Does Firm Type Affect Factor Payments and Growth?

By JAMES M. HOLMES, JOHN M. HOLMES, AND PATRICIA A. HUTTON*

Abstract

When advanced technologies increase productivity as delays between inputs

and output increase, a new reason for the existence of corporations appears. A

corporation, like government, is potentially deathless and can rationally adopt

technological advances with output delays longer than a proprietor’s lifespan be-

cause it can make long-term contracts that transfer the value of increased produc-

tivity in the distant future into present equity owner income, independent of paying

factors their marginal product, or either human or physical capital acquisition, or

contracting, transactions, information, and risk costs. Corporations can cause

economic growth and income equality that cannot occur with unincorporated

firms.

JEL: E25, O33, D23

Key Words: Output Delays, Contracting, Corporations.

*Holmes, James. M., Dept. of Economics, SUNY, Buffalo, NY 14260 (email: eco-

holme@gmail.com). Holmes, John M., Dept. of Mathematics, Ohio State Univer-

sity, Columbus, OH 43210 (email: holmes.782@osu.edu). Hutton, Patricia A.,

Dept. of Economics & Finance, Canisius College, Buffalo, NY 14208 (email: hut-

ton@canisius.edu).

We are indebted to Goncalo Monteiro for especially insightful comments and sug-

gestions. Copyright © 2015 by James M. Holmes. All rights reserved.

1

The advantage of the corporation in contributing to economic growth has

been largely attributed to its superior ability to accumulate capital. This paper

demonstrates that the distinctive contracts that a corporation can make,1 which an

unincorporated firm cannot, influence the adoption and efficient use of technolo-

gies with productivity delays. Production processes involving lags between inputs

and the resulting output have been noted by economists since Jevons (1871) and

higher productivity can be associated with longer delays in output. Examples in-

clude the technology of ocean transportation between Europe and England to and

from the Far East created about 1600,2 railroads in the 1800s and, according to

Brynjolfsson and McAfee (2014), even digital technology. The development and

use the new ocean transportation technology around the Cape of Good Hope about

1600AD also corresponded with the creation of the modern corporation as a type

of firm and, over the last century and a quarter, corporations have taken over a

dominate role in creating and adopting advanced production technologies.

Whereas in 1880 only 4% of utility patents (which are associated with production)

in the U.S. were granted to corporations, by 2014, the proportion was 92%, and

corporations now account for 75% of U.S. domestic business income. Yet, dating

back to Coase (1937, 1992), relatively little has been made of the differences in le-

gal organization on the kind of contracts that different firm types can make. For

example, Jensen and Meckling (1976) stated that a firm is a “nexus of contractual

arrangements” between individuals and Gibbons (2005) characterizes a firm as “a

... particular cluster of otherwise ordinary contractual relationships” that exist to

reduce contracting, transactions, information, or risk costs. Although Acemoglu

1 The contract may be implicit, and in the model with no uncertainty there is no distinction

between a contract, plan, or commitment. 2 Dari-Mattiacci, Gelderblom, Jonker, and Perotti 2013, Gelderblom and Jonker 2004, and

Parthesius 2010.

2

observed “how differences in the ability to write contracts between firms and their

suppliers (or firms and their workers) may have first order effects upon technologi-

cal adoption decisions” (2009, p. 631), he did not mention a firm’s ability to write

contracts of long duration between a firm and its future equity owners, as well as

its future employees.

This paper demonstrates the advantages of the ability to write contracts of

long duration when production technologies exist for which productivity increases

with delays in output. A corporation has the ability to make a complex, long-term

contract, plan, or commitment to adopt, create, and use advanced technologies that

have delays between inputs and the resulting output that are longer than the work-

ing lifetime of any proprietor, because a corporation is potentially deathless. This

is a fundamental “supply side” advantage of a corporation that is independent of

the contacting, transaction, or information costs hypothesized by Coase or of a cor-

poration’s ability to acquire capital.

A fundamental aspect of such a, perhaps implicit, complex contract, which

can span several generations, is that it specifies both the intertemporal allocation of

inputs to production and the distribution of the resulting output to its stakeholders

that are independent of the marginal product of the inputs used in each period.

This is a radical departure from conventional economic analysis, which is based

upon output being produced by proprietor and assumes inputs must be paid their

marginal product.

We model the effect of the choice of firm type on economic growth and in-

come inequality using the familiar 2-generation OLG model with a production

function expanded to include the possibility of delayed output associated with in-

creased productivity. In order to clearly distinguish our new rational for a corpora-

tion and remove the reasons for a firm to exist given by Coase, we assume no un-

certainty, no transactions, on information costs, and, if equity is marketable, no

possible appreciation in stock price because its price is a constant slightly above

3

zero. We also assume the only productive input is labor so that the acquisition of

capital cannot be relevant. We contrast the choices of a corporation with those of a

self-employed proprietor (SEP).

We demonstrate four findings when technologies exist for which increased

total factor productivity (TFP) is associated with increased delays in output. (1) A

proprietor will always use production technologies inefficiently. (2) Because it

need not pay factors their marginal product and can make contracts of long dura-

tion, a corporation will always use production technologies efficiently and distrib-

ute its output optimally. (3) A proprietor can increase their income/welfare by

forming a corporation. (4) Once formed, a corporation can adopt and use most effi-

ciently any further advances in production technology involving yet longer delays.3

From these findings we deduce that a corporation can be an important cause of

economic growth due to the contracts it can make. Thus, the creation of the mod-

ern corporation can reasonably be viewed as a major reason for the Industrial Rev-

olution because of their contracts as well as their ability to accumulate capital.

