Efficiency between large and small banks: the role of

Introduction The Model Analytical Analysis Numerical Analysis MonteCarlo simulations Conclusions

Efficiency between large and small

banks: the role of nonlinearity in

presence of capital regulation

Serena Brianzoni1 Giovanni Campisi2 Annarita Colasante3

1Department of Management

Polytechnic University of Marche

2Department of Economics Marco Biagi

University of Modena and Reggio Emilia

3Laboratory of Experimental Economics

Universitat Jaume I

The 11th Nonlinear Economic Dynamics conference - Kyiv

School of Economics (KSE), September, 4-6, 2019


Table of contents

Introduction

The Model

Analytical Analysis

Numerical Analysis

MonteCarlo simulations

Conclusions


Banking industry in economics

In this work we focus on the cost-e�ciency of the italian banking

system considering banks of di�erent size (large and small) as

Brianzoni et al.(2018).

We are going to stress the higher e�ciency of small banks with

respect to large banks following empirical evidence.


Empirical literature on italian banking

sector

• the role of small banks in supporting local �rms and families

thanks to their long and stable relationship (Alessandrini et

al.(2018));

• In period of crisis local banks increased their loans supply in

favour of families and small �rms (Stefani et al.(2016));

• thanks to soft information local banks mitigate information

asymmetries in credit markets allowing them a lower credit

rationing (Barboni and Rossi (2019)).


further empirical results

• e�ciency tends to decrease with size;

• cooperatives perform better than others

• geographic localisation matter (Aiello et al. (2013)).

• regulation plays a predominant role in period of �nancial

distress (Alessandrini et al. (2016,2018)).


Assumptions and motivations

Taking into account the italian banking sector we make the

following assumptions:

• We consider large banks and small banks.

• We assume a quadratic cost function for large banks and a

linear cost function for small banks.

• We introduce a nonlinear demand function as in

Tramontana(2010).

• Empirical evidence shows that e�ciency tends to decrease with

size.


A first banking model with capital regulationFanti(2014)

• banks of the same size;

• identical marginal costs;

• focus on the role of regulation;

• di�erent bifurcation structure.


The benchmark modelBrianzoni et al. (2018)

• large and small banks;

• quadratic costs for large banks and linear costs for small banks;

• Justi�cation of the speci�c use of expectations;

• focus on the role of e�ciency;

• An interesting economic scenario emerges from the local

stability analysis of �xed points.


Further extensionsOur model

We start from the work of Brianzoni et al. (2018) and we introduce

nonlinearity in the demand function too as in

Tramontana(2010).

Given that the model is hard to solve analytically we study it from a

numerical point of view.

We focus on di�erences and similarities with respect the work of

Brianzoni et al. (2018).


The principal ingredients of the model

Inverse demand function for loans

rL(L1,t + L2,t) = a− b

(1

L1,t + L2,t

)Moreover we have:

Li ,t = Ki ,t + Di ,t

and

Ki ,t = γLi ,t

for every i = 1, 2


The Map

L1,t+1 = L1,t + αL1,t

[a− bL2,t

(L1,t+L2,t)2− γ(rk − c1)− c1(2L1,t + 1)

]L2,t+1 =

√bL1,t

a−γ(rk−c2)−2c2− L1,t

(1)

where

• Li ,t , i = 1, 2, represents the loans of the i-th bank during

period t.

• α1 is a positive parameter capturing the speed of adjustment

of bank i's loans.

• c1 and c2 are positive parameters representing the marginal

costs.

• 0 < γ < 1 is a �xed percentage determined by the regulator.


bounded costs

We underline that in our model both the costs are upper bounded:

c1 < cu1 := a−γrk1−γ (because of the positivity of equilibrium loan

levels) and c2 < cu2 := a−γrk2−γ (for the de�nition set of the map T )

hence cu2 < cu1


The boundary equilibrium scenario

Di�erently from Brianzoni et al.(2018) the model never admits a

boundary solution.

From an economic point of view the market is better served if both

types of banks exist.

Now both the costs are upper bounded.


Small banks

Figure 1: Parameter values: γ = 0.16, α = 1.9, c1 = 1.5, rk = 2.5,a = 3.2, b = 0.205. Initial conditions L1,0 = 0.36, L2,0 = 0.3.


Large banks

Figure 2: Parameter values: γ = 0.16, α = 1.9, c2 = 1.35, rk = 2.5,a = 3.2, b = 0.201. Initial conditions L1,0 = 0.36, L2,0 = 0.3.


The role of regulation

Figure 3: Parameter values: α = 1.8, c1 = 1.52, c2 = 1.35, rk = 2.5,a = 3.2, b = 0.33. Initial conditions L1,0 = 0.39, L2,0 = 0.31.


Controverse effects of regulation



Economic Scenarios

• the e�ciency of small banks w.r.t large banks is con�rmed;

• Di�erently from Brianzoni et al.(2018) the cost e�ciency for

small banks holds for certain values of their marginal costs;

• for large banks we obtain similar results;

• regulation matters but it should be considered with others

measures together.


Cost efficiency 1the simulation

Figure 5: Parameter values: α = 1.9, rk = 2.5, a = 3.2, b = 0.205,γ = 0.16. Initial conditions L1,0 = 0.36, L2,0 = 0.4.


Cost efficiency 2



Cost efficiency with regulation



Large banks and regulation



Small banks and Regulation



Final remarks

The model focuses on the cost-e�ciency between large and small

banks.

Our work seems to con�rm empirical evidence of a greater

e�ciency of small banks w.r.t. small banks.

Di�erently from Brianzoni et al. (2018) the e�ciency of small

banks holds if their costs are not too low w.r.t. that of large banks.


Future extensions

Some issues and further development of the model:

• a direct role of regulation introducing quadratic costs for small

banks too;

• introduce a functional form for regulation parameter.

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Efficiency between large and small banks: the role of