Monopolistic and Oligopolistic Bankers
Monopolistic and Oligopolistic Bankers
Abstract and Keywords
This chapter covers models with either monopolistic or oligopolistic private banking. The three main models posed and solved are: 1) a fiatmoney model with a monopolistic “corporate bank” that makes loans to the traders, with the bank’s objective being solely to maximize its monetary profit; 2) a fiatmoney model with a monopolistic “individually owned bank” that makes loans to traders AND bids to consume good 1 and good 2; 3) a fiatmoney model with a (small) finite number of corporate banks who form an oligopoly in the lending market; each such bank is large enough to influence the market interest rate for loans by changing the amount that it lends.
Keywords: corporate bank, individually owned bank, monopolistic banking, oligopolistic banking
13.1 The Many Models of Banking
In this chapter and the two following, we examine several variations of banking. We begin with models of a single central bank and then consider corporate banks, stockholderheld corporate banks, and individually held private banks. In this chapter we consider monopolistic and oligopolistic market structures; Chapters 14 and 15 cover competitive banking markets.
Banks are complex institutions with many different functions. Here we model only one such function—shortterm lending—and we model the lending without uncertainty. Thus our models are gross oversimplifications of institutional reality, but they are justified because they represent first steps beyond static general equilibrium models and into the realm of process models. The process models here are designed to answer only a few questions concerning competition, ownership, purpose, and structure.
(p.164) Types of Banks
In our models in this chapter and Chapter 14, we consider several types of banks. A central bank is the government bank. As pointed out in Section 13.2, it can have many motivations, but usually it aims to maximize some measure of social welfare and/or perform some regulatory function. Any bank which is not a central bank we call a private bank. Private banks can be either individually owned or corporate. An individually owned bank (or utility maximizing bank) is one that is controlled by a single individual or family;^{1} hence its motives are similar to those of our private traders (i.e., it derives utility directly from consumption of the goods and services of the economy and indirect utility from monetary profit). Alternatively, a corporate bank (or profit maximizing bank) is run primarily for monetary profit. A special type of corporate bank (considered in Chapters 14 and 15) is a stockholderheld (100 percent dividend) corporate bank or “corporate bank with full payout to owners”; there we assume that the bank is actually owned by the traders, and so its monetary profits are cycled back to them.
The private banks can exist in markets that are monopolistic, oligopolistic, or competitive. Also, we model cases where one or more private banks are considered in the presence of a central bank.
Nine Simple Models
There are at least nine related simple oneperiod models involving trade with money and with various forms of money market, credit, and banking that merit consideration. Suppose n denotes the number of trader types (not financial institutions or government entities) in a model. Table 13.1 lays out the various cases together with some of their distinguishing properties.
The reader will note that in all of these models, the number of nonmonetary commodities (s) is equal to n.^{2} The symmetry s = n (together with the condition that all agent types have the same symmetric utility function) is useful because it cuts out concerns with complex problems in consumption, so we can concentrate on the financing of exchange and on the boundary (p.165)
Table 13.1 Many models of trade with money and credit
Model 
No. of goods 
No. of goods markets 
No. of fin. markets 
Money 
Financial instru. 
Agent types 

1. Cash market 
n + 1 
n 
0 
Tea^{a} 
0 
n 
2. Money market 
n + 1 
n 
1 
Tea 
1 
n 
3. Dummy central bank 
n 
n 
1 
Fiat 
1 
n + 1 
4. Profit max. central bank 
n 
n 
1 
Fiat 
1 
n + 1 
5. Altruistic central bank 
n 
n 
1 
Fiat 
1 
n + 1 
6. Tax & public goods central bank 
n + 1 
n 
1 
Fiat 
1 or 2 
n + 1 
7. Corp. bank, profit max. 1 
n 
n 
1 
Fiat 
3 
n + 1 or n + 2 
8. Corp. bank, profit max. 2 
n + 1 
n 
1 
Tea 
3 
n + 1 
9. Private bank, utility max. 
n + 1 
n 
1 
Tea 
1 
n + 1 
(^{a}) The term “tea” stands for a storable consumable money such as tea, cocoa beans, barley, or rice.
The following comments describe further each of the nine models listed above:
1. The cash market is characterized by there being n agent types, each with an endowment of one real good and each with a symmetric utility function, and each endowed with the same amount of a storable consumable (point consumption) commodity money which we call tea. There are n markets and all transactions are in terms of goods in exchange for a commodity money. No financial institutions are needed. Note that the case with n = 2 is just our basic model from Chapter 3.
(p.166) 2. The money market model is characterized by n + 1 markets (n goods markets and a money market between IOU notes and money). The individual IOU note^{4} is a new financial instrument. The financial institution is the money market. Again, we note that we have analyzed this model before, where n = 2, in Chapter 4.
3. The model with a dummy central bank introduces a fiat that is worthless to all traders as a consumption good, but is required for transactions. The traders are not trapped by the Hahn paradox, as they may rid themselves of residual fiat at the end of the period by appropriate borrowing from the bank. In the oneperiod model, as the bank only lends (i.e., increases the money supply), the new financial instrument is still only the individual IOU note.^{5} If there were more than one period, the bank might wish to borrow as well as lend, in which instance we would need to invent the government IOU note or bond. Either way, the bank must be added to the list of player types, as it may have a strategy set, even though it is not a natural person. Again we note that we have studied such a model before, this time in Chapters 7 and 8.
4. The model with a profit maximizing central bank assigns a utility function and a motivation to the bank. Here it is a “cashconsuming bank.” We have examined such a model in Section 5.2.2.^{6} However, there we did not consider how the bank utilizes the money it earns—and this should be a critical part of any analysis. In fact, central banks may earn a profit which might be paid to the government.
5. A different motivation for the bank might call for it to optimize some form of welfare function for the society as a whole. Whatever the goal or routine may be (such as ensuring enough money), formal modeling requires that it must be well defined. See the discussion in Section 5.2.1.
(p.167) 6. A richer and more realistic model calls for the introduction of taxation and a public good. The minimal model requires at least one public good^{7} and an extra financial instrument, the tax bill (Karatzas, Shubik, and Sudderth, 2008). Institutionally, the central bank and the tax authority both pertain to the government; thus they could be modeled utilizing one institutional player called “government.” However, one may wish to consider them as two financial institutions, with goals and policies which may or may not be highly correlated. Economics alone does not justify a monetary policy–fiscal policy distinction, but there could be political and bureaucratic reasons that call it forth.
7. A simple corporate banking model with fiat has the corporation maximize profits,^{8} which may be paid out as dividends. This model requires the specification of shares and dividends. The simplest model, as presented in Debreu (1959), has markets for neither shares nor dividends. A more complex multistage model could have both a stock market and a financial market for shares stripped of dividends. Beyond the money there are three financial instruments: IOU notes, stock, and dividends. Terminal conditions are required to specify the worth of any goods or financial assets left over after the end of the single period. Thus the bank must liquidate and a salvage value must be attached to any goods, debts, or fiat remaining. The salvage value may be interpreted as an expectation of future worth. If the corporate banks use fiat a central bank needs to be included as the bank of issue. Hence there are at least two financial institutions, the corporate banks and the central bank.
8. A second corporate banking model would be with a point consumption commodity money such as tea, beans, or chocolate bars. Here the banks’ assets at the end of the game need to be liquidated and must be flowed through to the natural persons who are stockholders. Thus the oneperiod game may end with a tea party or chocolate fest as the money is consumed. No central bank is needed.
(p.168) 9. The individually owned bank model has a special class of monied individuals—the capitalists—who own and lend a commodity money, who accept deposits, and whose only concern is their personal consumption. With commodity money there need be only one financial institution: the utility maximizing bank.
We do not cover all of these models; we solve several, but stress that all of these distinctions make a difference and suggest that in monetary control the devil may be in the details. The most realistic and complex of these models has corporate bank competition in an economy with a central bank and fiat money.
