Recovering Technologies That Account for Generalized Managerial

Preferences: An Application to Banks That Are Not Risk-Neutral 

 Joseph P. Hughes 

 Rutgers University 

 William W. Lang 

 Office of the Comptroller of the Currency 

 Loretta J. Mester 

 Federal Reserve Bank of Philadelphia 

 and 

 The Wharton School, University of Pennsylvania 

 Choon-Geol Moon 

 Hanyang University 

Office of the Comptroller of the Currency 

Economics Working Paper 97-11 

June 1997 
Recovering Technologies That Account for Generalized Managerial Preferences: 

An

Application to Banks That Are Not Risk-Neutral 

 Joseph P. Hughes 

 Rutgers University 

 William W. Lang 

 Office of the Comptroller of the Currency 

 Loretta J. Mester 

 Federal Reserve Bank of Philadelphia 

 and 

 The Wharton School, University of Pennsylvania 

 Choon-Geol Moon 

 Hanyang University 

Office of the Comptroller of the Currency 

Economics Working Paper 97-11 

June 1997 

Abstract Large banks involved in recent mergers often cite scale economies as a principal 

motive. Yet despite this belief that bigger is more economical, most economic studies find either 

constant or decreasing returns to scale for large banks. What accounts for the disparity between 

the beliefs of bank managers and the results of these studies? This paper suggests that the 

answer may lie with the assumptions underlying estimations of standard cost and profit 

functions. Typically these estimations assume that banks are risk-neutral -- that they ignore risk 

in their quest to maximize profits. This paper develops an alternative model which is sufficiently 

general to allow for active risk management. The resulting empirical estimates indicate large 

economies of scale that increase with bank size. 

 

The views expressed here are those of the authors and do not necessarily represent those of the Comptroller of the 

Currency, the Department of Treasury, Federal Reserve Bank of Philadelphia or the Federal Reserve System. 

Authors Hughes and Moon gratefully acknowledge the financial assistance of the Rutgers University Research 

Council on a preliminary project. Please address any comments to William Lang, Economics and Evaluation 

Division, Office of the Comptroller of the Currency, Washington, DC 20219. Phone: (202) 874-5386. E-mail: 

william.lang@occ.treas.gov. 

Additional copies of this paper, or other Economics Working Papers, can be obtained from the Communications 

Division, Office of the Comptroller of the Currency, 250 E Street, SW, Washington, DC, 20219. Phone: (202) 874-

5043. E-mail: kevin.satterfield@occ.treas.gov.

We develop a maximization model that allows for (but does not impose) preferences of 

bank management that are not risk neutral. Our estimated equations are fully consistent with the 

theoretical model underlying the estimation procedure. The model considers a bank managerial 

utility function in which profit is one of many elements. Since we do not observe the 

probabilities that a bank assigns to various sets of outcomes, our model considers how banks 
rank production plans given their choices of the quantity and quality of outputs and inputs. Risk 

neutrality then implies that a bank's preferences will depend only on profit and that other 

variables in the utility function have no impact on bank preferences. 

Our empirical model strongly rejects the hypothesis that banks are risk-neutral. 

Preferences over production plans are not invariant with respect to the tax rate on profits or on 

fixed revenues. Unlike previous studies, we find large increasing returns to scale even for the 

largest banks. This result is due to our allowance for behavior that is not risk-neutral. When the 

empirical restrictions implied by risk neutrality are imposed, our estimates of returns to scale are 

similar to those found in previous studies -- small returns to scale that decline with bank size. 
1 

1. Introduction

Managers who have participated in recent large bank mergers have often cited scale economies 

as a principal motive. Despite the apparent belief of some bank managers that bigger is more 

economical, most economic studies find either constant or decreasing returns to scale for large 

banks. What accounts for the disparity between the beliefs of bank managers and the results of 

these studies? This paper suggests that the answer may lie with the assumptions underlying 

estimations of standard cost and profit functions. Typically these estimations assume that banks 

are risk-neutral -- that they ignore risk in their quest to maximize profits. This paper develops an 

alternative model which is sufficiently general to allow for active risk management. 

The standard cost and profit functions using modern duality theory usually ignore risk. 

However, risk and risk management fundamentally shape bank costs and profits. Correctly 

accounting for behavior toward risk is crucial for uncovering the underlying production 

technologies in banking.

There are a number of reasons that banks might be concerned about the level of 

risk inherent in their production plans. If there is value in a bank charter, then bank 

owners might value production plans that reduce the risk of insolvency even if such a 

plan reduces earnings. In addition, banks may wish to avoid regulatory actions which 

punish excessive risk taking by banks. Alternatively, bank managers may be concerned 

about risk because a significant loss could threaten their jobs. Whatever the motive for 

managing risk, it is clear that aversion to risk should not be ruled out when estimating 

bank production decisions. 

Appropriately controlling for risk may have profound affects on empirical analysis 

of bank production. For example, if a larger scale of banking operations improves 

opportunities for diversification and hedging, this would produce two important effects -- 

a direct reduction in risk and a consequent reduction in the marginal costs of controlling 

risk. These are the direct or exogenous effects of greater risk diversification. These 

exogenous effects of diversification are accompanied by an indirect endogenous effect 

of diversification -- the bank's choice of risk in response to the reduced price of risk. 

The optimal amount of risk depends not only on the marginal cost of controlling risk, but 

also on the additional revenue or return. The net marginal benefit or compensation for 

risk-taking is the price of risk. 

If an increase in the scale of operations diversifies risk and increases the net 

marginal compensation for risk-taking, then banks may respond by assuming more risk 

and, hence, incur higher costs. Accurate measurement of scale economies must 

control for the endogenous component -- the component that is a matter of choice -- 

while allowing the exogenous component to adjust freely to scale variations. Hence, the 

cost-saving component of diversification of a loan portfolio is the exogenous reduction in 

credit risk at a given level of quality of assets and liabilities. 

This paper develops a maximization model that allows for (but does not impose) 

non-neutral risk preferences of bank management. Our estimated equations are fully 

consistent with the theoretical model underlying the estimation procedure. The model 

considers a bank managerial utility function in which profit is one of many elements. 

Since we do not observe the probabilities that a bank assigns to various sets of 
2 

outcomes, our model considers how banks rank production plans given their choices of 

the quantity and quality of outputs and inputs. Risk neutrality then implies that a bank's 

preferences will depend only on profit and that other variables in the utility function have 

no impact on bank preferences. 

Our empirical model strongly rejects the hypothesis that banks are risk-neutral. 

Preferences over production plans are not invariant with respect to the tax rate on 

profits or on fixed revenues. Unlike previous studies, we find large increasing returns to 

scale even for the largest banks. This result is due to our allowance for behavior that is 

not risk-neutral. When the empirical restrictions implied by risk neutrality are imposed, 

our estimates of returns to scale are similar to those found in previous studies -- small 

returns to scale that decline with bank size. 

1.1. Amending the Standard Cost and Profit Functions to Account for Risk

McAllister and McManus (1993) condition the cost function on financial capital and 

control for the risk of insolvency by adjusting each bank's level of financial capital so 

that all banks in their sample have the same probability of insolvency. They find 

essentially constant returns to scale. If improved diversification reduces return risk, it 

also reduces the risk of insolvency. Hence, adjusting capital to control for the 

probability of insolvency may obscure scale economies achieved by improving 

diversification. 

When the standard cost function omits any consideration of risk, it implicitly assumes that managers choose the return-maximizing level of risk for any given vector of 

outputs or, equivalently, that managers are risk-neutral. Hughes and Mester (1993) 

recognized that the level of financial capital may not minimize cost if bank managers are 

not risk-neutral, so they conditioned the bank's cost function on its level of financial 

capital. This conditional cost function permits the level of financial capital to deviate 

from the least-cost expansion path. However, given this level, their model assumes that 

banks choose levels of all other inputs to minimize cost. In measuring scale economies, 

they control for loan quality, but assume financial capital varies proportionately with the 

output vector. Not surprisingly, they obtain essentially constant returns to scale for their 

sample, U.S. banks whose assets surpass $1 billion. The equiproportionate variation in 

financial capital obscures any economies that might be obtained when better 

diversification allows banks to reduce their capital-to-asset ratios. 

When Hughes and Mester (1995) embed the conditional cost minimization 

problem into the problem of maximizing managerial utility, where managers choose the 

utility maximizing level of capital and profit, they obtain a utility-maximizing demand for 

financial capital that allows for non-risk-neutral preferences or, equivalently, a level of 

financial capital that does not minimize cost. They also control for loan quality. In 

measuring scale economies, they no longer need to assume that financial capital varies 

proportionately with output. In fact, using the same large bank sample, they obtain 

evidence of substantial economies in capitalization and of relatively large overall 

economies of scale. 

These attempts to account for risk in the measurement of technology raise an important question: Are managers neutral toward risk? There are good reasons to 
3 

believe that managers may in fact be risk-averse. Risk aversion poses grave difficulties 

for the standard cost and profit functions. When managers take on higher cost and 

reduced profit in return for less risk, their production decisions cannot be modeled by 

assuming that cost is minimized and profit is maximized. The theory of the firm draws 

important distinctions between the behavior of risk-neutral and risk-averse managers. 

