The Birth of Insurance Contracts - The Ataturk Institute for Modern ...

1 Introduction 

The Birth of Insurance Contracts 

Yadira González de Lara ∗ 

University of Alicante, Dep. of Economic Analysis 

Very incomplete and preliminry 

Comments are welcome 

“[T]he emphasis [of corporate finance] on large companies with dispersed investors has underemphasized 

the role that different financing instruments can play to provide investors better 

risk diversification. If all companies’ stock is held by well-diversified investors, there is little 

need for additional diversification. Unfortunately, this convenient assumption does not seem 

to hold in practice.” [Zingales, Journal of Finance, 55.4 (2000), p.1628] 

“To judge the extent to which today’s methods of dealing with risk are either a benefit or a 

threat, we must know the whole story, from its very beginnings.” 

[Bernstein, “Against the Gods. The Remarkable Story of Risk,” 1996, p.7] 

Premium insurance contracts first appeared in the Mediterranean trade during the fourteenth 

century. During the twelfth and the thirteenth centuries, the sea loan— a fixed payment loan 

with the particular feature that the investor took the “risk of loss arising from the perils of the sea 

or the action of hostile people”— and the commenda— a partnership agreement through which 

an investor supplied funds on which he both accepted the “risk of loss at sea or at the hands of 

hostile people” and received a share on profits— had securitized the marine risk. However, by the 

late thirteenth or the early fourteenth centuries credit and insurance were increasingly provided 

separately through risk-free bills of exchange and insurance contracts (de Roover, R. 1963). 

While some attention has been devoted to the transition from the sea loan to the commenda, 

very little research has been conducted on the emergence of insurance as an independent form of 

∗ My deepest debt of gratitude goes to Avner Greif for his constant suggestions, encouragement and support. 

I also want to thank Ramon Marimon for his excellent supervision of my Ph.D. dissertation, on which this paper 

is based. I have benefited from many stimulating comments of B. Arruñada, X. Freixas, L. Guiso, A. Lamorgese, 

R. Lucas, Lilya Malliar, Serguei Malliar, R. Minetti, A. Tiseno, G. Wright, two anonymous referees of a previous 

version of this paper, and various participants of seminars held at the Universidad de Alicante and Univesidade Nova. 

Funding for this research was made possible through grants from the Spanish Ministry of Science and Technology 

(CICYT: SEJ2004-02172/ECON). I acknowledge with gratitude the financial and intellectual support provided by 

the Ente Einaudi, the European University Institute and the Social Science History Institute at Stanford University. 

Very special thanks are due to Andrea Drago for his useful suggestions and patience. The usual caveats apply. All 

remaining errors are, of course, my own. yadira@merlin.fae.us.es 

1

usiness. Economic historians have noted the advantage of the commenda over the sea loan in 

terms of risk-sharing (Lane, 1966, pp.61-63, and Lopez, 1952, pp.76 and 104) and have associated 

the introduction of premium insurance both with the increased use of the bill of exchange for 

the financing of overseas trade and with the organization of this trade by means of permanent 

representatives abroad (de Roover, F.E., 1945, pp. 173 and 177). Yet, no satisfactory explanation 

for the observed selection of contracts have been so far provided. 

As the sea loan and the commenda resemble today’s debt and equity contracts, on the one 

hand, and fix-rental and sharecropping agreements, on the other hand, modern economic theory 

might shed light on their use. 1 Contrary to the Corporate Finance literature, Transaction 

Costs Economics, and the Historical Institutional Analysis, this paper, however, ignores the role 

of verification/falsification costs, multiple margins for moral hazard, adverse selection, signaling, 

incomplete contracting, transaction costs, and institutions for contract enforcement in shaping 

contracts. 2 Instead the choice between the sea loan or the commenda derives from simple, 

although historically backed, information assumptions: under hidden information, the sea loan 

emerges as a second-best because contracts cannot be contingent of the venture’s commercial 

profit; under full information, incentive-compatible constraints are relaxed, enabling the adoption 

of the first-best better risk-sharing commenda (Townsend, 1982). 

The paper, on the contrary, focuses on the role that different financial instruments play to 

provide the optimal allocation of risk, in particular, of marine risk. Like today’s debt and equity 

contracts, the sea loan and the commenda provided limited liability to both the merchant, who 

was exempted from repayment beyond the venture’s assets in case of loss at sea or at the hands 

1 The sea loan and the commenda indeed served the double function of financing and organizing overseas trade 

using someone else’s capital. In Venice, for example, both types of contract worked as financial instruments through 

which business capital together with personal savings, small incomes and legacies were mobilized into risky investments 

in overseas trade (Lane, 1973, and González de Lara, 2005). In twelfth-century Genoa, however, the 

commenda operated mostly as a labor contract to handle the investor’s business abroad (Greif, 1989 and 1993). 

Accordingly, the investing party have been referred to in the literature as sleeping partner, stay at home or 

sedentary merchant, principal, stans, and commendator. Similarly, the travelling merchant has been called managing 

partner, carrier, factor or agent, tractans or procertans, and accomendatarius. Since this paper focuses on the marine 

risk allocation that different contracts entailed, I shall use the terms “financier” and ”merchant.” 

2 The companion piece of this paper analyzes the enforcement and information problems underlying the transition 

from the sea loan to the commenda in late-medieval Venice and finds that, in this particular episode, institutional 

developments that enhanced the state’s ability to verify information enabled the adoption of the better risk-sharing 

commenda. EXPLAIN 

2

of hostile people, and to the investor, who risked loosing only the capital advanced. Commonly, 

merchants received funds from several financiers, while investing in other merchants’ ventures for 

the sake of diversification. Optimal Portfolio Selection theory can account for the merchant’s 

reliance on external funds on the basis of managerial risk aversion, but takes the specific form of 

contracts (securities) under which funds are raised as exogenously given. The Optimal Security 

Design literature derives endogenously the characteristics of debt and equity under the assumption 

that all contracting parties are risk neutral, but predicts the maximum investment participation 

of the merchant to the venture (for excellent surveys of Corporate Finance, see Harris and Raviv, 

1991, and Zingales, 2000). 3 However, the evidence indicates that medieval merchants both were 

risk averse and resorted to external funds before investing 100 percent of their own wealth in 

the sea venture they manage. 4 More importantly, neither the Traditional Capital-Structure 

theories nor the the Optimal Security Design literature have addressed the evolution from the 

securitization of the marine risk through debt-like sea loans and equity-like commenda contracts 

to the provision of insurance, not through the capital market, but through premium policies. 

The sea loan and the commenda differed from a combination of premium insurance and a 

risk-free bill of exchange in two respects. First, both the sea loan and the commenda conferred 

the merchant less protection against the marine risk than pure insurance contracts, which did not 

only exempt the merchant from repayment beyond what was saved from misfortune at sea but 

also provided a coverage payment to offset the merchant’s lack of gain in the event of a marine 

incident. Second, whereas the sea loan and the commenda exchanged funds for a claim on the 

venture’s returns, both the bill of exchange and the pure insurance contract established payments 

from the agents’ personal wealth. Historical evidence suggests that enforcing these payments was 

more difficult and costly than ensuring that the investor received a fair return from the venture. 

The paper proposes a historically-grounded mechanism-design model of corporate finance with 

3 The Costly State-Verification literatures explicitly recognized that standard debt contracts are not efficient if 

the parties are risk averse (Townsend, 1978, p. ?, and Gale and Hellwig, 1981, p. ?). Under risk-neutrality, contracts 

are optimally designed to minimize information and/or transaction costs. Therefore, the best a merchant can do 

is to finance the project he manages in its entirety, unless restricted to rely on external funds because of budget 

constraints and hence the prediction of a maximum inside participation rate to the firms’ investments. 

4 Zingales (2000) and (1995, table IV, p. 1439) provides equivalent evidence on today’s entrepreneurs/firms. 

3

two-side risk aversion and costly contract enforcement where the optimal contract choice depends 

on the risk and cost of a non-diversifiable investment project. In particular, if the project is highly 

risky and costly, the optimal risk allocation can be achieved either through letting the merchant 

finance a part of the venture and raising the remaining capital through a limited-liability sea loan 

or a commenda contract or through a combination of a risk-free bill of exchange and an insurance 

policy. Yet, the former is Pareto superior under the assumption that the agents’ personal wealth 

is contractible at a higher enforcement cost than the project’s returns (when the merchant lacks 

enough wealth to finance the project himself and/or collateral to repay in full in case of loss, 

the sea loan or the commenda uniquely sustains the optimal allocation of risk). However, if the 

project is relatively less risky and costly, the optimal risk allocation calls for more insurance than 

what limited-liability sea loans and commenda contracts provide to the merchant, thereby making 

insurance policies unavoidable. 

Consistent with the model’s predictions, we observe that the sea loan and the commenda 

actually gave way to marine insurance when commerce lost many of its adventurous features and 

wealth accumulation from commerce reduced the former scarcity of capital (Lopez, 1976, p. 97). 

The organization of the paper is as follows. Section 2 sets and solves the model, while sections 

3 evaluates empirically the assumptions and predictions of the model. Finally, section 4 concludes. 

2 The Model 

Consider a one-period two-dates contracting economy with a merchant and a financier who face 

the possibility to undertake a single welfare-enhancing trading venture. Both agents are risk 

averse with preferences represented by continuous and twice-differentiable utility functions Ui(ci) 

exhibiting a decreasing absolute risk aversion (DARA) coefficient with U ′ i 

(.) > 0, U ′′ 

i (.) < 0, 

where c denotes consumption and the subscript i = 1, 2 stands for the merchant and the financier, 

respectively. 5 Each agent i is initially endowed with ki units of good and maximizes his expected 

5 Note that Innada conditions are not assumed to hold, so that we do not force an interior solution. 

4

utility from second-date consumption. Goods can be costlessly self-stored or invested in the 

trading venture. Any amount of the good can be stored but the investment is lumpy. The venture 

requires k units of investment, plus the entrepreneurship of the merchant. It yields a random 

return s ∈ S = {y, x, ¯x} with probability ps, where y denotes the loss due to the marine risk 

and {x, ¯x} stand for the risky commercial return, with 0 ≤ y ≪ k ≪ x < ¯x and k < E[s]. 

The marine risk refers to the so-called “risk of sea and people” and includes the risk of natural 

shipwreck, piracy and confiscation of property by foreign rulers. The commercial risk accounts for 

wide variations in profits depending on the tariffs and bribes paid in customs, transportation and 

storage fees, the conditions of the goods upon arrival after hazardous trips, fluctuations in prices, 

and so on and so forth. 

To account for the optimality of undertaking the investment project we assume that 

E[Ui(s)] > Ui(k) ∀i, (1) 

which in combination with DARA ensures that the higher productivity of the trading venture 

(E[s] > k) compensates for its risk. Therefore, agents will only self-store and consume their 

autarkic endowments, ki, if they are unable to mobilize funds in long-distance trade. For ease of 

notation, let w(s) = k1 + k2 − k + s be the second-date total risky endowment of the economy 

when the venture is undertaken. We first study the case in which the merchant is constrained to 

rely on external funds because of budget constraints and, for simplicity, he is assumed to have 

zero initial wealth, k1 = 0. Then, the model is extended to capture the situation in which the 

merchant can finance the venture entirely on his own, k1 > k. The financier is always endowed 

with enough resources to fund the venture, k2 > k. 