In this chapter, the models we solve are summarized in Table 13.2:
Table 13.2. Models of monopolistic and oligopolistic banking
Profit max. bank 
Indiv. owned bank 
Olig. banking 


Type 1 traders 
(a, 0, m) 
(a, 0, m) 
(a, 0, m) 
Type 2 traders 
(0, a, m) 
(0, a, m) 
(0, a, m) 
Profit max. monopoly bank 
(0, 0, M − 2m) 
No 
No 
Individually owned monopoly bank 
No 
(0, 0, M − 2m) 
No 
Oligopolistic bank i (i = 1, …, n) 
No 
No 
(0, 0, B_{i}) 
13.2 The Monopolist Central Bank
Suppose there is a single central bank. The goals of this bank are modeled in several different ways and its strategic choices in two ways, that is, the choice between controlling either the money supply or the interest rate. In the following analysis the assumption is that the money is a fiat money that is recognized as having no intrinsic value beyond being accepted in trade and accepted in payment of debts by the central bank and the private banks. The parameter $\widehat{\Pi}$ denotes the expected per unit salvage value of fiat.
13.2.1 The Bank as a Strategic Dummy
The first and easiest way to model the central bank, which we have done in Chapters 7 and 8, is as a strategic dummy. For comparison purposes, we reproduce that model here (but not its solution).
Our model of the trader sector is the usual one with two consumable goods, and two trader types (each consisting of a continuum of agents) endowed with (p.169) (a, 0, m) and (0, a, m) as previously. We assume that the utility functions for all traders are $u(x,y,z)=2\sqrt{xy}+\widehat{\text{\Pi}}z$, where x and y are the amounts of goods 1 and 2 consumed, and z is the amount of fiat in hand at the end of the game. As defined above, $\widehat{\text{\Pi}}$ is the expected perunit salvage value of the fiat money.
A strategy for the Type 1 traders is denoted by (b, q, d), where b is the amount of money bid for good 2, q is the amount of good 1 offered for sale, and d is the amount of personal IOU notes bid for the fiat money offered by the central bank.^{9} The notation for the Type 2 traders is $\left(\overline{b},\overline{q},\overline{d}\right)$, with a similar interpretation.
The optimization problem for the Type 1 traders is then
Here p is the price at which the Type 1 traders sell their good 1, while $\overline{p}$ is the price at which they buy good 2. Also, ρ is the interest rate they pay on the IOU notes.
The Type 2 traders face a similar optimization problem.
The bank has an initial endowment of (0, 0, M − 2m = B), and may issue any amount g ∊ [0, B] for loan to the traders. The endogenous rate of interest is formed as
The strategic dummy bank then can be modeled in two ways:
13.2.2 The Central Bank Maximizing an Objective function
The “Cash Consuming Central Bank”
Modeling the central bank as a strategic dummy sidestepped the need to specify an objective function for it. But one of the two most important problems in understanding central banking is the specification of the goals of the bank (the other is delimiting its strategic powers). We consider several alternatives, noting that the efficacy of central banking depends on both its strategic power and its motivation. These properties are often empirically associated with government, but this is not a logical necessity. A central bank could be private or quasiprivate, as was the Bank of England.
For example, we may consider the goal of the central bank as that of making a profit from its issue of fiat money. Thus we can give it a welldefined objective function, namely, to maximize
The formal model, at this level of abstraction, will be equivalent to one with a profit maximizing monopolist bank. We investigate such a model in Section 13.3.1.
The Utility Maximizing Central Bank
Suppose that the central bank were owned by and operated for a king. He would wish to maximize his utility, derived from consuming perishables and owning cash. This is as though the central bank were a private monopolist individual banker. Again, an equivalent model (for a private bank) is treated in Section 13.3.3.
The Altruistic Central Bank
An altruistic central bank may wish to optimize some social welfare function, possibly subject to some constraints on its costs and revenues, such as being required to break even. Such goals are more naturally considered in the context of policies concerning taxation, subsidy, and/or public goods. Examples (p.171) are the control of inflation or employment goals. These are not considered further here.
Without considering public goods per se, a reasonable goal for a public bank might be to help to maximize the sum of individual welfare. This would imply an interpersonal comparison of welfare, such as a set of weights, to enable us to well define a sum. Although such a condition is strong, it is a matter of empirical investigation as to whether or not it is a reasonable approximation of the political process in evaluating social tradeoffs.
13.3 The Monopolist Private Bank
We now consider the possibility that instead of a government central bank there is a single private banker whose notes supply the transactions needs of the economy. At this level of abstraction, one can hardly tell a central bank apart from a private bank except by considering what it is trying to optimize—in the previous section, the central bank was a strategic dummy or was optimizing some social welfare function, while here the private bank will be optimizing its own utility function. In actuality, as is well known, the strategic variables under control may be different, as are the political links. Thus taxes, subsidies, and reserve ratios may be influenced strategically by a central bank, but far less by a private bank.
13.3.1 The Profit Maximizing Bank: Interest Rate
Suppose that the goal of the private (corporate) bank is to maximize monetary profits. The model is mathematically the same for a private monopolistic bank as for a central bank, although their consumption patterns may differ.
In Chapter 5, we presented and analyzed a related model using a consumable storable money. Here we solve a slight generalization, in which the money (in this case, fiat) has a salvage value of Π. To summarize, the Type 1 traders’ objective is
(p.172) Type 2 traders face a similar optimization problem. Prices are formed by $p=\frac{\overline{b}}{q}$ and $\overline{p}=\frac{b}{q}$, while the interest rate is formed by $1+\rho =\frac{d+\overline{d}}{g}$. For the bank, the rate of interest ρ is the strategic variable and the bank attempts to maximize ρg(ρ).
The model is solved in Appendix A. If there is “enough money w.d.” there is an equilibrium where the bank makes zero profit (see Case 1 of Appendix A). If the public has sufficiently small amounts of money, the bank can extract all 2m of it (Case 2). In this case, there is a continuum of equilibrium lending by the bank, ranging from all M − 2m of their endowment down to zero.
There is also a third intermediate case (Case 3). Here we display an example from this third zone, where demand for funds is elastic. In that case, the traders’ cashflow constraints are tight (i.e., $\lambda =\overline{\lambda}>0$) and their budget constraints are loose $(\mu =\overline{\mu}>0)$. Calculations in Appendix A yield $b=\overline{b}=\frac{a}{\Pi {(1+\rho )}^{3/2}+\Pi {(1+\rho )}^{1/2}},\text{\hspace{0.17em}}d=\overline{d}=\left(\frac{a}{\Pi {(1+\rho )}^{1/2}+(2+\rho )}m\right)(1+\rho )$. Thus, a nonzero rate of interest drives a wedge between buying and selling. As the interest rate increases, it cuts down trade.
The banker will choose ρ so as to maximize $\rho g(\rho )=\frac{2a\rho}{\Pi {(1+\rho )}^{3/2}+\Pi {(1+\rho )}^{1/2}}2m\rho $. This optimization can be done computationally. For example, if $m=\frac{a}{4}$ and Π = 1, we obtain ρ = 0.38317. This single solution point is selected because it illustrates something we saw with the monopolistic moneylender models of Section 5.2. In particular, for efficient trade, one needs more than just “enough” or “more than enough” commodity money in the system—it has to be welldistributed. If (as in Case 3) a monopolist has a sufficient part of the money supply, a shortage of money is created among the traders even with a loan market. Trade is inefficient. The positive rate of interest at the equilibrium may be purely due to monopolistic banking.
The final utility to the Type 1 traders in this example is 1.1961, which represents a large improvement over their valuation 0.25 for their initial endowments. However, in an “enough money w.d.” equilibrium with ρ = 0, the traders could realize a final utility of 1.25.
13.3.2 The Profit Maximizing Bank: Money Supply
Instead of utilizing the interest rate as the control variable, we could have the bank utilize the money supply as its strategic variable. As long as the boundary conditions do not interfere, ρ and g are dual variables and the maximization of gρ(g) will be equivalent to that of ρg(ρ). Mathematically, this means we work (p.173) with the inverse function ρ(g) of g(ρ). In Appendix B we calculate this inverse function for Case 3 of the model in Section 13.3.1. Indeed, we find that ρ(g) is a lot more complicated than g(ρ) in this case, making the computations much more difficult. However, in some of the cases of the next model,^{10} we will find that it is easier to work with ρ(g).
13.3.3 The Individually Owned Bank
A Modeling Issue
Now suppose we consider a monopolistic utility maximizing banker, who values money and consumes goods. Thus, the banker’s problem changes to
Here the new variables b* and $\overline{b}*$ represent the amounts bid by the bank for good 1 and good 2, respectively. Note that these variables will also enter the objective function for a third time, since prices are dependent on b* and $\overline{b}*$ via the new balancing conditions $p=\frac{\overline{b}+b*}{q}$ and $\overline{p}=\frac{b+\overline{b}*}{q}$.
