The Sustainability of Empire in a Global Perspective: the
Role of International Trade Patterns
Roberto Bonfatti∗
University of Nottingham
May 7, 2016
Abstract
I construct a model in which a colony trades raw materials for manufactures with the
mother country and the rest of the world, and can rebel at the cost of some trade disruption
with the mother country. Decolonisation is more likely when the rest of the world is scarcer
in raw materials, because its trade policy is more favourable to a rebel colony, and the
cost of trade restrictions under empire is higher. This has implications for the incentives
of colonisers to prohibit industrialisation in colonies. I use my results to explain some
important facts in the history of colonialism.
JEL Codes: D74, F1, N4
Keywords: Colonial trade, rise and fall of empires, economic legacy of colonialism.
1
Introduction
When France lost land-abundant Canada and Louisiana in the Seven Years’ War (1763),1 her
remaining North American empire became scarcer in food. As a consequence, French trade policy
became more open to foreign producers of food, such as the colonial USA. A few years later, in
1776, a coalition of US colonies found that the time was right to rebel against Britain. From a
trade perspective, they were right: although rebellion brought upon them the full cost of British
∗
An earlier version was circulated under the title: “Decolonisation: the Role of Changing World Factor
Endowments”. I am indebted to Tim Besley and Robin Burgess for their help and assistance. I also thank
Toke Aidt, Oriana Bandiera, Robert Bates, Dave Donaldson, Maitreesh Ghatak, Giovanni Facchini, Giammario
Impullitti, Guy Michaels, Gerard Padro-i-Miquel, Torsten Persson, Steve Redding and Fabrizio Zilibotti for
useful corrections and comments, as well as seminar and conference participants at LSE, Oxford, Cambridge,
Paris School of Economics, IAE Barcelona, WZB Berlin, IMT Lucca, University of Padua, Cattolica University,
the University of Edinburgh, and CESIfo Dresden. Research support from LSE, St John’s College Oxford, and
the University of Nottingham is gratefully acknowledged. All remaining errors are mine.
1
France regained Louisiana in 1800, only to sell it to the (by then independent) USA in 1803.
1
sanctions - which crippled their exports to the British Empire - buoyant exports to the French
Empire helped them recover soon after the revolution. These facts suggest that the change
of endowments of 1763, and the resulting change in trade policy, created a favourable global
environment for the American Revolution. Given the importance of trade for the revolutionaries,
this may well help to explain the timing of the revolution.
In this paper, I set up a model that clarifies the forces linking endowments, trade policy,
and colonial rebellion. A colony (C) trades “raw materials” for “manufactures” with the mother
country (M ) and a foreign country (F ). The economic fundamentals that shape the pattern of
trade are the relative abundance of raw materials in F versus M : if F is relatively abundant, it
is a competitor of C in selling raw materials to M ; otherwise, it is a competitor of M in buying
raw materials from C. In M and F , trade policy is set to maximise national welfare. In C,
trade policy is set by M to maximise its own national welfare. However, M cannot treat C too
harshly, since the colony can stage a successful revolution. Crucially, revolution entails a trade
cost for C, since it is assumed to disrupt trade between C and M . In equilibrium, the size of
this cost depends on F ’s trade policy. In this environment, the attractiveness of revolution come
to depend on M ’s trade policy before a revolution, and on F ’s trade policy after a revolution.
I show that there are two global environments that C can find herself in. First, when F is
relatively abundant in raw materials, its trade policy is hostile towards a rebel colony, since it
accommodates the interests of its own net sellers of raw materials (who are competitors of net
sellers located in C). In addition, trade between C and F is not very important in this case, and
M does not wish to impose strong trade restrictions upon C. This implies that C finds herself well
integrated in world trade before a revolution, but isolated after, making revolution unattractive.
Second, when F is relatively scarce in raw materials, its trade policy is friendly towards the rebel
colony, since it accommodates the interests of its net buyers of raw materials. Furthermore, trade
between C and F is important in this case, and M wishes to impose strong trade restrictions.
Thus, C finds herself isolated before a revolution, but well integrated after, making revolution
more attractive. I combine this economic mechanism with a simple political model of concessions
or repression, and find that the probability of peaceful or violent decolonisation is larger in the
second environment than in the first.
I use these results to clarify the international economic forces at play in the American Revolution. In a second case study, I then consider the rebellion of Latin America against Spain and
2
Portugal (1808-1827). In that case, the timing of rebellion can be explained by the Industrial
Revolution in Britain. This made the British Empire scarcer in raw materials. As a consequence,
British trade policy became more open towards foreign net sellers of raw materials, and the cost
of trade restrictions imposed by Spain and Portugal upon the Latin American colonies increased.
Both factors made rebellion a more attractive option.
My results imply that industrial leaders should have more stable empires, since their colonies
will more often find themselves in the first global environment: one in which they compete with
foreign countries to sell raw materials to the mother country. This may explain why Britain was
able to construct such a large empire in the 19th century. In comparative terms, it complements
Acemoglu, Johnson and Robinson (2005)’s argument on the different trajectories of the British
versus Spanish and Portuguese empires. According to those authors, the gains from empire
accelerated industrialisation in Britain, while they slowed it down in Spain and Portugal. My
results suggest that, in turn, industrialisation helped Britain build such a large and successful
empire, whereas lack of industrialisation led Spain and Portugal to lose much of their empires
early.
A second implication is that there is an incentive for M to block changes in endowments
that make it easier for C to rebel. In an extension, I show that M has an incentive to block
industrialisation in C, intended as a decrease in C’s relative abundance of raw materials. That
is because industrialisation would create a group of colonial citizens who would lose less, or even
benefit from trade disruption between C and M , and who would therefore find revolution more
attractive. This result may explain why colonisers discouraged industrialisation in colonies,
promoting international specialisation. It may also help explain the rise of pro-independence
movements in some African and Asian colonies after 1945. In these colonies as elsewhere, the
trade disruption of the 1930s and 1940s was a shock to the principle of international specialisation:
it led to the collapse of the price of raw materials, and to the emergence of groups who hoped
to profit from state-supported industrialisation (Findlay and O’Rourke, 2007). Relative to the
traditional export-oriented groups, these groups had less to lose from a weakening of imperial
trade relations. Consistent with the model’s predictions, they were a force behind the proindependence movements.
In a companion paper (Bonfatti, 2011), I argue that the value of controlling trade policy
in the colonies declined in the 20th century, as the rise of intra-industry trade made colonial
3
trade relatively less important. That paper addresses the claim, made by some historians, that
some European empires ended in the 1950s because the colonisers lost interest in them. It is
complementary to the present paper, which focuses on cases in which the end of empire was
driven by colonial rebellion.2
The paper is related to the literature on the endogenous size of nations (Alesina and Spolaore,
1997 and 2000), which finds that globalisation reduces local economic dependence, increasing the
equilibrium number of countries. In a similar vein, Martin et Al. (2008), find that more bilateral
trade decreases the probability of war between countries, while more multilateral trade increases
it. I contribute to this literature by constructing a new model of the link between endowments,
global trade policy, and revolution. This allows me to highlight the role of trade policy, and to
discuss new historical episodes. In addition, my results on colonial industrialisation are new.
Another related paper is Head et Al. (2010), who look at the impact of 20th century decolonisation on subsequent trade patterns. Their finding, that conflictual separation led to a
faster decline in trade between colonies and colonisers, is in line with my assumption of a trade
cost of revolution.3
The paper is also related to the literature on customs unions (for a survey, see Ornelas and
Freund, 2010). In my model, too, two countries (C and M ) may form a customs union, from
which the third country is excluded. The difference is that M selects trade policy for both C
and M , and it does so to maximise its own welfare.
Finally, the paper is related to the literature on trade and the economic legacy of empire
(e.g. Acemoglu et Al., 2005, Nunn, 2008), and to a historical literature on trade and war (e.g.
Findlay and O’Rourke, 2007).4
The paper is structured as follows. Section ?? describes the baseline model. Section ??
develops two extensions: one to account for repression and equilibrium revolution, and one on
colonial industrialisation. Section ?? contains the historical evidence. Section ?? concludes.
2
An alternative explanation for the fall of empires in the 20th century is provided by Gancia et Al. (2014),
who emphasise the increasing cost of culturally heterogeneous political entities, relative to looser unions who are
still able to co-ordinate on free trade.
3
On the “empire effect” on trade, see also Mitchener and Weidenmier (2008). The paper is also related to
the literature on natural resources and civil wars (see Blattman and Miguel, 2010). While this has focused on
fluctuations in world prices, I look at the role of trade patterns, for given world prices. This additional dimension
allows me to comment on the importance of foreign trade policy for secession.
4
Other models of colonial rebellion are Gartzke and Rohner (2011) and Grossman and Iyigun (1995, 1997).
These papers do not consider the role of trade in shaping rebellion, and are therefore very different from mine.
4
2
Baseline model
I first describe the model (Sections ?? and ??), and then solve for the equilibrium (??).
2.1
Trade model
There are three countries, a colony C, her mother country M , and a foreign country F .5 Two
goods x and y exist as endowments, and are traded and consumed. National endowments are:
xC = 1
yC = 0
xF = 1
yF = δ
(1)
xM = 1 y M = 1 − δ,
where δ ∈ 0, 23 . In other words, M and F are abundant in y relative to C, but F is not too
abundant relative to M . As clarified below, the latter assumption rules out that the colony
and the mother country are competing to export the same good, a case that is less historically
relevant.6 I interpret x and y as “raw materials” and “manufactures” respectively, but they could
represent any commodity that M and F are competing to buy from, or sell to, C.
