Varying the Money Supply of Commercial Banks

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.

Appendix: Supporting Algebra for Non-Cooperative Equilibria of Game Models

1.1 Steady Post-Innovation Output and Stable Money Supply Lead to Stable Bid Levels

This section shows from the first-order conditions for consumption that, if output levels converge to steady late-time values, and if the money supply converges to a steady value, then bid levels by both type-1 and type-2 agents also converge to steady values. This condition is not an accounting identity, but part of the optimization problem that agents must solve. It requires only one strategic degree of freedom to be met, which is the overall consumption asymmetry $\hat{\epsilon }$ that governs agents’ bid levels throughout the post-innovation consumption schedule.

From the notation of Eq. (30) in the main text, for the consumption asymmetries ε ₁ and ε ₂, and the fact that consumption rates c _i and $\tilde{c}_{\tilde{\imath }}$ are related to bid rates b _i and $\tilde{b}_{\tilde{\imath }}$ through the same prices p _i , the ratios of consumption levels of the same good by the two types of agents may be written in terms of the $\hat{q}_{i}$ and $\hat{\epsilon }$ as

$$\displaystyle\begin{array}{rcl} \frac{c_{1}} {\tilde{c}_{2}} = \frac{b_{1}} {\tilde{b}_{2}}& =& \frac{1 + 2\hat{\epsilon }/\hat{q}_{1}} {1 - 2\hat{\epsilon }/\hat{q}_{1}} \\ \frac{\tilde{c}_{1}} {c_{2}} = \frac{\tilde{b}_{1}} {b_{2}}& =& \frac{1 + 2\hat{\epsilon }/\hat{q}_{2}} {1 - 2\hat{\epsilon }/\hat{q}_{2}}.{}\end{array}$$

(56)

Introducing two further notational abbreviations

$$\displaystyle\begin{array}{rcl} x_{1} \equiv \frac{2\hat{\epsilon }} {\hat{q}_{1}}\qquad \qquad x_{2} \equiv \frac{2\hat{\epsilon }} {\hat{q}_{2}},& &{}\end{array}$$

(57)

the bid rates by either agent type are written in terms of the total bid rates B ₁ and B ₂ as

$$\displaystyle\begin{array}{rcl} b_{1}& =& \frac{1 + x_{1}} {2} B_{1}\qquad \qquad \tilde{b}_{2} = \frac{1 - x_{1}} {2} B_{1} \\ \tilde{b}_{1}& =& \frac{1 + x_{2}} {2} B_{2}\qquad \qquad b_{2} = \frac{1 - x_{2}} {2} B_{2}{}\end{array}$$

(58)

The bid rates B ₁ and B ₂ are then related to the total money supply by Eq. (34) in the main text.

As long as the values of the late-time interest rates are well-defined, $\hat{\epsilon }_{t}$ at late t has a fixed value, by Eq. (40). Then, as long as production levels q _i converge to steady values, the ratios of both B _i to the total money supply converge to steady values by Eq. (34). Finally, under these two conditions, the relations of all b _i and $\tilde{b}_{i}$ to the total money supply also converge, by Eq. (58).

This completes the result, and shows that steady credit and debt balances for the two agents a _1, T and a _2, T can be attained with a suitably chosen $\hat{\epsilon }$ by Eq. (23).

1.2 Solutions for the Utopia Economy

This section provides solutions for the non-cooperative equilibria of the Utopia model of Sect. 6.2. We begin with the output equations for good-2, which does not undergo an innovation shock.

1.2.1 The Unshocked Good Remains at Steady State Unperturbed

The main Eq. (51) for the response of output decisions to prices, under the condition (40) on shadow prices, becomes

$$\displaystyle{ \left [ \frac{d} {dt}\left ( \frac{d} {dt} -\rho _{\pi }\right ) + 2\gamma _{2}\bar{f}_{2}^{{\prime\prime}}\right ]\left (s_{ 2} -\bar{s}_{2}\right ) = 0. }$$

(59)

Since the initial condition from the pre-shock equilibrium was $s_{2,0} = \bar{s}_{2}$, the unique bounded solution is $s_{2,t} = \bar{s}_{2}$ for all t.

1.2.2 Recovery of the Shocked Good

$s_{1,t}^{\left (-\right )}$ denotes the stock of the type-1 firms that tried to innovate and failed, and $s_{1,t}^{\left (+\right )}$ denotes the stock of the type-1 firms that succeeded. The initial conditions for both stocks in the periods immediately following the innovation are $s_{1,0^{+}}^{\left (\pm \right )} \rightarrow \bar{s}_{ 1} - j$. The steady-state stock for failed-innovation firms is $\bar{s}_{1} \equiv \left (1/2\right )\log 2$, and the steady-state stock for successfully-innovating firms is $\tilde{s}_{1} = \bar{s}_{1} + \left (1/2\right )\log \left (1+\theta \right )$.

