Tell me where to stop: thresholds in the bank lending and output growth relationship

Abstract

The relationship between the growth of financial development and output growth is examined both analytically and empirically, using a long series of data for the seven largest economies in the world. The findings suggest that, other things being constant, a positive relationship between lending and output growth exists, however with diminishing returns after a country-specific threshold. These findings assist in explaining the findings of earlier studies and bear various policy implications.

Appendices

Appendix

For the first part of Proposition 1, start with the supply of deposits Eq. (7) and use Eq. (19) to eliminate $D_{t}$ and also use the fact that $\Gamma _{t}=N_{t}L_{t}$ to get

$$\begin{aligned} R_{t}^{d}=\frac{1}{\beta ^{2}}\frac{N_{t}L_{t}-V_{t}}{sY_{t}-N_{t}L_{t}+V_{t} } \end{aligned}$$

(40)

Taking the derivative gives

$$\begin{aligned} \frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}=\frac{1}{\beta ^{2}}N_{t}\frac{\frac{\mathrm{d}L_{t}}{\mathrm{d}V_{t} }sY_{t}}{\left( sY_{t}-N_{t}L_{t}+V_{t}\right) ^{2}} \end{aligned}$$

We need $\frac{\mathrm{d}L_{t}}{\mathrm{d}V_{t}}$ which we can get by first use Eq. (26) to find

$$\begin{aligned} \frac{\mathrm{d}L_{it}}{\mathrm{d}R_{t}^{D}}=-\frac{1}{1-\alpha }\left[ \frac{\alpha A}{2} \right] ^{\frac{1}{1-\alpha }}\left[ R_{t}^{D}\right] ^{-\frac{1}{1-\alpha } -1} \end{aligned}$$

(41)

The derivative $\frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}$ we can derive from Eq. (40)

$$\begin{aligned} \frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}=\frac{1}{\beta ^{2}}N_{t}\frac{-\frac{1}{1-\alpha } \left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha }}\left[ R_{t}^{D} \right] ^{-\frac{1}{1-\alpha }-1}\frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}sY_{t}}{\left( sY_{t}-N_{t}L_{t}+V_{t}\right) ^{2}}+\frac{1}{\beta ^{2}}\frac{-sY_{t}}{ \left( sY_{t}-N_{t}L_{t}+V_{t}\right) ^{2}} \end{aligned}$$

which we can solve for$\frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}$. That is,

$$\begin{aligned} \frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}=\frac{-sY_{t}}{\beta ^{2}\left( sY_{t}-N_{t}L_{t}+V_{t}\right) ^{2}+\frac{1}{1-\alpha }\left[ \frac{\alpha A }{2}\right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{D}\right] ^{-\frac{1}{ 1-\alpha }-1}sY_{t}}<0 \end{aligned}$$

(42)

We know that the output next period is

$$\begin{aligned} \frac{\mathrm{d}Y_{t+1}}{\mathrm{d}V_{t}}=\alpha N_{t}AL_{t}^{\alpha -1}\frac{\mathrm{d}L_{t}}{\mathrm{d}V_{t}} \end{aligned}$$

Differentiate it to get

$$\begin{aligned} \frac{\mathrm{d}Y_{t+1}}{\mathrm{d}V_{t}}=\alpha N_{t}AL_{t}^{\alpha -1}\frac{\mathrm{d}L_{it}}{ dR_{t}^{D}}\frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}} \end{aligned}$$

which by substituting in Eqs. (41) and (42) becomes

$$\begin{aligned} \frac{\mathrm{d}Y_{t+1}}{\mathrm{d}V_{t}}=\frac{\alpha N_{t}AL_{t}^{\alpha -1}\frac{1}{ 1-\alpha }\left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha }}\left[ R_{t}^{D}\right] ^{-\frac{1}{1-\alpha }-1}sY_{t}}{\beta ^{2}\left( sY_{t}-N_{t}L_{t}+V_{t}\right) ^{2}+\frac{1}{1-\alpha }\left[ \frac{\alpha A }{2}\right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{D}\right] ^{-\frac{1}{ 1-\alpha }-1}sY_{t}}>0 \end{aligned}$$