Each country is inhabited by a continuum (with mass 1) of citizens, who can only differ in
their endowments. I denote by xiJ and y iJ the endowments of citizen i in country J, and assume
that endowments are dispersed enough to make markets perfectly competitive. Preferences are:
1 iJ 12
iJ
iJ 2
u xiJ
,
y
=
x
yd
,
d
d
d
(2)
iJ
7
where xiJ
d and yd denote demand by citizen i in country J.
Use y as the numeraire, and call pJ the price of x in country J. Citizen i in country J then
J
iJ
iJ J
iJ
maximises (??), subject to the constraint xiJ
d p + yd ≤ x p + y . Her indirect utility is:
xiJ pJ + y iJ
v iJ pJ =
,
1
2 (pJ ) 2
(3)
where I have simplified the notation by writing indirect utility as a function of pJ only.
5
M should be seen as the mother country and the rest of her empire. F can be seen as the rest of the world.
I comment below on how my results change when δ ≥ 32 (see footnote ??).
7
x and y can be alternatively thought of as intermediate goods, and (??) as a production function.
6
5
Summing up across citizens, we find national indirect utility (welfare):
pJ + y J
v J pJ =
1 .
2 (pJ ) 2
2.1.1
(4)
Autarchy equilibrium
Let pJA denote the equilibrium autarchy price in J. With Cobb-Douglas preferences, the marginal
rate of substitution (M RS) equals relative demand,
yd
,
xd
which in autarchy must equal relative
domestic supply, y J (recall that xJ = 1 ∀J). But then, consumer optimisation (pJA = M RS)
requires:
pJA = y J .
(5)
Using (??), it is easy to see that welfare reaches a global minimum at pJA (countries gain from
trade).
2.1.2
Trade equilibrium
Trade policy is a stark decision: a country can be either “open” or “closed” to each of the other
two countries, and (free) trade takes place between two countries if and only if they are both open
to each other. There are then four possible trade outcomes: one in which all countries belong to
the same free-trade bloc, and three in which two countries belong to the same bloc, while the
third remains in autarchy. I use the notation {C, M, F }, {C, M, .}, {C, ., F } and {., M, F } to
denote these four cases.8
Given the assumption of an endowment economy, the equilibrium price within a free-trade
bloc can be calculated as the autarchy price of a fictitious country endowed with the sum of
endowments of members of the bloc. Thus, in {C, M, F }, all countries face the price:
pJ{C,M,F } =
yC + yM + yF
1
= ,
C
M
F
x +x +x
3
8
(6)
With no transportation costs, not all countries need to be open to all countries for {C, M, F } to be realised.
For example, {C, M, F } is realised if M and F are open to C but closed to each other, and C is open to both.
6
whereas the prices that they face in the other trade outcomes are:
pM
{C,M,·} =
1−δ
2
1−δ
2
pF{C,M,·} =
δ
pC
{C,M,·} =
δ
2
pC
{C,·,F } =
M
pM
{C,·,F } = 1 − δ = pA
= pFA
δ
2
pF{C,·,F } =
pC
{·,M,F } =
0
pM
{·,M,F } =
1
2
1
.
2
pF{·,M,F } =
Country J’s net imports of x when facing pJ can be shown to equal
J
pJ
A −p
:
2pJ
J
= pC
A
(7)
the country imports
1
3
< 23
if and only if pJ is lower than its autarchy price. Then, when facing p =
in {C, M, F }, C
always exports, F imports if and only if δ > 13 , M imports if and only if δ
(see Figure ??).
The restriction δ <
2
3
ensures that M always imports x. As anticipated, this rules out that C
and M are competing to sell the same good.
M imports x
0
M exports x
1
3
1
2
3
F exports x
F imports x
Figure 1: Trade in x in {C, M, F }, as a function of δ (C always exports).
How do countries rank trade outcomes? To answer this question, notice first that welfare is
increasing in pJ for pJ > pJA , decreasing for pJ < pJA - a net exporter of raw materials gains from
an increase in their price, a net importer loses. Then, using (??), national preferences are easy
to derive. Table ?? reports the trade outcome that maximises welfare in each country.9
δ∈
C’s optimum M ’s optimum
[0, 1 )
{C, M, .}
{C, M, F }
1 41 {C, M, .}
{C, M, F }
,
41 32 ,
{C, M, F }
{C, M, .}
3 3
Table 1: National welfare-maximising trade outcomes. It is
F ’s optimum
{., M, F }
{C, ., F }
{C, ., F }
1
4
= argδ [v F (pF{C,.,F } ) = v F (pF{.,M,F } )].
Table ?? has an intuitive interpretation. When δ is low, F (the third country) is a competitor
9
The full national preferences are reported in the Appendix. For values of δ such that a country is indifferent
between two trade outcomes, I use its preferences to the right of that value.
7
of C (the colony) in selling raw materials to M (the mother country). Thus, while C and F
prefer to trade with M exclusively, M prefers to trade with both. This is because C and F get
the highest possible price on exports by trading with M exclusively, while M gets the highest
possible price on exports by trading with both. Symmetrically, when δ is high, F is a competitor
of M in buying raw materials from C. Thus, in order to obtain the highest possible price on
exports, F and M want to trade with C exclusively, while C wants to trade with both. Notice
that, as δ increases, F ’s ideal world switches from one in which C’s trade is restricted, to one in
which M ’s trade is. We may then expect that F ’s actions will be aimed at hindering C’s trade
when δ is small, and at hindering M ’s trade when δ is high. The logic of this is quite clear. When
iJ
δ is low, F ’s economy is dominated by net sellers of raw materials ( xy iJ < 13 ). These citizens will
induce their government to hinder foreign net sellers, who are concentrated in C. Conversely,
iJ
when δ is high, F ’s economy is dominated by net buyers of raw materials ( xy iJ > 31 ), who will
induce their government to support foreign net sellers.
2.2
Political Model
I model empire in a very simple way: while M and F set policy freely, policy in C is set by M .
Each country sets policy to maximise its own payoff (defined below).
2.2.1
Policy
There are two policy instruments: trade policy, which is set in all countries, and a transfer from
C to M , which is set in C only. I discuss these instruments in turn. Trade policy is described by
a matrix τ , whose element τJI is equal to 1 if I is open to trade with J, zero otherwise. Mapping
from τ to trade outcomes, we can express equilibrium prices as functions of τ and δ only. Gains
from trade can be written as:
ΠJ (τ |δ) = v J pJ (τ |δ) − vAJ ,
(8)
where vAJ ≡ v J (pJA ) is autarchy indirect utility. Notice that vAC = 0 with these endowments.
M may extract wealth from C by selecting trade policy, but may also be able to impose a
lump-sum transfer (T ) from C to M . I consider two opposite cases, one in which such transfer
8
is technologically feasible (T = 1), and one in which it is not (T = 0).10 Formally, denote by T
the maximum feasible transfer. If T = 1, then T → ∞, whereas if T = 0 then T = 0.11
The case T = 1 does not need to be interpreted literally as a situation in which the mother
country can impose a monetary tax on the colony. Even though such situations were common
enough (for example, the shipment of treasury from Latin America to Spain in the 16th-18th
century), in many other cases the mother country could tax the colonies in kind, by granting its
investors the ownership of colonial assets. The tax T would then measure the amount of colonial
output that the mother country allocates to its investors. In this interpretation, the case T = 1
would refer to a colony whose assets are easy to appropriate (e.g., the mineral deposits of Africa),
whereas the case T = 0 would refer to a colony whose assets are hard to appropriate (e.g. the
farmer-based economy of the US North). Of course, the two should be seen as polar cases, more
useful to illuminate the effect of the tax structure on trade policy, than to realistically describe
any specific country.
2.2.2
Independence, Revolution and Sanctions
Before choosing policy, M decides whether to stick to empire, or to concede independence. In
the latter case, control of policy is transferred to C at no cost for either country. In the former,
C can stage a successful revolution. Revolution also transfers control of policy to C, but inflicts
two costs on it. The first is a stochastic cost µ, capturing the exogenous factors that determine
C’s relative military power.12 I take µ to be distributed uniformly over the interval [0, Nµ ], with
Nµ large enough. The second is a trade cost, since the mother country automatically enacts
C
= 0.13
trade sanctions against the rebel colony: it sets τM
I introduce sanctions by assumption, because they are not ex-post optimal in this model.
This can be justified in two ways. First, “sanctions” may actually capture the deterioration
in trade relations that is naturally associated with conflict. For example, if revolution leads to
10
Because indirect utility is linear in income, we can think of T as a transfer of indirect utility from C to M .
Historically, colonial extraction consisted of redistribution from the colony to the mother country, and from
the colonial masses to a colonial elite who were entrenched by the existence of empire. Colonies of settlement (e.g.
US North, Australia) and colonies where the Europeans only resided as temporary administrators (e.g. Ghana,
India) mostly featured the first type of extraction, whereas colonies where an elite of European descent co-existed
with large indigenous or slave populations (e.g. Peru, Bolivia, US South) featured both. The model is best suited
to describe the first type of extraction.
12
Such as the emergence of a successful leader or ideology that helps the colonists overcome their collective
action problem; or the occurrence of external events that weaken the military power of the mother country.