For example, the risk-averse manager's demand for inputs responds to changes in fixed 

costs and to changes in marginal tax rates on profits; the risk-neutral manager's 

demand does not. Equivalently, cost minimization and profit maximization equilibria are 

not affected by these variables, while the equilibrium of the firm that accepts lower profit 

and higher cost in return for reduced risk is influenced by them. Hence, by implicitly 

assuming risk neutrality, the standard cost and profit functions omit variables such as 

fixed costs and taxes on profits and, in the case of risk-averse behavior, fail to capture 

important aspects of production. 

To incorporate risk into a model of production, one should choose a model that is 

general enough to allow for non-neutral as well as neutral risk preferences. 

Additionally, generalized risk preferences should be incorporated within the context of 

an optimization process rather than as an ad hoc construction. Finally, as noted in the 

previous subsection, the model should also account for sources of risk, as opposed to 

the resulting levels of risk. 

1.2. Risk-Averse Managerial Preferences

Banking technology is characterized by fundamental information asymmetries between 

banks and their borrowers and between banks and their creditors and owners. Banks 

employ credit analysis and loan monitoring to reduce information asymmetries between 

themselves and their borrowers. In addition, a bank’s information on a borrower’s 

demand deposits gives it a low-cost source of credit information that historically has not 

been readily available to other lenders. This “inside” information concerning asset 

quality may generate another information asymmetry between bank management and 

those providing external financing, the owners and creditors of the bank. These 

outsiders have less information than insiders about the inherent risk of the loan portfolio 

or the resources that banks devote to credit analysis and loan monitoring. 

When owners and creditors of firms cannot perfectly monitor the actions of the 

firms' managers and managers' interests differ, agency problems arise. To protect their 

firm-specific human capital, or, perhaps, to appease their regulators, bank managers 

may accept higher cost and reduced profit in return for less risk. For example, to 

reduce liquidity risk, managers may fund the loan portfolio with more expensive but less 

volatile funding sources, such as core deposits. To reduce credit risk, managers may 

allocate extra resources to credit analysis and loan monitoring. To reduce interest rate 

risk, they may choose less profitable mixes of loans and funding sources to close the 

duration gap. Because bank owners are able to diversify risk in their own investment 

portfolios, they would prefer that their banks' managers make risk-neutral decisions. 

However, the owners' inability to perfectly monitor loan quality and risk management 

gives managers the opportunity to exercise risk-averse preferences. 

Even if managers are risk-neutral, the fundamental informational asymmetry be-
4 

tween managers and creditors may lead to phenomena that cannot be captured by the 

standard cost and profit functions. Since managers are better informed about loan quality and the resources devoted to risk management, they may signal loan quality to uninsured creditors to reduce the risk of a liquidity crisis by “overemploying” certain readily 

observed inputs, such as financial (equity) capital. Since loan losses are subtracted 

from financial capital, equity serves as a cushion against insolvency and represents a 

bank's “bet” on its loan quality. Consequently, the level of financial capital relative to the 

size of the loan portfolio is a credible signal of asset quality. Well-capitalized banks are 

more likely to have high quality loans and to allocate more resources to managing risk. 

If financial capital serves as a signal of risk to less informed, uninsured creditors, its 

level may not minimize cost or maximize profit. 

Risk-neutral managers might also “overemploy” financial capital to protect a valuable charter. The charter value constitutes a bankruptcy cost that is not included in the 

usual calculation of cost and profit. Hence, even risk-neutral managers might employ 

more than the cost-minimizing or profit-maximizing level of capital to protect the charter 

value. “Overemploying” capital to reduce the probability of liquidity crises and of 

bankruptcy would look like risk aversion. 

Regulators may also contribute to the appearance of risk aversion. For example, 

to protect the payments mechanism, regulators may induce or require bank managers 

to adopt production plans that reduce risk below cost-minimizing and profit-maximizing 

levels. Regulatory attention is routinely focused on maintaining “prudent” capital levels. 

Thus, there are many plausible theoretical explanations for managers choosing 

production plans that do not minimize cost or maximize profit. In this paper we do not 

attempt to uncover evidence for any particular theory. Rather, we formulate a general 

model of managerial preferences that allows for the possibility for departures from risk 

neutrality. 

1.3. Generalized Managerial Preferences

Managers who minimize cost and maximize profit prefer high profit production 

plans to low profit ones. The specific output and input mixes of the production plans 

matter only in terms of their implied profit. On the other hand, managers who accept 

higher cost and reduced profit in return for other objectives care about the particular 

composition of the output and input bundles insofar as their composition contributes to 

achieving the additional objectives. For example, they may care about the implied 

credit risk and liquidity risk of the bundles. Or they may rank plans by the implied work 

load and prestige for top management. Or they may rank plans by the incentives 

regulators provide. 

In any of these cases, managerial preferences can be characterized by a utility 

function defined by the inputs and outputs constituting the production plan and by the 

profit the production plan implies. The plan's profitability uses price information to 

aggregate the components of the plan. When managers simply minimize cost and 

maximize profit, only profit has marginal significance in their utility function. The 

production plan affects utility only indirectly through its effect on profit. However, when 

managers take on higher cost and reduced profit in return for other objectives, the 
5 

particular components of the production plan will influence utility individually as well as 

through their effect on profit. 

A managerial utility function defined over the production plan and profit is sufficiently general to represent a variety of objectives without having to be specific about 

any objective other than profitability. Similarly, it is sufficiently general that it could 

reflect the sources of a variety of risks without having to be specific about the nature of 

those risks. 

1.4. Modeling Risky Production

In the sections that follow, we draw on earlier work by Hughes (1989, 1990) on 

hospitals and education that allows managers to choose production plans that trade 

profit or net income for other objectives. We specify a generalized managerial utility 

function, defined over profit and the production plan, to model banking technology and 

to derive a measure of scale economies that accounts for the risky nature of production. 

 We assume that managers maximize utility with respect to the mix of inputs and profit, 

subject to the production constraint that the input mix must produce the given output 

vector. The problem's solution gives the manager's most preferred production plan. To 

the extent that managers have favored inputs and will increase their levels at the 

expense of profit, the most preferred production plan will not minimize cost or maximize 

profit. 

The cost function that follows from the utility-maximizing (or most preferred) production plan is sufficiently general to incorporate non-neutrality toward risk, and to 

account for the sources of risk, without masking the diversification effects of scale on 

risk. Moreover, the most preferred cost function allows tax rates and fixed charges and 

revenues to affect production and, thus, accommodates the comparative statics of a 

firm that is not risk neutral. Although the most preferred cost function allows departures 

from strict cost minimization, it does not sacrifice desirable duality properties. 

We empirically implement the model by adapting the Almost Ideal (AI) Demand 

System to allow for generalized managerial preferences. From this system we obtain 

input share equations and a profit (cost) function that, in the case of cost minimization, 

are identical to the translog cost function and input share equations. The model is 

estimated using 1989 and 1990 data from 286 U.S. banks with assets of at least $1 

billion. The AI Demand System fits the data well. Risk neutrality is conclusively 

rejected. The estimate of scale economies obtained from the most preferred cost 

function is larger than that obtained from more conventional cost functions; it increases 

with bank asset size, suggesting that the diversification economies enjoyed by larger 

banks allow them to reduce the share of resources used to control risk, which magnifies 

the diversification economies. These results may provide insights into the economic 

reasons for the recent wave of mergers among large banks in the United States. When 

risk neutrality is imposed on the model and sources of risk are deleted, the measure of 

scale economies is smaller and resembles that generally found in studies employing the 

standard framework. Our evidence helps to explain why the standard literature (see 

Berger and Humphrey [1992]) finds cost savings to be a poor rationale for recent 

mergers. 
6 

2. Risky Production and Managerial Preferences

To incorporate risk into the production model, two important principles should be 

observed. First, the framework for choosing the production plan should allow for nonneutrality toward risk. Second, the basic construction of the model should focus on the 

sources of risk rather than the levels of risk. To observe these principles, we represent 

managers' attitudes toward risk by a utility function whose arguments include profit, 

output levels, and input levels. In its application to banking, the utility function is 

expanded to include financial (equity) capital as an input in the production plan. In 

addition, to account for sources of credit risk, output prices and nonperforming loans are 

included to measure loan quality. 

If bank outputs are defined by asset categories such as commercial and 

industrial loans, consumer loans, commercial real estate loans, and securities, while 

inputs are labor, physical capital, financial capital, and other types of funding sources, 

the output vector, by comprising the bank's asset portfolio, reflects some of the sources 

of the bank's riskiness. In addition to asset composition, another determinant of risk is 

the quality of the assets and, in particular, of the loan portfolio. An important indication 

of loan quality is the interest rate charged on loans. In general, the higher the rate 

relative to the risk-free rate, the greater the credit risk and, thus, the lower the loan 

quality. Although an increase in the loan rate, given the risk-free rate, may improve the 

expected return on assets, it also reduces the quality of the loan applicants, since 

borrowers with better credit ratings will seek lenders offering lower rates. This suggests 

that the arguments of the bank managers' generalized utility function should include the 

vector of output levels, output prices, and the risk free interest rate. The risk premia 

inherent in the output prices are ex ante measures of output quality. The level of 

nonperforming loans, given the output levels, is an ex post measure of loan quality. 

