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Abstract

A topic of recent interest in the retail financial sector has been the growth of credit unions or “pure cooperatives”. Past credit union researchers built mathematical models of credit union operations. These models identified important operating characteristics but were modeled under assumptions of static operating environments. The model presented in this paper departs from the traditional static models and examines dynamic operation for a United States credit union. Its inter-temporal structure clarifies a number of issues—such as optimal equity retention and inter-temporal rate policy—not addressed by earlier studies. Given initial conditions, the model specifies equity retention and inter-temporal deposit and loan rate policies until an equilibrium state is reached.

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Notes

  1. http://www.woccu.org/memberserv/intlcusystem, accessed on the 14th May, 2012.

  2. http://cuna.org/public/press/credit-union-basics, accessed on the 14th May, 2012.

  3. http://www.cunamutual.com/portal/server.pt/document/2700462/june_2011_cu_trends_report, accessed on the 14th May, 2012.

  4. Overstreet and Rubin (1991) provide an early review oriented to both theory and empirical work with Cargill (1977) having presented an even earlier literature review.

  5. See Federal Credit Union Act 109.

  6. http://cuna.org/public/press/credit-union-basics, accessed on the 14th May, 2012.

  7. http://www.woccu.org/memberserv/intlcusystem, accessed on the 14th May, 2012.

  8. See the 2001 and 2008 Credit Union Directory, Callahan and Associates, 1.

  9. Smith (1986) denotes an entire paper to the discussion of objective functions.

  10. p can be thought of as the ratio of a constant rate of time preference, q, and the instantaneous probability of credit union survival, s(E,L). As E→0 or L→∞, s→0. As E→∞ or L→0, s→1. Blanchard (1985) uses a similar discount factor form.

  11. Assume, for example, that the loan rate is dropped slightly from such an equilibrium. Borrowers unequivocally benefit from such a change because both loan volume and the credit union/market rate spread increase. If these loans are made by shifting resources from lower-return investments, interest income might also increase, making additional saver allocations possible. Even if their benefit allocations are not identical after the rate change, both borrowers and savers prefer the change.

  12. The loan/investment decision is not a simple function of marginal yields and rates because credit unions often carry positive investments even though the marginal loan rate nearly always exceeds the investment rate. The ultimate rationale for investments is not readily apparent, but probably appeals to risk.

  13. The modern formulation of this technique can be found in Pontryagin et al. (1962).

  14. Blanchard and Fischer (1989) review the use of phase diagrams in optimal control theory.

  15. The Appendix explains the use of d cu and E, and the exclusion of l cu , in Fig. 1.

  16. Our model has no expectations component, but the effect of an anticipated tax change might have different effects. Forward-looking credit unions would probably smooth out the large anticipated drop in \(d_{\mathit{cu}}^{\ast}\) and increase in \(l_{\mathit{cu}}^{\ast}\) by moderately lowering the spreads before the tax change, increasing E, and then supporting these spreads after the tax change by drawing down the enlarged equity reserve.

  17. Concavity of the loan demand schedule ensures that marginal revenue (with respect to the loan rate) will eventually become negative. If the credit union rations credit based on risk, for example, an increase in loan volume is accompanied by higher default. Eventually, the cost of marginal loans will exceed revenue.

  18. The various non-negativity constraints are discussed in the final analysis rather than modeled explicitly for reasons of tractability.

  19. Notice that any non-negative level of equity is feasible, but that the deposit rate is constrained to lie above the market deposit rate.

  20. If steady state deposits and equity are monotonic in the loan rate (analysis in the following section suggests an indeterminant relationship between steady state deposit and loan rates), the loan rate which satisfies this equation is unique. A formal proof is omitted, but the basic logic is that an increase in the loan rate decreases loan demand and investment volume from (A.8), but also increases steady state deposits and equity. Assume that \(l_{\mathit{cu}}'\) satisfies (A.27). A higher rate decreases the left-hand side of (A.27) and increases the right-hand side, so is not admissible. A lower rate is similarly inadmissible since it causes an increase in the left-hand side of (A.27) and decrease in the right-hand side. Continuity ensures the existence of the unique \(l_{\mathit{cu}}'\) which satisfies (A.27) and is denoted \(l_{\mathit{cu}}^{\ast}\).

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Acknowledgements

The authors are grateful to the anonymous reviewers, who provided many helpful comments and suggestions for improvement. Funding of this research work was, in part, supported by the University of Cape Town Research Office and the Carnegie Research Development Grant.

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Correspondence to Kanshukan Rajaratnam.

Appendix

Appendix

This appendix derives the steady state equilibrium solution to the system (1), (2), and (3), and the properties of convergence towards equilibrium.