13
Results are robust to modelling sanctions in a more continuous way.
11
9
war between C and M , trade relations between the two may have to be interrupted, at least
temporarily. In the longer run, war may also lead to a more rapid erosion of the trade-enhancing
networks of empire (Head et Al., 2010). Second, it is easy to think of real-world situations
- and corresponding extensions of the current model - in which M finds it optimal to erect
higher tariffs against a rebel C. On one hand, two independent countries will face issues of
co-ordination in trade policy, that will make it harder to achieve free trade. Such issues are
well known to the literature on trade policy, and are normally seen as a rationale for political
integration (e.g. Alesina and Spolaore, 1997). On the other, M could have multiple colonies,
and standard reputation arguments could be used to rationalise (ex-post suboptimal) punitive
sanctions as a signal to other colonies (e.g. Milgrom and Roberts, 1982). Punitive sanctions are
often used in the real world, and I provide a very clear example of this in Section ??.
For C, there are two advantages of breaking free from empire (either through independence or
through revolution). First, it obtains control of policy. Second, it obtains an exogenous benefit
B > 0,14 which I interpret as capturing a preference for self-determination, or a gain due to
an overall inefficiency of imperial rule.15 In this interpretation, the empire does not exist for
efficiency reasons (though it may provide some efficiency gains), but only to allow M to extract
wealth from C. It should therefore disappear - in an equilibrium where nations are of optimal
size (Alesina and Spolaore, 1997) - but it may well not do so because of the opposition of M .16
An alternative interpretation of B > 0 is that M faces a commitment problem a la Acemoglu
and Robinson (2006), and cannot therefore promise to reduce extraction below a certain level.17
14
B > 0 is required for decolonisation to ever occur in equilibrium. Intuitively, if breaking free from empire
did not imply an efficiency gain, revolution would necessarily imply an efficiency loss, given a positive cost of
revolution. It would then always be possible for M to regulate policy in such a way as to make revolution
unattractive. I illustrate this point more explicitly below, when commenting on Figure ??.
15
A preference for self-determination may be idealistic, or driven by the expectation that, post decolonisation,
domestic politics will be more favourable to the revolutionary groups (I thank an anonymous referee for suggesting
this last point). Imperial rule would be inefficient if C and M had very different preferences, or if the delegation
of policy to a faraway capital was technologically inefficient.
16
To avoid trivial solutions, I assume that B is non-contractible, so that C cannot pay its way out of empire.
This non-contractibility could originate from the fact that C cannot commit to future payments (due after empire
has been dismantled), and cannot therefore fully compensate M for the loss of future gains from empire.
17
For example, suppose M promised to set T = 0, but C anticipated that, with probability π, M would actually
set T = Tb > 0. If µ < B = π Tb, M would only be able to prevent a revolution by conceding independence.
10
2.2.3
Timing and equilibrium concept
Denote the three political states (empire, independence and revolution) by S = E, I, R. The
initial political state is empire, S = E. The timing of the game is:
1. Nature chooses µ.
2. M decides whether to concede independence, or stick to empire. Then, τ and T are
simultaneously set. If S = I, τ C and T are set by C. If S = E, they are set by M .
3. If S = E, C decides whether to stage a revolution. Otherwise, nothing happens.
C
4. If S = R, τ and T are simultaneously reset, with τM
= 0. Otherwise, nothing happens.
5. Production, trade and consumption take place. Payoffs are realised.
To get rid of a number of implausible co-ordination failure equilibria, I focus on the CoalitionProof Nash Equilibria (CPNE) of the policy-setting game: no coalition of countries can be able
to improve on the payoff of all its members by co-ordinating on a different policy vector.18
2.3
Equilibrium
I solve the model using backward induction.
Date 5. Payoffs depend on policy choices (τ, T ), conditional on endowments (δ):
V C (τ, T |δ) = vAC + ΠC (τ |δ) − T + φB + θ(B − µ)
V M (τ, T |δ) = vAM + ΠM (τ |δ) + T
V F (τ |δ) = vAF + ΠF (τ |δ),
(9)
(10)
(11)
where φ (θ) is an indicator variables for S = I (S = R).
Date 4. If C has staged a revolution, the policy equilibrium is (proofs in the Appendix):
Lemma 1. If the political state is revolution (S = R), in all CPNE:
• If δ ∈ 0, 14 , the trade outcome is {·, M, F } (the colony falls into autarchy);
18
Without this refinement, τ = 0 (all being closed to all) could be realised in equilibrium.
11
• if δ ∈
2
,
, the trade outcome is {C, ·, F } (the colony can trade with the third country),
4 3
1
and T = 0.
By assumption, revolution disrupts trade between C and M . C’s trade prospects, then, must
depend on F ’s trade policy. Lemma ?? relates this to endowments. When δ is low, F is a
competitor of C in selling x to M , and its terms of trade are best when it trades with M alone.
Thus, it reacts to sanctions by closing down to trade with C. Conversely, when δ is high, F
is a competitor of M in selling y to C, and its terms of trade are best when it trades with C
alone. In this case, F reacts to sanctions by opening up to C. One implication of Lemma ??
is that revolution makes C and M worse off than they would be in {C, M, F }, while making F
better off. Intuitively, revolution generates trade diversion away from C and M , and this must
benefit F . Let τ (R|δ) denote equilibrium trade policy after a revolution. Gains from trade after
a revolution can be written as ΠJ (R|δ) = ΠJ (τ (R|δ)|δ).
Date 3. If S = E, C compares the policy equilibrium that has been realised in period 2 to
the one that would be realised after a revolution. It then stages a revolution if and only if its
payoff is greater in the latter equilibrium than in the former, or, using (??):19
ΠC (R|δ) + B − µ > ΠC (τ (E)|δ) − T (E),
(12)
where τ (E), T (E) denote policy under empire.
Date 2. To decide between empire and independence, M compares the policy equilibrium
that is realised in the two cases. If it concedes independence:
Lemma 2. If the political state is independence (S = I) in all CPNE the trade outcome is
{C, M, F }, and T = 0.
When countries set trade policy independently, the world is integrated in trade. Intuitively,
with two countries competing to sell x, the third must be open to both - or it would not obtain
the best possible terms of trade - and this is enough to lead to {C, M, F }. Let τ (I) denote
19
(??) is the relevant condition for revolution to maximise C’s payoff. If, in addition, I had assumed that
C’s citizens are homogenous (an assumption that would not change any of the results), (??) would also be the
condition for revolution to maximise the payoff of each individual citizen of C.
12
equilibrium trade policy after independence. Gains from trade after independence can be written
as ΠJ (I|δ) = ΠJ (τ (I)|δ). However C’s gains do not depend on δ (since pC
{C,M,F } does not), and
can be written simply as ΠC (I).
If M decides on empire, it sets τ M , τ C and T , while F sets τ F . It is easy to show that,
since M controls trade policy in two countries, its desired trade outcome must be realised in a
CPNE.20 This implies that we can describe M ’s trade policy in terms of the trade outcome that
it selects, simplifying our task of describing policy under empire.21
When choosing policy, M anticipates that, if (??) is satisfied, C will stage a revolution. Since
revolution is worse than independence, M sticks to empire if and only if its payoff under empire,
conditional on revolution not taking place, is higher than its payoff after independence. M ’s
payoff at this constrained optimum must depend on the cost of revolution µ, since the higher
µ is, the slacker the constraint in (??) is. We now want to find a threshold for µ such that, if
µ is above it, M is as well off at the constrained optimum as it would be at the unconstrained
optimum, that is when able to impose its favourite policy, with no fear of a revolution. Let
τeT (E), TeT (E) denote M ’s favourite policy, which will depend on the technology of extraction
(T ). It must be Te1 (E) = ΠC (e
τ1 (E)|δ) - with a lump-sum transfer, M must extract the entire
value of the colony at the unconstrained optimum22 - and Te0 (E) = 0. Substituting τeT (E) and
TeT (E) into (??) and rearranging, we get, for the case T = 1 and T = 0 respectively:
µ < B + ΠC (R|δ) ≡ µ1 (δ)
(13)
µ < B + ΠC (R|δ) − ΠC (e
τ0 (E)|δ) ≡ µ0 (δ).
(14)
µT (δ) is the threshold that we are looking for, since µ > µT (δ) ensures that M can impose its
favourite policy, and still avoid a revolution. From C’s perspective, it represents the gain from
20
If M ’s desired outcome is {C, M, .}, this must be realised in a CPNE, since M can single-handedly obtain it.
If M ’s desired outcome includes F , such outcome can be realised in a CPNE (since M is at its desired outcome,
and F can only deviate into autarchy), but no other can (not one that includes F , since M could deviate to its
desired outcome; not one that excludes F , since F and M could jointly deviate to M ’s desired outcome).
21
Notice that F could set trade policy strategically, in order to trigger a revolution. For example, it could set
F
F
τC = τM
= 0, to create the expectation that only if policy is reset will C get to trade with F . Of course, such
a choice of trade policy would be a non-credible threat, since F would like to renege on it should C not stage a
revolution. I avoid these implausible complications by assuming that F sets policy with the goal of maximising
welfare under current political institutions. From a modelling perspective, this would be equivalent to assuming
that F sets policy in period 4, after political institutions have been chosen.
22
Nothing would change by assuming that C must be left with a positive minimum level of utility.