Credit risk is important to a bank because it affects the bank's risk of insolvency. 

Financial capital is the cushion that absorbs the impact of loan losses. For a given loan 

portfolio, the risk of insolvency increases as the level of financial capital decreases and 

as the amount of nonperforming loans increases. Thus, the generalized utility function 

should also include financial capital and nonperforming loans. Finally, the risk 

characteristics of the input mix must also be taken into account. Although production 

techniques involving the greater use of overnight funding sources might be less costly, 

such a course might increase liquidity risk. And although a bank might save on costs by 

doing less intensive credit evaluations and loan monitoring, doing so would mean 

greater credit risk. Hence, the input mix is included among the arguments of the utility 

function. 

These generalized managerial preferences are represented by the utility function 

U(π,s), 

where π is 

real, after-tax 

accounting 

profit; s = (y, 
7 

x, p, r, n, k); 

y is the 

vector of 

output levels; x, the vector 

of input 

levels; p, the 

vector of 

output 

prices; r, the 

risk-free rate 

of return; n, 

the level of 

nonperformin

g loans; and 

k, the level of 

financial 

capital. Let 

w be the 

input price 

vector; w

k, 

the price of 

financial 

capital (rate 

of return on 

equity); and 

m, income 

from sources 

other than 

those 

accounted 

for by output. 

 Letting t be 

the tax rate 

on profit 

andp

π(=1

)

be the 

nominal 

"price" of a 

real dollar, 

the price of a 

dollar of real, 

after-tax

profit in 
8 

terms of 

nominal, 

before-tax 

dollars is 

pπ= pπ

1-t. 

Nominal, 

before-tax 

accounting 

profit is, thus, 

defined as 

pππ = pÀ___ + m - wÀ__

 

_ (1) 

Nominal accounting profit is composed of before-tax economic profit,pπ

π, and the 

 required payment to equity, wkk, which will depend on the riskiness of the bank. 

Hence, let 

1

The “price”,pπ , facilitates stating the homogeneity conditions: a proportional variation inpπimplies the 

same variation in pπ so homogeneity will be stated in terms of the latter.

wk = r  g(s), where, as noted above, r is the risk-free rate of return and g(s) > 1 is a risk 

premium. The risk premium, g(s), is assumed to be homogeneous of degree zero in 

(p,r). Thus, a proportional variation in the risk-free rate r and the asset returns p results 

in an equiproportional variation in wk. The other arguments that affect the premium are 

discussed below. Thus, 

pπ = pπ[

wk k

π

p

+π]

 = pπ[

rg(s)k

p

+π]

 (2) 

The nominal, before-tax return on equity is then

pππ

k = 

pπrg(s)

p

+

pπ

k

π

, which consists of 

the required return and the economic rent. 

The utility function U(π,s) allows for a variety of managerial objectives and 

sources of risk without having to be specific about the particular objectives and the 

particular kinds of risk. In the special case of risk neutrality, cost is minimized and profit 

is maximized. Given the level of financial (equity) capital, this corresponds to 

maximizing the rate of return on equity. In this special case, only profit has marginal 

significance in the utility function. The output levels and input levels affect utility only 

through their effect on profit. On the other hand, if managers have favored inputs that 

they prefer to employ more intensively, such as staff or core deposits, say, because 

larger staffs convey prestige or permit more careful credit evaluation and loan 
monitoring, or because core deposits reduce liquidity risk, then arguments in the utility 

function other than profit have marginal significance, and managers will accept higher 

cost (reduced profit) for greater prestige or reduced risk. 

3. Organizing Risky Production

The bank's technology is characterized by the transformation function T(y,x,n,k) < 0. 

Financial capital is included, since it is a source of loanable funds. The level of 

nonperforming loans, n, influences the mix of inputs through, for example, the labor 

required to respond to nonperformance. It may also indicate the labor intensity of credit 

analysis and loan monitoring. The standard cost function is derived from the cost 

minimization problem, 

C(y,w,n,k) = min wx s.t.T(x;y,n,k) < 0

 x

The optimal input levels are determined by the first-order conditions, which 

equate marginal rates of substitution and input price ratios: 

δT

δxi

δT

δxj

 = 

wi

wj

The standard problem of cost minimization implicitly assumes risk neutrality so 

that the optimal input mix must be on the boundary of the input requirement set. The 

boundary is, of course, the isoquant, and the input requirement set is the set of input 

combinations that can produce the given output vector. Any input combination on the 

boundary is defined as technically efficient, while the combination that satisfies (4) is 

termed allocatively efficient. 

But managers may prefer input mixes that achieve the risk level they desire or 

perhaps that confer comfort or prestige. They may prefer to “overemploy” an input to 

 2

Noting that the level of financial capital may not minimize cost if bank managers are not risk neutral, 

Hughes and Mester (1993) proposed this cost function as a means of controlling for such deviations from the leastcost expansion path; the levels of all inputs other than financial capital are chosen to minimize cost. A technique 

more general than this conditional cost function is required to account for the possibility that all inputs, not just 

financial capital, may contribute to production risk.

9 
reduce risk or to achieve another objective. From the managers' viewpoint, these input 

mixes are preferable to the least costly mixes; they lie either on the frontier of the input 

requirement set but are allocatively inefficient, or in the interior of the input requirement 

set and are technically inefficient from the viewpoint of risk neutrality. Thus, the input 

requirement set defines an infinite number of possible costs. One of the sets implies 

the globally least cost and would be chosen by someone who wishes to minimize cost 

and who is completely indifferent to risk. One of these input sets is the managers' most 

preferred, given their preferences that allow for objectives in addition to profit 

maximization. The interaction, then, of technology and generalized managerial 

preferences defines a most preferred cost function, which subsumes the standard least 

cost function as a special case. 

4. The Most Preferred Cost Function

10 

Error!

The most preferred (MP) cost function is derived from the solution to the managers' 

problem 

 (5) 

s.t. p  y + m - w  x - pππ = 0 (6) 

T(x;y,n,k)  0 (7)

Letting the price vector be represented by v = (w, p, r, pπ), the optimal production plan is 

defined by the solution x (y, n, v, m, k) and (y, n, v, m, k). As functions of the vectors of 

outputs and input prices, the input demand functions resemble the standard costminimizing ones. As functions of prices and income (revenue), they resemble consumers' 

demand functions. As functions of the tax rate on profit, they resemble neither. The profit 

equation reflects the optimal trade-off of profit for other objectives and will yield the MP cost 

function and its characterization in terms of scale and scope economies. 

These input demand and profit equations are conditioned on the level of financial 

capital, allowing us to investigate the effect of capital level on the organization of production. The choice of financial capital will be derived in a second-stage optimization 

discussed below. 

The expansion path defining the solution to (5)., which deviates from the standard 

one, is given by 

δT

δxi

δT

δxj

 = 

λwi

-

δU

δxi

λwj-

δU

δxj

That is, the marginal rate of technical substitution equals the ratio of shadow prices. The 

shadow price is the market price adjusted by the marginal utility of the input (A is the 

Lagrangian multiplier associated with the constraint [6]). Hence, in the case of a riskaverse manager, if the marginal utility of a risky input is negative, its shadow price will be 

increased by the element of risk. 
Error!

The most preferred (MP) cost function is defined by 

There are several notable features of the MP cost function. From (9), it is clear that the 

cost function is embedded in the utility-maximizing demand for profit. As will be discussed 

later, measures of technology such as scale and scope economies can be derived from the 

utility-maximizing profit equation. Second, when outputs and inputs as well as profit affect 

utility marginally, revenue influences cost. Not only will output-based revenue, p  y, affect 

the optimum, so will fixed revenue, m (and fixed cost). Additionally, the tax rate the bank 

pays on its profit will, in general, influence the optimum. Of course, in the special case of a 

risk-neutral manager, only profit has marginal significance in the utility function, and 

revenue and tax rates will not influence cost. Finally, notice that input mixes on the interior 

of the input requirement sets (isoquants) can be utility-maximizing. 

Unlike the standard cost function, the homogeneity of the MP cost function include 

output prices and fixed revenues. The input demand functions x(y, n, v, m, k) are 

homogeneous of degree zero in (v, m), while the nominal profit function pππ(y, n , v , m, k) 

is homogeneous of degree one in (v, m). Hence, the MP cost function w  x(y, n, v, m, k) is 

homogeneous of degree one in (v, m). 

5. Deriving the MP Cost Function from the Almost Ideal Demand System

The functional forms for the utility-maximizing input demands and profit equation can be 

obtained from the AI Demand System. In the special case where only profit has marginal 

significance in the utility function, these functional forms are identical to the translog cost 

function and share equations. Our strategy is to adapt the expenditure function of the AI 

Demand System to represent generalized managerial preferences, to apply Shephard's 

lemma to obtain the expenditure-minimizing demand system for inputs and profit, and then 

to substitute the indirect utility function in the demand system to convert it into the utilitymaximizing system that is to be estimated. 