We conclude the paper with the advocacy of a new view of a corporation as a

“social organization” and, independent of this, the implications of this research for

competitive market capitalism, when firms are corporations.

I. Model and Framework

Our analysis employs a canonical, competitive, 2-generation, over-lapping gen-

erations (OLG) model, similar to Samuelson (1958), in which agents have produc-

tive labor only in youth, and can choose to be self-employed proprietors (SEP) or

form a corporation. We assume that there is no cost for a proprietor to convert to,

3 We also demonstrate that a corporation can eliminate the inequality in income that occurs

when firms are SEPs.

4

or once corporations have been established to join, a corporation. Because popula-

tion and human capital growth are not important to our argument, we assume both

equal zero.

A. Production Functions with Delayed Output

In the typical specification of the production function, the output of the per-

ishable consumption good 𝑄(𝑡) produced in period t using the technology 𝐴(𝑡),

and inputs of capital 𝐾(𝑡), and labor 𝐿(𝑡) is

(1) 𝑄(𝑡) = 𝐹(𝐴(𝑡), 𝐾(𝑡), 𝐿(𝑡)),

where 𝐴(𝑡), 𝐾(𝑡), and 𝐿(𝑡) are cotemporaneous.

Consider an alternative in which a sequence of technologies for which the mth

technology has an output delay of 𝑚 = ∅, 1, … , 𝑀, … 𝑥 (∅ = 0 distinguishes it

from O for old) and a total factor productivity (TFP) of 𝐴𝑚,

(2) 𝐴∅ < 𝐴1 < 𝐴2 < …< 𝐴𝑥 ,

where x is the delay associated with the most productive technology. At time t,

only 𝑀 out of the set of possible technologies are available, but new technologies

can appear exogenously with the passage of time.4

Each young agent has an endowment of labor, 𝐿(𝑡), that is supplied inelas-

tically. Denoting proprietors and corporations with the superscript j = P or C, firm

j can partition the labor it has at time t into allocations to the 𝑀 ≤ 𝑥 technologies

available,

(3) 𝐿(𝑡) = ℓ𝑗(𝑡, ∅) + ℓ𝑗(𝑡, 1) + ⋯ ℓ𝑗(𝑡, 𝑀),

where ℓ𝑗(𝑡, 𝑚) ≥ 0, denotes firm j’s labor allocated in period 𝑡 to technology 𝑚

that results in output in period 𝑡 + 𝑚. When there is no physical or (additional)

4 We abstract from the issues of endogenous technological progress.

5

human capital, a firm’s output in period t associated with labor input ℓ𝑗(𝑡 − 𝑚, 𝑚)

and technology 𝑚 is,

(4) 𝑄𝑗(𝑡, 𝑚) = 𝐴𝑚𝑓𝑚(ℓ𝑗(𝑡 − 𝑚, 𝑚)).5

𝑓𝑚(ℓ𝑗(𝑡 − 𝑚, 𝑚)) denotes the production function that uses delay technology 𝑚

and inputs from period 𝑡 − 𝑚. Firm j’s total output at time t associated with past

and present allocations of labor to all technologies is,

(5) 𝑄𝑗(𝑡) = ∑ 𝐴𝑚𝑓𝑚(ℓ𝑗(𝑡 − 𝑚, 𝑚))𝑀 .

For our purpose, it suffices to assume that the production function is the same

for all technologies, 𝑓𝑚 = 𝑓, and only 𝐴𝑚 varies with the length of delay. We as-

sume 𝑓𝑚(∙) is homogeneous of degree 1, e.g. Cobb-Douglas. Thus, equations (4)

and (5) simplify to,

(6) 𝑄𝑗(𝑡, 𝑚) = 𝐴𝑚𝑓 (ℓ𝑗(𝑡 − 𝑚, 𝑚)) = 𝐴𝑚ℓ𝑗(𝑡 − 𝑚, 𝑚),

(7) 𝑄𝑗(𝑡) = ∑ 𝐴𝑚𝑓(ℓ𝑗(𝑡 − 𝑚, 𝑚))𝑀 = ∑ 𝐴𝑚ℓ𝑗(𝑡 − 𝑚, 𝑚)𝑀 .

The marginal products of labor are 𝜕𝑄𝑗(𝑡)/𝜕ℓ𝑗(𝑡 − 𝑚, 𝑚) = 𝑄𝑗(𝑡)/ℓ𝑗(𝑡 −

𝑚, 𝑚).

Regardless of firm type, there is an opportunity cost for adopting an ad-

vanced technology with delayed output. Current output, and presumable consump-

tion of some agent, must decrease as inputs are reallocated from no- or short- delay

to a longer delay technology. Sacrificing current consumption is a form of invest-

ment and creates intangible capital – the claim to the value of the future increased

output.

B. Firm Type

5 We investigate the role of the type of firm on the adoption of new technology and acqui-

sition of capital in a separate paper.

6

A young agent is born a self-employed proprietor (SEP). As such, they can

make contracts only in their own name and not as a firm as a separate legal entity

and, when they retire or die, the firm ceases to exist. Its ownership cannot be

transferred to subsequent generations, just the individual’s assets. It is well known

that if a competitive firm is a proprietor it must pay, when used, the marginal prod-

uct to the factors of production it employs. This becomes a problem when produc-

tion technologies involve long delays and is obvious when current input generates

output after the demise of the proprietor. Every young individual in the current

generation wants to sell the output they will produce in the future. Hence, there is

no one of this generation who will be a buyer, and future output had no value to

the current old because they will be dead. Hence a proprietor cannot benefit from

using the more productive technology.