13
rebelling against a mother country that treats it in the worst possible way. It is a function of
the difference between gains from trade after a revolution (ΠC (R|δ)) and gains from trade under
such an extractive empire (0 if T = 1, ΠC (e
τ0 (E)|δ) if T = 0). Since the technology of extraction
is more powerful for T = 1 than for T = 0, it is µ1 (δ) ≥ µ0 (δ).
Next, we want to find a second threshold for µ such that, if µ is below it, M is worse off at
the constrained optimum than it would be after conceding independence. This can be found by
replacing ΠC (τ (E)|, δ) = ΠC (I) and T (E) = 0 in (??), and re-arranging:
µ < B + ΠC (R|δ) − ΠC (I) ≡ µ(δ).
(15)
From C’s perspective, the threshold µ(δ) represents the gain from rebelling against a mother
country that treats it in the best possible way. It is a function of the difference between gains
from trade after a revolution, and gains from trade under such a generous empire (ΠC (I)). I
refer to this difference as to the trade cost of revolution.
It can be shown that:
Proposition 1. The political equilibrium is as follows. If T = 1:
• If µ ≥ µ1 (δ), M sticks to empire. In all CPNE, the trade outcome is {C, M, F }, and
T = T ∗ ≡ ΠC (I).
• If µ ∈ µ(δ), µ1 (δ) , M sticks to empire. In all CPNE, the trade outcome is {C, M, F },
and T = µ − µ(δ) < T ∗ .
• If µ < µ(δ), M concedes independence.
If T = 0:
• If µ > µ0 (δ), M sticks to empire. In all CPNE, the trade outcome is {C, M, F } if δ ∈ 0, 13 ,
{C, M, .} if δ ∈ 13 , 32 .
• If µ ∈ µ(δ), µ0 (δ) , M sticks to empire. In all CPNE, the trade outcome is {C, M, F }.
• If µ < µ(δ), M concedes independence.23
If δ ≥ 32 , and M is a competitor of C in exporting x, {., M, F } may be M ’s first best, and {C, ., F } may be
the first choice of C and M joint (since the two can obtain better terms of trade by restricting supply). In this
environment, M ’s unconstrained optimum would be {C, ., F } if T = 1, {., M, F } if T = 0.
23
14
If T = 1, M always keeps the empire open to external trade, even if it is in competition with F
to import raw materials from C (δ > 13 ). Intuitively, citizens from M are the residual claimant to
colonial raw materials, and their terms of trade are best when these can be exported everywhere.
As for the transfer, it is set to the maximum feasible level, conditional on revolution not taking
place. This is the entire value of the colony if µ ≥ µ1 (δ), a smaller amount if µ ∈ µ(δ), µ1 (δ) .
If µ < µ(δ), the transfer would have to be negative, and M prefers to concede independence.
If T = 0, M would like to close the empire to external trade if δ ≥ 13 , to extract wealth from
C by forcing upon it a lower price of raw materials. Since this choice of trade policy is costly
for C, however, it can only be made for µ > µ0 (δ). For µ ∈ µ(δ), µ0 (δ) , M must open up the
empire to external trade to avoid a revolution. For µ < µ(δ), trade policy would have to be more
favourable to C than after independence, and M prefers to concede independence.
The prediction that the mother country uses trade restrictions to extract wealth from the
colony fits well the historical experience: in most cases, colonisers restricted trade between their
colonies and the rest of the world. But the model also admits a case in which the mother country
keeps the colony open to external trade: this happens if either the citizens of the mother country
are the residual claimant to colonial raw materials, or if they are not, but the mother country
dominates colonial trade as a result of her being an industrial leader. Is there any piece of
historical evidence in support of this second prediction? Indeed, the case of the British empire
in the second half of the 19th century seems to provide one. In this period, the British empire
was largely open to external trade. Consistent with both mechanisms identified by the model,
this was a time when British industry was much more advanced than that of her European
competitors, and the stock of colonial assets owned by British investors massively expanded
(Piketty, 2014).
Notice that, when M closes the empire to external trade, revolution moves the colony from
outcome {C, M, .}, to outcome {C, ., F }, both of which are inferior to the colony’s optimal,
{C, M, F }. We can then divide the effect of revolution in two parts: a trade gain - from {C, M, .}
to {C, M, F }, and a trade cost - from {C, M, F } to {C, ., F }. I study below how these depend
on δ. What is immediately evident is that if δ >
1
2
- F is a more important trade partner than
M - the gain outweighs the cost, and revolution gives C better terms of trade.
To relate the equilibrium described in Proposition ?? to economic fundamentals, I plot µ(δ)
and µT (δ) in Figure ??, for the case T = 1 (left panel) and T = 0 (right panel). The lower
15
threshold is the same in the two cases. It is first constant, then jumps and up and becomes
increasing in δ. By Proposition ??, the probability that M concedes independence also follows
this pattern. Intuitively, the trade cost of revolution is maximum for δ < 14 , since C would find
herself isolated after a revolution. If δ > 41 , the trade cost is lower, since C would be able to
trade with F , and it is decreasing in δ, as F becomes a more important trade partner for C. If
δ = 32 , the trade cost of revolution is zero. The distance between the lower and upper threshold
represents the maximum cost of extraction for C. If T = 1, this is equal to the entire value of the
colony, and is constant in δ. If T = 0, it is only positive when M closes the empire to external
trade (δ ≥ 13 ). It is then increasing in δ, to reflect the fact that the cost of a closed empire which is also the trade gain from revolution - increases as F becomes an increasingly important
trade partner for C.24
Although the trade gain from revolution is increasing in δ, this does not contribute to making revolution more likely in this simple model. This is because, if imperial trade restrictions
become so costly on colonial citizens to make revolution imminent, the mother country is always
willing to relax such restrictions. However, the history of both the American Revolution and the
Latin American Independence Wars suggests that increasing dissatisfaction with imperial trade
restrictions was one key element of pre-revolutionary politics. I show in the next section that, if
M can react with repression to the threat of revolution, it may not always be willing to relax
trade restrictions. In that case, the model predicts a positive link between the trade gain, and
probability of revolution.
µ1 (δ)
B + ΠC (I)
µ0 (δ)
B
B + ΠC (I) − ΠC (pC
{C,M,.} )
B
B
µ(δ)
B − ΠC (I)
µ(δ)
B − ΠC (I)
1
4
2
3
1
4
2
3
Figure 2: µT (δ) and µ(δ) as functions of δ. The left panel is for T = 1, the right panel for T = 0.
Findings so far can be summarised in the following:
24
The figure confirms that, if B = 0, M never concedes independence (it is never µ < µ(δ)).
16
Proposition 2. The rebel colony obtains the commercial support of the third country if and only
if δ ≥ 14 . Ceteris paribus, the likelihood of decolonisation is higher for δ ≥
1
4
than for δ < 14 ;
furthermore it is constant in δ for δ < 41 , and increasing in δ for δ ≥ 41 . If T = 1, expected
extraction follows an opposite pattern to the likelihood of decolonisation.
Proposition ?? summarises the link between the sustainability of empire and trade patterns.
There are two global environments that C can find herself in. The first is a “hostile” environment,
when F is relatively abundant in raw materials δ < 14 . In this case, F ’s trade policy accommodates its net sellers of raw materials, who perceive C as a competitor. This results in F ’s trade
policy being hostile, and in C finding herself isolated after a revolution. The attractiveness of
revolution is then low, and so is the probability of decolonisation. The second is a “favourable”
environment, when F is relatively scarce in raw materials δ ≥ 14 . In this case, F ’s trade policy
accommodates its net buyers of raw materials, who perceive C as a trade partner. This results in
C being integrated in trade after a revolution, making revolution attractive and the probability
of decolonisation high. This second environment is more favourable the higher is δ, a parameter
that captures how well integrated is C after a revolution. If T = 1, expected extraction is lower
in the second environment than in the former, and the more so the larger is δ (if T = 0, this
is ambiguous).25 The proposition has sharp predictions for the link between endowments, trade
policy, and decolonisation, as well as for the role played by third countries in this process.26 In
Section ??, I relate these predictions to some important cases of decolonisation.
One interesting implication of Proposition ?? is that a mother country that is an industrial
leader should have a more stable empire, and one that is more profitable (at least if colonial
assets are easy to appropriate). This is because its colonies will more often find themselves
in the first global environment: one in which they compete with foreign countries to sell raw
materials to the mother country. This may explain why Britain was able to construct such a
large and successful empire in the second half of the 19th century. In comparative terms, it
complements Acemoglu, Johnson and Robinson (2005)’s argument on the different trajectories
25
If T = 0, an increase in δ increase the likelihood of revolution, but also the gain to M from imposing {C, M, .}.
The net effect of an increase in δ on expected extraction will then depend on the distribution of µ.
26
I have focused on F ’s trade policy, but this will typically be correlated to other forms of external support,
such as diplomatic recognition or military help. For example, during the Latin American Independence Wars,
Britain (F ) not only opened up to trade with the rebel colonies, but also deployed the Royal Navy to protect this
trade, further helping the revolutionary cause. In the context of the model, any further help that F gives to C
can be rationalised with the fact that, as discussed above, F benefits from a revolution.
17
of the British versus Spanish and Portuguese empires. According to those authors, the gains
from empire accelerated industrialisation in Britain, while they slowed it down in Spain and
Portugal. My results suggest that, in turn, industrialisation helped Britain build a large and
successful empire, whereas lack of industrialisation led Spain and Portugal to lose much of their
empires early.27
3
Extensions
I now extend the baseline model in two ways. First, I allow for the possibility that M uses
repression against the colony, making revolution possible in equilibrium. Second, I look at the
mother country’s incentives to block industrialisation in the colony, to make it more costly for
her to break away.