The expenditure function describes the amount of expenditure required to achieve a 

given level of utility Uo

. The managerial expenditure function is defined by the following 

problem: 

min wx+pπ

π

π,x

11 

The solution yields the constant-utility demand functions xu

(y, n, v, k, U0) and πu(y, n, v, 

k, U0). Substituting these demand functions into (10), the expenditure function E(y, n, v, 

k, U0

) is obtained. The expenditure-minimization problem (10) is dual to the utilityT(x;y,n,k)  0 

Error!
maximization problem (5) so that E(y, n, v, k, U0

) = p y + m. Additionally, the demand 

functions obtained from (5) and (10) are identically equal when the indirect utility 

function, V(y, n, v, m, k), derived by inverting the expenditure function, is substituted for 

the utility index in the expenditure-minimizing demands: 

Error!

Adapting the framework of the AI Demand System to accommodate the 

generalized managerial preferences yields the expenditure function, 

Error!

ln E() = ln P + Uβ0 

  Π 

i yi

βi 

  Π 

j wj

v

j psubπμkκ

ln P = α0 + α

p

lnp + 

i δi

ln yi + 

i ωj

ln wsubj 

+ ηπ

ln pπ + τln r + 'ln n + ρln k + 

1

2α

pp(ln p)2 + 

1

2



i



j δijln yi

ln yj

+ 

1

2



s



t ω*

ijln wsln wt

 + 

1

2ηππ(ln pπ)2 

 + 

1

2τrr(ln r)

2

 + 

1

2'

nn(ln n)

2

 + 

1

2ρkk(ln k)

2 

 + 

j θpjln p ln yj

 + 

s φps

ln p ln ws + ψpπ ln p ln pπ 

+ ψprln p ln r + ψpnln p ln n + ψpk ln p ln k 

+

j



s γjsln yj ln ws

 + 

j γjπ ln yj ln pπ + j γjrln yj ln r

+ 

j γjn ln yj ln n + j γjk ln yj ln k 

+ 

1

2



s ω*

sπ

ln ws ln pπ + 

1

2



s ω*

πs

ln pπ ln ws 

+ 

s ωsrln ws ln r + s ωsn ln ws ln n + s ωsk ln ws ln k

+ ηπrln pπ ln r + ηπn ln pπ ln n + ηπk ln pπ ln k 

+ τrn ln rln n + τrk ln rln k + 'nk ln n ln k . 

where 

12 
Hence, from (15) the indirect utility function is

V() = ln(py + m) - ln P

β0 

  Π 

i y

βi

i 

  Π 

j wisupvj p

μ

π kκ

Applying Shephard’s lemma to (15) to obtain the constant-utility input demand equations 

and profit equation and then substituting the indirect utility function (17) in these equations 

yields the utility-maximizing choice functions: 

ln E

ln wi

 = 

wixi

py + m

 = 

ln P

ln wi

 + vi[ln(py + m) - ln P]

= ωi + 

s ωsiln ws

 + φpiln p + 

j gammasubjiln yj

 + ωπiln pπ

+ ωirln r + ωin ln n + ωik ln k 

+ vi[ln(py + m) - ln P] 

ln E

ln pπ

 = 

pπ

py + m

π

 = 

ln P

ln pπ

 + μ[ln(boldpy + m) - ln P]

 = ηπ

 + ηππ ln pπ + ψpπ ln p + 

j γjpiln yj

 + 

s ωsπ ln ws

+ ηπrln r + ηπn ln n + ηπk ln k 

+ μ[ln(py + m) - ln P] 

where ωsi = 

1

2(ω*

si+ω*is) = ωisand ωsÀ!__ = 

1

2

(ωsÀ!__*

+ω

*

πs

 = ω

πs

)

. 

Symmetry requires αij = αji and δij = δji in addition to ωsπ = ωπs. The first two symmetry 

conditions must be imposed in the estimation of the share equations, since the constituent 

coeefficients cannot be separately identified. However, the latter two symmetry conditions 

involve coefficients of prices that are employed by Shephard’s lemma to obtain the share 

equations. Consequently, they appear in separate share equations and are, thus, identifiable and 

permit a test of symmetry when the condition is not imposed on the estimation. In summary, 

symmetry requires that 

S(1) αij = αji  i,j S(2) δij = δji  i,j 

S(3) ωsπ = ωπs  s S(4) ωsi = ωis  s,i 

The input and profit share equations are homogeneous of degree zero in (w,p,r,pπ,m), which 

implies the following conditions: 

13 
H(1) Συj + μ = 0 H(2) Σαi + Σωj + ηπ + τ = 1 

H(3) Σi αij + Σt φjt + ψjr + ψjπ = 0  j H(4) Σi φit + Σs ωst + ωtr + ωtπ = 0  t 

H(5) τrr + Σi ψir + Σsωsr + ηπr = 0 H(6) Σi θij + Σt γjt + γjπ + γjr = 0  j 

H(7) ηππ + Σi ψiπ + Σsωsπ + ηπr = 0 H(8) Σi ψin + Σs ωsn + τrn + ηπn = 0 

H(9) Σi ψik + Σsωsk + τrk + ηπk = 0 H(10) (1/2) [Σi Σjαij + ΣsΣtωst + τrr + ηππ] + 

 Σi(ψiπ+ψir)+ΣiΣtφit+Σs(ωsr+ ωsπ)+ ηπr = 0 

The input and profit shares sum to one, which implies the following adding up conditions: 

A(1) Σiωi + ηπ = 1 A(2) Σiωsi + ωsπ = 0  s 

A(3) Σiφji + ψjπ = 0  j A(4) Σiγji + γjπ = 0  j 

A(5) Σiωπi + ηππ = 0 A(6) Σiωir + ηπr = 0 

A(7) Σiωik + ηπk = 0 A(8) Σiωin + ηπn = 0 

A(9) Σjυj + μ = 0 (which is also a homogeneity condition) 

6. Managerial Objectives: Profit Maximization?

If banks maximize profits (which is equivalent to maximizing return on equity here, since 

financial capital is treated as exogenous), a variation in the tax rate and equivalently, in 

pπ (= [1-t]-1) will not affect the bank’s choice of before-tax profit. This implies that 

P(1) ηπ = ηππ = ψiπ = γjπ = ωsπ = ηπr = ηπn = ηπk = 0  i, j, s 

pππ

pÀ___+m = μ[ln(pÀ___+m) - lnP]

Thus , (19) is simplified to 

In addition, the revenue and risk characteristics of production represented by the output 

price vector will not influence the bank’s cost-minimizing production plan so that 

P(2) αi = αij = θij = φis = ψiπ = ψir = ψin = ψik = 0  i, j, s 

Similarly, variations in m have no marginal significance for the optimal input demand x. 

Hence, the numerators, wixi, of the shares (18) are unaffected by a variation in m. Instead, the 

variation in m solely affects profit so that

δpππ

δm

 = 1. Employing these results in differentiating 

equations (18) and (19) with respect to ln m yields the following parameter values in the case of 

profit maximization: 

P(3) 

14 
υi

 = δ





 wi

xi

pÀ___+m )δlnm = - 

wSUBixi

pÀ___+m

Therefore, we can test for profit maximization (and cost minimization) by testing the restrictions 

(P1), P(2), and P(3). 

Error!

μ = δ





 pπ

π

pÀ___+m )δlnm = 1 - 

pπ

pÀ___+m

π

Error!

C = pΐ___ + m - pππ = 

pÀ___ + m

1 + ln(pÀ___ + m) - lnP

Substituting (21) into (18) and (22) into (19) yields, 

and using (24), an expression for cost can be constructed, 

Substituting (25) into (23) and (24) shows that in the case of profit maximization, the share 

equations are cost shares (and are identical to the translog cost function and corresponding share 

equations for inputs): 

wixi

C = 

δlnP

δlnwi

7. Deriving the Demand for Financial Capital

Conditioning the MP cost function and input demands on the level of financial capital allows us 

to investigate how a bank’s underlying financial condition affects its production decisions. 

However, the more basic decision centers on the level of financial capital itself because, as a 

cushion against insolvency, this level signals the bank’s tolerance for risk. Thus, the utility-

-

pππ

C = lnP-ln(pÀ___ + m)

15 
maximization framework of (5) must be expanded to include a second stage where the financial 

capital level is determined. 

The utility-maximizing demands for inputs and profit derived from (5) are conditioned on 

the level of financial capital, k. It id straightforward to add a second stage to the maximization 

problem to determine the bank’s choice of capital. Writing the Lagrangian function for (5) and 

evaluating it at the first-stage optimum, conditional on k, one obtains the conditional indirect 

utility function: 

Error!

The demand for financial capital follows from maximizing (28) with respect to k. Using the 

definition from (2) that then pπ = pπ[

rg(s)k

π pSUBÀ!__+π], and differentiating (28) with respect to k 

yields the first-order condition: 

V()

k = 

U()

k -λ()

pπ[rg(s)+rkg(s)

k ]

pπ

 + γ()

δT()

δk = 0 

The solution to (29) is the demand for financial capital, k(y,n,v,m). 

