z^{\prime }+q^{*}z^{\prime \prime }-q^{*}\left( z^{\prime }\right) ^{2}\right] \right\} \sigma _{u}^{2} =\mu ^{\prime \prime }-\frac{1-\gamma }{2\mu }C\sigma _{u}^{2} \end{aligned}$$
(28)

where

$$\begin{aligned} C=2\left( 1-q^{\star }z^{\prime }\right) \left[ 1-\left( 2-\gamma \right) q^{*}z^{\prime }\right] -\left( 2-\gamma \right) \left( q^{\star }\right) ^{2}z^{\prime \prime }. \end{aligned}$$

Since \(\mu ^{\prime \prime }\) is negative, for B to be negative it is sufficient that \(C>0\) or that \(\frac{1-\gamma }{2\mu }C\sigma _{u}^{2}\) is small. The latter is certainly the case for \(\sigma _{u}^{2}\) small and/or \( \gamma \) close to (but not larger than) one.

In the subsequent example we show that in the case \(\phi \left( q\right) =q^{2}\) no constraints have to be imposed on \(\sigma _{u}^{2}\) and \(\gamma \) .

Example 3

Let us assume that the firm’s labour requirement function is \(\phi \left( q\right) =q^{2}\) as in Example 2. Then we can evaluate the expression C as follows. Since \(\phi ^{\prime }\left( q\right) =2q\) and \(\phi ^{\prime \prime }\left( q\right) =2,\phi ^{\prime }\left( q\right) q=\phi ^{\prime \prime }\left( q\right) q^{2}.\) Moreover, \(\mu ^{\prime }=1-\left( 1+r\right) w\phi ^{\prime }\left( q\right) \) and \(\mu ^{\prime \prime }=-\left( 1+r\right) w\phi ^{\prime \prime }\left( q\right) .\) Thus

$$\begin{aligned} q^{2}z^{\prime \prime }= & {} \frac{-\left( 1+r\right) w\phi ^{\prime \prime }\left( q\right) q^{2}}{\mu }=\frac{-\left( 1+r\right) w\phi ^{\prime }\left( q\right) q}{\mu }=\frac{\left[ 1-\left( 1+r\right) w\phi ^{\prime }\left( q\right) \right] q}{\mu }-\frac{q}{\mu }\\= & {} \frac{\mu ^{\prime }q}{\mu } -\frac{q}{\mu }=qz^{\prime }-z, \end{aligned}$$

with \(z=q/\mu .\) Therefore

$$\begin{aligned} C= & {} 2\left( 1-qz^{\prime }\right) \left[ 1-\left( 2-\gamma \right) qz^{\prime }\right] -\left( 2-\gamma \right) \left( qz^{\prime }-z\right) \\= & {} 2\left[ 1-\left( 2-\gamma \right) qz^{\prime }\right] \left( 1-qz^{\prime }\right) -\left( 2-\gamma \right) \left( qz^{\prime }-1+1-z\right) \\= & {} \left\{ 2\left[ 1-\left( 2-\gamma \right) qz^{\prime }\right] +2-\gamma \right\} \left( 1-qz^{\prime }\right) +\left( 2-\gamma \right) \left( z-1\right) \\= & {} \left[ 4-\gamma -2\left( 2-\gamma \right) qz^{\prime }\right] \left( 1-qz^{\prime }\right) +\left( 2-\gamma \right) \left( z-1\right) . \end{aligned}$$

Notice that

$$\begin{aligned} z-1=\frac{q-\mu }{\mu }=\frac{\left( 1+r\right) w\phi \left( q\right) -\left( 1+r\right) a}{\mu }>0. \end{aligned}$$

Therefore, a sufficient condition for \(C>0\,\)is that the expression in square brackets is positive which occurs if

$$\begin{aligned} \frac{4-\gamma }{2\left( 2-\gamma \right) }>qz^{\prime }. \end{aligned}$$