To keep the structure as simple as possible, the model assumes that the legal system can enforce 

contracts contingent on verifiable information at no cost, although this unrealistic assumption 

is relaxed in section 3.2. At the first date, when investment decisions are taken, there is no 

information whatsoever on the true realization of the venture return. At the second date the 

return is realized and revealed to the merchant. While losses “at sea or from the action of 

5

enemies” are assumed to be verifiable, the model considers two extreme scenarios regarding the 

ability of the State to verify the commercial return. Under hidden information the commercial 

returns is non-verifiable, at any cost, but the State can force the merchant pay out the minimum 

possible commercial return, x. Under the full information structure the commercial return is 

verifiable without any private cost for the financier. 

2.1 Contracts 

We consider allocations that can be obtained by means of a contract enabling the funding of 

the venture and specifying transfers from the merchant to the financier both at the first and 

at the second date. Whereas first-date transfers τ ∈ ℜ are prior to the realization of the state 

and, accordingly, are independent of it, second-date transfers τ(s) are contingent on the state. 

A contract will result in consumption schedules for the merchant and the financier, c1 and c2 

respectively, where c1 := c1(s) = k1 − k + s − [τ + τ(s)] and c2 := c2(s) = k2 + [τ + τ(s)]. 6 

Sometimes the merchant lacked the capital required to undertake a sea venture. Others, 

however, he was a man of substance who could fund the venture he managed in whole or in part 

if he so wished. 7 Let φ k, with φ ∈ [0, 1] denotes the capital contributed by the merchant to the 

venture, so that (1 − φ)k ∈ [0, k] is the capital raised from the investor. 

During the twelfth and thirteenth centuries, merchants typically relied on sea loans and com- 

menda contracts, even when they were endowed with the resources required to finance the venture 

6 These consumption schedules capture the fact that the venture’s returns first accrued to the merchant, who need 

to cover the fix cost k to generate the return s. This notation is better suited for the case in which the merchant 

owned the venture and hence was the residual claimant. Alternatively, consumption schedules can be expressed as 

c1(s) = k1 + ˘τ + ˘τ(s) and c2(s) = k2 − k + s − ˘τ − ˘τ(s), where ˘τ + ˘τ(s) = [−k − τ] + [s − τ(s)] are to be understood 

as a contingent salary the financier pays to the the merchant for handling his business abroad. It is worth noting 

that ownership is irrelevant to the model’s results. This is so because, on the one hand, the model is robust to the 

allocation of property rights (it considers all levels of bargaining power subject to individual rationality) and, on 

the other hand, there is no incomplete contracting and hence the allocation of control rights is unimportant. 

7 As the Consolat del Mar of Barcelona pointed out many merchants were propertyless men who owned nothing 

on their own among all that they carried (Pryor, 1983, p.162). We know, however, about many merchants with 

considerable assets. For example, according to the the estimo of 1207, Zaccaria Stagnario was among the richest 

men in Venice, yet he raised substantial amounts through commenda contracts to finance his numerous trips to 

the Levant (Buenger, 1999). Examples of merchants who invested in other merchant’s ventures abound. The 

extant Venetian documentation for he period 1021-1261 make reference to 435 sea loans and commenda contracts 

in which 233 merchants are mentioned. Excluding the merchants who appear only once, and could not hence show 

up performing both roles, 36 percent of the merchants invested in other merchants’ ventures and over 72 percent 

of the 83 merchant’s families who appear twice or oftener had members who acted as merchants and members who 

acted as investors (Morozzo della Rocca and Lombardo, 1940 and 1953, and González de Lara, 2005, p. 21). 

6

in his enterity. In return for some capital (τ = −(1 − φ)k), the sea loan and the commenda 

entailed the financier to a claim on the venture’s returns. Specifically, on the event of loss due to 

shipwreck, piracy, or confiscation of the merchandise by greedy rulers abroad, the sea loan and the 

commenda established a transfer equal to all what could be retrieved (τ(y) = y). The merchant 

was thus partially insured against the marine risk, for he was exempted from repayment beyond 

the amount saved from misfortune at sea or at the hands of hostile people— τ(y) = y < k— 

but he did not enjoy full insurance, which would have required a coverage payment— τ(y) < 0 

instead of τ(y) = y ≥ 0— to offset the merchant’s lack of gain in that event. The sea loan and 

the commenda differed only in the transfers they established in the case the ship arrived safe and 

sound in port. Whereas the sea loan paid out a fixed sum, the commenda divided the commercial 

profit according to a ratio agreed upon, and thus shared not only the marine risk but also the 

commercial risk between the merchant and his financier. In short, the sea loan and the commenda 

established the same income-rights as today’s debt and equity contracts. They were, however, 

silent about the allocation of control rights. 

Definition 1 The sea loan establishes: τ = − (1 − φ)k ∈ [−k, 0] and τ(y) = y < τ(x) = τ(¯x) 

Definition 2 The commenda establishes: τ = − (1 − φ)k ∈ [−k, 0] and τ(y) = y < τ(x) < τ(¯x) 

At some point during the late thirteenth or the early fourteenth century, however, credit 

and insurance began to be provided separately through risk-free bills of exchange and insurance 

contracts. 8 By the fifteenth century marine insurance was established on a secure basis on the 

Italian trade and expand to the Hanseatic and the English trade during the sixteenth century. 

The bill of exchange was an order to pay a fix amount in one place in one kind of money in 

return for a payment received in a different place in a different kind of money. There was always a 

time lag between receipt and payment, so that one of the parties was extending credit to the other 

8 There is no consensus on the origin of marine insurance. The earliest undisputed examples of premium insurance 

contracts are a series of policies drawn up in Palermo in 1350, but the accounts books of the Del Bene firm of Florence 

make clear reference to premium insurance as early as 1319. Already in the late thirteenth century insurance loans— 

loans repayable upon the safe arrival of a shipment that ship owners made to shippers who stayed on land — and 

fictitious sales— put options— shifted part of the burden of the marine risk from the merchant to the insurer ( for 

other early examples of marine insurance, see de Roover, F. E., 1945). 

7

in the meantime. Premium insurance established a coverage payment for loss due to shipwreck, 

piracy, or confiscation of the merchandise abroad in return for a premium payable in advance or, 

more generally, after the insured cargo was safely arrived to destination. Shipments, however, 

were rarely insured for more than half of their value or even less. 

Definition 3 The bill of exchange establishes: τ = − (1 − φ)k ∈ [−k, 0] and τ(y) = τ(x) = τ(¯x) 

Definition 4 A pure insurance contract establishes either 

τ > 0, τ(y) < 0, and τ(x) = τ(¯x) = 0 or τ = 0, τ(y) < 0, and τ(x) = τ(¯x) > 0 

2.2 Optimal Contracts 

A contract is optimal if and only if it sustains an optimal allocation of consumption, which is 

defined as the solution to the following (concave) problem for some given value of U 2. 9 Further, we 

restrict our attention to contractual forms that sustain the optimal allocations for all individually 

rational levels of U 2, i.e. regardless of the competitive structure of the financial market. Thus, 

the model accounts for the great variety of observed financial relations. Sometimes, an ambitious 

young merchant received funds and advice from an older merchant with considerable assets and 

political influence. This was the case of the Genoese merchant Ansaldo Bailardo, who concluded a 

series of contracts with the all powerful merchant Ingo da Volta, starting with noting in 1156 when 

he was still a minor (de Roover, R., 1963, pp. 51-52 and Greif, 1994). Many other times, however, 

an economically and politically prominent merchant took funds from ordinary members of society 

with some cash. For example, Domenico Gradenigo, who had been born into a centuries’ old 

Venetian patrician family, raised an small amount from a widow in 1223 for a round-trip voyage 

to Constantinople (Buenger, 1999, and González de Lara, 2005, p. 5). 

Program 1: 

max 

{ci(s)}i=1,2 

E{U1[c1(s)]} 

s.t. E{U2[c2(s)]} ≥ U 2 (2) 

9 The Revelation Principle ensures that any contingent allocation that satisfies the self-selection property, i.e. 

which is incentive compatible, can be achieved under a mechanism (a contract). This allows us to convert a problem 

of characterizing efficient contracts into a simpler one of characterizing efficient allocations, as in a pure exchange 

economy. Then, we only have to find a contract that sustains the optimal allocation and respects all the restrictions. 

8

ci(s) ≥ 0 ∀i, ∀s (3) 

 

ci(s) ≤ w(s) ∀s. (4) 

i 

Restrictions (3)-(4) are feasibility constrains. Consumption cannot be negative and cannot 

exceed total resources. Constrain (4) holds with equality, since resources are not spared optimally. 

Therefore, c1(s) = w(s) − c2(s) and the problem can be solve on c2(s) for s ∈ S. It also follows 

that (2) holds with equality, so that the concave utility possibilities frontier can be traced out as 

the parameter U 2 is varied. Individual rationality imposes both a lower and upper bound on U 2. 

Hereafter, this parameter is restricted to lie in this interval. 

The lower bound is given by the financier’s ex-ante participation constraint 

U 2 ≥ U2[k2], (5) 

which ensures that he is as well off under the contract as he is under autarky. The upper bound of 

the parameter U 2 varies depending on the merchant’s initial endowments. If he is endowed with 

zero wealth, k1 = 0, U 2 ≤ E{U2[k2 − k + s]} because the merchant will always agree to undertake 

the venture. His ex-ante participation constraint, E{U1[c1(s)]} ≥ E{U1[k1]}, can thus be ignored, 

for his best alternative is consuming nothing in all the states, which gives him no more utility 

than he can get through contracting, as constraint (3) reads. If, on the contrary, the merchant 

is initially endowed with the necessary resources to finance the venture himself, k1 > k, he can 

undertake the venture alone and consume c1(s) = k1 − k + s. In this case, his individually rational 

constraint 

E{U1[c1(s)]} ≥ E{U1[k1 − k + s]} (6) 

is more restrictive than his ex-ante participation constraint because of (1) for i = 1 and utility 

functions exhibit decreasing absolute risk aversion (DARA). This implies a smaller upper bound 

on the parameter U 2 for k1 > k than for k1 = 0. 

9

2.3 Hidden Information or Full Information: the Sea Loan or the Commenda 

Standard contract theory informs us that in a one-period model contracts can only be made 

contingent on states whose occurrence can be verified to the satisfaction of all contracting parties 

(Townsend, 1982). The commenda is thus ruled out under hidden information since incentive- 

compatibility 

c2(x) = c2(x) = c2(¯x) (7) 

leads to a fix-repayment transfer regardless of the venture’s commercial profit: τ(x) = τ(x) = τ(¯x). 

Hidden information also restricts the upper bound of the parameter U 2 

U 2 ≤ py U2[k2 − k + y] + [px + p¯x] U2[k2 − k + x] < E{U2[k2 − k + s]}. (8) 

As the merchant can expropriate the non-verifiable component of the profit, c1(¯x) ≥ ¯x − x > 0, 

contracts with τ(x) > x are non enforceable and the maximum utility that the financier can 

achieve is given by the value of U 2 in (8). 