The AI system’s conditional indirect utility function (17) implies that this first-order 

condition is 

990 990V()

k = 

V()

ln k

ln k

k

= -

1

k

  β0 

  Π 

i y

βi

i 

  Π 

j ωvj

j p

μ

π

kκ 







 ln P 

ln k

 + κ[ln(py + m) - ln P] = 0

 ρ + ρkk ln k + ψpk ln p + 

j γjk ln yj

 + 

s ωsk ln ws

 + ηπk ln pπ

+ τrk ln r + 'nk ln n + κ[ln(py + m) - ln P] = 0

Not surprisingly, the demand for financial capital follows readily from the 

parameters of the conditional system of input demands (18) and profit (19) and, thus, 

constitutes additional parameter restrictions. 

The output vector can be made endogenous in an analogous fashion; however, a 

simpler system can be derived by eliminating the portion of the two-stage problem that 

results from conditioning the utility maximization on the output vector. This modified 

procedure is developed in Hughes, Lang, Mester, and Moon (1993). 

In the special case of profit maximization, variations in m should not affect the 

optimal demand for capital so that another restriction is added to (P3) above: 

 

16 

3

If the regulatory capital constraint were binding on banks, changes in m would not affect even a 
(P3) κ = 0. 

8. Deriving Scale Economies from the MP Cost Function

Scale economies are defined by the inverse of the elasticity of cost with respect to 

output. Using the definition of the MP cost function (9) and substituting the utilitymaximizing demand for financial capital into (9), the degree of scale economies is given 

by 

SCALE = 

C

i yi









 C 

yi

C

k

k

yi

 = 

py + m -pπ

i













pi - 

pπ

yi

π

π

 - 

pπ

k

πk

yi

= 

py + m -pπ

i













piyi-(py+m)











 pπ 

py+m

π

π

lnyi

 - 







 pπ 

py+m

π

piyi - (py+m)











 pπ 

py+m

π

lnk

lnk

lnyi

The final expression in (31) is stated in terms of derivatives of the profit share equation 

(19). 

9. The Data

The AI Demand System is estimated using data on U.S. banks that reported at least $1 

billion in assets as of the last quarter of 1988. The data are taken from the 

Consolidated Reports of Condition and Income for the fourth quarters of 1989 and 1990. 

 Banks in unit-banking states and special-purpose banks chartered under Delaware’s, 

Financial Center Development Act and Consumer Credit Bank Act are excluded from the 

sample. A total of 286 banks, ranging in size from $1.025 billion to $69.612 billion, are 

included in the data set. The data are summarized in tables 1 and 2. 

We specify five outputs, each measured as the average dollar amount in the fourth quarters of 

1989 and 1990: y1, real estate loans, including commercial as well as noncommercial, y2, 

commercial and industrial loans, lease financing receivables, and agricultural loans; y3, loans to 

individuals for household, family, and other personal expenditures; y4, other loans (such as loans 

17 

 non-risk-neutral bank's demand for capital. But in our sample, and in general, the capital constraint is not binding. 
18 

 

for purchasing and carrying securities, unplanned overdrafts to deposit accounts, loans to 

nonprofit institutions, and loans to individuals for investment purposes); and y5, securities, assets 

in trading accounts, federal funds sold, securities purchased under agreements to resell, and total 

investment securities. 

Financial capital, k, is the average amount of equity capital, loan-loss reserves, and 

subordinated debt in 1990. In addition to financial capital, five other inputs are incorporated into 

the model: x1, labor, whose price, w1, is measured by salaries and benefits paid in 1990 divided 

by the average number of employees in 1990; x2, physical capital, whose price, w2, is proxied by 

the ratio of occupancy expense in 1990 to the average dollar value of net bank premises in 1990; 

x3, insured deposits, whose price, w3, is the ratio of interest paid in 1990 on deposits under 

$100,000, net of service charges received by the bank, to the average amount of interest-bearing 

deposits net of CDs over$100,000; x4, other borrowed money, whose price, w4, is the ratio of the 

total expense of federal funds purchased, securities sold under agreement to repurchase, 

obligations to the U.S. Treasury, and other borrowed money in 1990 to the average amount of 

these funds in 1990; and x5, uninsured deposits, whose price, w5, is the ratio of the interest 

expense in 1990 of deposits greater than $100,000 to the average amount of those deposits. 

Although some formulations have assumed that deposits are outputs, Hughes and Mester 

(1993) derived a test for determining whether deposits are inputs or outputs. Using a data set 

very similar to the one here, they concluded that insured and uninsured deposits are inputs. We 

treat them as inputs here as well. 

In addition to financial capital, another indicator of a bank's underlying financial condition is 

its amount of nonperforming loans, n, which is measured by the sum of the average level of 

loans past due 90 days or more and still accruing interest and the average level of nonaccruing 

loans. 

The price or yield, pi, on the i-th output is measured by the ratio of total interest income from 

the i-th output to the average amount of the i-th output that is accruing interest. This price is not 

just a component of revenue. Its magnitude relative to the risk-free rate indicates the risk 

premium incurred by the output and, hence, suggests the average quality of the asset. 

The variable, m, is measured by the amount of noninterest income received in 1990. 

 Revenue is the sum, py + m, and accounting profit is p  y + m - wx. The risks of the 

production plan (y, x, k) are suggested by the risk premia found in the output prices, p, 

by nonperforming loans, n, and by realized profit. The nature of these risks is dictated 

by the position the bank has chosen on the risk-return frontier. Note that actual or 

realized profit may be quite different from the expected profit that motivated the 

production plan. Hence, rather than measure revenue by actual earnings, we use 

potential revenue as a proxy for expected revenue. Potential revenue is the revenue 

that would be earned if all assets performed (that is, accrued interest). Since p 

measures the average interest rates on accruing assets and y includes all assets, 

accruing and nonaccruing, the product py captures potential interest income; total 

potential income is py + m; and potential profit is py + m - wx. 

4

Hughes and Mester (1993) show that when deposits are inputs (outputs), variable costs (i.e., the cost of all 

nondeposit inputs) will be decreasing (increasing) in the level of deposits. 
Banks pay both federal and state taxes on their income. The federal tax rates are 

similar for all banks in the data set. Thus, the main variation comes from the state tax 

component of p,. The state tax rates are obtained for each state from The Book of the 

States, published by the Council of State Governments, and from Significant Aspects of 

Fiscal Federalism, published by the U.S. Advisory Commission on Intergovernmental 

Relations. 

10. Estimation

The system to be estimated consists of the share equations, (18) and (19), given the 

definition of ln P found in (16), and the first-order condition (30), which defines the level 

of capitalization. Because a cross-section is employed, there is no variation in the riskfree interest rate, r, so it is dropped from the estimating equations. However, its 

parameters can be recovered by using the homogeneity conditions. 

To reduce the number of parameters to be estimated, the vector of output returns p 

is reduced to its weighted average 

p = i pi











yi 

jyj

ln P = α0 + α

p

lnp + 

i δi

ln yi + 

i ωj

ln wsubj + ηπln pπ + + 'ln n + ρln k + 

1

2α

pp(ln p)2

+ 1oversm2

i



j δijln yi

ln yj + 

1

2



s



t ωstln wsln wt

 + 

1

2ηππ(ln pπ)2 + 

1

2'

nn(ln n)2 

 + 

1

2ρkk(ln k)

2 + 

j θpjln p ln yj

 + 

s φps

ln p ln ws + ψpπ ln p ln pπ + ψpnln p ln n

+ ψpk ln p ln k +

j



s γjsln ysubj ln ws

 + 

j γjπ ln yj ln pπ

 + 

j γjn ln yj ln n 

+ 

j γjk ln yj ln k + 

1

2



s ωsπ ln ws ln psubpi + 

1

2



s ωπsln pπ ln ws

 + 

s ωsn ln ωs ln n

+ 

s ωsk ln ωs ln k + ηπn ln pπ ln n + ηπk ln psubpi ln k + 'nk ln n ln k 

The result of these amendments to ln P is the following: 

19 
The amended share equations are: 

420ln E

ln wi

 = 

wi

xi

py + m

 = 

ln P

ln wi

 + vi[ln(py + m) - ln P]

420 = ωi + 

s ωsiln ws

 + φpiln p + sumfromjγjiln yj + ωπiln pπ

+ ωin ln n + ωik ln k + vi[ln(py + m) - ln P] 

ρ + ρkk lnk + ψpklnp + 

j γjklnyj

 + 

s ωsklnωs

 + ηπklnpπk

+ υnklnn + κ[ln(py + m) - lnP] = 0 

The first-order condition defining the bank’s optimal capital level is estimated as 

The symmetry conditions are given above. Condition (S1) becomes moot once (32) 

is imposed. We impose (S2) in the estimation, but do not impose (S3) and (S4) so that 

these conditions can be tested. The homogeneity conditions (amended, since we are 

using the weighted average of the pis) are used to recover the coefficients on variables 

involving the risk-free rate in (15). These homogeneity conditions are (H1), as given 

above, plus the following: 

ln E

ln pπ

 = 

pπ

py + m

π

 = 

ln P

ln pπ

 + μ[ln(boldpy + m) - ln P] 

= ηπ + ηππ ln pπ + ψpπ ln p + 

j γjπ ln yj

 + 

s ωsπ ln ws

+ ηπn ln n + ηπk ln k + μ[ln(py + m) - ln P] 

H(2') αp + Σωj + ηπ + τ = 1 H(3') αpp + Σt φpt + ψpr + ψpπ = 0 

H(4') φpt + Σs ωst + ωtr + ωπt = 0  t H(5') τrr + Σi ψpr + Σsωsr + ηπr = 0 

H(6') θpj + Σt γjt + γjπ + γjr = 0  j H(7') ηππ + ψpπ + Σsωsπ + ηπr = 0 

H(8') ψpn + Σs ωsn + τrn + ηπn = 0 H(9') ψpk + Σsωsk + τrk + ηπk = 0 

H(10') (1⁄2)[αpp + ΣsΣtωst + τrr + ηππ] + Σtφpt + Σs(ωsr+ ωsπ) + ψpπ+ ψpr + ηπr = 0 

The adding-up conditions are (A1), (A2), (A4), (A5), (A7), (A8), and (A9) as given 

above, plus the amended condition: 

A(3') Σiφpi + ψpπ = 0  j [Note that (A6) is dropped.] 