1.1 A.1 Objective function

The credit union chooses functions d cu (t), l cu (t), and E(t) to maximize

$$ \int_0^{\infty} \bigl[U(B)+V(S) \bigr]e^{-p(E,L)t}dt $$
(A.1)

where

(A.2)
(A.3)
  • l(t)=l m l cu (t)>0; market loan rate less credit union loan rate, which is assumed positive

  • d(t)=d cu (t)−d m >0; credit union deposit rate less market deposit rate, which is assumed positive

  • U(⋅)= borrower utility function; U′(⋅)>0; U″(⋅)<0

  • V(⋅)= saver utility function; V′(⋅)>0; V″(⋅)<0

  • p(⋅,⋅)= discount rate of typical credit union members;

    (A.4)
  • E(t)= credit union equity; E>0; E(0)=E 0

  • D(t)= credit union deposit volume; D>0

  • L(t)= credit union loan volume; L>0.

1.2 A.2 Constraints

$$ L=\epsilon(l); \qquad \epsilon'(l)>0; \qquad \epsilon''(l)<0 $$
(A.5)

Loan demand is an increasing, concave function of the difference between the market and credit union rates.

$$ D=\sigma(d); \qquad \sigma'(d)>0; \qquad \sigma''(d)=0 $$
(A.6)

Deposit demand is an increasing linear function of the difference between the credit union and market rates.

$$ L+I(t) = D +E $$
(A.7)

The balance sheet constraint ensures that credit union loans plus investments, I(t), equal deposits plus equity.

$$ I=\alpha(L); \qquad \alpha'(L) >0 $$
(A.8)

This constraint relates loan and investment volume.

$$ \dot{E}= \bigl[l_{\mathit{cu}}L + rI -d_{\mathit{cu}}D - c(L,D) \bigr]\tau $$
(A.9)

where

$$ \dot{E}=\frac{d E}{d t} $$

r is the investment rate of return with r<d m ,

and τ=1—credit union tax rate; 0<τ≤1.

This is the income statement constraint, which states that addition to equity equals interest income plus investment income less dividends and cost. Operating costs net of fee income, c, is increasing and concave in both loans and deposits, reflecting the increasing returns to scale found by a number of researchers (see, e.g., Fry et al. 1982; Murray and White 1983). For model tractability, we assume cost increases linearly with respect to deposit volume, i.e., \(\frac{\partial^{2} c}{\partial D^{2}}=0\). Presently, τ is unity but for purposes of analysis is allowed to range between zero and one.

1.3 A.3 Solution

The solution method employed is optimal control. First, (A.7) and (A.9) are combined by substituting out I to form the new constraint

$$ \dot{E} = \bigl[ (l_{\mathit{cu}} - r)L + rE - (d_{\mathit{cu}} - r)D - c(L,D) \bigr] \tau $$
(A.10)

E is the state variable and l cu and d cu are the controls. I is solved as a function of l cu via (A.8). The resulting Hamiltonian is

$$ H= \bigl[U(B) +V(S) \bigr] e^{-pt} + \tau \lambda(t) \bigl[ (l_{\mathit{cu}} - r)L + rE - (d_{\mathit{cu}} - r)D - c(L,D) \bigr] $$
(A.11)

where λ(t) is the costate variable associated with the combined constraint.

The necessary conditions areFootnote 18

(A.12)
(A.13)
(A.14)
(A.15)

Since U(⋅), V(⋅), and c(⋅,⋅) are all strictly concave, the following transversality condition is sufficient to show that (A.13), (A.14), (A.15), and (A.16) yield a unique maximum (see Mangasarian 1966 for proof).

(A.16)

Next we make the following substitutions for notational convenience.

$$ \everymath{\displaystyle }\left \{ \begin{array} {l} \frac{\partial p}{\partial E} \equiv p_E; \qquad {-}\frac{\partial p}{\partial L}\epsilon'(l) \equiv P_l; \qquad \frac{dU}{dB} \frac{dB}{dl_{\mathit{cu}}} \equiv U_l;\qquad \frac{dV}{dS} \frac{dS}{dd_{\mathit{cu}}} \equiv V_d \\ \noalign{\vspace{2pt}} L-\biggl(l_{\mathit{cu}} - r - \frac{\partial c}{\partial L}\biggr)\epsilon'(l) \equiv \bar{L} \\ \biggl(r - d_{\mathit{cu}} -\frac{\partial c}{\partial D}\biggr)\sigma'(d)-D \equiv \bar{D} \end{array} \right . $$
(A.17)

solving (A.14) for λ

(A.18)

which can be differentiated with respect to time

(A.19)

Rearranging terms in (A.13) yields

(A.20)

substitution (A.18) into (A.20), then (A.18), (A.19) and (A.20) into (A.12) gives

(A.21)

Rearranging terms gives

(A.22)

which, along with (A.15), forms the system of two differential equations which characterizes the optimal evolution of equity and the deposit rate.