In contrast, two attributes of the corporation enable it to make long-term deci-

sions and contracts that are irrational or unfeasible for a proprietor. A corporation

is a separate legal entity from any individual. The contracts, plans, and commit-

ments it makes are legally separate from those made by its equity owners or em-

ployees, even though they may act as agents for the corporation.6 Second, the eq-

uity ownership in a corporation is transferable from one generation to succeeding

ones. This can be by either public sale on a stock exchange,7 private sale, inher-

itance, or gift to non-relatives, as can be specified in its articles of incorporation.

These attributes enable a corporation to exist, make decisions, and carry out con-

6 This is the basis for the limited liability of its equity owners. 7 Some researchers consider that corporate equity being publically traded so important that

they require it, e.g. Levine (1997, p. 692), and Magill and Quinzii (2002, p. 331). The cre-

ation of the first joint-stock companies, the Dutch and English East India corporations,

simultaneously with the first stock exchange (Amsterdam) in 1602 certainly appears to

strongly support this view (Petram 2011).

7

tracts long after the retirement or demise of any finite-lived proprietor. A corpora-

tion can make a complex intertemporal contract, plan, or commitment that speci-

fies in every period not only the allocation of inputs to the technologies with dif-

ferent delays, but also in the initial and all future periods the wage paid a new,

young employee, how they becomes an equity owner and receives income in their

old age in either dividends, increased pension benefits or pay for unproductive em-

ployment. Such a long term contract frees the corporation from paying labor their

marginal product of labor in the period in which it is employed, and it can contract

to pay the sequence of its employees and equity owners during the period of transi-

tion from one technology to a more productive one less in their youth than their old

age, but always increasing the lifetime welfare of each succeeding generation until

a new steady state is reached (so that each succeeding generation is better off join-

ing the corporation than choosing to be a SEP). At that point it can, and has an in-

centive to, pay its employees and stockholders in any pattern that optimizes their

lifetime welfare, subject to it being feasible, e.g. equally in the case of the Samuel-

son agents. Again, this is independent of the marginal product produced by either

set of stakeholders.

C. Income, Consumption, and Lifetime Welfare

Denote the young or old generation by the superscript 𝑖 = 𝑌 or 𝑂. The con-

sumption in period t of an agent i, whose income comes from firm type 𝑗, is

𝐶𝑗,𝑖(𝑡). Following Samuelson, current utility is simply 𝑈𝑗,𝑖(𝑡) = 𝑙𝑛𝐶𝑗,𝑖(𝑡) and,

where β measures the time preference/discount factor for consumption, lifetime

welfare is,

(8) 𝑈𝑗(𝑡) = 𝑈𝑗,𝑌(𝑡) + 𝛽𝑈𝑗,𝑂(𝑡 + 1) = 𝑙𝑛𝐶𝑗,𝑌(𝑡) +

𝛽𝑙𝑛𝐶𝑗,𝑂(𝑡 + 1).

8

An individual agent’s intertemporal budget constraint is

(9) 𝑊𝑗,𝑌(𝑡) = 𝐼𝑗,𝑌(𝑡) + (𝑡)𝐼𝑗,𝑂(𝑡 + 1) ,

where 𝑊𝑗,𝑌(𝑡) is the lifetime wealth of a young individual born in period t whose

income, 𝐼𝑗,𝑌(𝑡) and 𝐼𝑗,𝑂(𝑡 + 1), comes from firm type 𝑗, and 𝑅(𝑡) is the discount

rate between goods of period t for goods of period t +1. This implies the optimal

allocation of consumption of 𝐶𝑗,𝑌(𝑡) + 𝛽𝑅(𝑡)𝐶𝑗,𝑂(𝑡 + 1) ≤ 𝑊𝑡𝑗,𝑌

. When the only

durable asset is fiat money, it yields 𝑅(𝑡) = 1, if it does not change in purchasing

power. The amount of output, and how output is allocated to income and income

to consumption, depends on the type of firm and the production technology it uses.

II. Optimal Technology Adoption

A. Self-Employed Proprietors

A.1. Technology 𝐴∅

Assuming that the only production technology available prior to 𝑡 = 0 is with

no output delay, equation (7) becomes 𝑄𝑃(𝑡) = 𝑄𝑃(𝑡, ∅) = 𝐴∅ℓ𝑃(𝑡, ∅). A young

SEP’s income and consumption are 𝐼𝑃,𝑌(𝑡) = 𝐶𝑃,𝑌(𝑡) = 𝑄𝑃(𝑡), e.g. if 𝐿(𝑡) =

ℓ𝑃(𝑡, ∅) = 20, 𝐶𝑃,𝑌(𝑡) = 20.8 When old, they will have no income or consump-

tion. They would clearly benefit by exchanging some of their current income for

consumption next period, but, as Samuelson observed, there is no one with whom

they can make a contract. The current old will be dead and their contemporaries

will also have nothing to exchange when old. Samuelson argued that this was a

“competitive market failure” that could not be remedied by the voluntary, rational

8 There is no reason in this framework to assume that an unincorporated firm has more

than one employee, or is a partnership.

9

actions of individuals and suggested it justified the coercive government interven-

tion of either fiat money or a social security system of taxation and subsides. He

assumed, as do we, fiat money with stable purchasing power is created. The con-

sumption when old is equal to what the young proprietor saves in money,

𝐶𝑃,𝑂(𝑡 + 1) = 𝑆𝑃(𝑡). Consumption when young then is 𝐶𝑃,𝑌(𝑡) = 𝐼𝑃,𝑌(𝑡) −

𝑆𝑃(𝑡).

