# Net Worth Ratio and Financial Instability

- Toshio Watanabe
^{1}Email author

**2**:3

https://doi.org/10.1186/2193-2409-2-3

© T. Watanabe; licensee Springer. 2013

**Received: **31 July 2012

**Accepted: **19 February 2013

**Published: **15 March 2013

## Abstract

In order to better understand relationships between the real economy and financial economy, it is necessary to formulate a model of financing. New Keynesian theory emphasizes that a firm’s net worth influences investment decisions and business cycles under an imperfect capital market. We have constructed a dynamic model from the standpoint of post Keynesian economics. We incorporate a dynamic equation of a firm’s net worth ratio and investigate financial factors, which give rise to economic instability. Our results demonstrate that a steady state can be a saddle point when the dividend rate is low and the bank’s lending reaction to the net worth ratio is more elastic than investment reaction. When the steady state is the saddle, the change in the basic discount rate is likely to shift the economy from an unstable path to a convergence path. Financial policy has a stabilizing effect in the long-run as well as a positive effect in the short-run.

**JEL Classification:**E12, E44, E52.

## Keywords

## 1 Introduction

Financial markets in market economies gain in complexity and have significant effects on the real economy. From the standpoint of new Keynesian economics, Greenwald and Stiglitz [8] as well as Bernanke and Gertler [3] pointed out that the investment and output of rational firms depend on the firms’ balance sheet factors under an imperfect capital market.

It is hardly new to discuss the interactions between financial markets and the real economy. Keynes [10] rejected classical economics and indicated the instability of the real economy; especially, he emphasized the instability of financial markets. In general, a firm’s behavior depends not only on balance sheet factors, which new Keynesian pointed out, but also on bank behavior and other factors.

Minsky [12] analyzed the effects of financial factors on the economy. He alleged that financial booms and collapses become inevitable in market economies. In this respect, he is in the post Keynesian group, which emphasizes instability of the economy. He investigated the financing of investments and the cyclical behavior of the real economy. He developed a new theory called the financial instability hypothesis, which involved the argument of Fisher [7] as well as Keynes.

Minsky’s ideas lead to the development of various mathematical models.^{a} For example, Taylor and O’Connell [15] and Adachi [1] focused on long-run expectations and household portfolios. Though Adachi constructed a model with explicit inclusion of bank behavior, his model is too complex to show concisely the dynamic economy and stabilizing effects of monetary policy.

In this paper, we highlight net worth, which is line with new Keynesian theory. However, we focus on the net worth ratio, which is the ratio of net worth to capital of a firm. One of the purposes of this paper is to build a simple macro model on the net worth ratio from the standpoint of post Keynesian economics. We investigate financial factors, which give rise to instability of the economy. We formulate a model of investment decisions, bank behavior, and financing for better understanding of the relations between real economy and financial economy.

Our model has assumptions that are similar to Greenwald and Stiglitz. The equity market is imperfect. Firms cannot raise new equity and normally pay a dividend on existing shares. Meanwhile, there is a bank lending market. The firms can borrow as much as they want, but they must pay interests. Hence, the evolution of net worth is represented by retained earnings.

The second purpose of this paper is to analyze the effectiveness of monetary policy. We first construct a static model, and then extend to a dynamic model that incorporates dynamic equations of both the net worth ratio and bank lending rate.^{b} This dynamic model is similar to that of Taylor and O’Connell and Adachi. We investigate the effects of monetary policy on the short-run and long-run equilibrium; especially, we consider a stabilizing effect of monetary policy when the steady state is unstable.

Our results demonstrate that the economy can be unstable when the dividend rate is low and the bank’s lending reaction to the net worth ratio is more elastic than investment reaction. The latter factor is a feature of our model. When the steady state is a saddle point, the change in the basic discount rate is likely to shift the economy from an unstable path to a convergence path.^{c} We can see that financial policy has a stabilizing effect in the long-run as well as a positive effect in the short-run.

This paper is organized in the following manner. Section 2 presents an overview of the model we are proposing. Section 3 discusses the respective behavior of firms, banks, and households. We derive the investment function, bank lending function, and so on. In Sects. 4 and 5, we consider the equilibrium of commodity market and bank lending market, respectively. Section 6 analyzes the equilibrium of the short-run economy. Section 7 investigates the instability of economy. We construct a dynamic system and consider a stabilizing effect of monetary policy. Finally, Sect. 8 summarizes the results.

## 2 Description of the Model

The economic system in this paper consists of 4 sectors (firms, banks, households, and central bank) and 5 markets (commodities, bank lending, deposits, cash currency, and central bank advances).

The firm maximizes the present value of returns from investment and decides a capital accumulation rate. The investment is financed through either retained earnings or borrowings from the bank. There is a time-lag between borrowing and interest payment. The firm pays interest on previous borrowings during present period. The firm decides the capital accumulation rate and new borrowings from bank during present period given the existing capital stock and previous bank lending rate.

