Quantity versus Price Bank Competition and Macroeconomic Performance Given Bank Concentration

Erotokritos Varelas

doi:10.1515/roe-2015-1002

Article

Quantity versus Price Bank Competition and Macroeconomic Performance Given Bank Concentration

Erotokritos Varelas

Published/Copyright: April 7, 2016

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From the journal Review of Economics Volume 66 Issue 3

Abstract

This paper elaborates upon the following three theses: First, given bank sector concentration, the other aspect of this sector that matters for the overall economy is that of price vs. quantity competition by itself. Second, the macroeconomic performance of price competition is superior, enhancing the tax base and bank profit, capitalizing additionally the banks upon public debt induced instability, which the policymaker can minimize through Taylor rule. And, third, the ultimate link between banking competition and macroeconomic performance is the bank regulation shaping bank operation in accordance with the financial needs of fiscal policy.

Keywords: bank competition; bank concentration; public debt; macroeconomic stability; monetary policy

JEL: G21; L11; E32; E44; E63

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Appendix

Under quantity bank competition, i. e. under D=D∗N/N+1 and L=L∗N/N+1, the system becomes:

(A1)MP−11+π=NΓrλ−bμN+1,

(A2)Y=c1−τY+NAN+1r1+bε+G,

and

(A3)bQNN+1−b−1Q−1N−1N−1+1=P−11+πG−τY

where:

Q=A+Γrε+μ1+bελ−bμrε+11+bε

while Q−1 is as Q but with A−1,Γ−1, and b−1. The system becomes complete with 16′ Four more exogenous variables come up: A−1,Γ−1,N, and N−1. That is, we allow for changes in the bank market structure and regulatory environment.

Under price competition, relations 13, 14, and 15, become in view of equations (2″) and (3″):

(A1′)MP−11+π=NΓμrλ−bμ1+μμ,

(A2′)Y=c1−τY+NAε−1εεr1+bε+G,

and

(A3′)bGN−b−1G−1N−1=P−11+πG−τY

where:

G=Aε−1ε1+μμ+Γμrλ−bμεr1+bεrεr1+bε1+μμ

while G−1 is as G but with A−1,Γ−1, and b−1. The system become again complete with equation 16′.

Next, if b¯''>b″¯¯ holds, i. e. if

27F23+12−9F1327F13>27J23+12−9J1327J13

then

27F23−9F1327F13>27J23+12−9J1327J13⇒27F13−927>27J23+12−9J1327J13⇒

27F13−9>27J23+12−9J13J13⇒27F13−9>27J13−9+12J13⇒

27F13−27J13>12J13⇒9F13−J13>4J13⇒J13F13−J13>49

which is true given that J>0 and

F−J=108Nr32A32Γr3+27A+2332N27A+2332N32Γr31−r−108r12N64Γr4+27A>49⇒

243Nr32A32Γr3+27A+2332N27A94+2332N72Γr31−r>1+243r12N64Γr4+27A

which is also true, because even

243Nr32A32Γr3+27A>243r12N64Γr4+27A⇒

rNA32Γr3+27A>64Γr4+27A⇒

32NAΓr5+27NA2r2>64Γr4+27A⇒32Γr4NAr−2+27ANAr2−1>0

when 2>NA.

Let us next examine whether or not bˉ′>b′‾‾⇒

27F23−1±3i+12−1∓3i−18F1354F13>27J23−1±3i+12−1∓3i−18J1354J13⇒

27F23−1±3i−18F13F13>27J23−1±3i+12−1∓3i−18J13J13⇒

27F13−1±3i−18>27J13−1±3i−18+12−1∓3iJ13⇒

J13F13−J13>4−1∓3i9−1±3i⇒J13F13−J13±0i>3i±129,

which is true as a lexicographic, of course, total order, because it has already been shown that J13F13−J13>0.