3.1
Repression and equilibrium revolution
So far, I have assumed that M can only reduce extraction or concede independence, and that
revolution is always avoided. But imperial powers also used repression against rebel colonies,
and revolutions did happen. I extend the baseline model to account for this.28 Faced with a low
µ, M can now accommodate colonial requests as in the baseline, or can scale up repression. In
the latter case, the probability that C can stage a successful revolution drops to r ∈ [0, 1]. To
scale up repression has a stochastic cost η to M , distributed uniformly over the interval [0, Nη ],
with Nη large enough.29 The new timing of the game is (changes highlighted in bold):
1. Nature chooses µ and η.
2. M decides whether to concede independence, or stick to empire. In the latter case, it
27
One interesting extension would be to consider how colonisation can help resolve a hold-up problem in the
trade relations between C and M . Suppose that C is an independent country that can invest in the capacity to
export x. If δ is very low - so that C and F are competing to export x to M - M could extract much of the
surplus created by the investment by erecting an import tariff against C. Anticipating this, C may invest less
than optimally. Colonisation of C by M would then represent a way to solve this under-investment problem.
The additional prediction of this extended model would be that relatively successful industrial powers also have
stronger efficiency reasons to build an empire. I thank an anonymous co-editor for suggesting this point.
28
I follow closely Acemoglu and Robinson (2006)’s model of repression.
29
For example, exogenous factors (such as war elsewhere) may determine the cost of sending reinforcements to
the colony.
18
decides whether to scale up repression or not. Then, τ and T are simultaneously
set. If S = I, τ C and T are set by C. If S = E, they are set by M .
3. If S = E and M has not scaled up repression, C decides whether to stage a revolution. If
S = E and M has scaled up repression, with probability r C decides whether
to stage a revolution; with probability 1 − r nothing happens. If S = I, nothing
happens.
4-5. As in the baseline.
If S = R, I, or if S = E and M has not scaled up repression, the policy equilibrium is
the same as in the baseline. If S = E and M has scaled up repression, it sets extraction to its
maximum: intuitively, having defied revolution, it may as well try to extract as much as possible.
Consider first the case where T = 1. If µ < µ(δ), M can either concede independence, or scale
up repression and impose maximum extraction. The latter is optimal if and only if:
(1 − r) vAM + ΠM (I|δ) + T ∗ + r vAM + ΠM (R|δ) − η > vAM + ΠM (I|δ)
η < (1 − r)T ∗ − r ΠM (I|δ) − ΠM (R|δ) ≡ η 1 (δ, r),
(16)
where I have used the fact that, with T = 1, the trade outcome under empire is the same as
with independence.30 If µ ∈ [µ(δ), µ1 (δ)), M can either reduce extraction, or scale up repression
and impose maximum extraction. The latter is optimal if and only if:
(1 − r) vAM + ΠM (I|δ) + T ∗ + r vAM + ΠM (R|δ) − η > vAM + ΠM (I|δ) + µ − µ(δ)
η < η 1 (δ, r) − [µ − µ(δ)].
(17)
From (??) and (??), repression is more likely (η 1 (δ, r) is higher) the lower the trade cost of
revolution for M is. This is a new link between trade patterns and the equilibrium political
state.
30
Since η impacts upon payoffs linearly, M ’s preferences over policy are unchanged.
19
Next, consider the case where T = 0. If µ < µ(δ), M scales up repression if and only if:
(1 − r) vAM + ΠM (e
τ0 (E)|δ) + r vAM + ΠM (R|δ) − η > vAM + ΠM (I|δ),
(18)
η < (1 − r) ΠM (e
τ0 (E)|δ) − ΠM (I|δ) − r ΠM (I|δ) − ΠM (R|δ) ≡ η 0 (δ, r).
If µ ∈ [µ(δ), µ0 (δ)), M scales up repression if and only if its expected payoff is greater than
its payoff when conceding the trade policy that prevails under independence. So, (??) is still the
relevant condition. From (??), repression is more likely the lower the trade cost of revolution for
M is, and the higher its trade gain from extraction is. There are then two new links between
trade patterns and the equilibrium political state. Proposition ?? can be generalised as follows:
Proposition 3. The political equilibrium is as in Proposition ??, except that:
• If T = 1 and µ < µ1 (δ), and η < η 1 (δ, r) − max[µ − µ(δ), 0], M scales up repression. In
all CPNE, the trade outcome under empire is {C, M, F }, and T = T ∗ . With probability r,
there is a revolution, and C becomes independent.
• If T = 0 and µ < µ0 (δ), and η < η 0 (δ, r), M scales up repression. In all CPNE, the trade
outcome under empire is {C, M, F } if δ ∈ 0, 13 , {C, M, .} if δ ∈ 31 , 23 . With probability
r, there is a revolution, and C becomes independent.
There are now two distinct paths to decolonisation: the peaceful one found in the baseline
(for µ low and η high), and a violent one featuring revolution and trade disruption between C
and M (for µ and η low).
Before proceeding, I introduce some new terminology:
Definition 1. “Decolonisation” is the state in which C separates from M , either peacefully or
violently (S = I, R). “Punishment” is the state in which M represses, and keeps the empire.
Proposition ?? can be generalised as follows:
Proposition 4. The rebel colony obtains the commercial support of the third country if and only
if δ ≥ 41 . The political equilibrium is as follows:
• If T = 1, there exist r, r (with 0 < r ≤ r < 1) such that:
20
– If r ≥ r, there exists δb (with δb > 14 ; increasing in r; equal to 23 for r = 1) such that, if
b the equilibrium is as in Proposition ??. If δ ≥ δ,
b the probability of revolution,
δ < δ,
and the joint probability of decolonisation and punishment, are increasing in δ.
– If r ≤ r, the probability of revolution, and the joint probability of decolonisation and
punishment, are increasing in δ.
– If r ∈ (r, r), the probability of revolution, and the joint probability of decolonisation
and punishment, are constant or increasing in δ for δ 6=
1
,
4
may be increasing or
decreasing for δ = 41 .
• If T = 0, for all r, there exist δ, δ (with
1
3
< δ ≤ δ ≤ 23 ; increasing in r; equal to
2
3
for
r = 1) such that, if δ < δ, the equilibrium is as in Proposition ??. If δ ≤ δ < δ, the
probability of revolution, and the joint probability of decolonisation and punishment, are
positive and increasing in δ. If δ > δ, they are positive and may be increasing or decreasing
in δ.
The baseline link between the probability of colonial requests and trade patterns largely
survives in a more general setting, and carries over to the probability of revolution. Intuitively,
not only is C more likely to make requests as δ increases, but also, for most parameter values, M
is more likely to react with repression. When these two factors act together, they must lead to a
higher probability of revolution.31 But why does an increase in δ make M more likely to repress?
As mentioned above, there are two channels. The first - and only one at play when T = 1 - has
to do with the trade cost of repression. Put simply: as δ increases, not only M becomes a less
important trade partner for C, but also C becomes a less important partner for M . This force
decreases the cost of repression for M , because it decreases its cost of disruption.32 The second
channel has to do with the gain from repression, and is active, together with the first channel,
31
The probability of revolution always moves together with the joint probability of decolonisation (S = I, R)
and punishment. Except for r = 1, however, when the model collapses to the baseline, we cannot make predictions
for the probability of S = I (and thus decolonisation). Intuitively, an increase in δ increases the probability that
C asks for independence, but also, for most parameter values, the probability that M reacts with repression.
What happens to the probability of S = I, then, depends on the distribution of µ and η.
32
This force decreases the trade cost of repression ∀δ 6= 41 . However, the trade cost of repression jumps up at
δ = 14 , reflecting the fact that F ’s trade policy becomes more unfavourable to M . This jump is not visible if
repression is very inefficient, r > r (because it is then only used when δ > δb > 14 ), or when it is very efficient,
r < r, (because its trade cost is then very small, and the jump in η 1 (δ, r) is more than compensated by the jump
in µ1 (δ)). It may, however, be visible for intermediate values of r.
21
when T = 0. Recall that the gain from repression results from being able to impose maximum
extraction on C, which in this context means excluding F from colonial trade. By making F a
stronger competitor of M , an increase in δ increases the value of that exclusion.33 In summary,
when T = 0, an increase in δ makes M both less worried about the trade cost of repression, and
more keen on preserving its power to exclude F from colonial trade. This must lead to a higher
probability of repression, and thus revolution.
We already know that, if T = 0 and δ > 13 , an increase in δ both decreases the trade cost of
revolution, and increases its trade gain. However in the baseline model, only the former effect
contributed to making decolonisation more likely. A closer look at the new equilibrium reveals
that, when δ > δ > 13 , both effects contribute to making revolution more likely. Intuitively, if
imperial trade restrictions become so costly on colonial citizens to make revolution imminent,
there is now a positive probability that the mother country reacts with repression. From an exante perspective (before µ is realised), then, an increase in the cost of imperial trade restrictions
must make revolution more likely. In summary, an increase in δ shifts C to an environment in
which the colony is both more isolated in trade before revolution, and more integrated after. I
argue in Section ?? that such a shift occurred in the colonial US and in Latin America in the
years before their respective revolutions.