A.2. Technologies 𝐴∅ and 𝐴1

Assume at t = 0 a more productive technology, 𝐴1 > 𝐴∅, with an output delay

of one period becomes available for adoption. A young SEP can allocate a propor-

tion, 𝑃, of their labor to 𝐴0, ℓ𝑃(𝑡 = 0, ∅) = 𝑃𝐿(𝑡) and the remainder to 𝐴1,

ℓ𝑃(𝑡 = 0, ∅) = (1 − 𝑃)𝐿(𝑡). This produces 𝑄𝑃(𝑡, ∅) = 𝐴∅ℓ𝑃(𝑡 = 0, ∅) and

𝑄𝑃(𝑡 + 1, 1) = 𝐴1ℓ𝑃(𝑡 = 0,1), and eliminates the need for saving and money in

youth. Lifetime welfare is given by

(10) 𝑈𝑃(𝑡) = 𝑙𝑛[𝐴∅𝑃𝐿(𝑡)] + 𝑅𝑡𝑙𝑛[𝐴1(1 − 𝑃)𝐿(𝑡)].

The optimal use of these technologies by a proprietor is stated as a Theorem.

Theorem 1. When a technology with no delay and a more productive technol-

ogy with a short delay are both available, i.e. 𝐴∅ and 𝐴1, the efficient allocation is

when all inputs are used in the most productive technology. However, a SEP will

use these inefficiently, always allocating some labor to the least productive, no-de-

lay technology.

Proof. The F.O.C. for maximizing equation (10) with respect to P is 0 =

𝐴∅𝐿(𝑡) 𝐴∅⁄ 𝑃𝐿(𝑡) − 𝑅(𝑡) 𝐴1𝐿(𝑡) 𝐴1(1 − 𝑃)𝐿(𝑡)⁄ , or P = 1/(𝑅(𝑡) + 1). As

10

𝑅(𝑡) is > 0 and finite, 0 < P < 1. So a proprietor will always use 𝐴∅ and be inef-

ficient, e.g. if 𝑅(𝑡) = 1, then 𝑃 = 0.5. Whatever value of Pis optimal in period

t remains the same in t ≥ 1, and therefore is a steady state. ■

It is insightful to present a numerical paradigm with which the behavior, output,

and resulting welfare produced by a SEP and a corporation can be compared.

When 𝐿(𝑡) = 20, 𝑅(𝑡) = 1 = 𝛽, and, prior to t = 0 only 𝐴∅ = 1 is available, a

young SEP would produce output and income of 𝑄𝑃(𝑡 < 0, ∅) = I𝑃,𝑌(𝑡 < 0) = 20

and, by saving money, consume 𝐶𝑃,𝑌(𝑡 < 0) = 10 =

𝐶𝑃,𝑂(𝑡 < 0). This would produce lifetime welfare of 𝑈𝑃(𝑡 < 0) = 4.605. When

both 𝐴∅ and 𝐴1 = 10 are available in t = 0, the optimal allocation of labor to 𝐴∅ is

ℓ𝑃(𝑡 = 0, ∅) = 10 and to 𝐴1 is ℓ𝑃(𝑡 = 0,1) = 20 − ℓ𝑃(𝑡 = 0, ∅) = 10, resulting

in outputs 𝑄𝑃(𝑡, ∅) = 10 and 𝑄𝑃(𝑡, 1) = 100, and income and consumption when

young and old of, respectively, 𝐼𝑃,𝑌(𝑡 = 0) = 10 = 𝐶𝑃,𝑌(𝑡 = 0) and

𝐼𝑃,𝑂(𝑡 = 1) = 100 = 𝐶𝑃,𝑂(𝑡 = 1). The resulting lifetime welfare of an agent born

in period t is 𝑈𝑃(𝑡) = 6.908. Notice that with the adoption of the advanced tech-

nology 𝐴1 by the SEP decreases the young’s income, but increases their welfare in

a steady state. It clearly create a very substantial inequality in income, 10 to 1.

A.3. Technologies 𝐴∅, 𝐴1, and 𝐴2

Now assume that at t =1 (after the steady state equilibrium with both 𝐴∅ and

𝐴1is achieved) that a new more productive technology with a delay of 2 periods

becomes available, 𝐴2. SEPs cannot benefit from this technology because they

will die before the output is realized and there is no other agent currently alive with

which they can make a contract to exchange their future output for current output.

The lack of incentive to adopt more productive technologies with long delays and

11

the inefficient use of the technologies which they do adopt limits the economic

growth that is possible in an economy consisting of unincorporated firms.

B. Corporations

For simplicity, we assume that if a proprietor forms a corporation, its stake-

holders will consist of one employee and one equity owner. A corporation distrib-

utes its output, 𝑄𝐶(𝑡), in the intertemporal pattern that maximizes the lifetime wel-

fare of its stakeholders, subject to it maximizing its profit and its continued exist-

ence. This requires it to be able to hire a new employee each period. The em-

ployee is rational and cares about the “employment package” over their lifetime,

including potential equity ownership, not just the wage paid when young. The best

“employment package” will provide income over their lifetime that corresponds to

their optimal consumption pattern, e.g. equal in their youth and old age, independ-

ent of the pattern of their marginal product of their labor, e.g. when they are only

productive when young. Competition in the labor market will ensure that any firm

that fails to offer/pay such an “employment package” will cease to exist.