Recall that in our solution to the “profit maximizing bank” model (see Section 13.3.2 and also Appendix A), we solved the model by first solving the traders’ problems parametrically in ρ, then using this to find g(p) (which is the amount of money the traders will borrow, as a function of ρ), and then finally substituting into the banks problem to optimize. Thus we were essentially modeling a twostage game, in which the banks first announce a ρ, and then the traders give their strategies (or “response functions” to ρ) afterward. The equilibrium that we found was a perfect Nash equilibrium for this game.
Here we take the same approach. We assume that the bank makes its decisions ($b*,\overline{b}*$, and g) first, and then the traders make their decisions afterward (regarding $b*,\overline{b}*$, and g as given parameters). Again we will find not merely a Nash equilibrium for the game, but a perfect Nash equilibrium.
When we solve the game in Appendix C, we still can use the traders’ firstorder conditions as before, but we will not write down explicit firstorder (p.174) conditions for the bank. Rather, once we find the traders’ response functions, we will substitute back directly into the banks decision problem and solve.
The analysis in Appendix C turns out to be quite complex. To save complication, we present our analysis for the specific case where Π = 1. We believe that the results for other values of Π would be qualitatively similar, in that we would still see the same four cases as we do below.
Model Results
Our model breaks into four cases, depending on the size and distribution of money. The analysis is presented in Appendix C.
Case 1:
If the trader types each have a lot of money, they have enough to achieve trading efficiency without needing to borrow from the bank.
Case 2:
If the traders have very little money and the bank also has little money (but still relatively more than the traders), the bank will lend, spend on consumption, and hoard (maintain reserves).
Cases 3/4:
If the traders have very little money and the bank has a lot of money, the bank will spend some of its money on bidding for consumption and hoard the rest. It will not lend anything to the traders. By not lending any money, it deprives the traders of a chance to bid much for the commodity— thereby allowing it (the bank) to consume relatively more of the commodity.
Case 5:
If the traders and bank both have very little money, but this time with the traders having relatively more money in comparison with the bank, the budget constraints of the traders will be loose $\left(\mu ,\overline{\mu}=0\right)$. A whole range of behaviors for the traders and bank is possible—see Appendix C.
13.3.4 A Comparison of the Profit Maximizing and Utility Maximizing Banks When m Is Small
It is interesting to compare the models of monopolistic banking when m is small. In the model with the profit maximizing bank, since the bank profits are simply the traders’ original monetary endowments, we have that bank profits approach 0 as m → 0. (We should note, though, that there is a continuum of equilibrium ρs when m is small but positive.) When m = 0, by the same logic, profits are zero (but now the only equilibrium interest rate is p = 0). Since it has no profit opportunity, the only motivation it has to lend is the (p.175) altruism associated with lending at a zero interest rate.^{11} If it does this, it will be of considerable benefit to the traders, who will able to achieve the efficient consumptions of $(\frac{a}{2},\frac{a}{2})$ and $(\frac{a}{2},\frac{a}{2})$.^{12}
Things are different in the case of the utility maximizing bank when m → 0. Since it is bidding against the traders, the bank has a definite interest in keeping money scarce for the traders. Hence ρ → ∞, and again the bank makes no profit from lending. However, the bank does well in this model, because it is able to extract all of the goods from the traders for its own consumption. Interestingly, this is due to the traders being altruistic: with m = 0 and ρ unboundedly high, they get zero utility no matter what they do; yet they still choose to put all of their commodity endowment up for sale.
To make all this more precise, we compare the two monopolistic bank model solutions and the general equilibrium model solution for the case where a = 1, M = 1, Π = 1, and m = 0. For the profit maximizing bank, this puts us in Case 2 of Appendix A. For the utility maximizing bank, we are in Case 4 of Appendix C. By “competitive equilibrium,” we mean a simple nondynamic general equilibrium model with no banking, in which all transactions are instantaneous.
Table 13.3 gives the comparisons. Of particular interest are the rows labeled “a – q,” “$\frac{\overline{b}}{p}$,” and “$\frac{b}{p}$.” These represent the consumption of good 1 by the Type 1 agents, by the Type 2 agents, and by the bank (if applicable), respectively. One can see that in the limiting case where m = 0, the traders’ consumption is efficient in the case of the profit maximizing bank, and zero in the case of the consuming bank.
In comparing the three situations, we note that the consumption of the bank will depend on the utility function ascribed to it. We made it the same as the traders’, which can be interpreted as the bureaucrats having the same preferences as others. A discussion of the consumption of government as payment to a bureaucracy for enforcing the rules is given elsewhere (Shubik and Smith, 2005).
We remark that in cases where the traders have little money (such as here), strategic default is a real possibility. Thus one could consider modifications to these models along the lines of those presented in Chapter 8.
Table 13.3. Comparing model solutions
Profit max. bank 
Utility max. bank 
Competitive equilibrium 


ρ 
0 
→ ∞ 
Not defined 
g 
g ∈ (0, 1] 
0 
Not defined 
b* 
Not defined 
∈ 0 
Not defined 
b 
$\frac{g}{2}$ 
∈ 0 
0.5 
d 
$\frac{g}{2}$ 
0 
Not defined 
q 
0.5 
∈ 1 
0.5 
p 
g 
∈ 0 
1 
a – q 
0.5 
∈ 0 
0.5 
$\frac{\overline{b}}{p}$ 
0.5 
∈ 0 
0.5 
$\frac{b*}{q}$ 
Not defined 
∈ 1 
Not defined 
13.4 A Model With Oligopolistic Banking
13.4.1 The Model
In this section we propose a model with oligopolistic banking, which is meant to be compared with the model of the monopolistic corporate bank from Section 13.3.1.
As in the earlier model, there are two continua of traders, endowed with (a, 0, m) and (0, a, m), respectively. However, now we replace the single monopolist banker with n bankers, each of which is an atomic player able to influence the interest rate via the amount it lends. Bank i (i = 1,…, n) is endowed with only B_{i} units of fiat. Hence M (the total amount of money in the game) is equal to $2m+{\displaystyle {\sum}_{i=1}^{n}{B}_{i}}$.
The traders’ optimization problems are the same as in 13.3.1, namely equation (13.2) for the Type 1 traders, with a similar problem for the Type 2 traders. For the banks, Bank i’s decision variable is g_{i}, the amount of money it puts up for loan. Its objective is simply to end up with the most cash possible; hence its optimization problem is
The balance conditions are the same as before, save for the one determining the interest rate:
(p.177) Note that the bankers’ optimization in (13.3) is more complex than it appears, because the selection of g_{i} directly affects ρ, which indirectly affects the amount of money the banks can lend via the traders’ optimization problems.
13.4.2 Results
The model is solved in Appendix D. Not surprisingly, the results mirror the cases with the money maximizing monopolistic bank, solved in Appendix A—there is an “m large” case, an “m very small” case, and an intermediate case:
Case 1: Enough money w.d.:
If m is large, that is, $m\ge \frac{a}{2\Pi}$, there is an equilibrium in which consumption is efficient and the traders can finance their trading without having to borrow. The interest rate on loans is zero, so the banks each earn profits of zero.
Case 2: Very little money:
Here we assume m is very small, so that the traders’ cashflow and budget constraints both are tight. As in the corresponding monopoly case (Case 2 from Section 13.3.1), the banks extract all of the traders’ money. However, here the unique equilibrium has all the banks lending out all of their money. Comparing this to the monopoly case, we find that the addition of more banks has eliminated the equilibria with reduced lending (and thus higher interest rates). Thus we have a version of the classical result from oligopoly theory, where a move from monopoly to oligopoly causes higher output and lower prices.
Case 3: An intermediate amount of money:
In between these two cases, there is a third, intermediate case, in which the traders are endowed with an intermediate amount of money. Not surprisingly, the solutions have the banks providing some (but not all) of their funds for loan. Mathematically, this is the hardest case, involving a system of equations which must be solved computationally.
13.5 Appendix A: Trade With a Money Maximizing Bank
The Type 1 traders are endowed with a units of good 1 and m units of fiat money. The Type 2 traders are endowed with a units of good 2 and m units of money. The monopolistic bank is endowed with M – 2m units of money (p.178) (where 0 ≤ 2m ≤ M). The bank’s objective is solely to end up with the most possible money.