The choice between the sea loan or the commenda can be simply rationalized in terms of the 

information structure: under hidden information, the parties are constrained to exchange through 

fix-payment sea loans, with τ(x) = τ(¯x); under full information, the incentive-compatibility con- 

straint (7) is relaxed, enabling financial contracting through better risk-sharing commenda con- 

tracts, with τ(x) < τ(¯x). 

Yet, an explanation for the optimality of the sea loan or the commenda and for their evolution 

towards marine insurance as an independent for of business needs to account for the fact that both 

the sea loan and the commenda entailed the investor to recoup all the capital retrieved from a 

marine loss (τ(y) = y), whereas a premium insurance contract entailed the merchant to a coverage 

payment for such a loss (τ(y) < 0 ≤ y). 

2.4 Optimal risk-sharing with k1 = 0 under hidden information 

Let us first examine the case in which the merchant is initially endowed with zero wealth and 

has hidden information on the true commercial return, so that he is constrained to resort to the 

10

financier for the entire capital requirements and cannot commit to pay any amount beyond the 

verifiable venture’s returns. The programming problem becomes 

Program 2: 

max 

{c2(y),c2(x)} 

E{U1[w(s) − c2(s)]} 

s.t. E{U2[c2(s)]} = U 2 (9) 

c2(y) ≤ w(y), c2(x) ≤ w(x), c2(s) ≥ 0, (10) 

where U 2, satisfying (5) and (8), defines each individually-rational optimal allocation and restric- 

tion (10) combines the feasibility constraints (3) and (4) with the information constraint (7). 

Under hidden information, there are two instead of three choice variables and hence the optimal 

consumption allocation can be represented in a two-dimensional Edgeworth Box, but with a trick: 

the merchant’s consumption has two different origins. Each point in figures 1 to 4 thus represents 

the two event-contingent consumption of the financier and the three state-contingent consumption 

of the merchant. The dimensions of the Edgeworth Box are given by the total resources of the 

economy in each state, w(s) = k2 − k + s. Yet, the merchant’s inability to commit to refrain from 

embezzling the non-verifiable component of the profit leads to c2(x) ≤ w(x) and accounts for the 

shaded noncontractible area. 

[Figure 1 around here] 

From figure 1 it is clear without further analysis that any feasible allocation that provides 

constant consumption to the financier, i.e. c2(y) = c2(x) ≤ w(y) = k2 − k + y, does not satisfy his 

ex-ante participation constraint— P C2 in figures 1 to 4. Since the merchant is short of resources 

to pay out any amount beyond the venture’s verifiable returns, the financier is constrained to bear 

liability for loss of at least a part of his capital: τ(y) ≤ y < k. Would he optimally assume as little 

marine risk as possible through a sea loan, with τ(y) = y, or would he provide more insurance to 

the merchant, for example, by setting a coverage payment τ(y) < 0

Proposition 1 The more risky and costly the venture is (the higher py and the lower y, the higher 

k relative to k2, and the lower E[x] = px x + p¯x ¯x given x ≥ t pc ), the more likely it is that the risk 

averse merchant raises funds from the risk averse financier through a sea loan, optimally. 

If the venture is highly risky and costly, the financier might suffer a substantial loss. On the 

likely event of a navigation incident (py high), he cannot recoup any amount beyond what is saved 

from misfortune, which is dreadfully insufficient (y very low) to reward his capital investment: 

τ(y) ≤ y ≪ k. This implies a very significant loss to him, for he has invested a major part of 

his scarce initial endowment in the costly venture (k high relative to k2). To compensate for the 

possibility of loss, the financier requires soaring payments when the merchandise arrived at port 

safe and sound. As a consequence, his consumption varies widely, whereas that of the merchant 

is invariably small. At any feasible and individually-rational allocation, 

c2(y) ≤ k2 − k + y, c2(x) ≥ k2 − k + ˜t, and c1(y) ≥ 0, E[c1(x)] ≤ E[x] − ˜t, 

where ˜t ∈ [t pc , x] denotes the transfer required by the investor to compensate him for assuming as 

less marine risk as possible, i.e. the very high transfer a sea loan establishes on the event of safe 

arrival of the ship and its cargo to destination. The second and fourth inequalities derive from 

the downward slope of the financier’s indifference curves (see figure 1 and 4). Thus, on the choice 

set the merchant’s consumption is smoother than the financier’s consumption 

max{E[c1(x)] − c1(y)} = E[x] − ˜t ≪ ˜t − y = min{c2(x) − c2(y)}. (11) 

Because both agents are assumed to have the same preferences, the merchant values insurance 

relatively less than the financier so that it is optimal that he receives as less marine insurance as 

possible. Since the higher the amount paid out in the case of loss, the lower the repayment required 

otherwise and the less volatile the investor’s event-consumption relative to the merchant’s, it is 

optimal that the investor recoup as much of his capital as possible from the venture’s return. A 

sea loan, establishing τ = −k, τ(y) = y, and τ(x) = τ(¯x) = ˜t, can sustain all the allocations in 

the core, with ˜t ∈ [t pc , x] taking different values for each individually-rational level of bargaining 

12

power U 2. 10 It is worth to note, though, that the sea loan is a corner solution. As depicted 

in figure 4, the financier’s indiference curve in the optimum is more sloppy than the merchant’s 

one, reflecting that the investor values consumption in the event of loss relatively more than the 

merchant. Both parties would prefer a contract with higher repayment in the case of loss, but 

this is not feasible because the merchant’s initial endowments are either zero or non-contractible. 

If the venture becomes less and less risky and costly, though, the sea loan would eventually 

stop being optimal. As depicted in figure 1, the slope of the financier’s indifference curve at point 

A would decrease (py lower) and the size of the Edgeworth Box would increase (y higher), so that 

˜t would substantially decrease. Furthermore, a reduction in the relative cost of the venture (k2 

higher and/or k lower) would also push ˜t down because of DARA. At some point inequality (11) 

would be violated an a pure insurance contract with τ(y) < y would emerge optimally. 

2.5 Optimal risk-sharing with k1 = 0 under full information 

It has been proved that, when ventures are highly risky and costly, the sea loan emerges optimally 

under hidden information. However, the (risk-averse) merchant undesirably undertakes all the 

commercial risk, τ(x) = τ(¯x) because of his inability to commit to reveal a commercial return 

other than the minimum verifiable (x by assumption). The provision of verifiable information 

relaxes incentive-compatibility constraints and enables better risk-sharing commenda contracts, 

with τ(y) < τ(x) < τ(¯x). Yet, to explain the optimality of the commenda, one needs to account 

for the fact that it shares the marine risk in the very same manner as the sea loan, with τ(y) = y. 

With a sea loan, the financier bears as little marine risk as possible, τ(y) = y, while taking none 

of the commercial risk, τ(x) = τ(¯x). If this is the efficient agreement under hidden information, it 

naturally follows that the financier will not optimally bear more marine risk once he is optimally 

assuming part of the commercial risk, τ(x) < τ(¯x) under full information. Thus the optimality of 

the commenda derives from that of the sea loan and can hence be linked to the venture’s risk and 

10 The core in this simple two-agent economy is standardly defined as the Pareto efficient individually-rational 

event-contingent consumption allocations. It is represented as the Pareto set that lies between the indifference 

curves that pass through the initial endowments, meaning the shaded interval of the Contract Curve CC or CC ′ in 

figures 1 to 4. 

13

cost characteristics. Proposition 2, which is proven in Appendix B, follows. 

Proposition 2 If the sea loan sustains all the allocations in the core under hidden information, 

the commenda supports the corresponding allocations under full information. 

2.6 Optimal Risk-sharing with k1 > k 

From the observation that debt-like sea loans and equity-like commenda contacts are corner so- 

lutions when the venture is highly costly and risky, it follows that a well-endowed entrepreneur 

will not raise funds for such a venture in whole through either of these contracts. Rather, he will 

optimally finance part of the costly and risky venture himself, bear the liability of loss for the 

corresponding part, and resort to limited-liability sea loans or commenda contracts for the remain- 

ing part, thereby effectively receiving less insurance against the marine risk. In other words, the 

model delivers an optimal capital structure where inside equity held by the merchant, debt-like 

sea loans and outside-equity-like commenda contracts coexist. 

Proposition 3 If the venture is highly risky and costly, the optimal event-contingent allocation 

of consumption when k1 > k can be attained by letting the merchant finance part of the venture, 

φ ∗ ∈ (0, 1) and raising the remaining capital, τ ∗ = − (1 − φ ∗ ) k, through a debt-like sea loan or 

an equity-like commenda, one or the other depending on the information structure. 

This result is novel in the literature. For the last fifty years or so Corporate Finance has 

focused on various departures from the Modigliani and Miller’s theorem that make capital struc- 

ture relevant to a firm’s value under the assumption that economic agents are risk neutral (Harris 

and Raviv, 1991). Some models take the form of the securities (contracts) issued by the firm, 

mainly debt and equity, as exogenously given; others endogenously derive the characteristics of 

debt and equity in terms of cash flows and/or control rights; but all predict a maximum inside 

participation rate to the firms’ investments (φ = 1 in the model). Since contracts are chosen or 

designed exclusively to minimize the costs associated with agency relations, information asym- 

metries, signaling, incomplete contracting or transacting, the best an entrepreneur can do is to 

finance the project he manages in its entirety, unless restricted to rely on external funds because 

14

of budget constraints (k1 = 0). However, the evidence indicates that both medieval merchants 

and today’s entrepreneurs/firms are risk averse and resort to external funds before investing 100 

percent of their own wealth. 11 

A notable exception is Leland and Pyle (1977). They exploit managerial risk aversion to 

obtain a signaling equilibrium in which entrepreneurs retain a higher fraction of (inside) equity 

than they would hold in the absence of adverse selection, but still rely on external funds because of 

diversification. Leland and Pyle thus rationalize the observed fact that entrepreneurs self-finance 

only a part of their ventures, but do not address the fundamental question of why debt and equity 

are designed the way they are. 

2.6.1 Graphical analysis 

The proof of proposition (3) is developed in appendix 3. Lets here adressed it graphically. When 

the merchant is initially endowed with k1 > k > 0, there are more resources in the economy than 

when he does not have any initial wealth, k1 = 0, so that the corresponding Edgeworth Box is 

larger. Figure 5 represents both the Edgeworth Boxes for parameters λk1=0 (with k1 = 0) and 

λk1>k (with k1 > k > 0), although for ease of exposition the noncontractible area is not depicted 

under the hidden information scenario. 12 

Because of DARA, the indifference curves of the merchant become flatter as he receives more 

initial wealth, while those of the financier are unaltered by any increase of the merchant’s wealth. 

Therefore, the contract curve for interior points for parameters λk1>k lies below the contract curve 

for λk1=0. The previous subsections shows that the contract curve for k1 = 0 can be represented by 

CCλk 1 =0 

in figure 5 when a venture is risky and costly and both agents are risk-averse. Therefore, 

the contract curve for k1 > k will be characterized by CCλk 1 >k 

in figure 5. 

11 

For medieval evidence, see section 3 below. For today’s evidence, see Zingales (2000) and (1995, table IV, p. 

1439). 