The relationship between the income and consumption of stakeholders de-

pends upon how the equity ownership in the corporation is transferred. If it is by

inheritance or gift, then 𝐼𝐶,𝑌(𝑡) = 𝐶𝐶,𝑌(𝑡) and 𝐼𝐶,𝑂(𝑡) = 𝐶𝐶,𝑂(𝑡). If corporate eq-

uity is sold at the price 𝑃𝑆(𝑡) then 𝐶𝐶,𝑌(𝑡) = 𝐼𝐶,𝑌(𝑡) − 𝑃𝑆(𝑡) and 𝐶𝐶,𝑂(𝑡) =

𝐼𝐶,𝑂(𝑡) + 𝑃𝑆(𝑡). For notational simplicity we assume that the amount that a

young individual saves and pays for the purchase of the corporate equity from an

old agent is constant and extremely small, 𝑃𝑆(𝑡) = ɛ ≈ 0. Hence, there is no

meaningful distinction between the current income and consumption of either, i.e.

𝐶𝐶,𝑌(𝑡) = 𝐼𝐶,𝑌(𝑡) − ɛ and 𝐶𝐶,𝑂(𝑡) = 𝐼𝐶,𝑂(𝑡) + ɛ, and no possibility of the appre-

ciation of the stock price affecting the income of any agent.

12

B.1. Technology 𝐴∅

There is no incentive for a SEP to form a corporation because a corporation can-

not increase output and the distribution of output cannot be improved compared to

a SEP.

B.2. Technology 𝐴∅ and 𝐴1

If a corporation was already in existence when 𝐴1 becomes available, it would

only use this technology and, thereby, produce the maximum output to distribute to

its stakeholders. However, the availability of both 𝐴∅ and 𝐴1 in period 𝑡 does not

give a young SEP an incentive to form a corporation and be its equity owner and

hire a new employee in their old age, in 𝑡 + 1. To have such an incentive a corpo-

ration must be able to increase its output beyond the lifetime of its current stake-

holders and make a contract or plan to transfer some of the value of that increased

output into current equity owner’s income (via dividends). Because the output of

the labor allocated to 𝐴1 in period 𝑡 is realized in period 𝑡 + 1, the labor allocation

decision in period 𝑡 has no effect upon output produced beyond the lifetime of any

living agent. Hence, there is no labor allocation choice that the new corporation

can make that is superior to that made by the SEP of allocating (1 − 𝑃)𝐿(𝑡) to

technology 𝐴1.

B.3. Technologies 𝐴∅, 𝐴1, 𝐴2

Assume this economy is in a steady state equilibrium with the use of 𝐴∅ and 𝐴1

in t ≥ 1 as in Table 1 and 2. Then a sequence of more advanced technologies, each

with a longer delay and increased productivity, becomes available. Assume the

first new technology has a delay of 2 periods, 𝐴2 > 𝐴1.

13

In the steady state, when the technologies 𝐴∅, 𝐴1, and 𝐴2 are available to an

existing corporation, it can use these more efficiently and distribute the resulting

income to its stakeholders in a manner that makes their welfare greater than that

possible from the SEP from which the corporation was formed: a corporation as a

firm type can cause economic growth.

Theorem 2. When technologies 𝐴∅, 𝐴1, and 𝐴2 are available, a corporation

will only use the most productive technology in a steady state. In addition, a cor-

poration will distribute its output as income to maximize the lifetime welfare of its

stakeholders.

Proof. A corporation will maximize its output in a steady state by allocating all

labor to 𝐴2, because 𝐴2𝑓(𝐿(𝑡)) > 𝑀𝐴𝑋[𝐴∅𝑓(𝐿(𝑡)), 𝐴1𝑓(𝐿(𝑡))] >

𝐴∅𝑓 (𝑃𝐿(𝑡)) + 𝐴1𝑓((1 − 𝑃)𝐿(𝑡)).

A corporation can pay income equal to consumption in the pattern that maxim-

izes its stakeholders’ lifetime welfare. From equation (7), this is 𝐼𝐶,𝑂(𝑡)/

𝛽𝐼𝐶,𝑌(𝑡) = 1, or 𝐶𝐶,𝑌(𝑡) = 𝐴2𝑓(𝐿(𝑡))/(1 + 𝛽), and 𝐶𝐶,𝑂(𝑡) = 𝛽𝐴2𝑓(𝐿(𝑡))/(1 +

𝛽). When 𝛽 = 1, 𝑈𝐶(𝑡) = 2𝑙𝑛 [𝐴2𝑓(𝐿(𝑡))/2] > 2𝑀𝐴𝑋(𝑙𝑛[𝐴∅𝑓(𝐿(𝑡))/

2], 𝑙𝑛[𝐴1𝑓(𝐿(𝑡))/2]) > 𝑈𝑃(𝑡) = 𝑙𝑛 [𝐴∅𝑓 (𝑃𝐿(𝑡))] + 𝑙𝑛[𝐴1𝑓((1 − 𝑃)𝐿(𝑡))].■

Hence, for a SEP to have an incentive to form a corporation a technology with

a delay longer than the proprietor’s lifetime must be available.

B.4. A SEP’s Incentive to Form a Corporation

The inefficiency of a SEP versus that of a corporation raises the question of

whether a young SEP has the incentive and ability to form a new corporation. This

14

is complex because the adoption of an advanced technology with delayed output

has an opportunity cost for any firm. The current output and the consumption of

some agent must decrease. In addition, to make the corporation potentially death-

less, upon its founding the new corporation must be able to make a contract, plan,

or commitment that binds it to allocate future inputs to production technologies,

and distributions of the resulting output to all future stakeholders, both employees

and equity owners, that will make their lifetime welfare higher than if they chose

to be a SEP or to establish a new corporation. Due to the complexity of this, we

demonstrate in Theorem 3 the existence of one such contract using numerical

methods. Such a long-term contract is clearly not unique nor is it important for our

argument that it be optimal in the short-run period of transition. Rather, it is im-

portant that at least one such plan/commitment/contract exists.