The amended conditions for risk neutrality are then: 

20 
P(1') ηπ = ηππ = ψpπ = γjπ = ωsπ == ηπn = ηπk = 0  j, s 

P(2') αp = αpp = θpj = φps = ψpπ = ψpn = ψpk = ψik = 0  j, s 

and, omitting the restrictions on υj and μ, since testing whether they hold is not feasible, 

P(3') κ = 0 

We use nonlinear two-stage least squares to estimate the following system of non-linear 

simultaneous equations [subject to the parameter restrictions (A1)-(A2), (A3'), (A4)-(A5), (A7)-

(A9), and (S2)]: 

ft(yt,nt,vt,mtΘ)















 s1t

 - (r.h.s. of (34) with i = 1)

s2t

 - (r.h.s. of (34) with i = 2)

s3t

 - (r.h.s. of (34) with i = 3)

s4t

 - (r.h.s. of (34) with i = 4)

sπt - (r.h.s. of (35))

l.h.s. of (36)

 = ut

where r.h.s. designates the right hand side of the indicated equation and l.h.s. the left hand side; t 

is the bank index, ranging from 1 to T; sit is the i-th input's revenue share at the t-th bank, 

i.e.,sit 

wit xit

pt

yt+mtfor i = 1 ... 5; sπt is the profit share at the t-th bank, i.e., sπt 

pπt πt

pt

yt+mt; 

ut are i.i.d. over t with the cross-equation covariance matrix Σ; and Θ is the set of all identifiable 

parameters excluding those in Σ. The standard errors and t-statistics we report are based on the 

asymptotic covariance matrix of the estimate of Θ, which penalizes for not using cross-equation 

dependence (see Gallant [1977]). The contemporaneously correlated error terms ut reflect 

optimization errors (i.e., errors in utility minimization). 

11. The Empirical Findings

The parameter estimates are reported in table 3. To check for model adequacy, we used a Wald 

test to test the 15 symmetry conditions, (S3) and (S4). The value of the test statistic was 39.70, 

implying a p-value of 0.05%. Thus, these symmetry conditions are rejected, and the results 

reported below are based on the estimation in which they are not imposed. (In general, the results 

are qualitatively similar whether the symmetry conditions are imposed or not.) 

There are four striking primary findings. First, risk neutrality, or equivalently, cost 

minimization is conclusively rejected. Second, the measure of scale economies is larger than 

21 
22 

 

that found by most studies, which assume cost minimization. Third, the measure of scale 

economies increases with bank size, which is consistent with the wave of mergers among large 

banks. Last, banks with higher capital levels or higher levels of nonperforming loans rely less on 

volatile funding sources. 

11.1. Risk Neutrality

We used a Wald test to test the 31 restrictions (PI'), (P2'), and (P3') implied bv risk neutrality or 

profit maximization. The value of the test statistic was 294.01 with 29 degrees of freedom. 

Two of the restrictions are redundant because of the adding-up conditions that these parameters 

satisfy. Thus, the restrictions are strongly rejected, indicating that banks in the sample are not 

behaving in a risk-neutral manner. 

11.2. Scale Economies Measured by Other Studies

The many translog bank cost studies in the literature differ in the following ways: (1) how inputs and 

outputs are defined and, hence, how cost is constituted, (2) whether financial capital is ignored or 

included as a conditioning argument or included as an element of cost, and (3) whether an average 

practice cost function or best practice cost frontier is estimated. All of these differences might be 

expected to yield a variety of scale estimates, however, the estimates are quite similar. Most studies 

of large banks (whose assets surpass $1 billion) find either slight scale economies or slight 

diseconomies, and they usually find that scale economies decrease with bank size. 

Treating non-interest-bearing deposits as a quasi-fixed input while characterizing the other 

inputs and outputs as we do here, Noulas, Ray, and Miller (1990) examine large 

5

We also computed the test statistic after removing (P3'), the restriction related to m. The value of this test 

statistic was 52.83 with 28 degrees of freedom. Again, profit maxzation is rejected with a p-value of 0.003077. 

6

To maintain consistency with our measure of scale economies in (31), the discussion below transforms 

published measures of scale economies so that values greater than 1 imply scale economies, while values less than 

one imply diseconomies. 

banks in 1986 and find that scale economies decrease from 1.02 for the smallest banks ($1-3 billion) 

to 0.97 for the largest banks ($10 billion plus). In a similar study that differs primarily in controlling 

for the number of branches, Hunter and Timme (1991) find that in 1986 scale economies range from 

1.123 for the smallest group ($1-1.5 billion) to 0.950 for the largest group ($25 billion plus). When 

they omit branches, the measures drop to values very close to those of Noulas, Ray, and Miller: 

1.037 for the smallest banks to 0.977 for the largest banks. 

Berger and Humphrey (1991) calculate scale economies using a thick frontier and find mild 

diseconomies, 0.98, for banks with $1-2 billion in assets, decreasing to 0.97 for banks with more 
23 

 

than $10 billion in assets. When they use a conventional approach, the range drops to 0.96 for the 

smallest banks and 0.94 for the largest banks. 

Two other studies find approximately constant returns to scale overall but much wider ranges in 

scale economies. Hunter, Timme, and Yang (1990) obtain values ranging from 1.09 for the smallest 

banks ($1-1.5 billion) to 0.90 for the largest ($25 billion plus) using 1986 data. Excluding interest 

payments from cost, Evanoff and Israilevich (1991) find measures ranging from 1.11 at $0.72 billion 

to 0.76 at $30 billion. 

McAllister and McManus (1993) apply nonparametric estimation to 1984-90 data and find 

increasing returns to scale up to $0.5 billion and constant returns from $0.5-10 billion, the largest 

bank in their sample. Using 1988 data on all banks with more than $1 billion in assets, Pulley and 

Braunstein (1992) report an average measure ranging from 1.04 to 1.06 depending on the estimation 

procedure. Evanoff, Israilevich, and Merris (1990) estimate a shadow cost function to account for 

regulatory distortions from 1972 through 1987 and obtain values of 1.07 for multibank holding 

companies and 1.10 for one-bank holding companies. 

Studies that find scale economies increasing with bank size are rare. Mester (1992) defines 

outputs to capture the information producing and information processing role of banks and, using 

1988 data, finds slight scale economies that increase with bank size. Banks in the smallest group 

($1-1.5 billion) exhibit slight economies, 1.0305, and this increases to 1.0426 for banks in the largest 

group ($5 billion plus). Clark (1996), who examines the period from 1988 through 1991, treats core 

deposits as outputs and estimates a thick frontier. He finds economies of around 1.05 for the 

smallest banks (up to $4 billion) and constant returns in all larger size categories (the largest of 

which is $20 billion plus in assets). 

11.3. Scale Economies Measured Without Imposing Risk Neutrality

These studies all assume cost minimization. When sources of risk are included in the structural 

model of production and when cost minimization is not imposed, scale economies are much larger, 

as shown in Table 4 and Figure I. These larger economies are more consistent with the explanations 

given by bank managers for the recent wave of large bank mergers. The measures range from an 

average of 1.101 in the smallest asset-size quartile, 1.128 in the second quartile, 1.146 in the third, 

and 1.208 in the fourth. All are strongly significantly different from one. In addition to being larger 

than those found in previous studies, the measures also increase with bank asset size, contrary to 

most previous studies. 

7

Some of the difference, but not all, is due to our not imposing the symmetry conditions (S3) and (S4). 

When these conditions are imposed, the scale estimates range from 1.036 in the smallest quartile to 1.120 in the 

largest. While smaller than the estimates when symmetry is not imposed, they are still on the high side of estimates 

found in previous studies. 