12 

Alternatively, figure 5 can be thought of as representing a face of the Edgeworth Boxes for a given value of 

c2(¯x) under full information. In this case there are three events for both agents and the contract curve must be 

represented in a 3-dimensional Edgeworth Box. However, as we are looking at a binding restriction for only one 

event, we can abstract from the optimal relationship between c2(¯x) and c2(x) given by the Kuhn-Tucker conditions 

of program 3 (in the appendix) and draw the optimal allocation in an intuitive 2-dimensional Edgeworth Box with 

coordinates ci(y), ci(x). 

15

[Figure 5 around here] 

The individually-rational allocations, in addition to belonging to the contract curve, must sat- 

isfy the individually-rational participation constraints, which, as noted before, are more restrictive 

than the ex-ante participation constraint for the merchant. The merchants’ individually rational 

constraint (6) takes into account the technological and financial ability of the merchant to under- 

take the project alone. Therefore, any individually rational allocation must provide him with at 

least the same utility he would get by financing the venture himself and taking all its profits and 

risks. Yet, the merchant will voluntarily exchange with the financier because financial contracting 

leads to mutually beneficial risk-sharing. 

The core of this simple economy will be thus characterized by the interval [BP1, BP2] on the 

contract curve, where the point BPi is the individually-rational optimal allocation when agent i has 

all the bargaining power (see figure 5). In other words, the lower bound of the interval to which U 2 

belongs remains U2(k2) but its upper bound is now given by the merchants’ individually-rational 

constraint (6). When the financier has all the bargaining power, the optimal event-consumption 

allocation BP2 lies on the indifference curve of the merchant providing him with his minimum 

individually rational expected utility E{U1[k1 − k + s]} (see point A and BP2 in figure 5). By 

construction of the contract curve, this merchant’s indifference curve intersects at BP2 with the 

financier’s indifference curve assigning him the individually-rational maximum value of U 2. 

From inspection of figure 5, it follows that the optimal allocation BP1 reached when the 

merchant exercises all the bargaining poweris such that that c ∗ 2 (y) > k2 − k + y (see appendix C). 

Also, the optimal allocation BP2 reached when the financier exercises all the bargaining power 

is such that c ∗ 2 (y) < k2. Thus, any allocation in the core satisfies lemma 1, which is proved in 

appendix C. 

Lemma 1 k2 − k + y < c ∗ 2 (y) = k2 − (1 − φ ∗ ) k + τ ∗ (y) < k2 

Yet, there is a nominal indeterminacy. 13 The optimal allocation of risk when the investment 

13 This indeterminacy derives from possibility to make transfers both at date 0 and 1. What matters, though, 

16

project is highly costly and risky can also be attained through, for example, letting the merchant 

fund the venture in its entirety and contracting an insurance policy against the marine risk or 

through a combination of a risk-free bill of exchange and premium insurance. Under the assump- 

tion that the agent’s personal wealth is contractible at a higher enforcement cost than the project’s 

returns, though, a capital structure with merchant’s equity holdings and limited liability debt-like 

sea loans and/or equity-like commenda contracts uniquely sustains the optimal risk allocation. 

Otherwise, the irrelevance theorem of Modigliani and Miller applies. 

When the venture is less costly and risky, the nominal indeterminacy remains but, regardless 

of the contract used to fund the project, an insurance contract with a coverage payment is needed 

to attain the optimal risk allocation. 

3 The evidence 

3.1 Two-side Risk Aversion 

Since the “merchant class” and the “financier class,” to the extent that such classes existed, over- 

lapped considerably, it would be misleading to assume different risk preferences for the merchant 

and the financier. Examples of individuals who acted as a merchant in one sea loan or com- 

menda contract and as financier in another sea loan or commenda contract abound (for Venice, 

see González de Lara, 2005, p. 21; for Marseilles, see Pryor, 1983, p. 180). According to de 

Roover, F.E. (1945, p.177) the same individuals who took insurance policies were generally the 

underwriters of other policies, for diversification was “the key note of the business policy followed 

by international merchant-bankers and the lesser merchants alike” (see also Tenenti, 1991, p.674). 

Propertyless merchants and small investors, though, could not diversify. The extant documen- 

tation indicates broad participation in long-distance trade by them. For example, in the cartulary 

of the Marseilles notary Giraud Amalric which spans the months March to July 1248, over 65 

percent of the financiers invested only once, very few being involved in more than three or four 

is no so much the particular security structure but the linear space it involves. Any linear combination of various 

points in the Edgeworth box that span the optimal event consumption allocation can be thought of as optimal 

contracts. 

17

commenda contracts. Furthermore, these investments were not always well-diversified: a Bérenger 

Eguezier sent six commendae to Acre on the Saint Espirit and two to Valencia on the Léopard 

(Pryor, 198?) 14 According to Greif (1993 and 1994), the reputation mechanism prevailing among 

the Genoese hindered the extent to which they could diversify. Although trading investments in 

the cartulary of John the Scribe for the years 1155-1164 are concentrated on the hands of a few 

noble families(10 percent of the financiers invested over 70 percent of the total amount registered), 

a financier used, on average only 1.57 merchants. 

- Sometimes, the financiers did diversify but even then they were soused to aggregate risk. For 

example, the Venetians voyaged on convoy for safety. Yet, the convoy presented a concentrated 

target. If the convoy were captured, the loss would affect a significant part of all the Venetian 

merchants. For example, in 1264 the Genoese, at war with the Venetians, lured away the escorting 

galleys of the yearly convoy to the crusaders’ states and caught the unprotected Venetian convoy 

at sea. Although the Venetians defied the Genoese by withdrawing to the very large round ship 

on the convoy, the other vessels and most of their cargos were lost, as well as a whole year’s trade 

with Cyprus, Palestine, Syria and Egypt. 

- Also, marine insurance remained in the hands of private underwriters, operating alone or 

with partners but with very limited capital, until the end of the eighteenth century when the first 

joint-stock companies emerged. 

3.2 Enforcement Costs 

In the light of the incomplete contracting literature (see Hart, 1995) one would expect, as observed, 

that optimal contracts ensure that, whatever happens, each side has some protection against 

opportunistic behavior by the other party. Because both the merchant and his potential financier- 

insurer can default on their promised future payments, the optimal contract will minimize those 

payments, while guaranteeing the optimal allocation of risk. This is achieved by making both 

14 Since Amalric practiced in the commercial center of Marseilles and acts were commonly receive by the investor’s 

notary, this information is quite representative. 

18

parties commit part of their wealth to the funding of the venture, φ ∗ ∈ (0, 1), and financing the 

rest through sea loan and commenda contracts. In this respect, it is meaningful that insurance 

premiums were agreed to be paid after the conclusion of the voyage. Not only could the merchant 

pay the premiums from the venture’s proceeds, but he also received some “power” against the 

possibility that the underwriters defaulted on the coverage payments. However, as the insured 

not seldom delayed paying the premiums due on their policies long after their ships were safely 

returned, so the insurers were dilatory about payment in case of loss (Tenenti, 1991, p.675). 

Furthermore, one can reasonable argue that enforcing payments from the parties’ initial wealth, 

k1 and k2, if any, was much more arduous than from the venture’s returns, for example, because 

the underwriter might have actually failed in the intervening time from the signing of the insurance 

contract to the payment of the coverage or because the absence of an effective registration system 

enabled defaulters to hide their real property, particularly if abroad. In Venice as elsewhere, 

marine insurance remained in the hands of private underwriters, operating alone or with partners 

but with very limited capital, until the end of the eighteenth century. They were commonly 

involved in all kind of trades and it was all but rare that they were unable or unwilling to meet 

the coverage payments, as indicated by the rulings of the Venetian magistracy on insurance, who 

found that “there was no money that more gladly is touched and more difficultly is paid than that 

of the maritime insurance” (Tenenti, 1991, p. 677; see also De Roover, 1945, p. 197). 

Since merchants could not use collateral to fulfill their obligations without incurring in a 

high enforcement cost, risk-free debt and more generally any contract with τ(y) > y ≥ 0 was 

hardly used. Similarly, because it was costly to identify and confiscate agents’ future wealth 

at date 1, pure insurance contracts with τ(y) < 0 did not develop until the changing character 

of long-distance trade made them the only means through which the optimal allocation of risk 

could be achieved. In sum,the model predicts uniqueness in the optimal use of the sea loan 

and the commenda (one or the other depending on the information structure) under the realistic 

assumption that the (minimum) venture’s returns are contractible at a lower cost than the agents’s 

19

initial wealth. 

TO DO: 

1. Review literature on the institutions for contract enforcement that motivated merchants to 

refrain from embezzling the investor capital and flee to avoid sanctions 

2. Evidence on the observability/verifiability of losses at sea or at the hands of enemy men 

3. Evidence on the changing information structure 

4. Evidence on the institutional arrangements that reduced the cost of enforcing payments 

from the venture’s returns 

3.3 Predictions 

1. For the sea loan or the commenda to be optimal, the inequality (11) must be satisfied. 

This implies 

(a) ˜t very high 

max{E[c1(x)] − c1(y)} = E[x] − ˜t ≪ ˜t − y = min{c2(x) − c2(y)} 

Evidence: In the twelfth century, sea loans typically charged yearly interest rates above 

33 percent. Charges of 40 or 50 percent were not uncommon for voyages from Italy or 

Constantinople to Alexandria or Syria (de Roover, 1963, p.53, and González de Lara, 

2005). Likewise, commenda contracts—which provided exactly the same downside 

insurance than the sea loan, τ(y) = y— customarily remunerate capital with three 

fourths of the commercial profit. (meaning the return minus the capital invested). 

(b) E[x] − ˜t 

Evidence: According to de Roover (1963, p.54), the high rates charged in sea loans 

”absorbed most of the merchant’s trading profit.” 

2. The severe drop on insurance rates lends empirical support to the theoretical explanation 

for the selection of alternative contracts. The model predicts the use of the sea loan (and the 

20

commenda) because the high risk and cost initially involved in sea ventures made insurance 

too expensive, thereby leading the merchant to buy as little insurance as he could, τ(y) = y ≥ 

0. Likewise, the model predicts the use of premium insurance, with τ(y) < 0, in response to 

a fall in the insurance cost. In real truth, underwriters during the fifteenth century charged 

much lower premiums than their counterparts from the twelfth and thirteenth centuries used 

to obtained through sea loan and commenda contracts. The insurance rates varied widely 

according to the type of vessel, the port of call, the state of war or peace, the season of the 

year, and other circumstances, but insurance policies remained consistently cheaper than 

the insurance rates paid above the safe interest on sea loans. As already mentioned, rates of 

return on sea loans, repayable upon safe arrival of the ship, varied from 25 to 50 percent for 

the duration of the voyage, while yearly interest rates on risk-free loans— although high— 

only amounted to 20 percent, thereby implying an insurance cost from around 15 to 40 

percent on sea loans. The premium paid during the fifteenth century for the safest voyages 

amounted to a mere 1-1,5 percent and raised up to 20 percent for exceptionally risky ventures 

(see, for example, De Roover, 1945, and Tenenti, 1991). 

3. A decrease in the cost and/or risk of trading ventures would causes the transition from the 

sea loan/the commenda to premium insurance 

Evidence 1: According to Lopez, the sea loan (and the commenda) lost their popularity 

when “commerce lost much of its adventurous and almost heroic features” (py lower and 

y higher) and “tended to become a routine” (E[x] = px x + p¯x ¯x higher), 15 and wealth 

accumulation from commerce reduced the former scarcity of capital (k2 higher, at least with 

respect to k). 16 

15 The quotations are from the great historian of the Commercial Revolution Lopez (1976, p. 97), who also noted 

that “high risks and high profits were predominant in the early stages of the Commercial Revolution; they were 

instrumental in forming the first accumulations of capital.” 