Theorem 3. There exists a feasible contract, plan, or commitment that a young

SEP can make as part of establishing a new corporation that specifies (a) the allo-

cation of labor to the production technologies in the initial and all future periods,

(b) The initial equity ownership and how equity will be transferred to future gener-

ations, and (c) how the output of the corporation will be distributed as income to

them and all future stakeholders in the corporation.

Proof: Tables 1 and 2, if considered a contract or plan, present an allocation of

labor to the alternative technologies for each period and the distribution of the re-

sulting output that demonstrates the validity of Theorem 3. ■

The contract specifies how equity ownership is transferred from the old upon

their demise to a young agent, and that the corporation provides income to an em-

ployee in their youth and the old equity owner sufficient to increase their expected

welfare over that they would have received if they chose to be a SEP, and larger

15

than or equal to the preceding generations welfare. This eliminates the incentive

for any stakeholder to save, be a SEP, or start their own corporation.

Assume 𝐿(𝑡) = 20, 𝑅𝑡 = 1 = 𝛽, and 𝐴∅ = 1, 𝐴1 = 10, 𝐴2 = 1000, and that a

SEP has achieved the optimal steady state in period t = 1 by allocating 10 units of

labor each to 𝐴∅ and 𝐴1 in their youth, obtaining 10 units of output as income in

youth and 100 in old age, and lifetime utility U𝑃(𝑡) = 6.908. This is reproduced

in bold text in the first row of Tables 1 and 2.

Table 1. Labor Inputs and Outputs by a Corporation Formed by One Young

Agent in t = 2 that Hires One Employee in t ≥ 3, and 𝑨∅ = 𝟏, 𝑨𝟏 = 𝟏𝟎, 𝑨𝟐 =𝟏𝟎𝟎𝟎.

𝐴2 becomes available in period t = 2. The old SEP in period t = 2 have nothing

to contribute to the new corporation and no incentive to join. They receive 100

units of income, which they consume.

Period ℓ𝑡𝐶,∅

ℓ𝑡𝐶,1

ℓ𝑡𝐶,2

𝑄𝑡𝐶,∅

𝑄𝑡𝐶,1

𝑄𝑡𝐶,2

∑𝑄𝑡𝐶,𝑚

t = 1 10.0 10.0 0 10.0 100 0 110

t = 2 10.7 9.2 0.1 10.7 0 0 10.7

t = 3 10.4 9.4 0.2 10.4 92 0 102

t = 4 0.0 19.0 1.0 0.0 94 100 194

t = 5 0.0 0.0 20.0 0.0 190 200 390

t = 6 0.0 0.0 20.0 0.0 0 1000 1000

t ≥ 7 0.0 0.0 20.0 0.0 0 20000 20000

16

In order to increase the welfare of the generation born in t = 2, 3, relative to

what they would have had as a SEP, the corporation must increase the labor allo-

cated to 𝐴∅ in periods in t = 2, 3. This creates a transition period (t = 2, 3, and 4)

during which the corporation is “locked” into the less productive technologies.

Thereafter, the corporation gradually discards the less productive technologies un-

til only the most productive technology is used. This is consistent with the evi-

dence that the adoption of new technologies has been historically associated with

“lock-in periods” (Comin and Hobijn 2010). In t = 3, total output of the corpora-

tion is less than that produced by a SEP. This is consistent with the hypothesis of

a productivity slump associated with the adoption of advanced technology with de-

layed output (Brynjolfsson and McAfee 2014). In t ≥ 4, increased productivity

from labor previously allocated to 𝐴2 is realized and output exceeds that of a self-

employed proprietor. The optimal steady state production and distribution by a

corporation is achieved in t = 7.

Table 2 presents a distribution of the outputs in Table 1 to the income and con-

sumption of young and old employees/equity owners that increases equity owner

welfare, 𝑈𝐶(𝑡), in each generation over that of its predecessor. In t = 3, income

paid to the young employee must be lower than that earned as an unincorporated

proprietor, 𝐼𝐶,𝑌(𝑡 = 3) = 𝐶𝐶,𝑌(𝑡 = 3) = 7.0 versus 𝐼𝑃,𝑌(𝑡 = 3) = 10, so that the

old equity owner, who in their youth chose to switch from a SEP to a corporation,

can be paid enough to realize lifetime welfare higher than they would have re-

ceived as an unincorporated proprietor. This is possible because the young em-

ployee is paid less than their MPL in t = 3, but will be paid much more when old in

t = 4 when output is much higher. The corporation essentially contracts to transfer

some of the value of the output realized in t = 4 into equity owner income in t = 2

and 3.

17

Table 2. Individual Outcomes for Table 1

The increase in the initial corporate equity owner’s welfare is small

(𝑈𝐶(𝑡 = 2) = 6.924 compared to 𝑈𝑃(𝑡 = 1) = 6.908 for a self-employed propri-

etor), due to the welfare constraints stated above that the corporation must satisfy

for all future agents. However, this sequence of welfare increases provide an in-

centive for each succeeding generation to join the corporation. For example, in t =

3, a young agent who joins the corporation realizes lifetime welfare of 𝑈𝐶(𝑡 = 3)

= 7.052 compared to 𝑈𝐶(𝑡 = 2) = 6.924 if they formed their own corporation.