I conclude this section by briefly discussing the robustness of my results to a richer economic
structure. I have considered an endowment economy, but my results seem robust to a situation in
which xJ and y J represent optimal production plans on (convex) production possibility frontiers.
These frontiers could differ across countries because of differences in technology (a la Ricardo)
or endowments (a la Heckscher-Ohlin). To see this, notice that prices and trade patterns in
{C, M, F } would be the same, implying that it would still be p{C,.,F } <
δ<
1
3
and p{C,.,F } , p{C,M,.} <
1
3
1
3
< p{C,M,.} , p{.,M,F } if
< p{.,M,F } otherwise.34 Furthermore, it must still be the case that
an importer (exporter) benefits from receiving a lower (higher) relative price of imports (exports),
33
This force implies a positive monotonic relation between δ and the gain from repression. A second force in
the model - the fact that M ’s endowment of y decreases as δ increases - ensures that the overall link between δ
and the gain from repression has an inverted U shape (decreasing in δ for δ > 85 ), which carries over to the link
between δ and the probability of repression for very high values of δ. Thus, this link is negative for δ > δ.
1
34
It would
still be p{C,M,F } = 3 , since that is the only price that makes relative demand equal to world relative
1
supply 3 . Trade patterns would be the same because countries (or blocs of countries) would still import x if
and only if their relative supply is smaller than world relative supply. Standard argument (available from the
1
author upon request) can be used to argue that the “autarchy” price of bloc B, pA
B , must be greater than 3 if the
1
bloc imports x in {C, M, F }, and smaller than 3 otherwise.
22
since this pivots the budget line further away from autarchy.35 If δ is low, faced with a choice
of trade outcomes after a revolution, F would still prefer {., M, F }, since it is an exporter and
p{.,M,F } is the highest price it can get. Symmetrically, if δ is high, it would still prefer {C, ., F }.
But then C would still face a higher trade cost of revolution when δ is low than when it is high,
as in the current model. Furthermore, the cost would be constant in the first case (since C’s
autarchy equilibrium does not depend on δ), and decreasing in δ in the second (since it must
be pC
{C,.,F } =
1
3
for δ = 23 ).36 Of course, there would be more adjustment in the general model -
faced with trade disruption, C would start producing less x - but that could only partially make
up for the fall in external demand.
Similarly, results seem robust to a richer structure of negotiations between C and M . I
currently assume that M moves first, by making a policy offer that C can either accept or reject.
This gives the mother country all bargaining power, leaving the colony at best indifferent between
empire and revolution. A more realistic model would allow for some bargaining over policy, with
revolution being the outside option if bargaining fails, and independence being an alternative to
entering bargaining at all. In this alternative model, it would still be true that M ’s concessions
are greater, the lower µ is, since a decrease in µ only worsens C’s outside option. Furthermore,
the threshold for µ below which independence is conceded would still be µ(δ): that’s because
the maximum that M is willing to concede is the same as before, and so is C’s payoff under
a revolution. However, the way in which concessions under empire depend on δ would change,
since it would have to be taken into account that not only C’s payoff under revolution, but also
M ’s, depend on δ.
3.2
Colonial industrialisation
Endowments shape trade patterns, which in turn shape the sustainability of empire. Anticipating
this, M may want to block changes in C’s endowments - such as those deriving from industrialisation - that would make empire less sustainable. In now investigate this idea. For brevity, I
only consider the case T = 1 in this section. Let the baseline game start in t = 0, and let C be
35
This logic establishes that F prefers {., M, F } to {C, M, F }, and also to {C, ., F } if F is an exporter in this
last outcome. It does not establish that F prefers {., M, F } to {C, ., F } if F is an importer in this outcome. In
that case, we can only use the informal argument that, when δ is low, C must have a very similar economy to
F ’s, while M must have a very different one (since, in {C, M, F }, they respectively specialise on very similar and
different production plans): it should then be better for F to trade with C, than with M .
36
Since the net imports of the union of C and F in {C, M, F } are zero when δ = 32 .
23
a colony of M from the very beginning. In t = 0, the citizens of the two countries may invest
in converting their endowment of x into endowment of y. For simplicity, the marginal rate of
transformation (M RT ) is constant, and equal to 31 . Let φJ denote total investment undertaken
in country J. Endowments in t = 1 are then:
xC = 1 − φC
y C = 31 φC
xF = 1
yF = δ
xM = 1 + b − φM y M = 1 − δ − 31 (b − φM ),
where b > 0 is small.
Consider the decentralised investment equilibrium in period 0. Anticipating that, under all
political institutions, the price will be pJ{C,M,F } , agents must continue to invest until pJ{C,M,F } =
M RT , or
y C +y M +y F
xC +xM +xF
= 31 , or:
M
φC
d + φd = b.
C
If φC
d = 0, we are back to the baseline. If φd > 0, C is less abundant in x than in the baseline,
and M more so. With these new endowments, C continues to be an exporter of x,37 but M is
an importer for a smaller range of parameters (Figure ??). To continue to focus on the case in
which M is a net importer of x, I assume δ < 23 (1 − φC ) from now on.
M imports x
0
1
3
M exports x
2
3
F exports x
1 − φC
d
2
3
1
F imports x
Figure 3: Trade in x in {C, M, F }, as a function of δ (C always exports for b small).
C
M
Because of a constant MRT, the model does not pin down φC
d , but only φd + φd . For
simplicity, I assume that all agents correctly anticipate φC
d before it is realised. At this early
stage, M imposes a cap φbC on colonial investment. If φbC ≥ φC
d , the cap is not binding, and
φC = φC and φM = b − φC (a situation that I also describe as “no cap”). If φbC < φC , the cap is
d
37
d
C’s autarchy price is now
d
φ
3
1−φ ,
which, given φC ≤ b and b is small, is always lower than the world price ( 31 ).
24
binding, and I assume it is exactly φC = φbC and φM = b − φbC . I assume that, unless M strictly
benefits from imposing a cap, it imposes none. To summarise, the new timing is:
0. M can impose a cap on colonial investment, φbC . After this, the citizens of C
and M decide how much to invest, and φC and φM are realised.
1-5. As in the baseline.
Dates 1-5. For any φC (and φM = b − φC ) that has been realised in period 0, the game
unfolds as in the baseline. However, C’s gains from trade after a revolution, and thus the key
thresholds for µ, now also depend on φC :
µ(δ, φC ) = B + ΠC (R|δ, φC )
µ(δ, φC ) = B + ΠC (R|δ, φC ) − ΠC (I).
It can be shown that, for given trade policy of F , ΠC (R|δ, φC ) is increasing in φC . Intuitively,
investment creates a group of citizens in C, who are endowed with positive y. These citizens are
less penalised by a disruption in trade relations that increases the relative price of y. In fact,
those of them who have enough y to be net sellers (net buyers of x) stand to benefit from such
disruption. Since ΠC (R|δ, φC ) is increasing in φC , so are µ(δ, φC ) and µ(δ, φC ) (Figure ??). The
only exception is for δ just above 14 , since the point above which F ’s trade policy is favourable
to C shifts to the right. The direction of this shift seems to be driven by functional forms.38
The equilibrium political state is as described in Proposition ??, but with µ(δ, φC ) and
µ(δ, φC ) as key thresholds. I use the notation T (µ, δ, φC ) to denote the transfer that M can
impose. Using (??) in (??), and the fact that pM (τ |δ) =
written as V M (µ, δ, φC ) = v M 13 + T (µ, δ, φC ).
1
3
in all cases, M ’s final payoff can be
Date 0. Suppose φC
d > 0 is the expected decentralised equilibrium. What cap in the range
38
At the breaking point, F is indifferent between {., M, F } (where it exports x to M ) and {C, ., F } (where it
exports y to C). As φC increases, M becomes less important an importer of x, while C becomes less important
an importer of y. With these functional forms, the second effect is stronger, resulting in a rightward shift in the
breaking point.
25
µM (δ, φC )
B + ΠC (I)
B
B
M
C
µ (δ, φ )
B − ΠC (I)
1
4
2
3
Figure 4: µM (δ, φC ) and µM (δ, φC ) as functions of δ. The grey lines represent the case φC = 0.
C
bC
0, φC
d - with φ = φd representing the “no cap” case - will M impose? M ’s expected payoff is:
Eµ [V
M
bC
1
1 (T ∗ )2
∗ Nµ − µ(δ, φ )
(µ, δ, φ )] = v
+T
+
3
Nµ
Nµ 2
1
1
= vM
+ π(δ, φbC )v C
,
3
3
bC
M
bC )
φ
1 T∗
∗
C 1
+
,
and
I
have
used
the
fact
that
T
=
v
where π(δ, φbC ) ≡ Nµ −µ(δ,
. The term
Nµ
Nµ 2
3
π(δ, φbC ) can be interpreted as a measure of the sustainability of empire: it is the probability
that M can impose maximum extraction, as opposed to no extraction. Notice that, because of
a constant MRT, neither v M 31 nor v C 13 depend on φbC .39 In other words, this is a model in
which a cap on colonial investment has no impact on M ’s payoff, other than via its effect on the
sustainability of empire.
Consider first a benchmark case in which concerns about the sustainability of empire are
not an issue. Such a case can be represented by letting Nµ → ∞, since then revolution never
happens. In this case, it is π(δ, φbC ) → 1. It immediately follows that:
Lemma 3. Suppose Nµ → ∞. Then, M sets no cap on colonial investment.