The Type 1 traders face an optimization described by:
The firstorder conditions wrt b, q, and d yield
Similarly, the Type 2 traders face the optimization below:
The optimization conditions here are
The banker’s optimization is expressed as:
Finally, the balance conditions are $p=\frac{\overline{b}}{q},\overline{p}=\frac{b}{\overline{p}}$, and $1+\rho =\frac{d+\overline{d}}{g}$.
Our general approach here is to first solve the traders’ problems, and then solve the banker’s problem once we know what the traders do as a function of ρ. In essence we solve the trading problem parametrically for ρ, then consider g(ρ) and optimize. Gametheoretically, we can imagine an extensive form game in which the lender moves first, followed by the traders. We then find the perfect equilibrium for the game.
We remark that here again the problems for Types 1 and 2 are isomorphic and so we may assume a symmetric solution where $b=\overline{b},\text{\hspace{0.17em}}d=\overline{d},p=\overline{p}$, and $q=\overline{q}$.
Case 1:
First we consider the case where m is large. This is the case where both the cashflow constraints ((λ) and $(\overline{\lambda})$) and the budget constraints ((μ.) and ($(\overline{\mu})$)) will be loose; that is, $\lambda =\mu =\overline{\lambda}=\overline{\mu}=0$. Our first observation is that this case could not occur if Π = 0, because then traders could raise b and/or lower q in such a way as to increase their utility while maintaining feasibility. Hence we may assume here that Π > 0.
Next, if $\lambda =\mu =\overline{\lambda}=\overline{\mu}=0$, equation (13.5) implies $\sqrt{\frac{aq}{b}}=\Pi \sqrt{\overline{p}}$. Also (13.6) implies $\sqrt{\frac{b}{aq}}=\Pi p\sqrt{\overline{p}}$. Hence $\Pi \sqrt{\overline{p}}=\frac{1}{\Pi p\sqrt{\overline{p}}}$, which (using symmetry) gives $p=\overline{p}=\frac{1}{\Pi}$.
Next, the balancing conditions give $\frac{1}{\Pi}=p=\frac{\overline{b}}{q}=\frac{b}{q}$, so q = Πb. Since also $\sqrt{\frac{aq}{b}}=\Pi \sqrt{\overline{p}}=\sqrt{\Pi}$, we have $\frac{aq}{b}=\Pi $, or q = a − Πb. Hence $q=\overline{q}=\frac{a}{2}$ and $b=\overline{b}=\frac{a}{2\Pi}$.
(p.180) Next, we note that equation (13.7) (with λ = μ = 0) implies ρ = 0. Hence the bank gains zero profits.
At the end of Case 3, we argue that if $0<m<\frac{a}{2\Pi}$, the bank can earn positive profits by choosing a positive interest rate and forcing the traders’ cashflow constraints to be tight. Hence, the above results (in which the bank gains zero profits) cannot possibly be part of a perfect equilibrium. This in turn implies that the mathematical condition for Case 1 is $m\ge \frac{a}{2\Pi}$.^{13}
Finally, we note that none of this analysis has placed any limits on $d=\overline{d}$. In fact, there is a degree of freedom here—d can take on any value in the interval $\left[0,\frac{M2m}{2}\right]$, with $\overline{d}=d$ and g = 2d.
Case 2:
The second case is where m is very small: In this instance both the cashflow and budget constraints will hold with equality, so $m+\frac{d}{1+\rho}b=0$ and pq= d. In addition, $\lambda ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mu ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{\lambda}$, and $\overline{\mu}$ are all positive.
First, we note that pq = d (see above) and $pq=\overline{b}=b$ (balancing condition plus symmetry). Hence $b=d(=\overline{b}=\overline{d})$. Hence, $m+\frac{d}{1+\rho}b=0$ implies $m+\frac{d}{1+\rho}b=0$. If m ≠ 0, this gives $b+\frac{1+\rho}{\rho}m=d=\overline{b}=\overline{d}$. Furthermore,
Hence the banker maximizes $\rho g(\rho )=\rho (\frac{2m}{\rho})=2m$. In other words, the banks profits are 2m no matter what he does. Formally he can set g anywhere in (0, M – 2m], with $\rho =\frac{2m}{g}$, and attain profits of 2m. Another way to say this is that the function g(ρ) has unit elasticity.
Since the bank is indifferent among its feasible strategies, it may wish to choose a policy by which it would benefit the traders most. If so, it will set g as high as possible and ρ as low as possible; that is, g = M – 2m and $\rho =\frac{2m}{M2m}$.
Given the values of g and ρ the bank sets, we may now calculate the optimal values of the traders’ decision variables. Note that we’ve already calculated $b=d=\overline{b}=\overline{d}=\frac{1+\rho}{\rho}m$.
First, we have (13.6) implies $\sqrt{\frac{aq}{b}}=\frac{1}{p\sqrt{\overline{p}}(\Pi +\mu )}$. Together with (13.5) this implies $\frac{1}{p\overline{p}}=(\Pi +\lambda +\mu )(1+\mu )$, for which symmetry implies $p=\overline{p}=\frac{1}{\sqrt{(1+\lambda +\mu )(1+\mu )}}$. Now (13.7) implies λ = ρ(Π + μ), so $p=\overline{p}=\frac{1}{\sqrt{1+\rho}(1+\mu )}$. (p.181) Thus $p(1+\mu )=\frac{1}{\sqrt{1+\rho}}$. But then (13.6) $\Rightarrow \sqrt{\frac{aq}{b}}=\frac{1}{p\sqrt{\overline{p}}(1+\mu )}=\frac{\sqrt{1+\rho}}{\sqrt{\overline{p}}}$; that is, $\frac{aq}{b}=\frac{1+\rho}{\overline{p}}$. Hence $\frac{aq}{\overline{p}\overline{q}}=\frac{1+\rho}{\overline{p}}$. Rearranging, we have $q=\frac{a}{2+\rho}=\overline{q}$. Also, $p=\frac{\overline{b}}{q}=\frac{\frac{1+\rho}{\rho}m}{\frac{a}{2+\rho}}=\frac{(1+\rho )(2+\rho )m}{\rho a}=\overline{p}$.
In the case where m = 0, the assumption that the cashflow constraint is tight (and $b=d=\overline{b}=\overline{d}$) implies that either (a) $b=d=\overline{b}=\overline{d}=0$ or (b) ρ = 0. We discard (a) because it implies an outcome of no trade (see the discussion regarding the Hahn paradox in Chapter 7). But with (b), we have a continuum of symmetric equilibria, each with $\rho =0,q=\frac{a}{2},b=d=\overline{b}=\overline{d}\in \left(0,\frac{M2m}{2}\right]$, and $p=\frac{b}{q}=\frac{2b}{a}$. Note that the values for ρ and q represent limiting values from the m ≠ 0 case, but $b=d=\overline{b}=\overline{d}$ now has a degree of freedom.
Finally, for the multipliers, we have
We remark that the quantities above are valid so long as the multiplier μ is nonnegative. This gives a condition of
Note that the maximum value of $\frac{\rho}{{(1+\rho )}^{3/2}(2+\rho )}$ (on the interval ρ ∈ [0, ∞)) is about 0.12, which is much less than one half. Hence “Case 1” and “Case 2” do not cover all possibilities; that is, we must have at least one “intermediate value for m” case.
Case 3:
Now suppose that neither Case 1 nor Case 2 holds. Hence exactly one of the constraints (λ) and (μ) holds tightly. But just as with the borrowers’ problem in the money market model, it is impossible for (μ) to hold tightly but not (μ).^{14} Hence the only case to consider here is for λ > 0 (so m — b + d/(1 + ρ) = 0) and μ = 0. Also, $\overline{\text{\lambda}}\text{\hspace{0.17em}}\text{=}\text{\hspace{0.17em}}\text{\lambda}\text{\hspace{0.17em}}\text{>}\text{\hspace{0.17em}}\text{0}$ and $\overline{\mu}=0$.
(p.182) First, note that (13.7) implies λ = Πρ.