Notice that as output increases with the eventual allocation of all labor to 𝐴2,

the gap between the incomes of the young employee and the old equity owner de-

creases. This is because, although the equity owners of a corporation would prefer

to pay the least possible remuneration to its employees, they are prevented from

doing so by competition in the labor market and the prior contract(s) a corporation

has made. If they offer less than the competitive equilibrium “employment pack-

age” they will hire no new workers and cease to exist.

B.5. Technologies 𝐴∅, 𝐴1, 𝐴2 and 𝐴3

Period ∑𝑄𝐶,𝑚(𝑡) 𝐼𝐶,𝑌(𝑡) = 𝐶𝐶,𝑌(𝑡) 𝐼𝐶,𝑂(𝑡) = 𝐶𝐶,𝑂(𝑡) 𝑈𝐶(𝑡)

t = 1 110 10.0 100 6.908

t = 2 10.7 10.7 0 6.924

t = 3 102 7.0 95 7.052

t = 4 194 29 165 8.640

t = 5 390 195 195 11.488

t = 6 1000 500 500 15.425

t ≥ 7 20000 10,000 10,000 18.420

18

In Theorem 4 we demonstrate the existence of a contract/plan that provides an

incentive for a corporation, having been established and reached a steady state us-

ing technology 𝐴2 to adopt a new more productive delayed technology, e.g. 𝐴3 =

10,000, that becomes available.

Theorem 4. The stakeholders in a corporation that is initially in an efficient

steady state using the delay technology 𝐴2 have an incentive to adopt a more pro-

ductive technology with longer delays, 𝐴3, and to provide an incentive for future

stakeholders to transition from the less to the most efficient technology. Such in-

centives are provided in a plan or contract by the corporation, which specifies for

each future period a feasible allocation of the input of labor and the distribution of

the resulting output to the incomes of both young and old stakeholders, so that the

lifetime welfare of each generation is larger than or equal to the preceding gener-

ation’s welfare.

Proof: Assume that, in period t = 8, a more productive technology with 𝐴3 =

10,000 and a three-period output delay appears. Table 3 presents an example of a

feasible allocation of labor inputs and resulting outputs, and Table 4 presents a

welfare improving allocation of the resulting output to the equity owners and em-

ployees of the corporation.

19

Table 3. Labor Inputs and Outputs (in thousands, k) by a Corporation with

One Employee and a choice of Technologies 𝑨𝟎, 𝑨𝟏, 𝑨𝟐, and 𝑨𝟑 = 𝟏𝟎, 𝟎𝟎𝟎 in t

≥ 8.

Although output does not increase until t ≥ 11, the corporation’s ability to

transfer some of the output in t ≥ 11 into equity owner income in t = 8, 9, and 10

incentivizes the equity owner in t = 8 to adopt 𝐴3 because the corporation commits

itself to pay a wage to the young employees in periods t = 10 and t = 11 that is

considerably lower than that earned by workers in previous periods. In compensa-

tion, they will be paid much more when old in t = 11 and t = 12 when the delayed

output from adopting the more productive technology begins to be realized. The

most efficient steady state allocation of labor and optimal distribution of the output

to the income of the old and the young is achieved in period t = 14.

Table 4 presents the distribution of the corporation’s output in Table 3 to the

income of young employees and old equity owners during the transition period

which increases everyone’s welfare compared to a corporation that does not adopt

the new technology or to a SEP.

Period ℓ𝐶,∅ ℓ𝐶,1

ℓ𝐶,2 ℓ𝐶,3 Q𝐶,∅ Q𝐶,1 Q𝐶,2 Q𝐶,3 ∑Q𝐶,𝑚

t = 7 0 0 20 0 0 0 20k 0 20k

t = 8 2 1 16 1 2 0 20k 0 20,002

t = 9 0 1 16 3 0 10 20k 0 20,010

t = 10 0 0 15 5 0 10 16k 0 16,010

t = 11 0 0 0 20 0 0 16k 10k 26k

t = 12 0 0 0 20 0 0 15k 30k 45k

t = 13 0 0 0 20 0 0 0 50k 50k

t ≥ 14 0 0 0 20 0 0 0 200k 200k

20

Table 4. Individual Outcomes for Table 3

These two tables demonstrate convergence to the new steady state equilibrium

due to two factors. First is the reallocation of inputs to the new technology 𝐴3

from the less productive technologies which again involves a “lock-in” period in

the short-run during which some of the less productive technologies are used

which temporarily decreases total output, e.g. in period 𝑡 = 10. Second is the dis-

tribution of income between the young and the old, which was initially optimal

(and equal). This becomes unequal in periods 𝑡 = 9, 10, and 11, but eventually

equal again and optimal, thus increasing lifetime welfare in the new steady state in

period 𝑡 = 14 and beyond. Despite these complexities, the lifetime welfare of

each generation exceeds that of the preceding generation until the new steady state

in period t = 14 is attained. ■

III. Conclusions and Implications

In contrast to the current view in economics that a corporation is nothing more

than a “nexus of contracts” between individuals, this paper demonstrated that,

Period ∑𝑄𝐶,𝑚(𝑡) 𝐼𝐶,𝑌(𝑡) = 𝐶𝐶,𝑌(𝑡) 𝐼𝐶,𝑂(𝑡) = 𝐶𝐶,𝑂(𝑡) 𝑈𝐶(𝑡)

t = 7 20k 10,000 10,000 18.421

t = 8 20,002 10,001 10,001 18.421

t = 9 20,010 10,004 10,006 18.431

t = 10 16,010 5,910 10,100 18.482

t = 11 26k 9,000 18k 19.126

t = 12 45k 22.5k 22.5k 20.148

t = 13 50k 25k 25k 21.640

t ≥ 14 200k 100k 100k 23.026

21

when there is a delay between inputs and output and productivity increases with

the length of the delay, a corporation can make contracts of long duration which

are impossible for individual proprietors to make. This new view of a corporation

is consistent with the view of a corporation of Blair (2013) in the legal profession

as a “Persona” able to make long-term contracts, as well as Kenneth Arrow’s

(1969) view, neglected by economists, that a corporation is a “Social Organiza-

tion” similar to a government9.