16 That capital became relatively less scarce is indicated by the sharp drop in interest rates. During the twelfth 

century Venice’s “common use of our land” established a 20 percent yearly interest on risk-free loans, subjected to 

double penalty if payment was delayed and secured by a general lien on the debtor’s property (González de Lara, 

2004a). By the mid fourteenth century average interest rates on commercial loans within Venice had declined to 5 

percent (Luzzatto, 1943, pp. 76-79) and the 1482’s new issue of government bonds promised to pay 5 percent yearly 

interest on perpetuity (Mueller, 1997, p. 420). 

21

Evidence 2: According to Lane (1973), the Nautical Revolution (1290-1350) reduced the 

cost of trading (k lower) 

Evidence 3: The Black Death increased per-capita income (k2 higher) 

Evidence 4: Interest rates on bank deposits and government bonds decreased dramatically 

(from 20-25 percent in the 12th century to about 2 percent in the 14th century). This 

suggests an increase in the availability of capital (k2 higher) 

Evidence 5: Byrne, de Roover, Lane, and Lopez agrees on the decrease of the marine risk 

circa the 14th century. Munro (1991), however, argues that the widespread warfare and 

disruptions of trade around 1300, which raised the level of risk, may have also contributed 

to the emergence of marine insurance. 

Evidence 6: Evolution E[x] = p¯x ¯x + px x. There is consensus that ¯x began to decrease by 

the mid or late 13th century, but it seems that p¯x, so that the aggregate effect is unclear. 

According to Lopez (p.97): ”Later, competition forced down the average profit, but greater 

security and the widening of the market enabled careful managers to increase their assets 

more steadily than ever before.” 

4 Conclusions 

This paper combines a mechanism-design model and historical evidence to examine the evolution of 

financial contacts for overseas trade from the eleventh to the fifteenth century. Initially, merchants 

funded part of their ventures but raised additional capital through limited liability sea loans 

and commenda contracts. By the early fifteenth century, however merchants typically funded 

their ventures through risk-free bills of exchange and get insurance through premium policies. 

This changing financial structure can be rationalized as being determined to achieve the optimal 

allocation of risk while minimizing enforcement costs. The extent to which these factors are 

relevant today is indicated by the generality of the model’s assumptions and results. 

First, two-side risk aversion. Historically, significant indivisibilities and aggregate risk limited 

22

the agents’ ability to diversify, preventing them from effectively becoming risk neutral. Today, 

most companies have a large share-holder who is not well-diversified (Zingales, 2000, p. 1628). 

Risk-aversion, because of limited risk-pooling, is typically assumed when dealing with the myriad 

of young and small firms who do not have access to public markets, such as individual propri- 

etorships, partnerships and closely-held corporations. Even when the financial capital is held by 

well-diversified investors, large widely-held corporations might still act in a risk-averse manner 

because the human capital invested in the firm is not well-diversified (Zingales, 2000); the firm’s 

managers faces some kind of bankruptcy costs (Greenwald and Stiglitz, 1990); managerial com- 

pensation is aligned with shareholders wealth maximization (Smith and Stulz, 1985); or there exist 

proprietary information (De Marzo and Duffie, 1991). Furthermore, although various works on 

Transaction Costs Economics have recently criticized risk-sharing arguments on account of their 

lack of empirical significance (mainly for labor contracts), Ackerberg and Botticini (2002) shows 

that risk aversion appears to influence contract choice when controlling for endogenous matching. 

Second, limited contract enforceability. Historically, enforcing payments from the agents’ 

private wealth was more costly than from the venture’s returns since, on the one hand, neither 

effective registration systems nor joint-stock insurance companies had yet developed and, on the 

other hand, the State fairly controlled sea ventures. Today, enforcement costs arguably cause 

banks’ reluctance to go through bankruptcy procedures for recovering loans from the debtors’ 

personal wealth (collateral). Similarly, an insurance company runs administrative cost greater 

than those of a capital market where bonds and equity are traded. It can therefore be reasonably 

argued that credit and insurance are optimally provided through the capital structure, rather than 

separately through risk-free debt from a bank and premium policies from an insurance company, 

because of enforcement costs. Thus, the model complements the optimal portfolio selection theory, 

which explains the holding of equity by both entrepreneurs and outsiders in terms of managerial 

risk aversion but ignores other contracts through which an outsider can potentially provide the 

same risk allocation. 

23

Yet, the model ignores other important forces shaping contracts. It is therefore complementary 

to other capital-structure theories based on agency costs, multiple margins for moral hazard, 

adverse selection, signaling, transactions costs, and/or incomplete contracting. 

A Proof of proposition 1 

Let parameter values ˆ λ = ( ˆ θ, ˆ k1, ˆ ˆp¯x 

k, ˆy, ˆpy, ˆx, ˆ¯x, ˆpx+ˆp¯x ) with ˆ θ = 0 and ˆ k1 = 0 be such that the sea 

loan contract does not sustain program 2’s optimal allocation of consumption, (ĉ2(y), ĉ2(x)). Then, 

ĉ2(y) < ˆw(y), as draw in figure 1for CC ′ , and necessary and sufficient Kuhn-Tucker conditions 

lead to 

∂£(η,µy,µx,c2(y),c2(x);λ) 

∂η = − U 2 + E [U2[c2(s)]] = 0 

∂£(η,µy,µx,c2(y),c2(x);λ) 

∂c2(y) = −pyU ′ 1[w(y) − c2(y)] + η pyU ′ 2[c2(y)] − µy = 0 

∂£(η,µy,µx,c2(y),c2(x);λ) 

∂c2(x) = − pxU ′ 1[w(x) − c2(x)] + p¯xU ′ 1[w(¯x) − c2(x)] 

+η(1 − py)U ′ 2[c2(x)] − µx = 0 

evaluated at ˆη, ˆµ(y) = ˆµ(x) = 0, ĉ2(y), ĉ2(x), and ˆ λ, where η, µy, and µx are the Lagrangian 

multipliers associated with restrictions (9) and (10), respectively, and £(.) is the Lagrange function 

associated with Program 2. 17 

Let λ varies in an open neighborhood of ˆ λ such that the pattern of binding and slack constraints 

of program 2 does not change. Totally differentiating (12) and applying the Implicit Function 

Theorem, one can calculate the comparative statics effects of each parameter value λi on c2(y) 

at a solution point (ĉ2(y), ĉ2(x)) and parameter values ˆ λ with ˆη > 0, ˆµ(y) = ˆµ(x) = 0, and 

dµy = dµx = 0: 

where 

 

 

ˆ 

 

 

 

 

Hλi = 

 

 

 

 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

(∂η)2 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(y)∂η 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(x)∂η 

 

 

dc2(y) 

= − 

dλi 

ˆ 

 

Hλi 

 

 

ˆ , 

 

H 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂η∂λi 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(y)∂λi 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(x)∂λi 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂η∂c2(x) 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(y)∂c2(x) 

∂2 

 

 

 

 

 

 

 

£(η,µy,µx,c2(y),c2(x);λ) 

 

(∂c2(x))2 

and the (binding-constraints) border hessian ˆ H evaluated at ˆ λ and (ĉ2(y), ĉ2(x)) is negative 

 

 

definite: ˆ 

 

H 

> 0. 

17 This assumes that for parameter values ˆ λ the solution of program 2 is such that ĉ2(x) < ˆw(x). The case for 

which ĉ2(x) = ˆw(x) can be easily derived by adding restriction ∂£(η,µy,µx,c2(y),c2(x);λ) 

∂µx 

24 

= 0 to (12). 

(12)

It is proved below that at a solution point (ĉ2(y), ĉ2(x)), 

dc2(y) 

dx 

< 0, dc2(y) 

d¯x 

< 0, 

dc2(y) 

d(k2−k) > 0 but dc2(y) < dw(y) = d(k2 − k) 

dc2(y) 

dy > 0 but dc2(y) < dw(y) = dy 

dc2(y) 

dpy 

dc2(y) 

d p¯x 

px+p¯x 

> 0 and dw(y) = 0 

< 0 and dw(y) = 0. 

Therefore, beginning with parameter values ˆ λ for which the sea loan contract is not optimal 

(ĉ2(y) < ˆw(y)), a change in parameters such that the venture becomes more costly— d(k2 − k) < 

0— and/or risky— dy < 0, dpy > 0 and/or dE[x] < 0— leads the new solution to get closer and 

closer to the border c2(y) = w(y). At a point, the solution will reach the border and the sea loan 

will be optimal. 

However, this tricky point with c2(y) = w(y) and µy = 0 is on the boundary between two 

regions with different patterns of binding and slack constraints. As a result, deviations to the left 

side will have to be treated using the set of equations (12) with µy = µx = 0 and dµy = dµx = 

0 while totally differentiating, whereas those to the right side using a different set: equation 

∂£(η,µy,µx,c2(y),c2(x);λ) 

∂µy 

(13) 

= w(y) − c2(y) = 0 must be added to (12), which now hold for µy ≥ 0 = µx 

and dµy = 0 = dµx. If parameter values keep on changing so that the venture becomes even more 

risky and costly, the sea loan will keep on being optimal, because at a solution point (˜c2(y), ˜c2(x)) 

with ˜c2(y) = ˜w(y), and parameter values ˜ λ with ˜η > 0, ˜µ(y) ≥ 0, ˜µ(x) = 0, and dµx = 0, 

where 

 

 

˜ 

∂ 

 

 

 

 

 

Hλi = 

 

 

 

 

 

2 £(η,µy,µx,c2(y),c2(x);λ) 

(∂η) 2 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂µy∂η 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(y)∂η 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(x)∂η 

and 

 

 

dc2(y) 

= − 

dλi 

˜ 

 

Hλi 

 

 

˜ , 

 

H 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂η∂µy 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂η∂λi 

∂2 ∂ 

£(η,µy,µx,c2(y),c2(x);λ) 

∂η∂c2(x) 

2 £(η,µy,µx,c2(y),c2(x);λ) 

(∂µy) 2 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂µy∂λi 

∂2 ∂ 


∂µy∂c2(x) 

2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(y)∂µy 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(y)∂λi 

∂2 ∂ 


∂c2(y)∂c2(x) 

2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(x)∂µy 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

∂c2(x)∂λi 

∂2 £(η,µy,µx,c2(y),c2(x);λ) 

(∂c2(x)) 2 

 

 

 

 

 

 

 

, 

 

 

 

 

 

the relevant border hessian ˜ H evaluated at ˜ 

 

λ and (˜c2(y), ˜c2(x)) is negative definite— ˜ 

 

H 

> 0— 

dc2(y) 

dpy 

= dc2(y) 

dx 

= dc2(y) 

d¯x 

dc2(y) 

d(k2−k) > 0 but dc2(y) = dw(y) = d(k2 − k) 

dc2(y) 

dy > 0 but dc2(y) = dw(y) = dy 

= dc2(y) 

d p¯x 

px+p¯x 

= 0 and dw(y) = 0. 