The bank decides the lending to maximize profits. The household saves in the form of either deposits or cash currency. We assume that the deposit rate is regulated and that the bank accepts all deposits, which the household would like to make. In addition, in the market of central bank advances, we assume that the basic discount rate is exogenously determined as a means of monetary policy and the central bank supplies funds, which the bank requests. This type of monetary policy fits with the concept of “the horizontalist view.”

Simplified balance sheets

Central bank | Firm | ||
---|---|---|---|

Central bank advances ${A}^{s}$ | Bank reserves Cash currency ${M}^{s}$ | Capital | Debt Equity ${E}_{q}$ Earned surplus ${E}_{s}$ |

Private bank | Household | ||
---|---|---|---|

Lending ${L}^{b}$ Bank reserves | Deposit ${D}^{h}$ Central bank advances ${A}^{b}$ | Cash currency ${M}^{h}$ Deposit ${D}^{h}$ Equity ${E}_{q}$ | Wealth |

## 3 Structure

### 3.1 Investment Decisions of Firm^{d}

*τ*over the labor cost. The nominal wage is

*ω*, the labor is

*N*, the output is

*Y*, labor-output ratio is

*n*.

^{e}The price level

*p*is given by

The firm makes an investment, given the existing capital stock. The investment decisions of the firm are made based on expected returns over the periods during which the newly installed equipment will be used. But they cannot get returns when they make bad investments and go bankrupt. Let us denote the expected returns of investment by *Q*, and the possibility of bankruptcy by *σ*. The sequence of returns from investment is given by $\{Q,(1-\sigma )Q,{(1-\sigma )}^{2}Q,\dots \}$.

where *r* is the bank lending rate at which the firm borrows during present period.

*σ*depends on the net worth ratio

*z*, which is the ratio of net worth

*Z*to the existing capital stock

*pK*. The firm’s net worth is the sum of the value of equity ${E}_{q}$ and earned surplus ${E}_{s}$. The earned surplus is the accumulation of retained earnings. When the net worth ratio is low, the possibility of bankruptcy is high. The risk premium is represented by

*k*which is the investment per unit of capital. It can be represented by

where *I* is the investment. We assume that the marginal efficiency of capital ${\varphi}_{k}$ decreases as *k* increases.

*k*yields

*k*can be expressed as

The capital accumulation rate is a decreasing function of the bank lending rate and an increasing function of the net worth ratio.

### 3.2 The Financing of Investment

When the capital market is imperfect, the Modigliani–Miller theorem [13] is not true. In other words, under an imperfect capital market, the independence of investment and financing is lost. We have to consider the relations between investment and financing.

We assume that the equity is issued only when the firm is established. The firm cannot raise new equity and pay a dividend on existing shares. On the other hand, there is a bank lending market. The firm borrows from a bank and pays interests.^{f}

*F*is retained earnings and $\mathrm{\Delta}{L}^{d}$ is new borrowings from the bank during the present period. The total borrowings ${L}^{d}$ are written as

where *L* is debt.

where $\overline{r}$ is the bank lending rate during the previous period and *V* is the dividend.

*v*of profit as a dividend to shareholders. We can express the dividend as

*pK*, we obtain

where *y* is output-capital ratio and *l* is debt-capital ratio.

*l*is equal to 1 minus net worth ratio. This is represented by

^{g}

The borrowings from bank are a decreasing function of the bank lending rate during the present period and an increasing function of the bank lending rate in the previous period. The effect of net worth ratio will be ambiguous, because the increase of net worth ratio has effects on increase of investment as well as decrease of existing debt.

### 3.3 Bank Behavior

Like firms, banks seek profits. The bank earns revenue from lending to firms and makes interest payments to depositors and the central bank. We take into account the transition costs *G*. The costs involve auditing and monitoring costs to firms and losses in the event of corporate failures.

where ${\Pi}^{b}$ is profit of bank, ${L}^{b}$ is bank lending, ${i}^{d}$ is deposit rate, ${D}^{h}$ is deposit from the household, ${i}^{a}$ is the basic discount rate, and ${A}^{b}$ is the central bank advances.

*g*. First, the costs in auditing and monitoring generally depend on the level of bank lending. We assume that the cost

*g*is an increasing function of bank lending-capital ratio. Second, the bank’s risk premium for lending is high when the bank estimates that the possibility of the firm’s bankruptcy is high. We assume that the bank’s risk premium depends on the net worth ratio of the firm. When the net worth ratio is low, the bank’s risk premium becomes high. The cost

*g*increases as the net worth ratio decreases. Then the cost function

*g*is written as

We assume that the marginal cost of bank lending increases more than proportionally as ${l}^{b}$ increases and decreases as *z* increases.