In order to sharply distinguish our new view of the corporation from that based

upon Coase’s famous hypothesis that a firm exists to lower contracting, transac-

tions, information, and/or risk costs, including those associated with financing its

acquisition of capital, we assumed a model economy with no uncertainty, complete

knowledge, and no capital. We demonstrated that a proprietor would never adopt

and use efficiently the most productive technology, that a corporation would, and

that in a steady state there exists an incentive for a proprietor to form a corpora-

tion. This incentive is embodied in a complex contract or commitment by the new

corporation to allocate its initial and future inputs to different delay technologies,

and to distribute the resulting output to the current and future employees and eq-

uity owners in such a manner that each generation is better off than the preceding

one, until a new steady state is attained in which the corporation allocates its inputs

most efficiently and distributes its output to its stakeholders in an optimal inter-

temporal pattern, e.g. equally, that corresponds to that which a benevolent social

planned would choose. It is noteworthy that the payments to the factors of produc-

tion by the corporation, both in the steady state and the period of transition, are in-

dependent of their marginal product.

9 “It is a mistake to limit collective actions to State actions…Indeed, firms of any com-

plexity are illustrations of collective actions” (with “social norms”). (1969, p. 14).

22

The fact that many corporations involved in producing output with long delays

have existed for several centuries supports the argument that a corporation’s ability

to make contracts of very long duration is important,10 and may explain why cor-

porations have, over the last 130+ years, become the primary creators of advanced

production technology as measured by utility patents. Our analysis supports

Brynjolfsson and McAfee’s (2014) argument that the recent slump in U.S. growth

is due to the adoption of digital technology starting in the 1980s. Their observa-

tion, and our finding, of a lock-in period during the early stages of the transition

from the use of one technology to a more productive one with a longer delay, is

consistent with the U.S. economy operating at only 18% of digital capacity

(Manyika et al. 2015). Our model’s finding that income inequality between em-

ployees and equity owners increases in the early part of the adoption period is con-

sistent with equity owner income exceeding the 100-year average by 1.4% be-

tween 1985 and 2014 (Dobbs et al. 2016), and falling real wages in the 1980s and

early 1990s. However, rising income inequality is a short-run phenomenon in our

model, which dissipates as increased output is realized, because a corporation has

an incentive to distribute income to its stakeholders in a manner that maximized

their lifetime welfare, e.g. equally. This is consistent with the evidence that work-

ers in the most highly digitized U.S. industries have been experiencing extraordi-

nary high wage growth (Manyika et al. op. cit.). In short, a corporation, as a type

of firm, can cause income equality as well as economic growth.

This research has three implications. First, it implies a new reason for the ex-

istence of a corporation – the duration of the contracts it can make - independent of

10 The Dutch and English East India corporations existed for many centuries. The longest

lived corporation involved the Toulouse watermills on the Garrone River, which according

to Gimpel (1977) and Le Bris, Goetzmann, and Pouget (2015) started in the 12th century

with an agreement for joint ownership and its transfer.

23

the ability to lower contracting, transactions, and information costs, and its ad-

vantage in accumulating capital compared to a proprietor. Second, a public policy

goal of promoting economic growth through the creation, adoption, and efficient

use of advanced technologies with output delays should promote the formation and

growth of corporations, e.g., by lowering or eliminating the corporate income tax.

Third, it suggests that a corporation should be viewed as having a “Persona” or as

a “Mini-Society”, as for example in the case of Lincoln Electric Corporation, with

its famous focus upon the welfare of all of its stakeholders, or the even more ex-

treme example of the (many) Hutterite Farm Corporations, which are explicitly

communistic11. In fact it appears that the majority of corporations are consistent

with Kenneth Arrow’s view that a corporation is a “social organization” designed

to take “Collective Actions” to achieve the increase in the welfare of the group of

stakeholders it serves.12 These benefits of a corporation, as a type of firm, provide

a coherent, new, stronger defense of a slightly modified version of Milton Fried-

man’s (2002) argument that competitive market capitalism is the best way to or-

ganize an economy with a minimum of government intervention, when it is com-

petitive corporate market capitalism. 13 Arguably, Adam Smith’s “invisible hand”

works quit well when production is by firms that are corporations.

11 www.hutterites.org/day-to-day/structure/. 12 In 2000 there were 7,338 publicly traded corporations, which presumable have the ob-

jective of maximizing profit for their equity owners, out of a total of over 4,744,000 cor-

porations. Hence, 99.8+% of corporations did not transfer their equity by public sale and

presumable have some objective other than just profit maximization (Davis, et.al. 2006).

These facts should assuage some of the recent concerns of Magill, Quinzii, and Rochet

(2015). 13 But not his earlier (1970) argument that the only goal for a corporation should be to

maximize profit.

24

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