Next, we see that (13.5) is $\frac{1}{\sqrt{\overline{p}}}\sqrt{\frac{aq}{b}}=\text{\Pi +\lambda}$ and (13.6) implies $\sqrt{\frac{aq}{b}}=\frac{1}{\text{\Pi}p\sqrt{\overline{p}}}$, hence $\frac{1}{\text{\Pi}p\sqrt{\overline{p}}}=\text{\Pi}+\text{\lambda}$. Using symmetry, we have $p=\overline{p}=\frac{1}{\sqrt{\text{\Pi}(\text{\Pi}+\text{\lambda})}}=\frac{1}{\text{\Pi}\sqrt{1+\rho}}$.
Next, since $m+\frac{d}{1+\rho}b=0$ and $m+\frac{\overline{d}}{1+\rho}\overline{b}=0$ we have $b+\overline{b}=\frac{d+\overline{d}}{1+\rho}+2m=g+2m$. But now, since $\frac{\overline{b}}{q}=p=\overline{p}=\frac{b}{\overline{q}}$, we have $\frac{b+\overline{b}}{q+\overline{q}}=\frac{1}{\text{\Pi}\sqrt{1+\rho}}$, which is $q+\overline{q}=\text{\Pi}\sqrt{1+\rho}(b+\overline{b})=\text{\Pi}\sqrt{1+\rho}(g+2m)$.
Next, we see that (13.5) implies $\frac{1}{\overline{p}}\frac{aq}{b}={(\text{\Pi}+\text{\lambda})}^{2}={\text{\Pi}}^{2}{(1+\rho )}^{2}$ so $\frac{aq}{b}=\text{\Pi}{(1+\rho )}^{3/2}=\frac{a\overline{q}}{\overline{b}}$. This implies $\Pi {(1+\rho )}^{3/2}=\frac{2aq\overline{q}}{b+\overline{b}}=\frac{2a\Pi \sqrt{1+\rho}(g+2m)}{g+2m}=\frac{2a}{g+2m}\text{\Pi}\sqrt{1+\rho}$. Rearranging gives
This implies
Also,
The banker will choose ρ so as to maximize $\rho g(\rho )=\frac{2ap}{\text{\Pi}{(1+\rho )}^{3/2}+\text{\Pi}{(1+\rho )}^{1/2}}2m\rho $. This can be done computationally. Two comments:
1. We remark that the maximization is valid only so long as M is large enough so that the ρ so obtained does not cause g(ρ) to be more than M – 2m. This is certainly true if $M\ge \frac{a}{\Pi}$. Otherwise, g(ρ) will stay at the bound of M – 2m and ρ will satisfy ${\scriptscriptstyle \frac{2a}{\Pi {(1+\rho )}^{3/2}+{(1+\rho )}^{1/2}}}=M$. (The formulas for the other variables follow, using this “modified” value of ρ.)
2. Suppose $m\in \left(0,{\scriptscriptstyle \frac{a}{2\Pi}}\right)$. If we are in Case 2 the bank gains positive optimal profits by choosing a positive ρ. And if we are in Case 3 there must exist a small positive ρ such that $\rho g(\rho )=\rho \left({\scriptscriptstyle \frac{2a}{\Pi {(1+\rho )}^{3/2}+\Pi {(1+\rho )}^{1/2}}}2m\right)>0$. This compares with ρg(ρ) = 0 if ρ = 0. Hence again the bank’s optimal strategy is to choose a positive ρ to gain positive profits. These conclusions are important in our analysis of Case 1.
(p.183) 13.6 Appendix B: The Value of ρ(g)
The inverse function of g(ρ) when in Case 3 can be obtained as follows. We start with equation (13.21) from Appendix A:
We substitute $x=\sqrt{1+\rho}$, giving
or
Now set h = Π(g + 2m). Solving (13.24) yields:
or
Substituting Π (g + 2m) back in for h gives an explicit expression for ρ as a function of g.
13.7 Appendix C: Commodity and Money Consuming Bank
In the case where the bank consumes commodity in addition to money, we must be careful to specify exactly the timing of decisions, by traders and the bank, about how much commodity is put up for sale, and how much money is borrowed, lent, and/or bid for the commodity. In other words, we must specify an extensive form for the game. As we discussed in Section 6.3.1, our choice is to assume that the bank makes its decisions $b*,\overline{b}*$ first, and then the traders make their decisions afterward (regarding $b*,\overline{b}*$ as given parameters). We will then find a perfect Nash equilibrium for this game.
(p.184) Mathematically, this implies that the bank must take into account the traders’ “response functions” to its strategies when formulating its own decision problem—this makes the banks firstorder conditions extremely complicated. On the other hand, the traders’ firstorder conditions remain as before.
When we solve the game, we still can use the traders’ firstorder conditions as before, but we will not write down explicit firstorder conditions for the bank. Rather, once we find the traders’ response functions, we will substitute back directly into the bank’s decision problem and solve.
Type 1 traders are endowed with a units of good 1 and m units of money.
Type 2 traders are endowed with a units of good 2 and m units of money. The bank is endowed with M – 2m units of money. The perunit salvage value of money is assumed to be Π = 1.
TYPE 1 TRADERS
Firstorder conditions:
Firstorder conditions:
Balancing conditions:
BANKER’S PROBLEM
We note here again that p, $\overline{p}$, and ρ are all indirectly functions of b*, $\overline{b}*$, and g (and so the banker’s problem is nontrivial to solve). But as before, we may assume by symmetry that $p=\overline{p},q=\overline{q},b=\overline{b},b*=\overline{b}*$, and $d=\overline{d}$.
Case 1:
(p.186) m large (M anything). In this case λ = 0 and μ = 0 (and $\overline{\lambda}=\overline{\mu}=0$). Hence (13.27) ⇒ ρ = 0. Also, we see from (13.26) that $\sqrt{\frac{aq}{b}}=\frac{1}{p\sqrt{\overline{p}}}$. Plugging back into (13.25), we get
But if $p=\overline{p}=1$, (13.25) implies $\sqrt{\frac{aq}{b}}=1$, or
Also
From these, we have $b=aq=\overline{b}$ and $b*=q\overline{b}=2qa=\overline{b}*$. Hence $q=\frac{a+b*}{2}=\overline{q}$ and $b=\frac{ab*}{2}=b$. Substituting these, ρ = 0, and the balancing conditions for price (13.35) into the banker’s problem simplifies it to
This problem is vacuous, solving with any value of $b*=\overline{b}*$ which satisfies (λ_{B}), including zero. Thus, we cannot solve for any more of the variables—indeed, there are a continuum of symmetric solutions, each with $\rho =0,p=\overline{p}=1$, and satisfying (13.36) and (13.37). Hence, for any $q\in [\frac{a}{2},a]$, we have $\overline{q}=q,b=\overline{b}=aq$, and $b*=\overline{b}*=2qa$, For instance, both $\rho =0,p=\overline{p}=1,b=b=q=\overline{q}=\frac{a}{2},b*=\overline{b}*=0$ and $\rho =0,p=\overline{p}=1,b=\overline{b}=\frac{a}{4},q=\overline{q}=\frac{3a}{4},b*=\overline{b}*=\frac{a}{2}$ are solutions. Comparing to our solution of Case 1 in Appendix B (the “m large” subcase of the “bank consumes only cash” case), we see that the presence of the commodity consuming bank gives a continuum of ways for the bidding process to yield an equilibrium price of 1.
Case 2:
m and M both very small. Here λ > 0, μ > 0 (and $\overline{\lambda}>0,\overline{\mu}>0$), and λ_{B} > 0.
