A DSGE Model with Government-owned Banks

Hamilton Galindo Gil; Alexis Montecinos

doi:10.1515/bejm-2023-0099

Published by De Gruyter April 9, 2024

A DSGE Model with Government-owned Banks

Hamilton Galindo Gil and Alexis Montecinos

From the journal The B.E. Journal of Macroeconomics

https://doi.org/10.1515/bejm-2023-0099

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Abstract

How relevant are government-owned banks in the economy, especially during recessions? We study the role of government-owned banks in a dynamic stochastic general equilibrium (DSGE) model with heterogeneous financial intermediaries, heterogeneous households, and minimum capital requirement constraints. We show that the capitalization of government-owned banks during recessions smooths the effects of a negative shock and helps the economy recover more quickly. However, these stabilizing effects could be partially offset by banks’ inefficiency in transforming one unit of capital into loans. Therefore, ignoring the heterogeneity between private and government-owned banks may lead to misleading assessments and conclusions regarding the effects of economic policies on the macroeconomic and banking variables. This is particularly important for evaluating the effectiveness of macroprudential policies.

Keywords: government-owned banks; general equilibrium; capitalization; inefficiency

JEL Classification: G20; G21; G28

Corresponding author: Hamilton Galindo Gil, Department of Finance and Economics, Monte Ahuja College of Business, Cleveland State University, 1860 E 18th St, Cleveland, OH 44114, USA, E-mail: h.galindogil@csuohio.edu

Acknowledgments

We thank Markus Brunnemeier, Doug Diamond, Victor Duarte, Julia Fonseca, Deborah Lucas, Robert Merton, and Jonathan Parker for helpful suggestions to earlier drafts. Hamilton Galindo Gil expresses his thankfulness to Jesus Christ for his steadfast love and faithfulness.

Research funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

A Agent’s Optimization Problem

A.1 Private Banks’ Dynamic Optimization Problem

The cash flow of the private bank is defined as

(48) ( 1 − τ ) ( i t L t p + π t f − r t D t p ) = d i v t p + R E t p ,

where d i v t p represents dividends. Using this cash flow equation and the Eqs. (20) and (21), we have the budget constraint of the private bank, given by

(49) d i v t p = ( 1 − τ ) ( i t L t p + π t f − r t D t p ) − L t + 1 p + D t + 1 p + L t p − D t p .

Reordering the terms

(50) d i v t p = ( 1 + ( 1 − τ ) i t ) L t p − ( 1 + ( 1 − τ ) r t ) D t p + ( 1 − τ ) π t f − L t + 1 p + D t + 1 p ,

where inflows are formed by loan interest payment ( 1 + ( 1 − τ ) i t ) L t p , and new deposits D t + 1 p . Outflow is formed by the debt payment (deposits), ( 1 + ( 1 − τ ) r t ) D t p , in which the interest rate is net of taxes, which reflects the tax shield, and the new loans, L t + 1 p .

Then, the optimization problem becomes the following.

(51) max D t + 1 p , L t + 1 p E 0 ∑ t = 0 ∞ ∏ j = 0 t q j d i v t p

subject to

(52) ( 1 − γ p ) L t p ≥ D t + 1 p .

B Representative Agent Model

In this section, we describe the model considering a representative household. His optimization problem is as follows

(53) max { c t , l t , D t + 1 , s t + 1 } t = 0 ∞ E 0 ∑ t = 0 ∞ β t c t − l t ω ω 1 − θ 1 − θ

subject to

(54) ( 1 + r t ) D t + w t l t + d i v t p s t + p t s t ≥ c t + D t + 1 + p t s t + 1

Following the same procedure to find first-order conditions and keeping invariant the other building blocks of the model, we have the following system of equations, which characterizes the representative-agent model.

Figure 11 illustrates the dynamics of real variables in the representative agent model. We assume that the government only injects capital into its banks and does not transfer goods to households. As the figure shows, the presence of government banks accelerates economic recovery. Furthermore, consumption does not drop significantly due to income effects, but the inefficiency captured by the parameter κ affects the effectiveness of government policy.