*R*is bank reserves. The bank reserves must be satisfied legal reserve. This is represented by

where *θ* is legal reserve rate.

*pK*, we obtain

Bank lending is an increasing function of the bank lending rate and net worth ratio and a decreasing function of the basic discount rate.

### 3.4 Household Behavior

*W*is the sum of wealth at beginning of period $\overline{W}$ and savings during the present period $p{S}^{h}$. This is represented by

*δ*and assume that

*δ*is an increasing function of deposit rate ${i}^{d}$. The demand functions of deposit and cash currency are respectively expressed by

Since the increase of net worth ratio decreases present household’s wealth, both deposits and currency demands will decrease. In contrast, both demands will increase responding to the increase of output-capital ratio and previous bank lending rate.

## 4 Equilibrium of Commodity Market

This economic system consists of 5 markets; commodities, bank lending, deposits, cash currency, and central bank advances. We assume that the deposit rate is regulated and that bank accepts all deposits, which the household would like to make. The supply of deposits is constrained by demand. Similarly, we assume that the basic discount rate is exogenously determined as a means of monetary policy and the central bank supplies funds, which the bank requests.

Therefore, we will consider the equilibriums of 3 markets: commodities, bank lending, and cash currency. Taking into account Walras’ law, we can suppress the analysis of cash currency market. We will consider the commodity market and bank lending market.

*pS*are equal to the sum of the household’s savings and retained earnings of the firm. This is represented by

*pK*, we have

*y*will go up. This adjustment process is represented by

## 5 Equilibrium of Financial Markets

*r*will go up. This adjustment process is represented by

*BL*curve. When the basic discount rate and the previous bank loan rate increase, the

*BL*curve will shift upward as shown in Fig. 2. On the other hand, the effect of the net worth ratio on the

*BL*curve is ambiguous. The increase of net worth ratio will result in an increase of both bank lending and investment. When the former effect is larger than the latter effect, the

*BL*curve will shift downward.

## 6 Short-Run Equilibrium and Comparative Statics

We will now consider short-run equilibrium. The system comprises Eqs. (36) and (39) and determines output-capital ratio *y* and bank lending rate *r*.

^{h}We can express the short-run equilibrium as

*IS*curve is higher than that of

*BL*curve when the stability conditions are satisfied.

Let us examine the effect of exogenous variables on short-run equilibrium. When the previous bank lending rate $\overline{r}$ and the basic discount rate ${i}^{a}$ rise, the *BL* curve will shift upward. The output-capital ratio will decrease and the present bank lending rate will increase.

We shall examine how the net worth ratio *z* affects the output-capital ratio and bank lending rate. Though the shift of *BL* curve is ambiguous when the net worth ratio rises, the output-capital ratio will increase. But the effect on bank lending rate will be ambiguous.^{i} When the effect on bank lending is larger than that on investment, the bank lending rate will decrease. As shown in the following section, the economy can become unstable when the increase of net worth ratio decreases the bank lending rate.

From the static model, we can see that the firm’s balance sheet elements and bank behavior significantly affect the real economy as Minsky alleged. In the next section, we will investigate the unstable economy by using a dynamic model.

**Proposition 1** *In the short*-*run*, *when the net worth ratio rises*, *the output*-*capital ratio will increase*. *Additionally*, *when the previous bank lending rate and the basic discount rate rise*, *the output*-*capital ratio will decrease and the present bank lending rate will increase*.

## 7 Dynamic Model

### 7.1 Stability of Steady State

In this section, we construct a dynamic model to investigate the mechanism of financial instability. We shall analyze the characteristics of steady state in our model.^{j}

In the discussions so far, we treated the net worth ratio *z* and previous bank lending rate $\overline{r}$ as exogenous variables. To analyze the dynamic system, we have to specify how the net worth ratio *z* and previous bank lending rate $\overline{r}$ evolve over time.^{k}

*Z*is equal to the retained earnings. Taking into account Eq. (4), we have

^{l}

At the steady state $({z}^{\ast},{\overline{r}}^{\ast})$, evolution of net worth and the capital are the same. The net worth ratio and bank lending rate remain constant.

*A*,

^{m}

^{n}

*U*is negative, the system is unstable. Taking into account condition (51), it is negative when the absolute value of ${l}_{z}^{b}$ exceeds the value of ${k}_{z}$ and

*v*is low. In other words, we can point out financial factors that make an economy unstable as follows:

- (1)
The bank lending reaction to the net worth ratio exceeds the investment reaction.

- (2)
Dividend rate is low.

From the static model, we can confirm that these factors make the bank lending rate decrease responding to an increase of the net worth ratio.

