Net worth ratio, bank lending and financial instability

Abstract

This paper extends a Minsky model by incorporating net worth ratio, which Steindl stressed the importance of in reference to financial markets. We construct a dynamic macroeconomic model comprising discrete equations of both the net worth ratio and interest rate. We investigate what factors bring the instability of the steady state. We show that the effect of asset income on consumption can contribute to economic stability following Lavoie (1995) and Hein (2007). On the other hand, the economy becomes unstable when a bank’s lending reaction is elastic with respect to the net worth ratio of the firm. When the steady state is a saddle point, the monetary policy is likely to shift the economy from an unstable path to a convergence path. It can be said that monetary policy may have a stabilizing effect in the long run, however, in practice, there would be considerable difficulties in accomplishing this.

Appendices

Appendix 1: The outlines of the process to derive Eq. (25)

The household earns wages ωN _t and dividends from the firm. We assume that the costs of the bank G _t−1 and the profit Π ^b _t−1 are distributed to the household via the bank’s labor cost and other factors. The household can earn the interest from deposit. Finally, we suppose that the revenue of the central bank also belongs to the household.

Then, taking into account equation of dividends (14), the revenue of the household involved financial assets can be expressed as

$$\begin{aligned} {TR} = \omega N_{t} + aZ_{t - 1} + v\left( {pY_{t} - \omega N_{t} - i_{t - 1} L - aZ_{t - 1} } \right) + G_{t - 1} + i^{a} A_{t - 1} + \hfill \\ \quad \varPi_{t - 1}^{b} + \left( { 1 + i^{d} } \right)D_{t - 1} + M_{t - 1} + E^{q} . \hfill \\ \end{aligned}$$

(43)

Substituting Eqs. (16) and (17) into equation (43), we have

$$TR = \omega N_{t} + v(pY_{t} - \omega N_{t} ) + (1 - v)i_{t - 1} pK_{t - 1} + D_{t - 1} + M_{t - 1} + E^{q} .$$

(44)

Taking into account the economy as a whole, the assets which one has are the liabilities which another has. The following equation is satisfied:

$$L_{t - 1} = pK_{t - 1} - Z_{t - 1} = D_{t - 1} + M_{t - 1} .$$

(45)

From equations (44) and (45), we have

$$TR = \omega N_{t} + v(pY_{t} - \omega N_{t} ) + (1 - v)i_{t - 1} pK_{t - 1} + L_{t - 1} + E^{q} .$$

(46)

On the other hand, the expenses of the household involved financial assets can be expressed as

$${{TE}} = pC_{t} + D_{t}^{d} + M_{t}^{d} + E^{q} .$$

(47)

We have the Eq. (25) from equations (46) and (47)

Appendix 2: The comparative statics analysis in short-run model and the outlines of the process to derive Eqs. (39a) and (b)

The mathematical results of comparative statics in short-run model expressed as

$$r_{{z_{t - 1} }} = - \frac{{k_{{z_{t - 1} }} l_{{i_{t} }}^{s} - k_{{i_{t} }} (1 + l_{{z_{t - 1} }}^{s} )}}{\varOmega } > 0,$$

(48)

$$r_{{i_{t - 1} }} = - \frac{{(1 - v)\{ (1 - c)k_{{i_{t} }} + cl_{i_{t}}^{s} \} }}{\varOmega }{ \lesseqgtr }0,$$

(49)

$$r_{{i^{a} }} = - \frac{{k_{{i_{t} }} l_{{i^{a} }}^{s} }}{\varOmega } < 0,$$

(50)

$$i_{{i_{t - 1} }} = \frac{(1 - v)(1 - c)}{\varOmega } > 0$$

(51)

$$i_{{i^{a} }} = - \frac{{(1 - cv)l_{{i^{a} }}^{s} }}{\varOmega } > 0.$$

(52)

Linearizing this dynamic model near the steady state to analyze stability, we have

$$T_{1} = \frac{{\{ (1 - v)r_{{z_{t - 1} }} + 1\} - (k_{{i_{t} }} \cdot i_{{z_{t - 1} }} + k_{{z_{t - 1} }} )z}}{1 + k},$$

(53)

$$T_{2} = \frac{{(1 - v)(r_{{i_{t - 1} }} - 1) - z(k_{{i_{t} }} \cdot i_{{i_{t - 1} }} )}}{1 + k}.$$

(54)

The values of each element are appreciated at the steady state. Taking into account the results of comparative statics, we can rewrite equations (53) and (54) as Eqs. (39a) and (b) and determine the signs.

Appendix 3: The stability of the steady state

The following equation is satisfied at the steady state from Eqs. (32), (38a), (38b),

$$\{ 1 - v - (1 - cv)z\} k = (1 - c)(1 - v)i.$$

(55)

Assuming that the accumulation rate and the interest rate are positive at the steady state, we have the Eq. (40).

Substituting the results of comparative statics and equation (55) into the factors of coefficient matrix M _D , we have

$$T_{1} + R_{2} = \frac{1}{1 + k} + \frac{(1 - c)(1 - v)}{\varOmega }\left[\frac{1}{1 + k}\frac{i}{k}\{ k_{z_{t-1}} l_{i_{t}}^s - k_{i_{t}} (1 + l_{z_{t-1}}^s )\} + 1\right],$$

(56)

$$T_{1} \cdot R_{2} - T_{2} \cdot R_{1} = \frac{(1 - c)(1 - v)}{\varOmega }\frac{1}{1 + k}\{ k_{z_{t-1}} (1 - z) - l_{z_{t-1}}^s \} .$$

(57)

For convenience, we omit the asterisks which express the values of the steady state. Substituting these equations (56) and (57) into the Eqs. (41), we can derive the factors which bring about the stable steady state.