At this point, one should note the absence of a differential equation in l cu . In the analysis that follows, working with two rather than three differential equations is more revealing because two-dimensional phase diagrams are more graphically palatable than three-dimensional versions. Convenience and pedagogy underlie the selection of equity and the deposit rate. Solving (A.13) for λ and differentiating, and substituting (A.14) into (A.13) would have produced differential equations in equity and the loan rate. Reducing the system to two differential equations mildly restricts the results of later analysis, but the alternative is to solve both (A.13) and (A.14) for λ, differentiate both with respect to time, and then manipulate the three equation system. The significant computational advantage of working with two differential equations justifies the methodology.

1.4 A.4 Steady state analysis

Functions E(t), I(t), d cu (t), and l cu (t) that satisfy (A.15), (A.22), (A.7), (A.8), and (A.16) given the initial condition E(0)=E 0 are optimal policy for the credit union. The phase diagram in Fig. 1, plotted in (E, d cu ) space for a given l cu , illustrates the system.Footnote 19 The two thick curves are two equilibrium loci. The first equilibrium locus (the deposit rate locus) is found by setting \(\dot{d}_{\mathit{cu}} = 0\) and \(\dot{p}=0\) in (A.22) and solving for p(E)

(A.23)

The set of points in (E,d cu ) space which satisfy this equation comprise the equilibrium locus. Slope is given by

(A.24)

(A.24) shows that p increases in the d cu direction along this locus. This, combined with the property of decreasing p in E from (A.4) yields a decreasing deposit rate locus in (E, d cu ) space. The locus approaches the vertical axis asymptotically because as E→0 in (A.23), p→∞ so d cu →∞. By plugging in d cu =d m in (A.23), we see that the locus intersects the horizontal axis at some finite point E a when p is positive.

Similarly, setting \(\dot{E}= 0\) in (A.15) and solving for E gives

(A.25)

which defines the equity equilibrium locus. Slope and curvature are found by differentiating

$$ \everymath{\displaystyle }\left \{ \begin{array}{l} \frac{dE}{dd_{\mathit{cu}}}= r^{-1} \biggl[\biggl( \frac{\partial c}{\partial D}+d_{\mathit{cu}}-r\biggr)\sigma'(d)+D \biggr]>0 \\ \noalign{\vspace{4pt}} \frac{d^2 E}{d d_{\mathit{cu}}^2}=2 r^{-1} \sigma'(d)>0 \end{array} \right . $$
(A.26)

Thus, the equity locus is increasing and convex in (E, d cu ) space.

Equations (A.23), (A.25), and the constraints (A.7) and (A.8) characterize the optimal solution to the four variable model. For any given credit union loan rate, a phase diagram like in Fig. 1 yields the optimal steady state equity and deposit rate. Thus, we can write E and \(d_{\mathit{cu}}^{\ast}\) as functions of l cu . Solving (A.7) for I, substituting in E and \(d_{\mathit{cu}}^{\ast}\), and equating to (A.8) yields

(A.27)

The credit union loan rate and investment volume which satisfy this equation are optimal steady state values and are denoted \(l_{\mathit{cu}}^{\ast}\) and I .Footnote 20 As a result of the reduction to two differential equations, this solution method incorporates only steady state values of the loan rate and investment volume. Given \(l_{\mathit{cu}}^{\ast}\), the phase diagram in Fig. 1 describes the evolution of the deposit rate and equity level.

Behavior of the system in each of the four regions formed by the loci is characterized by the perpendicular arrows. In Eq. (A.15), we see that \(\frac{d \dot{E}}{dE} = \tau r > 0\), so for all points to the right of the equity locus, \(\dot{E} > 0\). The behavior of the deposit rate over time is given by differentiating (A.22) with respect to the deposit rate

(A.28)

(A.28) cannot be signed with certainty.

1.5 A.5 Stability and uniqueness of T

The stability of the equilibrium point (E , \(d_{\mathit{cu}}^{\ast}\)) in Fig. 1 is demonstrated here. The general methodology, taken from Ekman (1982), is to linearize the differential equations (A.15) and (A.22) around a neighborhood of the equilibrium point (E , \(d_{\mathit{cu}}^{\ast}\)) and then find the eigenvalues of the resulting system. Negative real eigenvalues correspond to stable manifolds, while positive real eigenvalues are associated with unstable manifolds. The same is true for the real parts of imaginary eigenvalues, but here the dynamic path towards the equilibrium point has a cyclical component. We restrict our attention to real eigenvalues.