We start by solving the Type 1 traders’ problem. Specifically,
Hence,
Next, $(13.28)\Rightarrow b=\frac{d}{1+\rho}+m=\frac{d}{(d+\overline{d})/g}+m=\frac{g}{2}+m$. Similarly, $(13.33)\Rightarrow \overline{b}=\frac{g}{2}+m$. Hence, $d=\overline{d}=(1+\rho )(bm)=(1+\rho )\frac{g}{2}$. Next, $\overline{\lambda}>0$ and $\overline{\mu}>0$ together imply $\overline{p}\overline{q}=\overline{d}$, and
Then
Next, since λ_{B} > 0 (the bank’s constraint is tight), we have $M2mgb*\overline{b}*=0$, which implies
Hence
At the same time, we had $p=\overline{p}=\left(1+\rho \right)\frac{g+m}{a}$, so
So $\rho =\frac{M}{g}1$. We can now use this, (13.38), and (13.39) to rewrite the bank’s problem
as
which simplifies to
To find the optimal g for general values of a, m, and M, we need to find W′(g) and set it equal to zero:
This is equal to zero when mM – 2mg – 2m^{2} – g^{2} = 0, i.e., when g^{2} + 2mg + (2m^{2} – Mm) = 0. The quadratic formula gives
(p.190) The utility actually attained by the banker is
At this point, it is a simple matter to calculate values for all the other variables:
Keeping in mind that $0\le m\le \frac{M}{2}$ by definition, it is clear that with one exception, all of the above expressions must be nonnegative. The lone (p.191) exception is the expression for μ. Hence the formulas are valid so long as ($\left(m\le \frac{a}{2}and\right)\mu \ge 0$ and) μ ≥ 0; that is,
which is
As an example of parameter values which fit into Case 2, consider a = 1, m = 0.01, and M = 0.1 (so the bank’s initial holding of money is 0.08). The formulas above yield the following equilibrium values: $g=0.02,\rho =4,b=\overline{b}=0.02,d=\overline{d}=0.05,p=\overline{p}=0.15,q=\overline{q}=\frac{1}{3},{b}^{*}={\overline{b}}^{*}=0.03,\text{\lambda}=\overline{\text{\lambda}}=\frac{16\sqrt{5}}{3}$, and $\mu =\overline{\mu}=\frac{4\sqrt{5}}{3}1$.
“Case 3”:
λ > 0, μ > 0, $\overline{\text{\lambda}}>0$, $\overline{\mu}>0$ and λ_{B} = 0. This is a “case” in which the bank has a lot of initial money and the traders do not. The reader will notice that we have put the word “case” in quotations; the reason is that our analysis will shortly show that this case cannot occur.
Since this set of multipliers is the same on the traders’ side, everything from the Case 2 analysis holds, up to equation (13.38). Hence, in particular,
Hence
We can now write the bank’s optimization problem
Using the symmetry assumption ${\overline{b}}^{*}={b}^{*}$, this reduces to
Substituting in our expression for ρ and simplifying, we have
Taking the derivative of the objective function with respect to b*, we get
This is positive everywhere, which implies that the banker should raise b* as high as possible. But if the banker does this, this will cause the constraint (λ_{B}) to be tight. Hence it was impossible for λ_{B} = 0 to begin with.
What has happened here? If we recall Case 2 of Appendix A (the case equivalent to this except that the bank consumes only cash), the bank in that model extracted all of the traders’ 2m of money no matter what strategy it chose. Here again the traders’ budget constraints are tight, so the bank must end up with all of the cash in the game no matter what it does. Hence, it might as well try to earn the highest possible utility from consumption of the commodity, via bidding as much as possible for the commodity. Hence b* (and ${\overline{b}}^{*}$) are set as high as possible.
Case 4:
λ > 0, μ = 0, $\overline{\text{\lambda}}>0$, $\overline{\mu}>0$ and λ_{B} = 0. We now move to the case where the bank has a lot of money and the traders have little, except the traders do end up with a positive amount of money at the end of the game.
Again we start by analyzing the traders’ problems. This time (since μ = 0), (13.27) implies λ = ρ. And (13.26) gives $\frac{1}{\sqrt{\overline{p}}}\sqrt{\frac{b}{aq}}=p\Rightarrow \frac{1}{p\sqrt{\overline{p}}}=\sqrt{\frac{aq}{b}}=\sqrt{\overline{p}}\left(1+\text{\lambda}\right)=\sqrt{\overline{p}}\left(1+\rho \right)$, where the penultimate equality follows from (13.25). Hence, using symmetry, we have $p=\overline{p}=\frac{1}{\sqrt{1+p}}$.
(p.193) Next, $\left(13.28\right)\Rightarrow b=\frac{d}{1+\rho}+m=\frac{d}{\left(d+d\right)/g}+m=\frac{g}{2}+m$. Similarly, $\left(13.33\right)\Rightarrow \overline{b}=\frac{g}{2}+m$. In addition, $d=\overline{d}=\left(1+\rho \right)\left(bm\right)=\left(1+\rho \right)\frac{g}{2}$.
Finally, $\left(13.25\right)\Rightarrow \frac{1}{\sqrt{\overline{p}}}\sqrt{\frac{aq}{b}}=1+\text{\lambda}=1+\rho $. Squaring both sides and rearranging gives $\frac{aq}{b}=\overline{p}{\left(1+\rho \right)}^{2}={\left(1+\rho \right)}^{\frac{3}{2}}$. So $q=ab{\left(1+\rho \right)}^{\frac{3}{2}}$.
Next we proceed to use these results to solve the bank’s problem. Starting from
we substitute using symmetry $(\overline{b}*=b*)$ and the above expressions forp and $\overline{p}$ to get
The balancing constraint gives $b*=pqb=\frac{ab{(1+\rho )}^{3/2}}{\sqrt{1+\rho}}(\frac{g}{2}+m)=\frac{a(\frac{g}{2}+m){(1+\rho )}^{3/2}}{\sqrt{1+\rho}}(\frac{g}{2}+m)$. So our optimization problem is now down to two variables, namely, ρ and g:
We denote the objective function by W(ρ, g). Taking the derivative of this with respect to g, we get
We claim that this expression is negative for all nonnegative values of ρ (and g). To prove the claim, we show that; (1) $\frac{\partial W}{\partial g}(\rho =0,g)=0$ for all g; and (2) $\frac{{\partial}^{2}W}{\partial g\partial \rho}(\rho ,g)<0$ for all (ρ, g). Now (1) is obvious. As for (2), we calculate the indicated second derivative and get $\frac{{\partial}^{2}W}{\partial g\partial \rho}(1\sqrt{1+\rho})\frac{2+\rho}{2\sqrt{1+\rho}}+1$. (p.194) Here the first term is negative; hence (2) is proved if we can show that $\frac{2+\rho}{2\sqrt{1+\rho}}$ is always greater than 1. But this follows because (a) $\frac{2+\rho}{2\sqrt{1+\rho}}$ is equal to 1 when ρ = 0, and (b) $\frac{d}{d\rho}\left(\frac{2+\rho}{2\sqrt{1+\rho}}\right)=\frac{\rho}{4(1+\rho )\sqrt{1+\rho}}$, which is positive for all nonnegative ρ.
The above claim implies that the bank’s problem will be optimized at some (ρ, g) where g = 0. Hence we can reduce the bank’s problem to the following onevariable optimization:
This can be solved by computational techniques. When the optimal ρ is found, together with g = 0 we can substitute back to find the values of all of the other variables.
Perhaps the reader will find it strange that the optimal value of g turns out to be zero. The reason is that by not lending money, the bank prevents the traders from getting their hands on much cash. This in turn prevents the traders from bidding much for the commodity. The upshot is that the bank can then buy more of the commodity for itself, at a cheaper price.
As an example of parameter values for this case, consider an example where a = 1, m = 0.05, and M = 1. Note that (13.41) is not satisfied, so we are not in Case 2. Using graphing software, we found that the optimization (13.42) is attained at approximately ρ = 2.028. With ρ = 2.028 and g = 0, we can work backward to get the other optimal variable values: $p=\overline{p}=\frac{1}{\sqrt{1+\rho}}=0.575,b=\overline{b}=\frac{g}{2}+m=0.05,d=\overline{d}=(1+\rho )\frac{g}{2}=0,q=\overline{q}ab{(1+\rho )}^{3/2}=0.737$, and $b*=\overline{b}*=pqb=0.374$. The bank consumes $\frac{b*}{p}=\frac{\overline{b}*}{p}=0.65$ of each good. Finally, we have λ = ρ = 2.028.
If m is lowered to 0.005, we obtain ρ = 8.04, $p=\overline{p}=0.333,b=\overline{b}=0.005,d=\overline{d}=0,q=\overline{q}=0.864$, and $b*=\overline{b}*=0.283$. The bank consumes $\frac{b*}{p}=\frac{\overline{b}*}{p}=0.85$ of each good.
If m is lowered to 0.0005, we obtain ρ = 26.45, $p=\overline{p}=0.191,b=\overline{b}=0.0005,d=\overline{d}=0,q=\overline{q}=0.928$, and $b*=\overline{b}*=0.177$. The bank consumes $\frac{b*}{p}=\frac{\overline{b}*}{p}=0.925$ of each good.
To summarize our conclusions from “Noncase” 3 and Case 4: If the bank is endowed with a lot of money and the traders little, it is in the bank’s interest not to lend the traders anything!