Use in the above

$$\begin{aligned} \beta ^{2}\left( sY_{t}-N_{t}L_{t}+V_{t}\right) =\frac{D_{t}}{R_{t}^{d}} \end{aligned}$$

to get

$$\begin{aligned} \frac{\mathrm{d}Y_{t+1}}{\mathrm{d}V_{t}}=\frac{\frac{2}{1-\alpha }\left( \frac{\alpha A}{2} \right) ^{\frac{1}{1-\alpha }}sY_{t}N_{t}R_{t}^{d}}{D_{t}^{2}\left( R_{t}^{d}\right) ^{\frac{-2-\alpha }{1-\alpha }}+\frac{1}{1-\alpha }\left( \frac{\alpha A}{2}\right) ^{\frac{1}{1-\alpha }}sY_{t}N_{t}}>0 \end{aligned}$$

which is Eq. (37) in the text.

Appendix

For the second part of Proposition 1, take the second derivative of Eq. (37)

$$\begin{aligned}&\frac{\mathrm{d}^{2}Y_{t+1}}{\mathrm{d}V_{t}^{2}} =-\frac{2}{1-\alpha }\left[ \frac{\alpha A }{2}\right] ^{\frac{1}{1-\alpha }}N_{t}sY_{t} \nonumber \\&\quad \frac{2D_{t}\left( R_{t}^{d}\right) ^{\frac{-3-2\alpha }{1-\alpha }}\frac{ dD_{t}}{\mathrm{d}V_{t}}{-}\left[ \frac{3+2\alpha }{1-\alpha }\left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-4-\alpha }{1-\alpha }}{+}\frac{1}{ 1-\alpha }\left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{D}\right] ^{-2}sY_{t}\right] \frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}}{\left( \left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-3-2\alpha }{1-\alpha }}+ \frac{1}{1-\alpha }\left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha } }N_{t}\left[ R_{t}^{D}\right] ^{-1}sY_{t}\right) ^{2}}\nonumber \\ \end{aligned}$$

(43)

Since Eq. (7) when $\Pi _{t}=sY_{t}$ is

$$\begin{aligned} R_{t}^{d}=\frac{1}{\beta ^{2}}\frac{D_{t}}{sY_{t}-D_{t}}, \end{aligned}$$

(44)

the derivative $\frac{\mathrm{d}D_{t}}{\mathrm{d}V_{t}}$ becomes

$$\begin{aligned} \frac{\mathrm{d}D_{t}}{\mathrm{d}V_{t}}=\beta ^{2}\frac{\left( sY_{t}-D_{t}\right) ^{2}}{sY_{t} }\frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}} \end{aligned}$$

Use it to substitute $\frac{\mathrm{d}D_{t}}{\mathrm{d}V_{t}}$ in Eq. (43) which implies

$$\begin{aligned} \frac{\mathrm{d}^{2}Y_{t+1}}{\mathrm{d}V_{t}^{2}}= & {} \frac{2}{1-\alpha }\left[ \frac{\alpha A}{ 2}\right] ^{\frac{1}{1-\alpha }}N_{t}sY_{t} \\&\frac{\left[ \begin{array}{c} \frac{3+2\alpha }{1-\alpha }\left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-4-\alpha }{1-\alpha }}+\frac{1}{1-\alpha }\left[ \frac{\alpha A}{2} \right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{d}\right] ^{-2}sY_{t} \\ -2D_{t}\left( R_{t}^{d}\right) ^{\frac{-3-2\alpha }{1-\alpha }}\beta ^{2} \frac{\left( sY_{t}-D_{t}\right) ^{2}}{sY_{t}} \end{array} \right] \frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}}{\left( \left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-3-2\alpha }{1-\alpha }}+\frac{1}{1-\alpha }\left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{D}\right] ^{-1}sY_{t}\right) ^{2}} \end{aligned}$$

Call the expression in the parenthesis $\Upsilon _{t}$. Therefore,

$$\begin{aligned} \Upsilon _{t}\equiv & {} \frac{3+2\alpha }{1-\alpha }\left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-4-\alpha }{1-\alpha }}+\frac{1}{ 1-\alpha }\left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{d}\right] ^{-2}sY_{t} \\&-2D_{t}\left( R_{t}^{d}\right) ^{\frac{-3-2\alpha }{1-\alpha }}\beta ^{2} \frac{\left( sY_{t}-D_{t}\right) }{sY_{t}}\left( sY_{t}-D_{t}\right) \end{aligned}$$

