Money creation system in the Real Business Cycle theory

Economics Letters 0165-1765/93/$06.00

42 (1993) 367-371 0 1993 Elsevier

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Money creation system in the Real Business Cycle theory Yasushi

Ohkusa*

Department of Economics, Keceived Accepted

Ritsumeikan

University, 56-1, Kitamachi Tohjiin Kita-ku, Kyoto 603, Japan

24 February 1993 1 June 1993

Abstract This paper attempts to provide the first model to include the money creation system in the Real Business Impulse response functions of equilibrium show that financial intermediation is a very powerful propagation

Cycle theory. mechanism.

1. Introduction In Real Business Cycle (RBC) theory, some empirical research [King and Plosser (1984) and the role of inside money, which is created through the money Ohkusa (1993)] h as emphasized creation system, in the monetary side of the economy. The theoretical parts of RBC, however, have not yet developed a model incorporating inside money or the money creation system. While some researchers have tried to incorporate money into their model [Cooley and Hansen (1989), Lucas (1990), Christian0 (1991) and Fuerst (1992)], their research does not include the money creation system, and thus treats only the outside, or helicopter, money. This paper provides the first model to include the money creation system in RBC theory. This model has two features. The first feature is the shareholder’s/firm’s utility, whose purpose is profit and deposit. I call this utility function, deposit-in-utility (DIU). The background of DIU is that (i) the amount of deposit is used to signal the firm’s condition, and (ii) the deposit is closed to the concept of reminder. The second feature is that I use the first-order condition approach instead of the value function approach to solve the decentralized decision-making problem. This method does not calculate the value function and obtains the optimal behavior only through the first-order condition. ’ The impulse response functions of equilibrium in this model show many counter-intuitive aspects. The most impressive one is that financial intermediation is a very powerful propagation mechanism.

* I have received helpful comments from Kenn Ariga and Akihisa all errors in this paper are mine. ’ There are some variations of this method, as suggested by Taylor Coleman (1990).

Shibata.

I wish to thank

and Uhlig (1990).

I adopt

both of them.

Needless

one type of stereometry

to say, as in

368

Y. Ohkusa I Economics Letters 42 (1993) 367-371

2. The model The shareholder’s/firm’s utility function, V, is DIU, deposit, D. Hence, the problem for the shareholder/firm

s.t.

zz,,,=p,y,

in which arguments at period 0 is

+ (1 + r)D, - (1 + R,)B, - (p,Z, + w,L:) + Mj >

are profit,

17, and

(2)

r, =A,F(Kt,L:) ,

(3)

rr, + B, =

D, + M; ,

(4)

M+p,Z,+

w,L:,

(5)

K ,+,=(I-a)Kt+Z,, log A,+1 = p + p log A, + E, ,

(6) lr - N(0, (T’), i.i.d. ,

(7)

where Z&+, is profit at the end of period t, p is the discount factor, A,, is the information set at period 0, E[.] is the mathematical expectation, Y is output, B is debt, K is capital stock, Z is investment, Lf is labor input, Mf is the firm’s amount of money at the beginning of the period, F is the production function, p is the good’s (nominal) price, w is (nominal) wage, r is the (nominal and fixed) rate of return for the safe asset (deposit), R is the (nominal) interest rate for debt, and A is technological level. The information set assumes the inclusion not only of the information about firm/shareholder, but also about the other parts of the model; household/worker, financial intermediation and monetary authority. Equations (2)-(6) describe the definition of profit at the end of period t, the production function, the allocation of money at the beginning of period t, cash-in-advance (CIA) constraints for input, and accumulation of capital stock with depreciation rate, 6, respectively. Equation (7) shows the technological level following the stationary log AR(l). In this equation E stands for the technological shock. V and F are assumed to satisfy the Inada condition. Hereafter, I analyze this problem where (5) is satisfied with an equality. The household’s/worker’s utility, U, consists of consumption, C, and leisure, H - Lh, where H is disposable time in units. Hence the household’siworker’s problem is max (C,.S,,L;,M:),=”

x

E[(; p’“(cz> H- L3

lAoI

s.t. S, + M;+, = (1 + r)S,_, + w,L: -p,C,

M: 2 p,C,

>

.

(8)

+ Mr ,

(9) (10)

where S is the deposit of the household/worker and Mh is the household’siworker’s amount of money at the beginning of period t. Equations (9) and (10) stand for the allocation of money at the end of period t and the CIA constraint for consumption, respectively. Like the components for shareholders/firms, I assume that U satisfies the Inada condition and that (10) is satisfied with an equality. Financial intermediation collects deposits from shareholders/firms and households/workers.

Y. Ohkusa I Economics Letters 42 (1993) 367-371

369

rate) of collected deposits plus money Then it lends firm (1 - T) 100% (T is the preparation authority. At the end of the period the financial intermediary injection, AM, from the monetary charges R to the lending firm and pays I to the depositing firm and household. For convenience, all deposits and loans are assumed to clear up for the period. Moreover, the financial intermediary is constrained with zero profit. Hence, I have (1 - T)(S, + D,) + A; = B, )

(11)

(1 + r)(S, + 0,)

(12)

and = (1 + R,)B, .