It is worth noting that for parameters values such that µy > 0, the comparative statics ef- 

fects given by (14) holds for both parameters’s increases and decreases. Therefore, the sea loan 

contract is optimal for a robust set of parameters: let parameter values ˜ λ be such that the sea 

25 

(14)

loan is optimal, with ˜c2(y) = ˜w(y), then for parameter values in an open neighborhood of ˜ λ, 

˜c2(y) + dc2(y) = ˜w(y) + dw(y). 

 

 

Show that ˆ 

 

H 

> 0 evaluated at ˆ λ and (ĉ2(y), ĉ2(x)). 

 

 

ˆ 

 

 

0 py U 

 

H 

= 

 

 

 

 

′ 2 [c2(y)] (1 − py) U ′ 2 [c2(x)] 

py U ′ 2 [c2(y)] h22 0 

(1 − py) U ′ 2 [c2(x)] 

 

 

 

 

 

 

= 

 

 

0 h33 

because both h22 and h33 are negative. 

and 

= − [py U ′ 2 [c2(y)]] 2 h33 − [(1 − py) U ′ 2 [c2(x)]] 2 h22 > 0 

h22 = py U ′′ 

1 [w(y) − c2(y)] + η py U ′′ 

2 [c2(y)] = 

= − py R1[w(y) − c2(y)] U ′ 1[w(y) − c2(y)] − η py R2[c2(y)] U ′ 2[c2(y)] = 

= − py R1[w(y) − c2(y)] U ′ 1[w(y) − c2(y)] − py R2[c2(y)] U ′ 1[w(y) − c2(y)] = 

= − py [R1[w(y) − c2(y)] + R2[c2(y)]] U ′ 1[w(y) − c2(y)] < 0 (15) 

h33 = px U ′′ 

1 [w(x) − c2(x)] + p¯x U ′′ 

1 [w(¯x) − c2(x)] + η (1 − py)U ′′ 

2 [c2(x)] = 

= − R1[w(x) − c2(x)] U ′ 1[w(x) − c2(x)] − η (1 − py) R2[c2(x)] U ′ 2[c2(x)] = 

= − R1[w(x) − c2(x)] U ′ 1[w(x) − c2(x)] − R2[c2(x)] U ′ 1[w(x) − c2(x)] = 

= − [R1[w(x) − c2(x)] + R2[c2(x)]] U ′ 1[w(x) − c2(x)] < 0, (16) 

where the first equalities are simply the definitions of h22 and h33; the second equalities derive 

from applying the definition of absolute risk-aversion coefficient, R(z) = − U ′′ (z) 

U ′ (z) , and defining 

U ′′ 

1 [w(x) − c2(x)] 

R1[w(x) − c2(x)] = − 

U ′ 1 [w(x) − c2(x)] = − px U ′′ 

1 [w(x) − c2(x)] + p¯x U ′′ 

1 [w(¯x) − c2(x)] 

px U ′ 1 [w(x) − c2(x)] + p¯x U ′ 1 [w(¯x) − c2(x)] 

to make the notation more compact; the third equalities follow from, respectively, the second and 

third row in (12) with µy = µx = 0; the fourths, from simply operating; and the inequality, from 

U ′ (.) > 0 and U ′′ (.) < 0, so that Ri(.) > 0, for i = 1, 2. QED 

 

 

Note that ˆ 

 

H 

can be expressed as 

 

 

ˆ 

 

H 

= [py U ′ 2 [c2(y)]] 2 U ′ 1 [w(x) − c2(x)] R1[w(x) − c2(x)] + 

[py U ′ 2 [c2(y)]] 2 U ′ 1 [w(x) − c2(x)] R2[c2(x)] + 

[(1 − py) U ′ 2 [c2(x)]] 2 py U ′ 1 [w(y) − c2(y)] R1[w(y) − c2(y)] + 

[(1 − py) U ′ 2 [c2(x)]] 2 py U ′ 1 [w(y) − c2(y)] R2[c2(y)]. 

26 

(17) 

(18)

Show that dc2(y) 

 

 

d(k2−k) = − 

ˆ H (k2 −k) 

 

 

ˆ 

 

H 

 

 

 

∈ (0, 1) evaluated at ˆ λ and (ĉ2(y), ĉ2(x)). 

 

 

ˆ 

 

H (k2−k) = (1 − py) U ′ 2[c2(x)] 2 py U ′ 1[w(y) − c2(y)] R1[w(x) − c2(x)] − 

 

(1 − py) U ′ 2[c2(x)] 2 py U ′ 1[w(y) − c2(y)] R1[w(y) − c2(y)] < 0 (19) 

because U ′ (.) > 0, w(y) − c2(y) < w(x) − c2(x) < w(x) − c2(x), 18 and utility functions exhibit 

DARA, so that R1[w(y) − c2(y)] > R1[w(x) − c2(x)]. 

 

 

Comparing (19) with (18) , it follows that 0 < ˆ 

 

H 

+ ˆ 

 

H (k2−k) . QED 


dy 

 

 

= − 

 

ˆ 

 

H(y) 

ˆ 

 

H 

∈ (0, 1) evaluated at ˆ λ and (ĉ2(y), ĉ2(x)). 

 

 

ˆ 

 

H(y) = (1 − py) U ′ 2[c2(x)] 2 py U ′′ 

1 [w(y) − c2(y)] = 

= − (1 − py) U ′ 2[c2(x)] 2 py U ′ 1[w(y) − c2(y)] R1[w(y) − c2(y)] < 0 (20) 

 

 

Comparing (20) with (18), it follows that 0 < ˆ 

 

H 

+ ˆ 

 

H (k2−k) . QED 


dpy 

 

 

= − 

 

ˆ 

 

Hpy 

ˆ 

 

H 

> 0 evaluated at ˆ λ and (ĉ2(y), ĉ2(x)). 

 

 

ˆ 

 

Hpy = [U2[c2(y)] − U2[c2(x)]] py U ′ 2 [c2(y)] U ′ 1 [w(x) − c2(x)] [R1[w(x) − c2(x)] + R2[c2(x)]] 

− (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] U ′ 1 [w(x)−c2(x)] 

1−py 

because Ri(.) > 0, Ui ′(.) > 0, and c2(y) ≤ c2(x), so that [U2[c2(y)] − U2[c2(x)]] < 0. QED 


dx 

 

 

= − 

 

ˆ 

 

H(x) 

ˆ 

 

H 

< 0 and dc2(y) 

d¯x 

 

 

= − 

 

ˆ 

 

H(¯x) 

ˆ 

 

H 

< 0 

< 0 evaluated at ˆ λ and (ĉ2(y), ĉ2(x)). 

 

 

ˆ 

 

H(x) = − (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] px U ′′ 

1 [w(x) − c2(x)] > 0. 

 

 

H(¯x) = − (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] p¯x U ′′ 

1 [w(¯x) − c2(x)] > 0. 

These follow from U ′ (.) > 0 and U ′′ (.) < 0. QED 


d p¯x 

 

 

 

 

= − 

px+p¯x 

ˆ 

 

H p¯x 

px+p¯x 

 

 

ˆ 

< 0 evaluated at 

H 

ˆ λ and (ĉ2(y), ĉ2(x)). 

 

 

 

ˆ H p¯x 

 

 

 

px+p¯x 

= (1 − py) U ′ 2 [c2(x)] py U ′ 2 [c2(y)] [U ′ 1 [w(x) − c2(x)] − U ′ 1 [w(¯x) − c2(x)]] > 0 

18 The optimal sharing rule gives some insurance to both parties, for both are risk-averse. Appendix B proves this 

result under full information. The proof under hidden information runs similarly. 

27

ecause U ′ (.) > 0, U ′′ (.) < 0 and w(x) < w(¯x), so that U ′ 1 [w(x) − c2(x)] − U ′ 1 [w(¯x) − c2(x)] > 0. 

QED 

 

 

Show that ˜ 

 

H 

> 0 evaluated at ˜ λ and (˜c2(y), ˜c2(x)). 

 

 

˜ 

 

0 0 py U 

 

 

 

 

H 

= 

 

 

 

 

 

′ 2 [c2(y)] (1 − py) U ′ 2 [c2(x)] 

0 0 − 1 0 

py U ′ 2 [c2(y)] − 1 h22 0 

(1 − py) U ′ 2 [c2(x)] 

 

 

 

 

 

 

 

= 

 

 

 

 

0 0 h33 

(1 − py) U ′ 2[c2(x)] 2 > 0, 

where h22 and h33 are defined by (15) and (16), respectively. QED 


 

 

 

d(k2−k) = − 

˜ 

 

H (k2−k) 

 

 

˜ 

 

H 

= 1 evaluated at ˜ λ and (˜c2(y), ˜c2(x)). 

 

 

˜ 

 

H (k2−k) = − [(1 − py) U ′ 2 [c2(x)]] 2 < 0. QED 


 

 

 

dy = − 

˜ 

 

H(y) 

 

 

˜ 

 

H 

= 1 evaluated at ˜ λ and (˜c2(y), ˜c2(x)). 

 

 

˜ 

 

H(y) 


dpy 

= dc2(y) 

dx 

dc2(y) 

= d¯x 

 

 

B Proof of proposition 2 

= − [(1 − py) U ′ 2 [c2(x)]] 2 < 0. QED 

dc2(y) 

= 

d p¯x = 0 evaluated at 

px+p¯x 

˜ λ and (˜c2(y), ˜c2(x)). 

˜ 

 

Hpy = ˜ 

 

H(x) = ˜ 

 

H(¯x) = 

˜ 

 

H p¯x 

= 0. QED 

The commenda contract solves the relevant optimization problem under full information with 

k1 = 0 

Program 3: 

max 

[c2(s)] s∈S 

px+p¯x 

E [U1[w(s) − c2(s)]] 

s.t. E [U2[c2(s)]] = U 2 (21) 

w(s) − c2(s) ≥ 0 ∀s (22) 

c2(s) ≥ 0 ∀s. (23) 

Step 1. Show that all individually-rational efficient allocations are characterized by 

c2(y) < c2(x) < c2(¯x) 

Let η and µs be the Kuhn-Tucker multipliers of restrictions (21) and (22) for each s, respec- 

tively. Define the Lagrangian function for program 3 as 

£(c2(s), η, µs) = E [U1[w(s) − c2(s)]] + η(− U 2 + E [U2[c2(s)]]) + 

µs[w(s) − c2(s)]. 

28 

s

Program 3 is concave so that Kuhn-Tucker conditions are necessary and sufficient for optimality: 

∂£(.) 

∂c2(s) = −psU ′ 1 [w(s) − c2(s)] + η psU ′ 2 [c2(s)] − µs ≤ 0 c2(s) ≥ 0 c2(s) ∂£(.) 

∂c2(s) = 0 ∀s 

∂£(.) 

∂η = − U 2 + E [U2[c2(s)]] ≥ 0 η ≥ 0 η ∂£(.) 

∂η = 0 

∂£(.) 

∂µs = w(s) − c2(s) ≥ 0 µs ≥ 0 µs ∂£(.) 

= 0 ∀s. 

∂µs 

Let c2(s) > 0∀s, then complementary slackness in (24) ensures that ∂£(.) 

