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2015 | OriginalPaper | Buchkapitel

Varying the Money Supply of Commercial Banks

verfasst von : Martin Shubik, Eric Smith

Erschienen in: Dynamics, Games and Science

Verlag: Springer International Publishing

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Abstract

We consider the problem of financing two productive sectors in an economy through bank loans, when the sectors may experience independent demands for money but when it is desirable for each to maintain an independently determined sequence of prices. An idealized central bank is compared with a collection of commercial banks that generate profits from interest rate spreads and flow those through to a collection of consumer/owners who are also one group of borrowers and lenders in the private economy. We model the private economy as one in which both production functions and consumption preferences for the two goods are independent, and in which one production process experiences a shock in the demand for money arising from an opportunity for risky innovation of its production function. An idealized, profitless central bank can decouple the sectors, but for-profit commercial banks inherently propagate shocks in money demand in one sector into price shocks with a tail of distorted prices in the other sector. The connection of profits with efficiency-reducing propagation of shocks is mechanical in character, in that it does not depend on the particular way profits are used strategically within the banking system. In application, the tension between profits and reserve requirements is essential to enabling but also controlling the distributed perception and evaluation services provided by commercial banks. We regard the inefficiency inherent in the profit system as a source of costs that are paid for distributed perception and control in economies.

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This is true for taps at the same elevation; we leave aside corrections for gravity which are not central to the point of this illustration.

The coupling is in linear proportion to the spread at sufficiently small spreads.

An added condition is that prices are stationary when the real goods distribution is stationary. This raises further complications involving incentives and information conditions in an economy where all laws are not indexed against inflation or deflation. This problem is not considered further here.

This abstraction is easy to define in models. Validating the abstraction for actual economies may be more or less difficult depending on the sectors considered.

Our models resemble the von Neumann growth model, restricted to a single good. However, in our production function the rate of output is a non-linear rather than a linear function of the input stock.

We do not digress to derive the solution for Robinson Crusoe here, because its important features are subsumed in the solutions we demonstrate. A more systematic introduction to this class of models, including a separate solution for Robinson Crusoe as a reference, will be given elsewhere.

There are essentially three levels of models that require consideration for a complete exposition of basic distinctions. They are

Crusoe without money,
the price-taking individual firm with money,
the oligopolistic firm without money.

The first two should produce the same physical allocations but differ in the presence or absence of money.

The source of the simplification is that difference equations and discrete series reduce to differential equations and integrals, though the structure and meaning of the Bellman equations remains unchanged.

We could introduce a k:1 gearing ratio here with a little extra work, but our illustration does not need it.

These forms are smoothed versions of a linear production function with a limiting output and corner solutions, developed by Shubik and Sudderth [6, 7]. Corner solutions provided a convenient way to truncate discrete-period models to a single period, but in the continuous-time setting, the smoothed production rate produces a simple decomposition of solutions.

The form (2) is the smoothed counterpart to a combination of “cost innovation” and “capacity innovation” in the terminology introduced by Shubik and Sudderth [6, 7]. The rate of production for $s_{1,t} \lesssim 1/2$ is larger by the factor $\left (1+\theta \right )$, generating the same output at less input cost. The saturation level f _1, ∞ likewise increases by the factor $\left (1+\theta \right )$, so that maximum output capacity likewise increases. This combination is simpler, for the smoothed production function, than either cost innovation or capacity innovation alone.

Thus, in the discrete-period model, the amounts consumed in one period are c ₁ Δ t and c ₂ Δ t.

The absolute magnitude of this constant does not matter for the definition of $u\!\left (c_{1},c_{2}\right )$; only the dimension of a rate is required. We use the rate ρ in the discount factor as this avoids introducing a further arbitrary parameter.

To express this more didactically, $\tilde{\mbox{ }}$ is used to indicate exclusion, or opposition in binary sets: $\tilde{\imath }$ means whichever value in $\left \{1, 2\right \}$ that is not the value taken by index i. $\tilde{c}_{i}$ indicates the consumption rate of the good that is not the consumption rate c _i.

This construction avoids most of the concerns with corporate financing.

Many alternative rules are well-defined: interest on deposits could accrue one period later than interest charged on loans, etc. Nothing depends on the intra-temporal order of interest charges and payments, in the continuous-time limit.

Under conditions when the bank is actively used, a _t = 0 occurs only on time intervals of measure zero, so the results are not sensitive to the way the interest rate is regularized. Because, in this model, we assume initial conditions prior to the accumulation of bank balances, it is convenient to choose a regularization condition that will be consistent with the other simplifying assumptions made in the model.

The residual terms at $\mathcal{O}\!\left (\varDelta t\right )$, which we denote explicitly despite the fact that they approach zero as Δ t → 0, come from time lags between the making of bids and the delivery of profits. As long as the rates are continuous (differentiable at order one) functions, these effects contribute terms $\sim \left (db_{i}/dt\right )\varDelta t$ in Eq. (20).

Without uncertainty it calls for the rate ρ defining the utilitarian rate of discount in Eq. (16) to equal the average of the two interest rates faced by the agents, as shown in Eq. (42) below. (In the worked example of the following sections, this will be the average of the borrowing and the lending rates.) With uncertainty there is a delicate correction depending on the variance.