If incorporating sources of risk and allowing for non-neutrality toward risk has revealed 

economies previously hidden, imposing neutrality on this model should yield the more commonly 
24 

found magnitudes. When we drop financial capital and nonperforming loans and impose the 

coefficient values implied by risk neutrality, so that the resulting functional form is identical to the 

standard translog cost function and share equations used in previous studies, we find that the 

estimated scale measures are all significantly different from one but are considerably smaller and 

remarkably familiar: ranging from 1.022 for the smallest quartile, 1.029 for the second quartile, 

1.035 for the third, and 1.050 for the fourth. These values are surprising only in that they increase in 

magnitude with asset size. 

Since there is no simple difference between the MP cost function and the risk-neutral 

formulation, it is difficult to determine what leads to the larger scale measures. However, we gain 

some insight by examining the model of Hughes and Mester (1995) that allows for non-neutrality 

only in the choice of capital level. They estimate a cost function conditioned on the level of 

financial capital to allow for the possibility that banks may choose a higher capital level than the 

cost-minimizing one to reduce their risk of insolvency. The bank's capital choice is derived from 

utility maximization so that banks can accept higher costs in exchange for greater safety. But this 

formulation is not as general as the one used here, since it imposes risk neutrality on the choice of 

inputs other than financial capital. Nevertheless, the estimated demand for financial capital indicates 

that banks in all four asset-size quartiles are not risk neutral, and the resulting scale economies are in 

the 1.13 to 1.15 range across the quartiles, much larger than models that impose risk neutrality 

completely. 

11.4. Other Results

Table 4 shows that for banks of all sizes, capital level, k, and volatile funding, x4, are inversely 

related. Thus, banks with higher capital are less likely to rely on volatile sources to fund their assets. 

 It appears that banks choosing lower insolvency risk also choose lower liquidity risk. And in all 

size quartiles except the smallest, the higher a bank’s level of nonperforming loans, n, which is an ex 

post measure of asset quality, the less it relies on volatile funding sources. Because nonperforming 

loans are likely to be less liquid than other assets, these banks may prefer more stable funding 

sources. To the extent that the prices of some of the components of volatile funding include a risk 

premium, this may also be a cost-reducing strategy, since the risk premium is likely to be higher for 

banks with poor loan quality. 

Table 4 also reports that, for the smallest banks, the most preferred level of capital responds 

positively to the level of nonperforming loans. These banks may be acting to protect their solvency 

from the ex post realization of poor loan quality by increasing their capitalization. Table 4 also 

indicates that financial capital, k, and the average return on asset, p, are inversely related (although 

not significantly so). Lower quality assets have an ex ante higher return, but this return is a gamble. 

 Our results suggest that the greater the gamble, that is, the higherpis, the lower the amount of 

financial capital, k, that is bet. 

12. Conclusions

When financial intermediation is studied, risk cannot be ignored. Whether managers are risk neutral 

or risk averse, it is important to control for sources of risk so that the potentially cost-saving effects 

of scale-related diversification can be identified. But, equally important, allowance must be made 
25 

for non-neutral risk preferences. 

This paper has developed a theoretical model which amends the standard framework to allow 

for non-neutral risk preferences. While our approach provides a theoretically consistent method for 

addressing non-neutral risk preferences, our amendments to the standard framework add obvious 

complexity to the model. However, the importance of incorporating sources of risk and nonneutrality toward risk into models of production can be confirmed only by comparing the empirical 

adequacy of explanations that incorporate risk with those that abstract from it. 

The most preferred cost function we develop in this paper seems to explain important 

production phenomena in banking that elude approaches using the standard cost function. Most 

notably, our approach reveals large scale economies, while previous studies that assumed risk 

neutrality found only small scale economies or constant returns to scale. Our results may be useful 

in explaining the economics behind the recent wave of mergers among very large banks. Our 

findings are consistent with the rationale frequently cited by participants in these mergers who argue 

that significant cost savings can be achieved by enlarging the scale of operations. 
26 

References

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Cost Efficiency as an Antitrust Defense,” Antitrust Bulletin, 37, 541-600. 

[2] Berger, Allen N. and David Humphrey (1991), “The Dominance of Inefficiencies over 

Scale and Product Mix Economies in Banking,” Journal of Monetary Economics, 28, 

117-148. 

[3] Clark, Jeffrey A. (1996), “Economic Cost, Scale Efficiency and Competitive Viability in 

Banking,” Journal of Money, Credit, and Banking, 28, 342-364. 

[4] Evanoff, Douglas D. and Phillip R. Israilevich (1991), “Scale Elasticity and Efficiency for 

U.S. Banks,” Working Paper 91-15, Federal Reserve Bank of Chicago. 

[5] Evanoff, Douglas D., Philip R. Israilevich, and Randall C. Merris (1990), "Relative Price 

Efficiency, Technical Change, and Scale Economies for Large Commercial Banks," 

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[6] Gallant, A. Ronald (1977),“Three-Stage Least-Squares Estimation for a System of 

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[7] Hughes, Joseph P. (1989), “Hospital Cost Functions: The Case Where Revenues Affect 

Production,” Working Paper No. 1990-01, Rutgers University, Department of 

Economics, November. 

[8] Hughes, Joseph P. (1990), “The Theory and Estimation of Revenue-Driven Costs: The 

Case of Higher Education,” unpublished manuscript, Rutgers University, Department 

of Economics, February. 

[9] Hughes, Joseph P., William Lang, Loretta J. Mester, and Choon-Geol Moon (1993), 

“Risk-Taking by U.S. Thrifts: Modeling Risk-Indicative. Endogenous Interest Rates,” 

Working Paper (revised January 1994). 

[10] Hughes, Joseph P. and Loretta J. Mester (1995), “Bank Capitalization and Cost: 

Evidence of Scale Economies in Risk Management and Signaling,” Federal Reserve 

Bank of Philadelphia, Working Paper 96-2 (revised December 1995). 

[11] Hughes, Joseph P. and Loretta J. Mester (1993), “A Quality and Risk-Adjusted Cost 

Function for Banks: Evidence on the ‘Too-Big-To-Fail’ Doctrine,” Journal of 

Productivity Analysis, 4, 292-315. 

[12] Hunter, William C. and Stephen G. Timme (1986), “Technical Change, Organizational 

Form, and the Structure of Bank Production,” Journal of Money, Credit, and 

Banking, 18:2, 152-166. 
27 

[13] Hunter, William C. and Stephen G. Timme (1991), “Some Evidence on the Impact of 

Quasi- Fixed Inputs on Bank Scale Estimates,” Economic Review, Federal Reserve Bank 

of Atlanta, May/June, 12-20. 

[14] Hunter, William C., Stephen G. Timme, Won Keun Yang (1990), “An Examination of 

Cost Subadditivity and Multiproduct Production in Large U.S. Banks,” Journal of 

Money, Credit, and Banking, 22:4, 504-515. 

[15] McAllister, Patrick H. and Douglas McManus (1993), “Resolving the Scale Efficiency 

Puzzle in Banking,” Journal of Banking and Finance, 17:2-3, 389-406, 

[16] Mester, Loretta J. (1992), “Traditional and Nontraditional Banking: An InformationTheoretic Approach,” Journal of Banking and Finance, 16, 545-566. 

[17] Mester, Loretta J. (1994), “How Efficient Are Third District Banks?” Business Review, 

Federal Reserve Bank of Philadelphia, January/February, 3-18. 

[18] Noulas, Athanasios G., Subhash C. Ray, and Stephen M. Miller (1990), “Returns to 

Scale and Input Substitution for Large U.S. Banks,” Journal of Money, Credit, and 

Banking, 22:1, 94-108. 

[19] Pulley, Lawrence B. and Yale M. Braunstein (1992), “A Composite Cost Function 

for Multiproduct Firms with an Application to Economies of Scope in Banking,” Review 

of Economics and Statistics, 74:2, 221-230. 

[20] Shaffer, Sherrill and Edmond David (1991), “Economies of Superscale in Commercial 

Banking,” Applied Economics, 23, 283-293. 