This inequality is always satisfied in the case of non-negative relative risk aversion. In fact, the LHS is larger than or equal to 1 while the RHS is smaller than 1. To see the latter, recall that

$$\begin{aligned} qz^{\prime }=\frac{q-\left( 1+r\right) w\phi ^{\prime }\left( q\right) q}{ q-\left( 1+r\right) w\phi \left( q\right) +\left( 1+r\right) a}. \end{aligned}$$

But \(\left( 1+r\right) w\phi ^{\prime }\left( q\right) q > \left( 1+r\right) w\phi \left( q\right) - \left( 1+r\right) a\) whenever \(\left( 1+r\right) w \times \left[ \phi ^{\prime }\left( q\right) q-\phi \left( q\right) \right] +\left( 1+r\right) a\) is positive which is true since \(\phi \left( q\right) \) is convex and \(\phi \left( 0\right) =0.\) Therefore \(C>0.\) From this follows that the RHS of (28) and thus B and \(\partial q^{\star }/\partial \sigma _{u}^{2}\) are negative.

C)Derivation of condition (19) From (17) and (18) the first-order condition is

$$\begin{aligned}&\mu ^{\gamma -1}\left( \frac{1}{\sigma _{u}}-2\left( 1+r\right) w\frac{ \sigma }{\sigma _{u}^{2}}\right) \\&\quad -\frac{1-\gamma }{2}\frac{\mu ^{2-\gamma }2\sigma -\sigma ^{2}\left( 2-\gamma \right) \mu ^{1-\gamma }\left( \frac{1}{ \sigma _{u}}-2\left( 1+r\right) w\frac{\sigma }{\sigma _{u}^{2}}\right) }{ \mu ^{4-2\gamma }}=0 \end{aligned}$$

which is equivalent to

$$\begin{aligned}&\mu ^{\gamma -1}\frac{1}{\sigma _{u}}\left( 1-2\left( 1+r\right) w\frac{ \sigma }{\sigma _{u}}\right) \\&\qquad -\frac{1-\gamma }{2}\mu ^{\gamma -3}\sigma \left[ 2\mu -\left( 2-\gamma \right) \frac{\sigma }{\sigma _{u}}\left( 1-2\left( 1+r\right) w\frac{\sigma }{\sigma _{u}}\right) \right] =0 \end{aligned}$$

\(\Leftrightarrow \)

$$\begin{aligned} \mu ^{2}\frac{1}{\sigma _{u}}\left( 1-2\left( 1+r\right) w\frac{\sigma }{ \sigma _{u}}\right) -\frac{1-\gamma }{2}\sigma \left[ 2\mu -\left( 2-\gamma \right) \frac{\sigma }{\sigma _{u}}\left( 1-2\left( 1+r\right) w\frac{\sigma }{\sigma _{u}}\right) \right] =0 \end{aligned}$$

\(\Leftrightarrow \)

$$\begin{aligned}&\mu ^{2}\frac{1}{\sigma _{u}}\left( 1-2\left( 1+r\right) w\frac{\sigma }{ \sigma _{u}}\right) \\&\qquad -\left( 1-\gamma \right) \sigma \mu +\frac{\left( 1-\gamma \right) \left( 2-\gamma \right) }{2}\frac{\sigma }{\sigma _{u}} \left( 1-2\left( 1+r\right) w\frac{\sigma }{\sigma _{u}}\right) =0 \end{aligned}$$

\(\Leftrightarrow \) (19). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tulli, V., Gallegati, M. & Weinrich, G. Financial conditions and supply decisions when firms are risk averse. J Econ 128, 259–289 (2019). https://doi.org/10.1007/s00712-019-00655-x

Download citation

Keywords

  • Portfolio possibilities locus
  • Financing gap
  • Net worth
  • Price volatility
  • Risk aversion

JEL Classification

  • D21
  • D81
  • G11