Case 5:
(p.195) λ > 0, μ = 0, $\overline{\lambda}>0,\overline{\mu}=0$ and λ_{B} > 0. Again we start by analyzing the traders’ problems. Since the multipliers for the traders are unchanged from Case 4, we again have λ = ρ, $p=\overline{p}=\frac{1}{\sqrt{1+\rho}},b=\overline{b}=\frac{g}{2}+m$, and $d=\overline{d}=\left(1+\rho \right)\frac{g}{2}$.
Next, since λ_{B} > 0, we have $M2mgb*\overline{b}*=0\Rightarrow b*=\overline{b}*=\frac{Mg2m}{2}$. Hence
And since $p\overline{p}=\frac{1}{\sqrt{1+\rho}}$, we have $\frac{1}{\sqrt{1+\rho}}=\frac{M}{2q}$, or $q=\frac{M\sqrt{1+\rho}}{2}$.
Finally, (13.25) ⇒ $\frac{1}{\sqrt{\overline{p}}}\sqrt{\frac{aq}{b}}=1+\lambda =1+\rho $. Squaring both sides and rearranging gives $\frac{aq}{b}=\overline{p}{(1+\rho )}^{2}={(1+\rho )}^{3/2}$. Substituting in our known expressions for q and for b gives
Note that this is an equation forg in terms of ρ, rather than the reverse. So to finish off the solution, all we need do is find the bank’s optimal ρ. To do this, we rewrite the bank’s problem, using ρ as the strategic variable instead of g. Start with the bank’s problem
and substitute our known expressions for $b*\overline{b}*$ and for $p\overline{p}$. We get
(p.196) Finally, substitute in for g:
or
This can be solved computationally. Once we have found ρ, we can go back and compute g from (13.43), and then all of the rest of the variables easily.
An example of parameter values that fit Case 5 would be a = 1, m = 0.05, and M = 0.4. Using graphing software, we find that the optimal ρ in the above is approximately 2.141. Substituting into (13.43), we get g = 0.132 (approximately). Then $p=\overline{p}={\scriptscriptstyle \frac{1}{\sqrt{1+\rho}}}=0.564,\text{\hspace{0.17em}}b=\overline{b}={\scriptscriptstyle \frac{g}{2}}+m=0.116,\text{\hspace{0.17em}}d=\overline{d}=(1+\rho ){\scriptscriptstyle \frac{g}{2}}=0.207,\text{\hspace{0.17em}}q=\overline{q}={\scriptscriptstyle \frac{M\sqrt{1+\rho}}{2}}=0.354$, and $b*=\overline{b}*=pqb=0.084$. Finally, we have λ = ρ = 2.141.
We remark that within Case 5 there is a wide range, qualitatively, of equilibrium behaviors for the bank. For instance, if the optimal ρ from above yields a g from (13.43) which is greater than M – 2m, the equilibrium solution will be to reset g = M − 2m, that is, to have the bank lend all of its money and use none in bidding for the commodity^{15}. On the other hand, if we obtain a g from (13.43) which is negative, the equilibrium solution will have g = 0; that is, the bank lends nothing and uses all of its endowment in bidding for the commodity.
To demonstrate, consider the case in which a = 1, m = 0.05, and M is slowly lowered from 1. At M = 1, we are in Case 4 and the bank does not lend anything. When M reaches a value of about 0.85, we cross over from Case 4 to Case 5—but the bank still does not lend.^{16} Finally, within Case 5, when M reaches a value of roughly 0.55, the equilibrium solution has the bank lending. As M decreases further, the bank lends more. Finally, at a value (p.197) of about M = 0.31, the bank lends all it has, and continues to do so all the way down to the lowest possible value M = 0.1.
13.8 Appendix D: Oligopolistic Banking
The traders’ optimization problems here are identical to those in Appendix A. For ease in following the math, we reproduce them here.
The Type 1 traders face an optimization described by:
The firstorder conditions wrt b, q, and d yield
Similarly, the Type 2 traders face the optimization below:
(p.198) The optimization conditions here are
The difference between this model and the one in Appendix A is that here we have n bankers (n ≥ 2) instead of one. Suppose Bank i (i = 1, …, n) is endowed with B_{i} units of fiat. The optimization problem for this bank is:
Finally, the balance conditions are $p={\scriptscriptstyle \frac{\overline{b}}{q}},\overline{p}={\scriptscriptstyle \frac{b}{\overline{q}}}$ and $1+\rho ={\scriptscriptstyle \frac{d+\overline{d}}{{g}_{1}+\cdots +{g}_{n}}}$.
Case 1:
m large. In this case both the traders’ cashflow and budget constraints are loose; that is, $\text{\lambda}=\overline{\text{\lambda}}=\mu =\overline{\mu}=0$.
In this case the analysis follows that of Case 1 of Appendix A. First we note that Π can’t be zero here, because otherwise traders could raise b and/or lower q in such a way as to increase their utility while maintaining feasibility. Hence we may assume here that Π > 0. Next, (13.45) implies $\sqrt{{\scriptscriptstyle \frac{aq}{b}}}=\text{\Pi}\sqrt{\overline{p}}$. Also, (13.46) implies $\sqrt{{\scriptscriptstyle \frac{b}{aq}}}=\text{\Pi}p\sqrt{\overline{p}}$. Hence $\text{\Pi}\sqrt{\overline{p}}={\scriptscriptstyle \frac{1}{\text{\Pi}p\sqrt{\overline{p}}}}$, which (using symmetry) gives $p=\overline{p}={\scriptscriptstyle \frac{1}{\text{\Pi}}}$.
Next, the balancing conditions give ${\scriptscriptstyle \frac{1}{\text{\Pi}}}=p={\scriptscriptstyle \frac{\overline{b}}{q}}={\scriptscriptstyle \frac{b}{\overline{q}}}$, so q = Πb. Since also $\sqrt{{\scriptscriptstyle \frac{aq}{b}}}=\text{\Pi}\sqrt{\overline{p}}=\sqrt{\text{\Pi}}$, we have ${\scriptscriptstyle \frac{aq}{b}}=\text{\Pi}$, or q = a – Πb. Hence we have q = Πb and q = a – Πb, which together imply $q=\overline{q}={\scriptscriptstyle \frac{a}{2}}$ and $b=\overline{b}={\scriptscriptstyle \frac{a}{2\text{\Pi}}}$.
Also, condition (13.47) (with λ = μ = 0) implies ρ = 0. This implies the banks each get profits of zero.
(p.199) An argument in footnote 17, below, states that the mathematical condition for this case is $m\ge {\scriptscriptstyle \frac{a}{2\text{\Pi}}}$.^{17}
None of the analysis above puts any restriction on d. Indeed, as long as $m\ge {\scriptscriptstyle \frac{a}{2\text{\Pi}}}$ we have a continuum here, with $d=\overline{d}\in \left[0,\frac{{\displaystyle {\sum}_{i}{B}_{i}}}{2}\right]$ and g_{1}, …, g_{n} satisfying ${\sum}_{i}{g}_{i}=2d$ and g_{i} ∈ [0, B_{i}] for all i.
Case 2:
Very little money (m very small). Here both the traders’ cashflow and budget constraints are tight, that is, λ, $\overline{\text{\lambda}}$, μ, and $\overline{\mu}>0$.
In this case the tight cashflow and budget constraints together imply pq = d. Since the balancing condition (plus symmetry) implies pq = b, we have $b=d=\overline{b}=\overline{d}$. So then our tight cashflow constraint can be written as
If m ≠ 0, (13.57) implies $b=d=\overline{b}=\overline{d}={\scriptscriptstyle \frac{1+\rho}{\rho}}m$. But then ${g}_{1}+\cdots +{g}_{n}={\scriptscriptstyle \frac{d+\overline{d}}{1+\rho}}={\scriptscriptstyle \frac{2{\scriptscriptstyle \frac{1+\rho}{\rho}}}{1+\rho}}={\scriptscriptstyle \frac{2m}{\rho}}$. This then gives $\rho ={\scriptscriptstyle \frac{2m}{{g}_{1}+\cdots +{g}_{n}}}$, so Bank i’s problem (13.56) becomes
This is optimized (for any values of m and {gk}_{k≠i}) by settingg_{i} = B_{i}. So, in conclusion, each bank lends out all of its money and the interest rate becomes (p.200) $\rho ={\scriptscriptstyle \frac{2m}{{B}_{1}+\cdots +{B}_{n}}}$. One can then work backward, as in the analysis from Case 2 in Appendix A, to get the rest of the values: $b=d=\overline{b}=\overline{d}={\scriptscriptstyle \frac{1+\rho}{\rho}}m,q=\overline{q}={\scriptscriptstyle \frac{a}{2+\rho}}$, and $p=\overline{p}={\scriptscriptstyle \frac{(1+\rho )(2+gr)m}{\rho a}}$. The results are valid so long as ${\scriptscriptstyle \frac{m}{a}}\le {\scriptscriptstyle \frac{gr}{\text{\Pi}{(1+\rho )}^{3/2}(2+\rho )}}$ (again following the analysis from Appendix A, Case 2).