In the general case, however:
Proposition 5. If the decentralised equilibrium φC
d > 0 is anticipated:
e M imposes a cap φbC = 0 on colonial investment;
• If δ < 41 , or if δ > δ,
39
It is v C
1
3
=
bC ) 1 + φ
(1−φ
3
3
2
√1
3
b
1
= √3 1 and v M =
2
3
bC
bC ) 1 +1−δ− φ
(1+φ
3
3
√1
2
26
3
=
1
3 +1−δ
2 13
√
.
• If δ ∈
where δe ∈
h
i
1 e
,δ
4
1 1
,
4 3
, M imposes a cap φbC < φC
d on colonial investment.
. The effect of a cap is to shift investment to the mother country, to increase
expected extraction, and to decrease the probability of independence.
Although M is the residual claimant to the C’s wealth, concerns about the sustainability of
empire induce it to block industrialisation in the colony. By doing so, M prevents the formation
of a group of colonial citizens who are natural “enemies” of the empire, in the sense that they
are less fearful of a deterioration in imperial trade relation, or even stand to benefit from it. The
result is a “less developed” but more stable empire, from which the mother country can extract
more. In terms of Figure ??, the cap shifts down µ(δ, φC ) and µ(δ, φC ).40
It can also be shown that, for any φC
d > 0, the optimal cap pays off more (it leads to a
greater shift in investment, etc) for a mother country who is an industrial leader (high 1 − δ).
In terms of Figure ??, the cap shifts the threshold more, the lower δ is.41 Intuitively, the threat
of trade disruption is a more effective tool in the hands of an industrial leader: it must then be
more valuable to preserve. This result implies that, if caps were expensive to impose, industrial
leaders should have imposed them more often. Referring back to my discussion of the opposite
trajectories that the British and Spanish empires may have been on (see end of Section ??), this
last result suggests that, as part of that story, policies that favoured colonial dependence may
have been stronger in the British empire than elsewhere. This is admittedly a surprising result,
given the common belief that British colonialism was a relatively benign one.
Finally, notice that a cap promotes international specialisation, as it leads to M exporting
more manufactures, and to C exporting more raw materials. Furthermore, once international
specialisation is in place, the cap becomes redundant, because the possibility to import cheap
manufactures from M reduces any incentive for C to industrialise.
Results in this section are important for two reasons. On one hand, they help interpret
the historical evidence suggesting that colonisers restricted colonial industrialisation, a fact that
seems important to understand the economic legacy of colonialism. On the other, they help
explaining the rise of pro-independence movements in some African and Asian colonies after
1945. In these colonies, as elsewhere in the developing world, trade disruption in the 1930s and
40
e
If M could force C to invest more than φC
d , it might do so for δ just below δ. That result would hold for a
narrow range of δ, and would rely on the functional forms that make the breaking point shift to the right.
41
This result is true in the current model, and more in general if C is not too large an exporter of x.
27
1940s was a shock to the principle of international specialisation. It led to a collapse in the
price of raw materials, and to the emergence of groups of import-substituting industrialists or
other groups (e.g. state bureaucrats) who hoped to profit from state-supported industrialisation
(Findlay and O’Rourke, 2007, p. 477). Relative to traditional export-oriented groups, these
groups had less to lose from a deterioration in imperial trade relations. In fact, they stood
to gain from trade restrictions that would put up the price of manufactures. As predicted by
the model, they were a force that pushed for decolonisation: they supported the decolonisation
movements, and came to dominate politics in the post-colonial era (Bates, 1981).
4
Historical evidence
I have built a theory of the sustainability of empire in a global perspective. I now put the
theory to work to shed some light on two important cases of colonial rebellion. What follows is
a summary account, based on detailed case studies available in an Online Appendix.
4.1
The American Revolution (1776)
The American Revolution is the first example of successful rebellion against a European empire.
I argue that, by transferring Canada from France (F in the model) to Britain (M ), the peace
provisions of the Seven Years’ War (1756-1763; 7YW from now on) created a more favourable
global environment for the US (C) to rebel against Britain, because it made the remaining French
Empire scarcer in raw materials (it increased δ). Even before the 7YW, parts of the US South
faced a high δ. But the 7YW also gave a high δ to the US North, creating a coalition of colonies
that could look at trade prospects outside of the British empire with some confidence. US trade
was subject to a series of restrictions, designed to transfer part of the trade surplus to Britain
(in terms of the model, Britain had imposed {C, M, .}).42 Failure by Britain to adapt to the new
environment (by relaxing these restrictions) triggered the revolution.
42
The bulk of British extraction was represented by the Navigation Laws, whose most significant provision
required most US exports/imports to be intermediated through Britain, independent on their final destination/origin (Thomas, 1965; Sawers, 1992). The US colonies were also required not to trade with the foreign West
Indies. These restrictions biased US trade in favour of the British empire, something that in the model would be
captured by M choosing {C, M, .}.
28
Consider the trade of the US colonies in greater detail. The Lower South faced a low δ,
because the Spanish and French Empire (F ) had a large domestic supply of indigo, a key export
commodity of the Lower South, and were thus trade competitors: US indigo could only be sold
in Britain (and that only thanks to a preferential tariff), and conditions could not be expected
to improve following a revolution. Conversely, the Upper South faced a high δ, since its tobacco,
being far better than that found anywhere else (Price, 1964), was largely exported to continental
Europe: it could then be expected that F (and particularly France, whose fiscal revenues relied
on the sale of US tobacco) would adopt a favourable trade policy after a revolution. In the North,
δ was low before the 7YW. The colonies’ food exports were largely sent to the West Indies, where
they fed large populations of slaves.43 However, the French West Indies - by far the richest islands
in the region - were closed to US exports, because of both British and French restrictions. The
latter were due to the fact that France wanted Canada to develop as an exporter of food (Gould,
1939; Goebel, 1963; Dewar, 2010).
With the loss of Canada in the 7YW, the exclusion of US food from the West Indies became
increasingly untenable. Cut out from the North American trade, the French sugar planters saw
their terms of trade deteriorate sharply, resulting in acute economic distress (Goebel, 1963).
Perhaps more importantly, the loss of Canada meant that the dream of a self-sufficient French
empire was gone, removing much of the rationale for a protectionist policy. Despite opposition
by French food producers, restrictions were relaxed in the mid 1760s, resulting in a sharp increase
in illegal US trade with the French islands.44 In terms of the model, the 7YW increased δ for the
US North, transforming the French empire from a trade competitor into a trade partner. Such
an increase in δ probably also increased the cost of British extraction for the US North, because
it increased the cost of not being allowed to trade with the French West Indies (the cost of being
in {C, M, .}).
The argument that trade shaped the American Revolution is consistent with the economic
background, expectations, and early actions of the revolutionary leaders. The most prominent
among the Founding Fathers were, for the most part, tobacco planters (net sellers of x), and
merchants involved in the Atlantic trade (Sawers, 1992). These people would care about the trade
43
The West Indies absorbed 42% of the exports of the Middle Colonies in 1768-1772, and 62% of the exports
of New England. This compared with the 23% and 18% of the British Isles (McCusker and Menard, 1985).
44
Data is lacking, but abundant anecdotal evidence suggests that US exports to the foreign West Indies increased
significantly in the second half of the 1760s (Greene, 1980; Magra, 2006).
29
prospects of a rebel US.45 Interestingly, while they expected that revolution would disrupt trade
with Britain (see for example the diaries of John Adams; Hutson, 1980), they also expected that
other European countries would intervene in their support, given the importance of the American
trade for the Europeans (Hoffman and Albert, 1981). A number of diplomatic missions were sent
to Europe in the 1770s, with the primary goal of seeking commercial alliances.
Consistent with my argument, many of the groups who supported the revolution faced a good
trade environment, whereas those who opposed it did not. New England and Virginia were among
the most radical supporters of revolution, whereas the Lower South was more prudent. Virginia’s
tobacco planters and New England’s merchants were, as a group and as individual leaders, among
the most fervent supporters of the revolution (Sawers, 1992). And so was New England’s fishing
industry, who according to Magra (2006) had greatly benefited from an expansion of foreign
markets following the expulsion of the French from the Canadian fisheries. Interestingly, the
merchants who traded outside of the British Empire were particularly fervent supporters of the
revolution, and increasingly so in the 1760s (Sawers, 1992; Tyler, 1986). The latter fact is
consistent with an increase in δ leading to a higher cost of being in {C, M, .}, and in a much
higher probability of revolution if M is not willing to compromise on trade policy restrictions.
The American Revolution inflicted large domestic costs (µ) on the US, and a trade cost due
to disruption of trade with Britain (ΠC (R|δ) − ΠC (e
τ0 (E)|δ)). But the impact of the latter was
alleviated by increased trade with the rest of the world. Already during the war of 1776-1783
(when all trade between the US and Britain was forbidden), as a result of formal trade treaties,
the French, Spanish and Dutch empires became important trade partners (Shepherd and Walton,
1976). After the war, the US faced many new restrictions on British markets.46 As shown by
Shepherd and Walton (1976)’s study of trade before and after the revolution, US exports to
the British Empire suffered because of these restrictions, but were more than compensated by
expansion in other markets (Figure ??).47 Different colonies were hit differently, however, with
45
Sawers (1992) argues that the revolutionary leaders were much more involved in trade than average.
Most notably, it lost the preferential tariff on indigo, and was shut out of the British West Indies on a number
of key products.