To order Y’s, we start with b¯'' and b″¯¯. Inserting b¯'' in equation A1 and solving for r:

(A4)r=MN+1P−11+πNΓλ−b=MN+127F13P−11+πNΓ9F131+3λF13−27F23−12

Replacing the r and b in equation A2 by equations A4 and 17′, respectively, one obtains the following quadratic equation in Y,

Y2U2h1−y2−Y1−c1−τ−2Uh1−y+U+G=0

with roots

(A5)Y=V±V2−4U2h1−y2U+G2U2h1−y2

where

U=NA27F13P−1NΓ9F131+3λF13−27F23−122N+1MN+1729F23+27F23+12−9F132,

V=1−c1−τ−2Uh1−y

Repeating the same procedure for the case of price competition, in which case,

(A6)r=2MP−11+πNΓλ−b=2M27J13P−11+πNΓ9J131+3λJ13−27J23−12

these results are duplicated as:

(A7)Y=Z±Z2−4W2h1−y2W+G2W2h1−y2

with

W=NA27J13P−1NΓ9J131+3λJ13−27J23−1224M1458J23+27J23+12−9J132,

Z=1−c1−τ−2Wh1−y

We have to show that

Z±Z2−4W2h1−y2W+G2W2h1−y2>V±V2−4U2h1−y2U+G2U2h1−y2

or that

2UZ±2UZ2−4W2h1−y2W+G>2WV±2WV2−4U2h1−y2U+G,

which would be true for sure iff: 2UZ>2WV⇒UZ>WV⇒

NA27F13P−1NΓ9F131+3λF13−27F23−1221−c1−τ−2Wh1−yN+1MN+1729F23+27F23+12−9F132>NA27J13P−1NΓ9J131+3λJ13−27J23−1221−c1−τ−2Uh1−y4M1458J23+27J23+12−9J132

To prove this, is the formidable task, which is why has prompted in the text the ordering of Y’s based on the sign of ∂Y/∂b. In the case of quantity competition, proceeding to solve for Y using equation A4 but not equation 17′, equation A5 becomes

(A5′)Y=[1−c(1−τ)]±[1−c(1−τ)]2−4Б2NA[P−1NΓ(λ−b)]2[h(1−y)]2(N+1)[M(N+1)(1+b)]2

where

Б=4NA2P−1NΓλ−b4h1−y2N+12MN+11+b4+4NAP−1NΓλ−b2h1−y2GN+1MN+11+b2

Next, given the spreads

rL−rD=1+2b−λr

rL−rD=4−λ+5b2r

substituting b’s yields under quantity competition

rL−rD=2MN+127F23+12−9F13+1−λMN+127F13P−11+πNΓ9F131+3λF13−27F23−12

rL−rD=227F23−1±3i+12−1∓3i−18F13MN+1+M1−λN+154F13P−11+πNΓ18F131+3λ−27F23−1±3i−12−1∓3i

while under price competition

rL−rD=5M27J23+12−9J13+4−λM27J13P−11+πNΓ9J131+3λJ13−27J23−12

rL−rD=5M27J23−1±3i+12−1∓3i−18J13+M4−λ54J13P−11+πNΓ18J131+3λ−27J23−1±3i−12−1∓3i

with the first in order of presentation spread referring in either competition case to Taylor rule, and the second spread to discretionary monetary policy. The corresponding derivatives with respect to N are:

∂rL−rD∂N=−2M27F23+12−9F13+1−λM27F13P−11+πΓ18J131+3λ−27J23−1±3i−12−1∓3iN2<0

∂rL−rD∂N=−2M27F23−1±3i+12−1∓3i−18F13P−11+πΓ36F13−27F23−1±3i−12−1∓3iN2<0

∂rL−rD∂N=−227F23−1±3i+12−1∓3i−18F13M+M1−λ54F13P−11+πΓ18F131+3λ−27F23−1±3i−12−1∓3iN2<0

and

∂rL−rD∂N=−5M27J23−1±3i+12−1∓3i−18J13+M4−λ54J13P−11+πΓ18J131+3λ−27J23−1±3i−12−1∓3iN2

Finally, it is too difficult technically to obtain the derivative ∂Y/∂N to see through its sign the impact of bank concentration on output.

Published Online: 2016-4-7

Published in Print: 2015-12-1

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Articles in the same Issue

https://doi.org/10.1515/roe-2015-1002

Keywords for this article

bank competition; bank concentration; public debt; macroeconomic stability; monetary policy