## Abstract

We construct an overlapping generations growth model, where young consumers choose how to allocate resources among real investment (deposits), acquisition of bank ownership, and young-age consumption. At old age, consumers sell bank ownership and collect their bank deposits to support consumption. The model shows that an increase in banks’ market power stimulates bank profit and bank value, thereby raising the resources required for young consumers to acquire bank ownership. This causes a crowding-out effect on real investment, the magnitude of which is amplified with higher endowment growth rate and real investment return. Finally, we conduct a welfare analysis of the investment crowding-out effect.

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## Notes

- 1.
Crowding-out refers to the feature that productive, growth-promoting investments are reduced because the young generation has to spend more resources on the acquisition of bank ownership. Not only relaxed competition, but also other activities that raise bank profits and, hence, bank equity market value, such as relaxed regulation, may also contribute to the crowding out of deposits that fund real investment.

- 2.
- 3.
The overlapping generations framework has been applied before to model investment failures in the banking industry. Gersbach and Wenzelburger (2008) show that risk premia built into loan prices are insufficient to prevent banking crises. Gersbach and Wenzelburger (2011) analyze stability issues in the banking sector from a macroeconomic perspective.

- 4.
Focusing only on safe returns for bank investments greatly simplify the derivations of our main results. Shy and Stenbacka (2017) develop an OLG model where banks actually fail and may be bailed out with taxpayer money. Also, to simplify, the model assumes that banks are not subjected to reserve requirements. The concluding section provides a discussion of possible model extensions involving money creation by banks.

- 5.
To simplify, the analysis relies on constant marginal investment productivity \(\rho \). One could assume diminishing returns or that banks run out of productive investments to fund after collecting a certain amount of deposits. To avoid introducing additional notation we do not incorporate these aspects into our analysis.

- 6.
In fact, an anonymous referee has pointed to us that the steady-state equilibrium property is actually implied by the assumption of time-invariant deposit rate, \(r_t = r\) for all

*t*.

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## Acknowledgements

We are deeply indebted to two anonymous referees for their extensive guidance and proposed modifications that greatly improved this article. We also thank seminar participants at Tufts University and the Bank of Canada for valuable comments. Special thanks to Sofia Priazhkina and Maarten van Oordt for valuable suggestions. Rune Stenbacka acknowledges financial support from Suomen Arvopaperimarkkinoiden Edistamissaatio. The work on this project has started while Oz Shy was teaching at MIT Sloan School of Management.

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## Appendix A Derivations

### Appendix A Derivations

**Derivations of Result** 1 Derivations involve differentiation of the equilibrium values (10) and follow from Assumption 1.

**Derivations of Results** 2, *and* 3 Derivations involve differentiations of the equilibrium measure of crowding out (12) and Assumption 1.

**Derivation of Result** 5*(a)* Suppose that we maintain the consumption of the young at the equilibrium level given in the left term of (13). We start with period \(t=0\). Subtracting total consumption spending (young of generation \(t=0\) and old of generation \(t=-1\)) from total resources available in period \(t=0\). yields

From left to right, the first two terms are aggregate resources available in period \(t=0\). These are the sum of the endowment of the young \(\omega _0\) and the investment gross return on the investment made by the young of generation \(t=-1\), given by \(d_{-1}(1+{\rho })\). The third and fourth terms are consumption of the young in \(t=0\) (whose constancy is assumed) and the consumption of the old in \(t=0\) that we need to maintain in order to construct a Pareto-superior allocation.

Expression (A.3) shows that the difference between available resources and consumption equals exactly the equilibrium amount of deposits and investment level by generation \(t=0\) as derived in (10). This shows that trying to maintain the equilibrium consumption of the young and old in period \(t=0\) restricts the economy from growing beyond the equilibrium level.

Moving on to period \(t=1\), the difference between available resources and aggregate consumption becomes

Note that the second term in (A.4) is the amount invested in period \(t=0\) (\(d_0\)) taken from (A.3) and multiplied by \(1+{\rho }\). Once again, keeping the same consumption levels of the young and old in \(t=1\) leaves the same amount \(d_1\) allocated to investment which is given in (10).

Moving on to periods \(t=2, 3,\ldots \), we can show that investment in each period *t* cannot be increased without reducing the consumption of the young, while maintaining the same consumption levels of the old.

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Shy, O., Stenbacka, R. Bank competition, real investments, and welfare.
*J Econ* **127, **73–90 (2019). https://doi.org/10.1007/s00712-018-0634-0

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### Keywords

- Investment crowding-out
- Size of the banking sector
- Deposit market competition
- Economic growth

### JEL Classification

- G21
- O41