∂c2(s) = 0. Then, c2(y) < 

c2(x) because 

if µy = µ1(x) = 0 Ψ(c2(y), c2(x); λ1) = U ′ 2 [c2(y)] 

if µy > 0 and µ1(x) = 0 Ψ(c2(y), c2(x); λ1) = U ′ 2 [c2(y)] 

if µy = 0 and µ1(x) > 0 c2(x) = w(x) > w(y) ≥ c2(y). 

Ψ(c2(y), c2(x); λ1) = U ′ 2 [c2(y)] 

U ′ 2 [c2(x)] − U ′ 1 [w(y)−c2(y)] 

U ′ 1 

conditions for µy ≥ 0 and µ1(x) = 0. From applying U ′ i 

U ′ 2 [c2(x)] − U ′ 1 [w(y)−c2(y)] 

U ′ 1 

U ′ 2 [c2(x)] − U ′ 1 [w(y)−c2(y)] 

U ′ 1 

[w(x)−c2(x)] = 0 

[w(x)−c2(x)] > 0 

(24) 

[w(x)−c2(x)] ≥ 0 derives from operating the Kuhn-Tucker 

(.) > 0 and U ′′ 

i (.) < 0 to this expression, 

it follows that optimal event-contingent consumption must satisfy c2(y) < c2(x). The equality 

c2(x) = w(x) in the third expression derives from complementary slackness in (24) with µ1(x) > 0; 

w(x) > w(y) by assumption; and w(y) ≥ c2(y) is restriction (22) for s = y. Likewise, it can be 

proved that c2(x) < c2(¯x). 

The case c2(y) = c2(x) = c2(¯x) = 0 lies in the contract curve, but does not satisfy the partic- 

ipation constraint of agent 2. The case c2(s) = 0 and c2(s ′ ) > 0 for s, s ′ ∈ S and s > s ′ leads to 

contradiction. 

Step 2. Show that all individually-rational efficient allocations are characterized by 

c2(y) = w(y) = k2 − k + y. 

Assume they are not and, then, let (k2 − k + ˆt y , k2 − k + ˆt x , k2 − k + ˆt ¯x ) be the solution of 

program 2, with ˆt y = y. From imposing non-negative consumption, it follows that ˆt y < y and it 

is possible to write 

ˆt x = a + d 

ˆt ¯x = ā + d 

with ā = 0 = a, E(a) = 0 and d = p¯x 

px+p¯x 

ˆt ¯x + px 

px+p¯x 

ˆt x . For notational purposes, let us define 

E [u1[τ(y), τ(x), τ(¯x)]] = E [U1[s − τ(s)]] and E [u2[τ(y), τ(x), τ(¯x)]] = E [U2[k2 − k + τ(s)]]. 

Let the event-contingent allocation (k2 −k +y, k2 −k + ˜t, k2 −k + ˜t) be the solution to program 

2. Then, for d = ˜t, it holds that if 

 

E u2[ˆt y 

, d, d] ≥ E u2[y, ˜t, ˜t] = U 2, 

29

E u1[ˆt y 

, d, d] < E u1[y, ˜t, ˜t] . 

Let also define (k2 −k +y, k2 −k + ˜t x , k2 −k + ˜t ¯x ) as the solution to a restricted version of program 

3 in the sense that c2(y) = k2 − k + y is imposed. If (k2 − k + ˆt y , k2 − k + ˆt x , k2 − k + ˆt ¯x ) is efficient, 

then, by definition, 

 

E u2[ˆt y 

, a + d, ā + d] = U 2 = E u2[y, ˜t, ˜t] 

(25) 

 

E u1[ˆt y 

, a + d, ā + d] > E u1[y, ˜t x , ˜t ¯x ] . (26) 

It can be proved that (26) leads to a contradiction and, therefore, the optimal allocation of 

consumption establishes c2(y) = k2 − k + y. Indeed, 

 

E u1[ˆt y 

, a + d, ā + d] < E u1[ˆt y 

, d, d] < E u1[y, ˜t, ˜t] < E u1[y, ˜t x , ˜t ¯x ] , (27) 

where the first inequality derives from the utility function being concave and E(a) = 0. The 

second one follows from (k2 − k + y, k2 − k + ˜t, k2 − k + ˜t) solving program 1 and from 

 

E u2[ˆt y 

, d, d] > E u2[ˆt y 

, a + d, ā + d] = U 2 = E u2[y, ˜t, ˜t] , 

where the inequality derives from the utility function being concave and E(a) = 0 and 

the equalities are imposed by (25). The last inequality of (27) holds because the allocation 

(k2 − k + y, k2 − k + ˜t, k2 − k + ˜t) could have been chosen—program 2 is a restricted version of 

program 3— but it was not. Therefore, it must give less expected utility to agent 1 than the 

optimally chosen allocation (k2 − k + y, k2 − k + ˜t x , k2 − k + ˜t ¯x ). 

Step 3. The individually-rational efficient allocations of consumption 

(k2 − k + y, k2 − k + ˜t x , k2 − k + ˜t ¯x ) 

can be sustained by a commenda contract establishing τ = −k, τ(y) = y, τ(x) = ˜t x , and τ(¯x) = ˜t ¯x . 

Just note that contracts are defined such that c2(s) = k2 + τ + τ(s). QED 

C Proof of proposition 3 and lemma 1 

The relevant problem when the merchant is initially endowed to fund the venture on his own 

(k1 > k) becomes a restricted version of program 2 (under hidden information)or program 3 

(under full information) in which restriction (6) is added. The proof is developed for the hidden 

information structure, but can be extended to full information easily. 

Step 1: Show that c ∗ 2 (y) > k2 − k + y. 

30

When the entrepreneur lacks the wealth to finance his project in its entirety (k1 = 0), subsec- 

tion 2.4, and proposition 1 in particular, states the conditions under which the optimal allocation 

is a binding solution with a positive Kuhn-Tucker multiplier µy associated with the restriction 

w(y) − c2(y) ≥ 0. This means that for the (binding) solution (˜c2(y), ˜c2(x)), 

Υ(˜c2(y), ϕ(˜c2(y), λ1); λ1) > 0, (28) 

where ˜c2(y) = k2−k+y and ˜c2(x) = k2−k+˜t, and λ1 represents parameter values representing 

a risky and costly venture, and θ = 0, k1 = 0. From (28) and utility functions exhibiting DARA, it 

follows that the allocation (˜c2(y), ˜c2(x)) is not optimal for parameters λ2 such that k1 > k either. 

Lemma 2 Υ(˜c2(y), ˜c2(x); λ2) > 0. 

Proof of Lemma 2: Υ(˜c2(y), ˜c2(x); λ1) > 0, and 

∂Υ(c2(y),c2(x);λ) 

∂k1 

= − 

= 

= 

U ′′ 

1 [w(y)−c2(y)] [p U ′ 1 [w(¯x)−c2(x)]+(1−p)U ′ 1 [w(x)−c2(x)]] 

[p U ′ 1 [w(¯x)−c2(x)]+(1−p)U ′ 1 [w(x)−c2(x)]] 2 + 

+ U ′ 1 [w(y)−c2(y)] [p U ′′ 

1 [w(¯x)−c2(x)]+(1−p)U ′′ 

1 [w(x)−c2(x)]] 

[p U ′ 1 [w(¯x)−c2(x)]+(1−p)U ′ 1 [w(x)−c2(x)]] 2 = 

R1[w(y)−c2(y)] U ′ 1 [w(y)−c2(y)] 

[p U ′ 1 [w(¯x)−c2(x)]+(1−p)U ′ R1[w(x)−c2(x)] U 

1 [w(x)−c2(x)]] − 

′ 1 [w(y)−c2(y)] 

[p U ′ 1 [w(¯x)−c2(x)]+(1−p)U ′ 1 [w(x)−c2(x)]] = 

U ′ 1 [w(y)−c2(y)] 

p U ′ 1 [w(¯x)−c2(x)]+(1−p)U ′ 1 [w(x)−c2(x)] [R1[w(y) − c2(y)] − R1[w(x) − c2(x)]] > 0, 

where the first equality derives from simple derivation; the second one from applying the defini- 

tion of absolute risk-aversion coefficient, R(z) = − U ′′ (z) 

U ′ (z) , and (17); the third equality derives from 

operating and the inequality from U ′ (.) > 0, U ′′ (.) < 0 and the utility function exhibiting DARA, 

with w(y)−c2(y) < w(x)−c2(x) < w(x)−c2(x), so that R1[w(y)−c2(y)] > R1[w(x)−c2(x)].QED 

Therefore, the optimal allocation is not (˜c2(y), ˜c2(x)) but (c ∗ 2 (y), c∗ 2 (x)) = (˜c2(y) + ɛ, ˜c2(x) − 

δ(ɛ)), where different values of ɛ define distinct points in the core. 

Lemma 3 There exists an ɛ > 0 and a δ(ɛ) > 0 such that all the restrictions are satisfied and 

Υ(˜c2(y) + ɛ, ˜c2(x) − δ(ɛ); λ2) = 0. 

Proof of Lemma 3: The result follows from applying the sign of the following derivatives to 

lemma 2. ∂Υ(c2(y),c2(x);λ) 

∂c2(y) 

< 0 and ∂Υ(c2(y),c2(x);λ) 

∂c2(x) 

> 0. The proof is straightforward. Just find the 

derivatives and note that U ′ (.) > 0 and U ′′ (.) < 0. QED 

Lemma 4 The contract curve for interior points, defined by ϕ(c2(y), λ) for parameters λ2, such 

that k1 > k > 0, lies below the contract curve for λ1, with k1 = 0: ϕ(c2(y), λ2) < ϕ(c2(y), λ1) ∀c2(y). 

31

Proof of Lemma 4: The Implicit Function Theorem also gives the first-order comparative 

statics of any parameter on ϕ(c2(y)), for any c2(y) at a solution. In particular, 

d ϕ(c2(y), λ) 

d k1 

= − 

∂Υ(c2(y),c2(x);λ) 

∂k1 

∂Υ(c2(y),c2(x);λ) 

∂c2(x) 

< 0 ∀c2(y) in the restricted set, (29) 

since U ′ (.) > 0, U ′′ (.) < 0 and the utility functions exhibit decreasing absolute risk-aversion 

(DARA). QED 

Therefore, the optimal allocation establishes c ∗ 2 (y) = k2 − k + y + ɛ with ɛ > 0. 

Step 2: Show that c∗ 2 (y) < k2. 

This follows from imposing the optimal sharing rule c∗ 2 (y) < c∗2 (x) to the ex-ante participation 

constraint of agent 1 holding with equality and noting that all individually rational allocations of 

agent 2’s consumption are upper-bounded by the ex- ante participation constraint of agent 1. 

Let c1(y) = w(y) − c2(y) ≤ k1−k+y, and, consequently, c2(y) ≥ (k1+k2−k+y)−k1+k−y = k2. 

Then, for E [U1[w(s) − c2(s)]] = E [U1[k1 − k + s]] to be satisfied, c1(¯x) = w(¯x) − c2(x) > 

k1 − k + ¯x. These, in turn, imply that c2(x) ≤ k2, which is in contradiction with the optimal 

sharing rule, c ∗ 2 (y) < c∗ 2 (x). 

