Net Worth Ratio and Financial Instability
© T. Watanabe; licensee Springer. 2013
Received: 31 July 2012
Accepted: 19 February 2013
Published: 15 March 2013
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.
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  as well as Bernanke and Gertler  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  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  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  as well as Keynes.
Minsky’s ideas lead to the development of various mathematical models.a For example, Taylor and O’Connell  and Adachi  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 advances
Bank reserves R
Bank reserves R
Central bank advances
3.1 Investment Decisions of Firmd
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 .
where r is the bank lending rate at which the firm borrows during present period.
where I is the investment. We assume that the marginal efficiency of capital decreases as k increases.
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  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
where L is debt.
where is the bank lending rate during the previous period and V is the dividend.
where y is output-capital ratio and l is debt-capital ratio.
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 is profit of bank, is bank lending, is deposit rate, is deposit from the household, is the basic discount rate, and is the central bank advances.
We assume that the marginal cost of bank lending increases more than proportionally as increases and decreases as z increases.
where θ is legal reserve rate.
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
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.
5 Equilibrium of Financial Markets
6 Short-Run Equilibrium and Comparative Statics
Let us examine the effect of exogenous variables on short-run equilibrium. When the previous bank lending rate and the basic discount rate 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 as exogenous variables. To analyze the dynamic system, we have to specify how the net worth ratio z and previous bank lending rate evolve over time.k
At the steady state , evolution of net worth and the capital are the same. The net worth ratio and bank lending rate remain constant.
The bank lending reaction to the net worth ratio exceeds the investment reaction.
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 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 .
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.
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  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
We can confirm that Eq. (A.2) is satisfied.
Appendix 2: The Process to Derive Eqs. (50a) and (50b)
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.
2There are various types of dynamic models. For example, Kaldor  is one of the typical dynamic models. Asada  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  and Adachi .
3We describe the interest payment of the central bank advances as the basic discount rate.
4This formulation follows Adachi .
5We assume that τ, ω, and n are constant throughout this paper.
6As discussed in the Introduction, these assumptions are based on Greenwald and Stiglitz .
7We suppose that investment exceeds retained earnings. In other words, .
8Appendix 1 provides the mathematical proof of the stability of short-run equilibrium.
10We assume that there is a unique steady state.
11Although 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.
12We suppose that the retained earnings F, the net worth ratio , and capital accumulation rate are positive at the steady state.
13For convenience, we omit the asterisks which express the values of the steady state. Appendix 2 outlines the process to derive Eqs. (50a) and (50b).
14From this assumption, becomes negative.
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.
- Adachi H: Unstable economy and financial factors. In The theory of macro dynamics. Yuhikaku, Tokyo; 1994:293–324. (Japanese edn) (Japanese edn)Google Scholar
- Asada T: Government finance and wealth effect in a Kaldorian cycle model. J Econ 1987, 47: 143–166. 10.1007/BF01237549View ArticleGoogle Scholar
- Bernanke BS, Gertler M: Agency cost, net worth, and business fluctuations. Am Econ Rev 1989, 82: 901–921.Google Scholar
- Delli Gatti D, Gallegati M: Financial instability, income distribution and the stock market. J Post Keynes Econ 1990, 12: 356–374.Google Scholar
- Dos Satos C: A stock-flow consistent general framework for formal Minskian analyses of closed economies. J Post Keynes Econ 2005, 27: 711–735.Google Scholar
- Dos Satos C: Keynesian theorising during hard times: stock-flow consistent models as an unexplored ‘frontier’ of Keynesian macroeconomics. Camb J Econ 2006, 30: 541–565.View ArticleGoogle Scholar
- Fisher I: The debt-deflation theory of Great Depression. Econometrica 1933, 1: 337–357. 10.2307/1907327View ArticleGoogle Scholar
- Greenwald B, Stiglitz JE: Financial market imperfection and business cycles. Q J Econ 1993, 108: 77–114. 10.2307/2118496View ArticleGoogle Scholar
- Kaldor N: A model of the trade cycles. Econ J 1940, 50: 78–92. 10.2307/2225740View ArticleGoogle Scholar
- Keynes JM: The general theory of employment, interest, and money. Macmillan & Co, London; 1936.Google Scholar
- Lavoie M: Interest rate in post-Keynesian models of growth and distribution. Metroeconomica 1995, 46: 146–177. 10.1111/j.1467-999X.1995.tb00375.xView ArticleGoogle Scholar
- Minsky HP: John Maynard Keynes. Columbia University Press, New York; 1975.Google Scholar
- Modigliani F, Miller M: The cost of capital, corporation finance and the theory of investment. Am Econ Rev 1958, 48: 261–297.Google Scholar
- Passarella M: A simplified stock-flow consistent dynamic model of the systemic financial fragility in the ‘New Capitalism’. J Econ Behav Organ 2012,83(3):570–582. 10.1016/j.jebo.2012.05.011View ArticleGoogle Scholar
- Taylor L, O’Connell SA: A Minsky crisis. Q J Econ 1985, 100: 871–885.View ArticleGoogle Scholar
- Tobin J: A general equilibrium approach to monetary theory. J Money Credit Bank 1969, 1: 15–29. 10.2307/1991374View ArticleGoogle Scholar
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