Figure 11:

Real sector variables (different values of inefficiency parameter κ). Impulse-response functions shown in the figure are calculated considering the economy suffers a recession, that is, a negative shock of one standard deviation in the total factor productivity.

C The Government Budget Constraint and the Aggregate Resource Constraint

In this section, we derive the government budget constraint (31) and the aggregate resource constraint (38).

C.1 The Government Budget Constraint

Equation (55) represents the budget constraint of the government. The expenses (left) side of the government budget constraint is formed by the total capital injection from the government to its banks (1 − ϕ)g _t, and total transfers to non-ricardian households (1 − λ)tr_t. The income (right) side is formed by tax revenues, dividends from its banks ( 1 − ϕ ) d i v t g , external resources B _t (such as savings from previous periods), and private benefits PB _t.

(55) ( 1 − ϕ ) g t + ( 1 − λ ) t r t = τ i t L t − r t D t + ϕ π t f + ( 1 − ϕ ) d i v t g + B t + P B t ,

where

(56) g t ≡ ( 1 − κ ) g t + κ g t ,

(57) t r t ≡ t r t b + t r t c .

Taking into account these two expressions in the budget constraint (55), it is now expressed as

( 1 − ϕ ) ( 1 − κ ) g t + ( 1 − ϕ ) κ g t + ( 1 − λ ) t r t b + ( 1 − λ ) t r t c = τ i t L t − r t D t + ϕ π t f + ( 1 − ϕ ) d i v t g + B t + P B t .

We next consider that countercyclical transfers are financed by B _t (expression 58) and the private benefit obtained by the government-owned bank manager is a fraction 1 − κ of capital injection (expression 59).

(58) B t = ( 1 − λ ) t r t c ,

(59) P B t = ( 1 − ϕ ) ( 1 − κ ) g t

Taking these two expressions into account in the government budget constraint (58), we have

( 1 − ϕ ) κ g t + ( 1 − λ ) t r t b = τ i t L t − r t D t + ϕ π t f + ( 1 − ϕ ) d i v t g ,

which is the budget constraint shown in Equation (31).

C.2 The Aggregate Resource Constraint

We begin with the individual household constraint for the Ricardian and non-Ricardian types, given by

( 1 + r t ) D t + w t l r , t + d i v t p s t + p t s t = c r , t + D t + 1 + p t s t + 1 w t l n r , t + t r t = c n r , t

Considering that s _t = 1 for all t, and the population of Ricardian (λ) and non-Ricardian agents (1 − λ), the aggregate budget constraints are

( 1 + r t ) D t + w t λ l r , t + ϕ d i v t p = λ c r , t + D t + 1 w t ( 1 − λ ) l n r , t + ( 1 − λ ) t r t = ( 1 − λ ) c n r , t ,

where ϕ d i v t p is the aggregate dividends distributed by private banks. We next sum every side of these equations, resulting

( 1 + r t ) D t + w t λ l r , t + ( 1 − λ ) l n r , t ︸ = l t + ϕ d i v t p + ( 1 − λ ) t r t = D t + 1 + λ c r , t + ( 1 − λ ) c n r , t ︸ c t ( 1 + r t ) D t + w t l t + ϕ d i v t p + ( 1 − λ ) t r t = D t + 1 + c t

Equivalently,

(60) D t + 1 + c t = ( 1 + r t ) D t + w t l t + ϕ d i v t p + ( 1 − λ ) t r t

We then consider the government budget constraint (55), given by

(61) ( 1 − ϕ ) g t + ( 1 − λ ) t r t = τ i t L t − r t D t + ϕ π t f + ( 1 − ϕ ) d i v t g + B t + P B t

Next, we sum the aggregate household budget constraint (60) and the government budget constraint (61), resulting the following expression

(62) ( 1 − ϕ ) g t + D t + 1 + c t = w t l t + ( 1 + r t ) D t + ϕ d i v t p + ( 1 − ϕ ) d i v t g + τ ( i t L t − r t D t + ϕ π t f + B t + P B t

Next, we work on some of the terms of Eq. (62) as follows.