Use the fact that

$$\begin{aligned} sY_{t}-D_{t}=\frac{D_{t}}{\beta ^{2}R_{t}^{d}} \end{aligned}$$

which is a rearrangement of Eq. (44) to get

$$\begin{aligned} \Upsilon _{t}= & {} \frac{3+2\alpha }{1-\alpha }\left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-4-\alpha }{1-\alpha }}+\frac{1}{1-\alpha }\left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{d}\right] ^{-2}sY_{t} \\&-2\left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-4-\alpha }{ 1-\alpha }}\frac{\left( sY_{t}-D_{t}\right) }{sY_{t}} \\= & {} \left( \frac{1+4\alpha }{1-\alpha }\right) \left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-4-\alpha }{1-\alpha }}+\frac{1}{1-\alpha }\left[ \frac{\alpha A}{2}\right] ^{\frac{1}{1-\alpha }}N_{t}\left[ R_{t}^{d}\right] ^{-2}sY_{t} \\&+2\left( D_{t}\right) ^{2}\left( R_{t}^{d}\right) ^{\frac{-4-\alpha }{ 1-\alpha }}\frac{D_{t}}{sY_{t}} \end{aligned}$$

Thus,

$$\begin{aligned}&\frac{\mathrm{d}^{2}Y_{t+1}}{\mathrm{d}V_{t}^{2}}=\frac{\frac{2sY_{t}N_{t}}{1-\alpha }\left( \frac{\alpha A}{2}\right) ^{\frac{1}{1-\alpha }}\left( R_{t}^{d}\right) ^{ \frac{-4-\alpha }{1-\alpha }}\left( \frac{1+4\alpha }{1-\alpha }D_{t}^{2}+ \frac{sY_{t}N_{t}}{1-\alpha }\left( \frac{\alpha A}{2}\right) ^{\frac{1}{ 1-\alpha }}\left( R_{t}^{d}\right) ^{\frac{6-\alpha }{1-\alpha }}+2\frac{ D_{t}^{3}}{sY_{t}}\right) }{\left( D_{t}^{2}\left( R_{t}^{d}\right) ^{\frac{ -3-2\alpha }{1-\alpha }}+\frac{1}{1-\alpha }\left( \frac{\alpha A}{2}\right) ^{\frac{1}{1-\alpha }}sY_{t}N_{t}\left( R_{t}^{d}\right) ^{-1}\right) ^{2}}\nonumber \\&\quad \frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}} \end{aligned}$$

because everything is positive but $\frac{\mathrm{d}R_{t}^{d}}{\mathrm{d}V_{t}}<0$. This is the expression in Eq. (38) in the text.

Appendix

In this appendix, we conduct a sensitivity analysis of the results presented in the main study. More specifically, the analysis is executed again using a slightly different definition for output growth (instead of real we use nominal GDP to proxy for output growth). Furthermore, we augment the sample by including three more European countries (Austria, Belgium and the Netherlands). The selection of the additional countries stems from data availability.

Table 3 presents the results from estimations in which the output growth is obtained as the difference of logarithm of nominal GDP. As the reader may observe, while the numerical values are different from the ones of Table 2 in the main analysis the same conclusions hold: conditional correlations are higher before the threshold value and decrease afterwards suggesting diminishing returns.

Table 4 reports the results for the three additional countries. Similar to the findings for the G7 countries, the additional countries exhibit similar behaviour, i.e. the conditional correlations are higher before the threshold and lower after it. In line with the figures in Table 2, the abruptness in the change of these correlations is not related to the size of the change. In the case of the Netherlands, there is a relatively smooth transition to the lower correlation regime, while in Austria and Belgium an abrupt change takes place. With regard to the threshold values, the change in correlation for the Netherlands occurs at a relatively low value, while for Austria and Belgium the change occurs at higher ones.¹⁸ Generally, from the above results, it can be inferred that the stock of loans does not hamper growth at any debt-to-GDP ratio even when additional countries are included in the sample or when the definition of growth is changed.