In an initial interval following the shock, only a measure $\left (1-\xi \right )$ of firms offer in markets. The recovery equation (51) for these firms becomes

$$\displaystyle{ \frac{\left (1-\xi \right )} {2\gamma _{1}} \frac{d} {dt}\left ( \frac{d} {dt} -\rho _{\pi }\right )\left (s_{1}^{\left (-\right )} -\bar{s}_{ 1}\right ) \approx -\bar{f}_{1}^{{\prime\prime}}\left (s_{ 1}^{\left (-\right )} -\bar{s}_{ 1}\right ). }$$

(60)

This solution will govern offers q _1, t until the shadow prices of successfully-innovating firms fall to intersect market prices. Thereafter the successfully-innovating firms also begin to offer.

Once both firms have entered, both relax to the new steady states with the converging solution to the equation

$$\displaystyle{ \frac{1} {2\gamma _{1}} \frac{d} {dt}\left ( \frac{d} {dt} -\rho _{\pi }\right )\left (s_{1}^{\left (-\right )} -\bar{s}_{ 1}\right ) \approx -\bar{f}_{1}^{{\prime\prime}}\left (s_{ 1}^{\left (-\right )} -\bar{s}_{ 1}\right ). }$$

(61)

These are both second-order linear equations, which possess growing and decaying solutions. We first introduce notations for characteristic rates in the two regimes:

$$\displaystyle\begin{array}{rcl} \omega _{+}^{2}& \equiv & -2\gamma _{ 1}\bar{f}_{1}^{{\prime\prime}}\quad \mbox{ evaluates on Eq. (1) to}\quad 4\gamma _{ 1}\rho _{\pi } \\ \omega _{-}^{2}& \equiv & - \frac{2\gamma _{1}} {\left (1-\xi \right )}\bar{f}_{1}^{{\prime\prime}} = \frac{\rho _{+}^{2}} {\left (1-\xi \right )} {}\end{array}$$

(62)

In terms of these, the solutions for the relaxation time constants are

$$\displaystyle\begin{array}{rcl} \frac{1} {\tau } & =& \pm \sqrt{\omega _{- }^{2 } + \frac{\rho _{\pi }^{2 }} {4}} - \frac{\rho _{\pi }} {2}\qquad \qquad t \leq t_{1} \\ \frac{1} {\tau } & =& \pm \sqrt{\omega _{+ }^{2 } + \frac{\rho _{\pi }^{2 }} {4}} - \frac{\rho _{\pi }} {2}\qquad \qquad t > t_{1}.{}\end{array}$$

(63)

Both the positive and negative roots are needed in the initial transient for t ≤ t ₁. Only the positive root is required for relaxation toward the turnpike solution in the initial transient for t > t ₁. The negative root in the second line of Eq. (63) will become important again, however, for the growing solution in the terminal transient.

Equivalent expressions exist for production by type-2 firms. In the numerical example, where the production and consumption parameters are set to equal values for the two types, the type-2 dynamics will depend on the same time constants as the dynamics for type-1 firms in the interval t > t ₁.

1.2.2.1 Relaxation and Matching Conditions

The timescale for relaxation shared among models is the discount rate in the profit criterion ρ _π . Therefore introduce a dimensionless coordinate

$$\displaystyle{ z \equiv \rho _{\pi }t. }$$

(64)

Two scale factors that define local timescales relative to z are given shorthand notations $\sqrt{\pm }$, which denote

$$\displaystyle\begin{array}{rcl} \sqrt{ \mbox{ +}}& \equiv & \sqrt{1 + \frac{4\rho _{+ }^{2 }} {\rho _{\pi }^{2}}} = \sqrt{1 + \frac{8\gamma _{1 } \bar{f} _{1 }^{{\prime\prime} }} {\rho _{\pi }^{2}}} = \sqrt{1 + \frac{16\gamma _{1 } } {\rho _{\pi }}} \\ \sqrt{\mbox{ -}}& \equiv & \sqrt{1 + \frac{4\rho _{- }^{2 }} {\rho _{\pi }^{2}}} = \sqrt{1 + \frac{16\gamma _{1 } } {\left (1-\xi \right )\rho _{\pi }}}. {}\end{array}$$