[21] Van Wissen, Leo and Thomas F. Golob (1992), "A Dynamic Model of Car Fuel-Type 

Choice and Mobility," Transportation Research, 26B, 77-96.
28 

Table 1: Summary Statistics of the Data

variable sample mean sample std. dev. minimum maximum 

y1† 1490521.75 2739095.05 1038.50 26541000.00

y2† 1606152.95 3114969.78 15956.00 23962500.00

y3† 672172.08 1070269.29 8762.50 11781500.00

y4† 470269.42 1486811 713.00 11868570.50

y5† 1326574.19 2232463.87 13549.50 20454834.50

p1‡ 0.109 0.017 0.026 0.205

p2‡ 0.107 0.02 0.022 0.187

p3‡ 0.123 0.027 0.028 0.279

p4‡ 0.085 0.065 0.001 0.533

p5‡ 0.088 0.014 0.055 0.174

p‡ 0.104 0.011 0.064 0.161

w1¶ 33.092 9.857 18.140 92.178

w2‡ 0.396 0.178 0.116 1.378

w3‡ 0.06 0.009 0.028 0.108

w4‡ 0.087 0.033 0.039 0.350

w5‡ 0.081 0.017 0.027 0.233

p~‡ 1.664 0.112 1.515 1.871

s1 0.146 0.041 0.043 0.309

s2 0.05 0.018 0.011 0.141

s3 0.277 0.095 0.01 0.542

s4 0.101 0.076 0.004 0.469

s5 0.314 0.128 0.109 1.208

sπ 0.337 0.071 0.122 0.749

pππ† 223763.03 360436.38 17125.04 3116164.55

p.y+m† 667979.16 1020863.51 90000.55 8419110.74

n† 157495.47 387914.46 1254.00 3629843.00

k† 561765.95 1069979.99 69516.50 8787000.00

m† 124239.66 277310.33 488.00 2060000.00

total assets† 5865120.40 8037258.81 1025143.00 69611500.00

†in thousands of dollars ‡in dollars per dollar ¶in thousands of dollars per employee 

y1 = real estate loans; y2 = C & I loans; y3 = loans to individuals; y4 = other loans (to purchase 

accounts; pi = price of output i; p~ = weighted average of output prices; w1 = price of labor; w2 = 

(repos, fed funds purchased, etc); w5 = price of uninsured deposits; pπ = price of real, after tax 

profit; si = share of input i; sπ = profit share; pππ = nominal, before-tax accounting profit; 

p.y+m = expected revenue; n = nonperforming loans; k = financial capital; m = noninterest 

income. 
29 

Table 2: Means of the Variables by Asset-Size Quartiles 

variable 1st Quartile 2nd Quartile 3rd Quartile 4th Quartile 

y1† 357760.68 657490.28 1281000.08 3680519.81

y2† 249271.21 467700.81 952710.22 4780167.5

y3† 188318.96 326874.65 639945.94 1538866.02

y4† 32843.49 80126.14 157397.63 1620612.07

y5† 325200.96 552054.45 964296.82 3480755.74

p1‡ 0.108 0.111 0.112 0.105

p2‡ 0.115 0.106 0.109 0.098

p3‡ 0.124 0.119 0.125 0.125

p4‡ 0.083 0.095 0.082 0.082

p5‡ 0.086 0.085 0.086 0.093

p‡ 0.105 0.103 0.105 0.101

w1¶ 29.372 30.886 33.052 39.09

w2‡ 0.382 0.387 0.393 0.421

w3‡ 0.06 0.06 0.059 0.06

w4‡ 0.08 0.08 0.086 0.102

w5‡ 0.08 0.081 0.084 0.081

p~‡ 1.651 1.662 1.669 1.675

s1 0.141 0.144 0.144 0.157

s2 0.047 0.05 0.048 0.053

s3 0.316 0.309 0.271 0.211

s4 0.072 0.072 0.107 0.155

s5 0.289 0.293 0.297 0.376

sπ 0.333 0.337 0.342 0.337

pππ† 47708.37 84558.38 169048.99 596467.64

p.y+m† 140029.89 249238.77 489462.14 1801597.91

n† 18162.45 49266.1 84958.04 480141.3

k† 106021.51 187606.21 359011.36 1602550.27

m† 18240.58 33365.32 68254.24 379166.96

total assets† 1335138.53 2393107.19 4607230.76 15191623.49

†in thousands of dollars ‡in dollars per dollar ¶in thousands of dollars per employee 

y1 = real estate loans; y2 = C & I loans; y3 = loans to individuals; y4 = other loans (to purchase 

accounts; pi = price of output i; p~ = weighted average of output prices; w1 = price of labor; w2 = 

(repos, fed funds purchased, etc); w5 = price of uninsured deposits; pπ = price of real, after tax 

p.y+m = expected revenue; n = nonperforming loans; k = financial capital; m = noninterest 

income. 
30 

Table 3: Coefficient Estimates 

parameter estimate parameter estimate parameter estimate 

α0 101.846 αp 24.576* δ1 0.199 

(69.98) (14.92) (2.00) 

δ2 -0.683 δ3 -2.199 δ4 3.045* 

(1.99) (1.74) (1.68) 

δ5 -0.705 ω1 -2.857* ω2 -1.235 

(2.00) (1.56) (0.76) 

ω3 -7.345 ω4 8.667 ω5 -1.187 

(5.23) (5.32) (1.23) 

ηπ 4.958* υ -0.342 ρ -0.979 

(3.00) (1.33) (0.62) 

δ11 0.08 δ12 0.147* δ13 -0.065 

(0.08) (0.08) (0.04) 

δ14 0.006 δ15 -0.024 δ22 0.088 

(0.04) (0.07) (0.07) 

δ23 0.163** δ24 0.001 δ25 -0.117 

(0.08) (0.04) (0.08) 

δ33 0.025 δ34 -0.022 δ35 0.015 

(0.06) (0.03) (0.05) 

δ44 0.045** δ45 0.008 δ55 0.00007 

(0.02) (0.04) (0.11) 

αpp 2.128 ω11 0.131*** ω12 0.040** 

(2.93) (0.04) (0.02) 

ω13 0.088 ω14 -0.155 ω15 0.025 

(0.10) (0.10) (0.03) 

ω21 0.022 ω22 0.019** ω23 0.061 

(0.02) (0.01) (0.05) 

ω24 -0.066 ω25 0.019 ω31 0.104 

(0.05) (0.01) (0.09) 

ω32 0.05 ω33 0.786** ω34 -0.523 

(0.05) (0.33) (0.33) 

ω35 -0.021 ω41 -0.179* ω42 0.077* 

(0.07) (0.09) (0.05) 

ω43 -0.595* ω44 0.667** ω45 -0.117 

(0.31) (0.32) (0.08) 

ω51 0.018 ω52 0.004 ω53 0.082 

(0.02) (0.01) (0.08) 

ω54 -0.091 ω55 0.074*** υnn 0.112* 

(0.08) (0.03) (0.07) 

continued on next page 
31 

Table 3 continued 

parameter estimate parameter estimate parameter estimate 

ρkk 0.012** θp1 -0.524 θp2 0.019 

(0.01) (0.50) (0.61) 

θp3 -0.192 θp4 0.816* θp5 -0.901 

(0.36) (0.46) (0.56) 

ψpπ 0.625 φp1 -0.268 φp2 -0.127 

(0.51) (0.26) (0.12) 

φp3 -0.993 φp4 0.740 φp5 -0.037 

(0.93) (0.95) (0.17) 

ψpn -0.036 ψpk -0.065 γ1π -0.036 

(0.44) (0.10) (0.08) 

γ2π -0.060 γ3π -0.015 γ4π 0.051 

(0.07) (0.06) (0.04) 

γ5π -0.048 γ11 0.009 γ12 0.005 

(0.08) (0.04) (0.02) 

γ13 0.042 γ14 -0.030 γ15 0.009 

(0.14) (0.15) (0.02) 

γ21 0.024 γ22 0.009 γ23 0.114 

(0.04) (0.02) (0.13) 

γ24 -0.119 γ25 0.032 γ31 0.022 

(0.14) (0.03) (0.03) 

γ32 0.009 γ33 0.073 γ34 -0.068 

(0.01) (0.11) (0.11) 

γ35 -0.020 γ41 -0.023 γ42 -0.011 

(0.02) (0.02) (0.01) 

γ43 -0.104 γ44 0.103 γ45 -0.016 

(0.08) (0.08) (0.02) 

γ51 0.018 γ52 0.007 γ53 0.047 

(0.04) (0.02) (0.15) 

γ54 -0.030 γ55 0.006 γ1n 0.077 

(0.15) (0.02) (0.06) 

γ2n 0.003 γ3n -0.031 γ4n -0.066* 

(0.06) (0.04) (0.04) 

γ5n 0.004 γ1k 0.000 γ2k 0.011 

(0.06) (0.02) (0.01) 

γ3k 0.006 γ4k -0.009 γ5k 0.007 

(0.01) (0.01) (0.02) 

ω1n -0.030 ω2n -0.009 ω3n -0.083 

(0.03) (0.01) (0.10) 

ω4n 0.090 ω5n -0.012 ω1π -0.130** 

(0.10) (0.02) -0.060 

continued on next page 
32 

Table 3 continued 

parameter estimate parameter estimate parameter estimate 

ω2π -0.055* ω3π 0000.395** ω4π 0000.302* 

(0.03) (0.19) (0.18) 

ω5π -0.087* ωπ1 -0.064 ωπ2 -0.036 

(0.05) (0.07) (0.03) 

ωπ3 -0.474** ωπ4 0.368** ωπ5 -0.099 

(0.19) (0.18) (0.08) 

ηπn 0.045 ηπk -0.009 ηππ 0.306** 

(0.06) (0.02) (0.14) 

ω1k 0.023 ω2k 0.009 ω3k 0.033 

(0.01) (0.01) (0.04) 

ω4k -0.075** ω5k 0.019 υnk -0.010 

(0.04) (0.01) (0.01) 

v1 -0.034** v2 -0.015** v3 0.123*** 

(0.01) (0.01) (0.04) 

v4 0.126*** v5 -0.019 μ 0.065*** 

(0.04) (0.01) (0.02) 

κ 0.013*** 

(0.005) 
33 

Table 4: Scale Economies and Other Elasticities