Needless to say, Eq. (11) is the balance sheet of the financial intermediary and Eq. (12) indicates a zero profit condition. The monetary authority can control AM, which follows a log Ar(1) process: log AM, = ~“+PMlogh~+e~,

E:--N(O,~~*),i.i.d.

(13)

The monetary authority controls eM, and cM is monetary shock indeed. I now show the shareholder’s/firm’s and household’s/worker’s optimal condition in the arbitrary period t. Although the shareholder’s/firm’s control variables are Z,, D,, B,, L: and M:, I can reduce them to I,, D, and B, by inserting between Eqs. (2)-(6). After easy computation, the shareholder’s/firm’s optimal conditions are

(14)

+,,+pE[

1-

(1 + +‘t+,l~i+l4+,F~r+l

(15)

and

(16) where superscripts indicate partial derivatives of functions. Equation (16) implies that the shareholder/firm wants unlimited lending. Thus B, is decided by the right-hand side of Eq. (11). Similarly, although the household’s/worker’s decision variables are C,, S,, L: and M:, I can reduce them to L: and M: by inserting between Eqs. (9) and (10). After easy computation, the household’s/worker’s optimal conditions are

(17) and

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Y. Ohkusa

I Economics Letters 42 (1993) 367-371

Equation (17) is the intertemporal substitution condition and Eq. (18) is the marginal condition between consumption and leisure. Due to the CIA constraint, the timing of consumption and leisure are different in Eq. (18). Note that this economy should be a decentralized perfect competition equilibrium. To solve this problem, I have to set the functional forms of U, V and F, and the parameters. Since the purpose of this paper is not necessarily to mimic an actual economy, I give the functional forms and parameters ad hoc from the existing RBC literature. The functional forms are given as I/(C,, H - L,) = C’P(H - L,)lPoL, V(II,+,, D,) = -exp[-Q,, -
4. Impulse

response

experimentations

To discover the features of the model obtained in the previous section, I experiment with impulse response. I use two experiments. The first experiment is that the shock, E, = (T, occurs only shock experiment. in the twentieth period and ,” = 0 for every period; I call this the monetary These two experiments would make clear differences in the response of the economy to the real shock and the monetary shock. The impulse responses of the output and money (M, = Mh + M’ and M, = M, + S + 0) are shown in Figs. 1, 2 and 3. In these figures, responses are shown by the percent of the divergence rate from the steady state. Solid lines indicate the real shock experiment and dotted lines indicate the monetary shock experiment. It is clear from the figures that the effect of the monetary shock on output shows stronger and larger waves than the effect of the real shock, in spite of the fact that the deviation rate in the

l

Fig. 1. Y. Key: shock.

’ Details

about

-

solving

real

shock;

the algorithm

- - -

monetary

are available

Fig. 2. M,. Key: __

from the author

upon

request

real shock;

- - monetary

shock.

Y. Ohkusa

I Economics

Letters

42 (1993) 367-371

371

l_ *.___

-OS0 Fig. 3. Ml. Key: __

20 real shock;

40 - ~ - monetary

60 shock.

monetary shock (about 0.4%) is far less than in the real shock (about 1.5%). This phenomenon seems to be the result of the fact that money injection expands lending to firms which then increases investment and output. Moreover, it needs a long duration (more than 4 periods) to reach the peak. This implies that the temporary and direct effects due to money injection would be dominated by the longer and indirect effect due to money creation. Similarly to the output variables, the money variables show long and amplified waves in the monetary shock. In contrast, the real shock almost never affects the amount of money. This phenomenon seems to be a reflection of the fact that money injection expands lending to firms, which is the supply side in the loanable fund market, directly.

5. Concluding

remarks

The analysis in this paper finds that money injection has a large impact on the economy. I conclude that this model has succeeded in clearing the role of inside money in spite of many ad hoc assumptions like DIU. However, the question remains about the consistency with the model’s implications and the actual economy. Thus I will estimate the functional forms and parameters in future research.

References Christiano, L.J.. 1991, Modeling the liquidity effect of a money stock, Federal Reserve Bank of Minneapolis Quarterly Review, Winter, 3-34. Coleman. W.J., II, 1990, Solving the stochastic growth model by policy-function iteration, Journal of Business and Economic Statistics 8, 27-29. Cooley, T.F. and G.D. Hansen, 1989, The inflation tax in a real business cycle model, American Economic Review 79. 733-748. Fuerst, T.S., 1992, Liquidity, loanable fund, and real activity, Journal of Monetary Economics 29, 3-24. King, R.G. and C. Plosser. 1984, Money, credit and prices in a real business cycle, American Economic Review 74, 363-380. Lucas, R.E., Jr., 1990, Liquidity and interest rate, Journal of Economic Theory 50, 237-264. Ohkusa, Y., 1993, An empirical study of the RBC theory, manuscript. Taylor, J.B. and H. Uhlig, 1990, Solving nonlinear stochastic growth models: A comparison of alternative solution methods, Journal of Business and Economic Statistics 8. 1-17.