(65)

The two trajectories in the initial interval after the innovation event are

$$\displaystyle\begin{array}{rcl} s_{1,z}^{\left (+\right )} -\tilde{s}_{ 1}& =& -j -\frac{1} {2}\log \left (1+\theta \right ) + \frac{f_{1,\infty }\left (1+\theta \right )} {\rho _{\pi }} \left (e^{z} - 1\right ) \\ s_{1,z}^{\left (-\right )} -\bar{s}_{ 1}& =& e^{z/2}\left [-j\mbox{ ch}\left (\frac{z} {2}\sqrt{\mbox{ -}}\right ) +\sigma \mbox{ sh}\left (\frac{z} {2}\sqrt{\mbox{ -}}\right )\right ].{}\end{array}$$

(66)

The trajectory for $s_{1,z}^{\left (+\right )}$ is fully determined by the production function because these firms are not responsive to markets. The trajectory for $s_{1,z}^{\left (-\right )}$ is determined by its initial conditions up to a single parameter σ which will be determined by matching conditions when successful innovators enter the markets.

The market prices and the shadow prices of successful type-1 firms become equal at some time z ₁, which we will identify numerically. (The existence of a unique intersection is assured because the shadow prices of successful firms are falling while the market prices that can be maintained by the unsuccessful firms are rising, during the initial post-innovation interval.)

When the successful type-1 firms have entered the markets, their stocks relax with a fixed offset equal to the difference of late-time steady-state stocks, according to the functions

$$\displaystyle{ s_{1,z}^{\left (+\right )} -\tilde{s}_{ 1} = s_{1,z}^{\left (-\right )} -\bar{s}_{ 1} = \left (s_{1,z_{1}}^{\left (-\right )} -\bar{s}_{ 1}\right )e^{\left (z-z_{1}\right )\left (\sqrt{\mbox{ +}}\,-1\right )/2} }$$

(67)

The undetermined parameter σ in Eq. (66) is set by the requirement that the total offering q _1, t be continuous through the transition at z = z ₁, because continuity of q ₁ is required for continuity of the price against which firms perform their discounting.

In the numerical solutions of Sect. 6.2.2, the radicals determining the relaxation time constants (65) evaluate to $\sqrt{ \mbox{ +}} = 3$ and $\sqrt{\mbox{ -}} = \sqrt{11} \approx 3.3166$. The resulting time constants (63) are given by $1/\rho _{\pi }\tau = \left (\pm \sqrt{11} - 1\right )/2$ for t ≤ t ₁; 1∕ρ _π τ = 1 for t > t ₁. The matching parameter that makes both prices and quantities continuous is σ ≈ −0. 14536. The remaining features of these solutions are presented as plots in the main text.

1.2.3 Terminal Transient

A terminal transient is solved in terms of the divergences of the three working stocks from their steady-state turnpike values. The functional forms (using properties of non-cooperative equilibria previously derived for stocks when all firms optimize against a shared price system) are given by

$$\displaystyle\begin{array}{rcl} s_{1,t}^{\left (+\right )} -\tilde{s}_{ 1} = s_{1,t}^{\left (-\right )} -\bar{s}_{ 1}& =& \left (s_{1,T}^{\left (-\right )} -\bar{s}_{ 1}\right )e^{\left (t-T\right )/\tau } \\ s_{2,t} -\bar{s}_{2}& =& \left (s_{2,T} -\bar{s}_{2}\right )e^{\left (t-T\right )/\tau }.{}\end{array}$$

(68)

When (as is the case in the numerical example) γ ₁ = γ ₂ = ρ _π ∕2, the time constant in both divergences is given by the negative root in the second line of Eq. (63), which evaluates to

$$\displaystyle{ \frac{1} {\tau } = -\frac{\left (\sqrt{+} + 1\right )} {2} \rho _{\pi } = -2\rho _{\pi }. }$$

(69)

The two parameters in the solution (68), $s_{1,T}^{\left (-\right )}$ and s _2, T , are determined by the requirements that $\left (a_{1,T} - a_{2,T}\right ) = 0$ and $\left (a_{1,T} + a_{2,T}\right ) = 0$. Initial conditions are $\left (a_{1,t} + a_{2,t}\right ) = 0$ as t → −∞, and $\rho _{C}\left (a_{1,t} - a_{2,t}\right ) = \tilde{b}_{1} -\tilde{b}_{2}$ of the turnpike solution for t → −∞. Results of numerical solution are shown in the figures of Sect. 6.2.3.