When the term in curly braces is exactly zero, the late-time steady-state relation becomes

$$\displaystyle{ \frac{\left (\rho _{B,1t} + \rho _{B,2t}\right )} {2} \frac{\left (a_{1,t} - a_{2,t}\right )} {2} = \left (\tilde{b}_{2,t} -\tilde{b}_{1,t}\right ). }$$

This expression is simply the interest paid to agents of type-1, plus their share of bank profits when profits are defined, which balances the deficit in the profits of type-1 firms relative to the bids made by type-1 agents (who will consume more). Thus a consistent circular flow is restored in the asymptotic steady state, in a context of asymmetric production, profits, depositing/borrowing, and consumption.

These multipliers are always nonzero, as the budget constraint is always tight.

If bounds were placed on the account balances, additional multipliers could arise within each period as shadow prices associated with these constraints.

This term must be corrected with a measure term to relate it to individual firms’ output levels if not all firms are active in markets, as we show below.

A continuum of solutions to the first-order conditions exists, in which the type-1 and type-2 firms deplete or hoard stocks in differing degrees so as to cancel the intra-economy debt $\left (a_{1,T} - a_{2,T}\right )$. This continuum includes a solution in which the type-2 firms continue to produce at the pre-innovation level, so they are buffered at all times. That solution, however, does not lead to a net aggregate balance $\left (a_{1,T} + a_{2,T}\right ) = 0$, if $\left (a_{1,t} + a_{2,t}\right )$ starts from a zero aggregate balance at t ≪ T. Therefore the solution with $s_{2,t} = \bar{s}_{2},\;\forall t$ can only be reached by leaving a finely tuned non-zero aggregate balance $\left (a_{1,t} + a_{2,t}\right )$ of $\mathcal{O}\!\left (e^{-\left (T-t\right )\rho _{\pi }}\right )$ at early times t following the transient. Such an initial condition would lead to a different terminal solution than ($s_{2,t} = \bar{s}_{2},\;\forall t$) at any slightly different value for T, and would be incompatible with any non-cooperative equilibrium solution at a value of T differing by more than $\mathcal{O}\!\left (1/\rho _{\pi }\right )$ from the value for T which $\left (a_{1,t} + a_{2,t}\right )$ was tuned.

Firms of type-1, in the period when both are offering in the markets, have equations identical in form to Eq. (55), for the deviations of their stocks from the Utopia solutions. For the firms that attempt to innovate and fail, we denote these deviations $\delta \left (s_{1}^{\left (-\right )} -\bar{s}_{1}\right )$, and for the firms that attempt to innovate and succeed, the corresponding quantity is $\delta \left (s_{1}^{\left (+\right )} -\tilde{s}_{1}\right )$. In the initial period, when firms that successfully innovated are sitting outside the markets, their inventory growth is governed only by internal production and they do not optimize against prices. The type-1 firms that failed to innovate satisfy a slightly modified equation given by

$$\displaystyle{\left [\left (1-\xi \right ) \frac{d} {dt}\left ( \frac{d} {dt} -\rho _{\pi }\right ) + 2\gamma _{1}\bar{f}_{1}^{{\prime\prime}}\right ]\left (s_{ 1} -\bar{s}_{1}\right ) = \pm \varTheta \!\left (t_{\mathrm{split}} - t\right )\gamma _{1}\left (\rho _{B,L} -\rho _{B,D}\right ),}$$

because their measure is (1 −ξ) and the level of output than can contribute scales by the same factor.

In a true small-parameter expansion with both ρ _π Δ t ≪ 1 and $\left (\rho _{B,L} -\rho _{B,D}\right )/2\rho \ll 1$, the value t _split would be shorter than the natural recovery time for stocks $s_{1}^{\left (\pm \right )}$, so that the output of the successfully-innovating firms would never even respond to the interest-rate spread. The resulting solution would be simpler in structure than the one presented here, as well as smaller in magnitude.

Bagehot, W.: Lombard Street, 3rd edn. Henry S. King, London (1873)

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Samuelson, P.A.: Structure of a minimum equilibrium system. In: Stiglitz, J.E. (ed.) The Collected Scientific Papers of Paul A. Samuelson, pp. 651–686. MIT Press, Cambridge, MA (1966)

Schumpeter, J.A.: The Theory of Economic Development. Harvard University Press, Cambridge (1955)

Shubik, M., Smith, E.: The control of an enterprise economy (2014)

Shubik, M., Sudderth, W.: Cost innovation: Schumpeter and equilibrium. part 1: Robinson crusoe, cFDP # 1786, pp. 1–32 (2012)

Shubik, M., Sudderth, W.: Cost innovation: Schumpeter and equilibrium. part 2: Innovation and the money supply, cFDP # 1881, pp. 1–41 (2012)

Smith, E., Shubik, M.: Strategic freedom, constraint, and symmetry in one-period markets with cash and credit payment. Econ. Theory 25, 513–551 (2005). sFI preprint # 03-05-036

Smith, E., Shubik, M.: Endogenizing the provision of money: costs of commodity and fiat monies in relation to the valuation of trade. J. Math. Econ. 47, 508–530 (2011)MathSciNetCrossRefMATH

Titel: Varying the Money Supply of Commercial Banks
verfasst von: Martin Shubik
Eric Smith
Verlag: Springer International Publishing
Buch: Dynamics, Games and Science
Print ISBN: 978-3-319-16117-4

Electronic ISBN: 978-3-319-16118-1

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-16118-1_36

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