The steady state ${E}^{\ast}({z}^{\ast},{\overline{r}}^{\ast})$ becomes a saddle point and there is a unique path that converges to it. For example, the economy *A* can converge to the steady state. But it is unusual for the economy to be on the saddle path. The gaps between the economies on the convergence path and the others will expand cumulatively as time goes by.

Let us consider economy *B* in Fig. 4. The net worth ratio tends to increase in the situation of economy *B*. The firm increases investment and bank lending increases. The output of the economy and firm’s profit will increase. As demand for funds increase, the bank lending rate rises. This in turn causes investment to decrease. After a while, the firm gets into the red and the net worth ratio will begin to decrease. The bank lending rate will become increasingly higher. The economy continues to decline.

**Proposition 2** *The dynamic system which consists of the net worth ratio of firm and the bank lending rate may be unstable when the dividend rate is low and the bank lending reaction to the net worth ratio is more elastic than investment reaction*.

### 7.2 Comparative Analysis of the Steady State

Finally, let us analyze the effect of monetary policy on the steady state. We will focus on the effect of the decrease of the basic discount rate ${i}^{a}$.

The change in the basic discount rate makes the saddle path shift up. Let us show Fig. 5. The economy *B* was on the divergence path at first as shown in Fig. 4, now is on the convergence path. The reason is that the decrease of the basic discount rate suppresses the excessive increase of bank lending rate. We can see that the monetary policy has a stabilizing effect on the economy.

**Proposition 3** *When the steady state is a saddle point*, *the change in the basic discount rate is likely to shift the economy from an unstable path to a convergence path*. *The monetary policy has a stabilizing effect*.

## 8 Conclusions

We constructed a simple macro model on the net worth ratio of the firm. As a result, we have shown that the increase of net worth ratio increases the output-capital ratio in the short-run. Additionally, we pointed out that the excessive reaction of bank lending to the net worth ratio constitutes a serious destabilizing factor for the economy.

We can reconfirm that these results agree well with aspects of Minsky’s theory. It is said that we have to have some regulations on bank behavior to stabilize the economy in the long run.

On the other hand, recently, from the view of stock-flow consistent (SFC) approach, it has been pointed out that traditional models such as Taylor and O’Connell often assume oversimplified hypotheses that do not do justice to Minsky’s literary analyzes and do not consider the logical implications of these hypotheses.^{o}

The SFC models are based on accounting frameworks that consistently integrate financial flows of funds with a full set of balance sheets. They show the logical interrelations between the transactions among the sectors. Passarella [14] analyzed the impact of both capital-asset inflation and consumer credit on the financial soundness of the business sector. We do not take into consideration these elements.

Our model is closer to a traditional model. We must construct a model that is applicable to the analysis of macroeconomic policies.

## Appendix 1: The Proof of the Stability of Short-Run Equilibrium

*T*:

*T*is positive. The latter condition is represented by

We can confirm that Eq. (A.2) is satisfied.

## Appendix 2: The Process to Derive Eqs. (50a) and (50b)

*A*,

The values of each element are appreciated at the steady state. Taking into account the results of static model, we can rewrite Eqs. (B.2) and (B.3) as Eqs. (50a) and (50b) in the main text.

## End note

^{1}Other than Taylor and O’Connell [15] and Adachi [1], discussed below, these include Delli Gatti and Gallegati [4] and Lavoie [11].

^{2}There are various types of dynamic models. For example, Kaldor [9] is one of the typical dynamic models. Asada [2] developed a Kaldorian cycle model and investigated the effects of government economic policy. Our dynamic model differs from them. Instead, we follow Taylor and O’Connell [15] and Adachi [1].

^{3}We describe the interest payment of the central bank advances as the basic discount rate.

^{4}This formulation follows Adachi [1].

^{5}We assume that *τ*, *ω*, and *n* are constant throughout this paper.

^{6}As discussed in the Introduction, these assumptions are based on Greenwald and Stiglitz [8].

^{7}We suppose that investment exceeds retained earnings. In other words, ${l}^{d}>0$.

^{8}Appendix 1 provides the mathematical proof of the stability of short-run equilibrium.

^{9}

^{10}We assume that there is a unique steady state.

^{11}Although we present a continuous model in this paper, we can also construct a discrete model. Whether the model is continuous or discrete, we can obtain similar results.

^{12}We suppose that the retained earnings *F*, the net worth ratio ${z}^{\ast}$, and capital accumulation rate ${k}^{\ast}$ are positive at the steady state.

^{13}For convenience, we omit the asterisks which express the values of the steady state. Appendix 2 outlines the process to derive Eqs. (50a) and (50b).

^{14}From this assumption, ${F}_{2}$ becomes negative.

## Declarations

### Acknowledgements

I am very grateful for the comments and suggestions of two referees of this Journal, which led to several improvements in the paper. Of course, all remaining errors are my own.

## Authors’ Affiliations

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