References 

[1] Ackerberg D.A. and M. Botticini. “Endogenous Matching and the Empirical Determinants 

of Contract Form.” Journal of Political Economy, 110.3 (2002): 564-591. 

[2] Bensa, E. “Il Contratto di Asicurazione nel Medio Evo. Studi e Ricerche.” Genova: Tipografia 

Marittima Edittrice, 1884. 

[3] Bernstein, P.L. Against the Gods. The Remarkable Story of Risk. New York: John Wiley & 

Sons, 1996. 

[4] Byrne, E. H. “Commercial Contracts of the Genoese in the Syrian Trade of the Twelfth 

Century”. The Quarterly Journal of Economics, 31 (1917): 128-70. 

[5] De Roover, R. “The Organization of Trade” in Cambridge Economic History of Europe, Vol. 

3, 42-118. Cambridge: CUP, 1963. 

[6] De Roover, F.E. “Early Examples of Marine Insurance.” The Journal of Economic History, 

5 (1945): pp-172-200. 

[7] De Marzo, P.M. and D. Duffie. “Corporate Financial Hedging with Proprietary Information”, 

Journal of Economic Theory 53 (1991): 261-286. 

[8] Gale, D. and M, Hellwig. “Incentive-Compatible Debt Contracts: The One Period Problem.” 

Review of Economic Studies, 52 (1985): 647-663. 

[9] González de Lara, Y. “Institutions for Contract Enforcement and Risk-sharing: from Debt 

to Equity in Late Medieval Venice.” Working Paper Ente Einaudi, ??? 2004. 

32

[10] González de Lara, Y. “The Secret of Venetian Sucess: the Role of teh State in Financial 

Markest” Working Paper IVIE ???, 2005 

[11] Greif, A. “Reputation and Coalitions in Medieval Trade: Evidence on the Magribi Traders”. 

Journal of Economic History, 49 (1989): 857-82. 

[12] Greif, A. 1994b. ”On the Political Foundations of the Late Medieval Commercial Revolution: 

Genoa during the Twelfth and Thirteenth Centuries” The Journal of Economic History, 54.4, 

271-287. 

[13] Greif, A. Institutions: Theory and History. Cambridge: CUP, forthcoming. 

[14] Harris, M. and A. Raviv. “The Theory of Capital Structure.” The Journal of Finance, 46.1 

(1991): 297-355. 

[15] Hoover, C. B. “The Sea Loan in Genoa in the Twelfth Century.” Quarterly Journal of Economics, 

40 (1926): 495-529. 

[16] Krueger, H. C. “The Genoese Merchant in the Twelfth Century.” Journal of Economic History, 

22.2 (1963): 251-283. 

[17] Lacker, J. M. and J. A. Weinberg. “Optimal Contracts Under Costly Falsification”. Journal 

of Political Economy, 97.6 (1989): 1345-63. 

[18] Lane, F.C. Venice. A Maritime Repubic. Baltimore: The Johns Hopkins University Press, 

1973. 

[19] Lane, F.C. Venice and History: the Collected Papers of Frederic C. Lane. Baltimore: The 

John Hopkins Press, 1966. 

[20] Leland H.E. and D.H. Pyle. “Informational Asymmetries, Financial Structure, and Financial 

Intermediation.” The Journal of Finance, XXXII.2 (1977): 371-387. 

[21] Lopez, R. The Commercial Revolution of the Middle Ages, 959-1350. Cambridge: Cambridge 

University press, 1976. 

[22] Luzzatto, G. “La Commenda nella Vita Economica dei Secoli XIII e XIV, con particolare 

risguardo a Venezia” in Atti della Mostra Bibliografica e Convegno Internazionale di Studi 

Storici dell Diritto Maritimo Medioevale, I, 139-164. Amalfi, and Napoli, 1943. 

[23] Marimon, R. “Mercados e Instituciones para el Intercambio” in Marimon, R. y X. Calsamiglia 

(Ed.), Invitación a la Teoría Económica. Barcelona: Ariel, 1991. 

[24] Mueller, R.C. The Venetian Money Market: Banks, Panics, and the Public Debt, 1200-1500. 

Baltimore: The Johns Hopkins University Press, 1997. 

[25] Pryor, J.H. “Mediterranean Commerce in the Middle Ages: A voyage under commenda”. 

Viator, 14 (1983): 132-194. 

[26] Rajan, R.R. and L. Zingales. “What do We Know about Capital Structure? Some Evidence 

from International Data.” The Journal of Finance, L.5 (1995): 1421-1460. 

[27] Smith, C.W. and R.M. Stulz. “The Determinants of Firms’ Hedging Policies”, Journal of 

Finance and Quantitative Analysis, 20.4 (1985): 391-405. 

[28] Tenenti, A. “L’assicurazione marittima” in Cracco G. and G. Ortalli. Storia di Venezia, 

Vol IV. Le Istituzioni, pp. 663-686. Roma: Istituto della Enciclopedia Italiana fondata da 

Giovanni Treccani, 1991. 

33

[29] Townsend, R. M. “Optimal Contracts and Competitive Markets with Costly State Verification”. 

Journal of Economic Theory, 21.2 (1979): 265-293. 

[30] Townsend, R. M. “Optimal Multiperiod Contracts and the Gain from Enduring Relationships 

under Private Information.” Journal of Political Economy, 90.6 (1982): 1116-86. 

[31] Williamson, O.E. “The New Institutional Economics: Taking Stock, Looking Ahead.” Journal 

of Economic Literature, 38 (2000): 595-613. 

[32] Zingales, L. “In Search of New Foundations.” The Journal of Finance, LV.4 (2000): 1623- 

1653. 

34

FIGURE 1 

(Hidden Information, k1 = 0, Two-side Risk Aversion ) 

c2(x) 

✻ 

✛ 

✛ ❆ 

✲ 

❄ 

❆ 

❚ 

❩❩ 

❩ 

❚ 

✉ 

❚ sea loan: τ(y) = y 

❅❚ ❏❏ 

❅❅ 

❏ 

❏ 

❭ 

❏ 

❭ 

❏ 

❭ 

❭ 

❭ 

❭ 

❧ 

❧ 

❧ 

❩ 

❩ 

❩P 

C2(12th − 13th) 

✡ ✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡✡ 

CC 

✡ 

✡ 

✡ 

✡ ✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ ✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ ✡ 

✡ 

✡ 

✡✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ ✡ 

✡ 

✡ 

✡✡ 

❏❏ 

❏ 

❏ 

❏ 

❏ 

❡ 

❡ 

❡ 

❧ 

❧ 

❧ 

❧ 

❩ 

❩❩ 

❍ 

❍ 

❍❛ 

❛ 

❛P C2(15 th ❩ 

τ(y) < y 

❅ 

❏ 

❆ 

❆ 

❆ 

) 

✉ 

CC ′ 

✔ ✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ 

✔ ✔ 

✔ 

✔ 

✔ ✔ 

✔ 

✔ 

✔ ✔ 

✔ 

✔ 

✔ ✔ 

Agent 1(the merchant) 

Risk-averse 

(0,0) if x =¯x 

c1(y) 

(0,0) if x =x 

c1(y) k2 − k + x 

✔ 

✔ 

 

c2(y) 

Agent 2 

k2 − k + y k2 

(0,0) 

(the financier) 

c1(x) 

Risk-averse 

//////////////////////// 

//////////////////////// 

//////////////////////// 

//////////////////////// 

//////////////////////// Noncontractible area 

//////////////////////// 

//////////////////////// 

//////////////////////// 

//////////////////////// 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

A 

35

FIGURE 4 

(Hidden Information, k1 = 0, Two-side Risk Aversion, 

c2(x) 

Higly Costly and Risky Venture) 

✻ 

✛ 

✛ 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ ♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ 

♣ ✲ 

❄ 

A 

❆ 

❆ 

❚ 

❚ 

❚ ❏❏ 

❏ 

❏ 

❏ 

❏ 

❭ 

❭ 

❭ 

❧ 

❧ 

❧ 

❩ 

❩ 

❩P 

C2 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ ✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

CCλk1 =0 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ ✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡ 

✡✡ 

✡ ✡✡ 

✡ ✡✡ 

✡ ✡✡ 

✡ ✡ ✡ 

✡ ✡✡ 

✡ ✡✡✡ 

✡ ✡✡ 

✡ ✡✡ 

Agent 1(the merchant) 

(0,0) if x =¯x 

c1(y) 

(0,0) if x =x 

c1(y) 

c2(y) 

Agent 2 (0,0) 

k2 − k + y 


c1(x) 

 

k2 − k + t 

 

k2 

cc 

k2 − k + t pc 

k2 − k + x 

sea loan: τ = − k 

τ(y) = y 

τ(x) = τ(¯x) = τ(x) = ˜t ≥ t pc 

//////////////////////// 

//////////////////////// 

//////////////////////// 

//////////////////////// 

Noncontractible area 

//////////////////////// 

//////////////////////// 

//////////////////////// 

//////////////////////// 

//////////////////////// 

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ✉ 

♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 

36

c1(y) ✛ 

FIGURE 5 

(k1 > k, Two-side Risk Aversion, Higly Costly and Risky Venture) 

k2 − k + x 

c2(x) 

Agent 2 (0,0) 


✻ 

k1 

k1 − k + y 

 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

BP2 

. 

✓ ✓✓ 

✓ ✓✓ 

✓ ✓✓ 

✓ ✓✓ 

✓ ✓✓✓✓ 

CCλk1 >k 

✓ ✓ 

✓ ✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ ✓✓ 

✓ ✓ 

✓ ✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ ✓✓ 

✓ ✓ 

✓ ✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ ✓✓ 

✓ ✓ 

✓ ✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ ✓✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

❝❝ 

❧ 

❧ 

❅ 

❅ 

❅ 

❙ 

❙ 

❙❙ 

❙ 

❚ 

❙ 

Ex-ante P C1 

❚ 

❙ 

❏ 

❧ ❏ 

❧ ❏ 

❧ 

❝ ❏ 

❝ ❆ 

❝ 

. . . . . . . . . . . . k1 

. 

BP1 

. 

. 

k2 

. . . . . . . . . . . . A. 

. . . . . . k1 − k + x 

. 

✡ 

. 

✡✡ 

✡ ✡✡ 

CCλk1 =0 

✡ ✡ 

✡ 

✡ 

✡✡ 

✡ ✡✡ 

✡ ✡✡ 

✓ ✓✓ 

✓ ✓✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

❩ ❆ 

❩❩◗◗◗❍❍Ex-ante 

✓ ❆ Individual Rationality 

❆ (k1 > k) 

❆ 

❇ P C2 ≡ 

❇❇ Individual Rationality 

✡ ✡✡ 

✡ ✡✡✡ 

✡ ✡✡ 

✓ ✓✓✓ 

✓ ✓✓✓ 

✓ ✓✓✓ 

✓ ✓✓✓ 

✓ ✓✓✓ 

✡ ✡✡ ✡ ✡ 

✡ ✡✡ 

✓ ✓✓ ✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

✓ 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

. 

(0,0) 

. ✲ 

❄ 

❄ 

k2 − k + y k2 k1 + k2 − k + y 

k2 − k + y < c ∗ 2(y) = k2 − k + y + ɛ < k2 

37 

Agent 1 

(the merchant) 

c2(y)

The Birth of Insurance Contracts - The Ataturk Institute for Modern ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?