w _t l _t: From the first-order conditions of the firm, the law of capital movement, and the fact that capital is totally financed with loans, the following holds
w t l t = Y t − ( 1 + i t ) L t + L t + 1 − I t
π t f : Using the definition of π t f and the previous expression of w _t l _t, we have
π t f ≡ Y t + L t + 1 − w t l t − ( 1 + i t ) L t − I t = 0

Using these two expression in (62) and reordering terms, we have

(63) ( 1 − ϕ ) g t + D t + 1 + c t + I t = Y t − ( 1 + i t ) L t + L t + 1 + ( 1 + r t ) D t + ϕ d i v t p + ( 1 − ϕ ) d i v t g + τ ( i t L t − r t D t ) ︸ Term A + B t + P B t

We now work on Term A.

(64) ϕ d i v t p = ϕ ( 1 − τ ) ( i t L t p − r t D t p ) − ϕ R E t p ( 1 − ϕ ) d i v t g = ( 1 − ϕ ) ( 1 − τ ) i t L t g − r t D t g − ( 1 − ϕ ) R E t g ϕ d i v t p + ( 1 − ϕ ) d i v t g = ( 1 − τ ) ( i t L t − r t D t ) − ( ϕ R E t p + ( 1 − ϕ ) R E t g ) ϕ d i v t p + ( 1 − ϕ ) d i v t g + τ ( i t L t − r t D t ) = ( i t L t − r t D t ) − ( ϕ R E t p + ( 1 − ϕ ) R E t g )

Introducing Eqs. (64) into (63), we have

(65) ( 1 − ϕ ) g t + D t + 1 + c t + I t = Y t − L t + L t + 1 + D t − ( ϕ R E t p + ( 1 − ϕ ) R E t g ) ︸ Term B + B t + P B t

Working on Term B,

(66) ϕ R E t p = ϕ γ L t + 1 p − L t p ( 1 − ϕ ) R E t g = ( 1 − ϕ ) γ L t + 1 g − L t g ϕ R E t p + ( 1 − ϕ ) R E t g = γ L t + 1 − L t ,

where γ ^p = γ ^g ≡ γ. Introducing Eqs. (66) into (65), we have

(67) ( 1 − ϕ ) g t + D t + 1 + c t + I t = Y t + D t + ( 1 − γ ) L t + 1 − L t + B t + P B t

We now consider private benefits PB _t = (1 − ϕ)(1 − κ)g _t, then Eq. (67) becomes

(68) D t + 1 + c t + I t = Y t + D t + ( 1 − γ ) L t + 1 − L t + B t − ( 1 − ϕ ) κ g t

In equilibrium at period t, we have

D t = ( 1 − γ ) L t − ( 1 − ϕ ) κ g t D t + 1 = ( 1 − γ ) L t + 1 − ( 1 − ϕ ) κ g t + 1 ,

Introducing these two expressions into Eq. (68), we obtain the aggregate resource constraint, given by

(69) c t + I t = Y t + B t + ( 1 − ϕ ) κ Δ g t + 1 − ( 1 − ϕ ) κ g t ,

where B _t is an external resource to finance countercyclical transfers, and Δg _t+1 ≡ g _t+1 − g _t. First, at the steady-state equilibrium, we have g _t = g _t+1 = B _t = 0, then

c s s + I s s = Y s s

Second, we implicitly assume that

B t + ( 1 − ϕ ) κ Δ g t + 1 = ( 1 − ϕ ) κ g t ,

resulting

(70) c t + I t = Y t .

We recognize that this is a strong assumption. However, since the effects of government policy (countercyclical transfers and banking capital injection) are already captured by c _t (related to transfers) and I _t (related to loans and hence capital injection), it is reasonable to make this assumption. Relaxing this assumption would be an important research avenue that we leave for future work.

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Received: 2023-06-14

Accepted: 2024-03-25

Published Online: 2024-04-09

A DSGE Model with Government-owned Banks

Abstract

Acknowledgments

A Agent’s Optimization Problem

A.1 Private Banks’ Dynamic Optimization Problem

B Representative Agent Model

C The Government Budget Constraint and the Aggregate Resource Constraint

C.1 The Government Budget Constraint

C.2 The Aggregate Resource Constraint

References

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