We can rewrite (A.15) and (A.22) as

$$ \left \{ \begin{array}{l} \dot{d}_{\mathit{cu}} = \varphi(d_{\mathit{cu}},E)\\ \noalign{\vspace{2pt}} \dot{E} = \psi(d_{\mathit{cu}},E) \end{array} \right. $$

Then the linear part of the Taylor series expansion about the point (E , \(d_{\mathit{cu}}^{\ast}\)) can be written

$$ \left [ \begin{array}{c} \dot{d}_{\mathit{cu}} \\ \dot{E} \end{array} \right ] = \left [ \begin{array}{c@{\quad}c} \varphi_{d_{\mathit{cu}}} & \varphi_E \\ \psi_{d_{\mathit{cu}}} & \psi_E \end{array} \right ]\left [ \begin{array}{c} d_{\mathit{cu}}-d_{\mathit{cu}}^{\ast} \\ E - E^{\ast} \end{array} \right ] $$

where

$$ \everymath{\displaystyle }\left \{ \begin{array}{l} \varphi_{d_{\mathit{cu}}}=\frac{d \dot{d}_{\mathit{cu}}}{d d_{\mathit{cu}}}\\ \noalign{\vspace{4pt}} \varphi_E = \frac{d \dot{d}_{\mathit{cu}}}{d E}\\ \noalign{\vspace{4pt}} \psi_{d_{\mathit{cu}}} = \frac{d \dot{E}}{d d_{\mathit{cu}}} \quad \mbox{and}\\ \noalign{\vspace{4pt}} \psi_E = \frac{d \dot{E}}{d E} \end{array} \right. $$

and \(\varphi_{d_{\mathit{cu}}}\), φ E , \(\psi_{d_{\mathit{cu}}}\), and ψ E are all evaluated at (E , \(d_{\mathit{cu}}^{\ast}\)).

The eigenvalue of the main matrix are

$$ \lambda_{1,2}=\frac{\varphi_{d_{\mathit{cu}}}+\psi_E}{2} \biggl[1 \pm \biggl[ 1 + 4 \frac{\varphi_E \psi_{d_{\mathit{cu}}} - \varphi_{d_{\mathit{cu}}} \psi_E}{(\varphi_{d_{\mathit{cu}}}+\psi_E)^2} \biggr]^{\frac{1}{2}} \biggr] $$

We see that the eigenvalues are real and of opposite signs if and only if

$$ \varphi_E \psi_{d_{\mathit{cu}}} > \varphi_{d_{\mathit{cu}}} \psi_E $$
(A.29)

From (A.15)

$$ \left \{ \begin{array}{l} \psi_{d_{\mathit{cu}}} = \tau \bar{D} < 0\\ \noalign{\vspace{2pt}} \psi_E = \tau r > 0 \end{array} \right. $$

Thus,

$$ \varphi_E\frac{\bar{D}}{r} > \varphi_{d_{\mathit{cu}}} $$

Assuming that (A.29) holds, the speed of convergence along the saddle point path T in Fig. 1 is given by the absolute value of the negative eigenvalue. The influences of the various parameters on the eigenvalue can then be ascertained. Unfortunately, the only signable differential was the change in the eigenvalue with respect to the discount rate. Higher discount rates, ceteris paribus, lead to faster convergence.

The proof of uniqueness is similar to Blanchard and Fischer (1989). For any E 0, the d cu (0) that places the system on T is optimal. The transversality condition (A.16) is satisfied because E(t)→E and λ(t)→−V d e pt τ −1D−1→0 as t→∞. In Fig. 1, if the credit union starts with E 0 and \(d_{\mathit{cu}}'(0)\) is chosen, the deposit rate and equity will evolve along the trajectory T′ and the vertical axis will be approached (or reached) in finite time. Approaching zero equity in finite time violates the first order conditions (A.13) and (A.14). Since members perceive p→∞ when E→0, benefits would have to be instantaneously unbounded to justify full discounting. If \(d_{\mathit{cu}}''(0)\) is chosen, T″ is followed and the credit union approaches d m in finite time, which prescribes zero saver benefit with only finite equity buildup. The transversality condition is violated because λ(t)→0 as t→∞. Similar arguments hold when E 0>E .

The difficulty with analysis of convergence and the strong assumption made in (A.28) highlight an important fact about optimal control theory—its elegant form can rapidly deteriorate under even the most simple specifications. Sydsaeter (1978) points out a number of pitfalls encountered by economists working with optimal control theory. As illustrated above, copious amounts of algebra and calculus are the price for using this powerful technique.

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Rubin, G.M., Overstreet, G.A., Beling, P. et al. A dynamic theory of the credit union. Ann Oper Res 205, 29–53 (2013). https://doi.org/10.1007/s10479-012-1246-7

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