If m = 0, condition (13.57) implies that either (a) $b=d=\overline{b}=\overline{d}=0$, or (b) ρ = 0. If (a) holds, there is no trade, so we disregard (a). If (b) holds, (13.47) implies that λ = 0. But then (13.45) implies $\sqrt{{\scriptscriptstyle \frac{aq}{b}}}=\sqrt{\overline{p}}\left(\text{\Pi}+\mu \right)$ and (13.46) implies $\sqrt{{\scriptscriptstyle \frac{b}{aq}}}=p\sqrt{\overline{p}}\left(\text{\Pi}+\mu \right)$. Substituting in for $\sqrt{\overline{p}}\left(\text{\Pi}+\mu \right)$, we get $\sqrt{{\scriptscriptstyle \frac{b}{aq}}}=p\sqrt{{\scriptscriptstyle \frac{aq}{b}}}$, which is $p={\scriptscriptstyle \frac{b}{aq}}$. But by symmetry and the balancing condition $p={\scriptscriptstyle \frac{b}{q}}$; hence ${\scriptscriptstyle \frac{b}{q}}={\scriptscriptstyle \frac{b}{aq}}$ or $q={\scriptscriptstyle \frac{a}{2}}=\overline{q}$. There is a degree of freedom regarding b, d, and p: any values work so long as: (a) $b=d=\overline{b}=\overline{d}$ (b) $p=\overline{p}={\scriptscriptstyle \frac{2b}{a}}$ and (c) ${g}_{1}+\cdots +{g}_{n}=d+\overline{d}$, with g_{i} ∈ [0, B_{i}] for i = 1, …, n.
Case 3:
An intermediate amount of money. Here we assume that the traders’ cashflow constraints are tight (λ, $\overline{\text{\lambda}}$ > 0) while their budget constraints are loose (μ, $\overline{\mu}$ = 0).
In this case let us define the quantity g as equal to g_{1} + ⋯ + g_{n}. We follow the analysis from Case 3 in Appendix A, obtaining
(These equations are (13.21), (13.22), and (13.23), above.) Then $d=(1+\rho )(bm)={\scriptscriptstyle \frac{(1+\rho )a}{\text{\Pi}{(1+\rho )}^{3/2}+\text{\Pi}{(1+\rho )}^{1/2}}}(1+\rho )m$. But the balance condition (plus symmetry) says $1+\rho ={\scriptscriptstyle \frac{2d}{g}}$; hence $d={\scriptscriptstyle \frac{2ad/g}{\text{\Pi}{(2d/g)}^{3/2}+\text{\Pi}{(2d/g)}^{1/2}}}{\scriptscriptstyle \frac{2md}{g}}$, or $g={\scriptscriptstyle \frac{2a}{\text{\Pi}{(2d/g)}^{3/2}+\text{\Pi}{(2d/g)}^{1/2}}}2m$. From this, we can derive d as a function of g, which we call d(g). So then Bank i’s problem (13.56) becomes:
(p.201) Let us suppose that the quantities B_{1}, …, B_{n} are all large, so that the conditions (λ_{B}) are all loose. Even so, the quantity g is itself dependent ong_{i}, so the firstorder conditions for solving (13.62) become quite complicated:
Even allowing for symmetry ${g}_{i}={\scriptscriptstyle \frac{g}{n}}\text{\hspace{0.17em}}(i=1,\dots ,n)$ does not make this easy, due to the complicated nature of the function d(g). But presumably the optimal g_{i}’s are interior to [0, B_{i}], so the banks are lending out some, but not all, of their money. And if we can find the equilibrium d and g, and hence ρ, we can find b and q using (13.60)–(13.61).
Notes:
(1) We do not deal here with the empirically important aspects of structures such as cooperatives, partnerships, or holding companies for corporate banks. These are critical for understanding the shadings in the motivations and goal structure of the institutions. In essence, we study only the simplified extremes.
(2) One issue is: Mathematically, does s being <, =, < n make any difference? Empirically, in the various aggregations of consumers and products there are fewer consumer types than there are products, i.e., s > n.
(3) Shubik and Smith (2007) and Smith and Shubik (2003) have investigated general models with n > 2.
(4) More strictly, the individual IOU note calls for n new financial instruments reflecting the differences in creditworthiness among all n agent types, but as we assume symmetry all agents are equally creditworthy and this distinction disappears.
(5) We face a definitional problem in the switch from a commodity money to fiat. Is fiat a new financial instrument or is it a synthetic or artificial commodity? If we regard it as a financial instrument then it is the only financial instrument for which there is not necessarily a second financial instrument that nets the two to zero. Here we treat fiat as an artificial asset.
(6) The model in Section 5.2.2 utilized a storable consumable money, not fiat. However, as discussed earlier, that model is essentially equivalent to a model with fiat.
(7) The minimal modeling of a public good requires a production function operated by the government or privately.
(8) If there is any form of uncertainty, the corporation will maximize some riskadjusted form of profits such as expected profits. If the stockholders have different risk profiles, then the corporation has a problem in fulfilling its fiduciary duty in selecting the most representative risk policy.
(9) In an economy involving several periods we would wish to consider the central bank as both a lender and a borrower. We do not deal with this in this book. Furthermore, we could introduce an internal money market. We also omit this complication at this point.
(10) Cases 2 and 3 of Appendix C (the model where the bank consumes goods as well as money).
(11) See the discussion in the subsection “Weak or Strong Pareto Optimality?” in Section 5.2.2.
(12) Keep in mind that in cases like this, in which there is an interest rate of ρ = 0, we again have a “coordination problem” (see Chapter 4).
(13) As we shall see, the case m = 0 yields a continuum of equilibria and doesn’t really fit neatly into any of the three cases. Somewhat arbitrarily, we analyze it below in Case 2.
(14) To repeat the argument from the money market model: (μ) tight and (λ) loose implies d > 0, which in turn implies Type 1 traders could improve by simultaneously raising b and lowering d.
(15) Once we reset g = M – 2m, ρ must be reset so as to satisfy (13.43), and then the other variables can be calculated from these.
(16) Remember, the distinction between Case 4 and Case 5 is not whether or not the bank lends money—but rather it is whether or not the bank hoards money.
(17) We do this by proving that if $m<{\scriptscriptstyle \frac{a}{2\text{\Pi}}}$, there cannot be a symmetric perfect equilibrium in which ρ = 0. The proof is by contradiction—so suppose there was such a symmetric equilibrium, with g_{1} = g_{2} = ⋯ = g_{n} = ĝ. Since ρ = 0, the banks are each getting a profit of zero from this equilibrium.
Now, if ĝ = 0, Bank 1 could change its strategy to g_{1} = g*, where g* is the positive loan level associated with the positive ρ whose existence we argued for in list item 2 at the end of Appendix A, Case 3. In this case the traders will demand exactly the loans which will put the interest rate at that ρ, and so Bank 1’s profits will be positive. This in turn impliesg_{1} = g_{2} = ⋯= g_{n} = 0 is not an equilibrium.
Now suppose ĝ > 0. We have $1+\rho ={\scriptscriptstyle \frac{d+d}{n\widehat{g}}}$, with ρ = 0. Now Bank 1 can change its strategy by lowering its loans to ĝ – ε, where ε is an infinitesimal amount which would raise ρ to a small positive level at which the traders would demand loans. (Essentially, equation (13.21) gives the total amount g of loans demanded by traders as a function of ρ; if $m<{\scriptscriptstyle \frac{a}{2\text{\Pi}}}$ and if ρ is small enough, this will be positive) This again would make Bank 1’s profits positive, and so again there is no equilibrium.