47
It is important to acknowledge that the re-orientation of exports towards Europe was largely due to the
cessation of the Navigation Laws, and the re-orientation towards the foreign West Indies must be overestimated
in the data (since it does not consider smuggling to the foreign West Indies in 1768-1772, and to the British
West Indies in 1790-1792). The important message of Figure ?? is that US exports were well received in foreign
markets.
46
30
the Lower South being hit much harder than the Upper South or New England.48 In terms of
the model, F chose {., M, F } in the former case, {C, ., F } in the latter.
Figure 5: US exports to the British Empire and to the Rest of the World, 1768-1772
and 1790-1792. Units: millions of pounds. Source: Shephard and Walton, 1976.
Of course, French support of the American Revolution was also driven by geopolitical calculations, most notably the desire to weaken Britain. Still, grass-root economic forces of the sort
illustrated by the model were at play within the French empire. “Dependent on the New England trade, the French [West Indies] colonials were to favor the American cause; French colonial
officials were to open the island ports to American agents and to urge on the home authorities
a liberal trade policy.” (Goebel, 1963). These forces must have mattered a lot on the ground,
since it was down to colonial officials to decide which goods to admit into their ports.
My argument is closely related to an historical literature that has linked the American Revolution to the collapse of the French North American Empire. This literature has focused on
the fact that the latter eliminated a political threat to the colonial US. I argue that the 7YW
also removed an economic threat. In terms of the model, not only did µ decrease (because of
the elimination of the political threat) but also µ0 (δ) and µ(δ) increased (because δ did). Even
though both arguments may be valid, it is important to distinguish between the two.49
48
While the trade of the Lower South suffered a sharp decline in the 1780s and 1790s (Bjork, 1964; Williamson
and Lindert, 2008), exports of the Upper South went through a period of real prosperity in the 1780s, and tobacco
exports reached an all-time high in 1790-1792 (Bjork, 1964; Shepherd and Walton, 1976). As predicted by the
model (see Section ??) revolution improved the terms of trade of the Upper South. Similarly, New England’s
main exports were much higher in 1790-1792 than in 1768-1772 (Shepherd and Walton, 1976).
49
The argument is also related to the debate on the economic origins of the American Revolution. For a
discussion, see the Online Appendix version of the case study.
31
4.2
The Latin American Independence Wars (1808-1824)
Latin America (C) rebelled against Spain and Portugal (M ) in the early 19th century, taking
advantage of Napoleon’s invasion of Iberia (a decrease in µ). By 1824, after bitter independence
wars, it had become independent. I argue that this was a favourable period for Latin American
emancipation (µ0 (δ) and µ(δ) were high), because the industrial revolution was making Britain
(F ) more scarce in raw materials (it was increasing δ), thus increasing the amount of trade
support that the rebel colonies could expect to receive.
In the second half of the 18th century, Latin America exchanged mainly raw materials (x)
for manufactures (y), with both the Iberian peninsula and the rest of Europe. In line with the
model’s prediction that M chooses {C, M, .}, the latter trade was severely restricted.50 As the
industrial revolution relocated European manufacturing to Britain from the late decades of the
century (an increase in δ),51 trade between Latin America and Britain became more and more
important. In particular, as predicted by the model for an increase in δ, Britain increasingly
specialised in the export of manufactures, whereas Spain and Portugal increasingly turned into
exporters of raw materials (Figure ??); Britain sourced an increasing portion of her raw materials
from outside of her own empire (Figure ??); and much of this increase was made possible by an
increase in imports from Latin America (Figure ??).
Consistent with the model’s predictions, this change in trade patterns altered the political
economy of British trade policy in favour of a rebel colony. In the age of mercantilism (16501780), European empires were typically hostile to foreign sources of raw materials, partly because
of the mercantilist ideology itself (which praised the achievement of self-sufficiency or export
capacity), partly because of the lobbying of imperial producers (net sellers of x).52 But the
industrial revolution made it increasingly unfeasible for Britain to source raw materials from her
own empire, and tilted the political economy of British trade policy in favour of foreign sources.
Emblematic is the case of cotton, a booming import that the British government wanted to be
50
Latin American exports were to be sent to Spain and Portugal, from where they could be re-exported (and
symmetrically for imports). This gave the mother country better terms of trade, and the possibility to tax (either
implicitly or explicitly, through intermediation fees) the trade surplus of the colonies.
51
The British textile sector grew at an estimated 7% per annum between 1770 and 1815 (Crafts and Harley,
1992). Bairoch (1989) estimates that the British share in European manufacturing went from 15% in 1800 to
28% in 1830, and per-capita industrialisation from 110% of the European average in 1800 to 250% in 1830.
52
For example, one of the reasons why Britain chose Canada over the French West Indies as a spoil of war in
1763 was the opposition of British sugar producers.
32
Figure 6: Pattern of specialisation by type of commodity, 1785-1827. Sources: Davis (1979)
and Prados de la Escosura (1984). Data points in the British series are three-year averages of t − 1, t and t + 1.
Figure 7: Britain, estimated share of empire in imports of raw materials, total and
selected commodities, 1773-1855. Sources: Davis (1962, 1979), McCusker and Menard (1985). The
construction of these estimates required making several assumptions, described in the Online Appendix.
imported from the British West Indies, but that Manchester manufacturers succeeded in keeping
freely importable from all sources (Harlow, 1964).
In this new economic environment, Britain provided substantial support to the rebel colonies,
even though political logic must have advised against doing so.53 It provided a safe heaven for
Latin American conspirators, some of whom had access to the top echelons of British government
(Harlow, 1964). It refused to help Spain and Portugal to restore order, and took various steps to
prevent other European powers from doing so (Graham, 1993). And it was first to recognise the
53
Spain and Portugal were war-time allies of Britain (Miller, 1993; Kaufman, 1951). Furthermore, to support
a revolution was deeply at odds with the spirit of reaction that prevailed in Europe in that period, and it was
feared that it could help spread Jacobin principles around the world (Paquette, 2004; Harlow, 1964).
33
Total imports from LA
Share of LA in total imports.
Figure 8: Britain, total imports from Latin America and share of Latin America in
total imports, 1785-1855. Unit: thousands of £. Sources: the series for total British imports from Latin
America is the sum of direct imports from Latin America as recorded in the British data (Davis, 1979) and of the
estimate provided by Prados de la Escosura (1984b) for imports from Latin America through Spain. The series
for the share of Latin American in British imports of raw materials is direct imports from Latin America only
(Davis, 1979). All data points are calculated as three-year averages of the values in t − 1, t and t + 1.
newly-formed republics.54 Considerations about trade - largely shaped by the lobbying of British
industrialists - were a primary motivation behind this support (e.g. Harlow, 1964). Consistently,
Britain rushed to sign trade treaties immediately after granting recognition (Palmer, 1990). In
terms of the model, F chose {C, ., F }.
There is evidence that the Latin American revolutionaries were aware of the favourable economic conditions. For example, they used the offer of trade alliances as a bargaining tool in
dealing with the British, and were able to use the threat of trade sanctions to extract important
concessions from them.55 At the same time, trade prospects must have been important for the
revolutionaries, who were supported by an elite with strong links to the international economy:
for example, the cattle ranchers and merchants of Rio del Plata, the export-oriented landowners
of Venezuela, and the coastal planters and mine owners of Mexico (Graham, 1993).
My interpretation is consistent with the view that the Latin American Independence Wars
were triggered by the Napoleonic Wars, but highlights the economic conditions that made this
chain of events particularly likely to take place.56 It is related to a historical literature that
54
In addition, the British merchants lent more than £1 million to Simon Bolivar (the liberator of Gran Colombia), contributing to determine his success after 1816 (Graham, 1993).
55
The promise of trade alliances was one key argument used by conspirators in London (Harlow, 1964). In 1822
the government of Gran Colombia was able to use the threat of trade sanctions to secure important concessions
concerning the rights of Latin American ships entering British ports (Palmer, 1990).
56
An interesting comparison is with the War of Spanish Succession (1701-1714), when Spain and Portugal were
34
has emphasised the importance of international factors in explaining the Independence Wars
(e.g. Dominguez, 1980). It is also related to an argument according to which increasing volumes
of trade between Latin America and Northwestern Europe increased the cost of Spanish trade
restrictions (the cost of being in {C, M, .}), making revolution more likely (Graham, 1993). As
discussed in Section ??, this is what the model predicts for an increase in δ, when M prefers to
repress than to compromise on trade policy restrictions.
5
Conclusions
This paper has emphasised the importance of trade for the sustainability of empire. If the rest
of the world is relatively abundant in raw materials, their trade policy is unsupportive of a rebel
colony, and empire is more stable. Conversely, if the rest of the world is relatively scarce in
raw materials, their trade policy is supportive, and empire is less stable. I have argued that
this simple mechanism may help to explain the timing of the American Revolution (1776) and
the Latin American Independence Wars (1808-1824). I have also drawn several implications of
the analysis. Perhaps most interestingly, a mother country has an incentive to block colonial
industrialisation, because this prevents the formation of a group who would benefit from a
deterioration in imperial trade relations. This helps explaining why colonisers restricted colonial
industrialisation, and suggests a link between global trade disruption in the 1930s and 1940, and
the emergence of pro-independence movements in some African and Asian colonies in the 1950s.
The model has sharp predictions for the link between economic fundamentals, the pattern of
trade, and the probability of colonial rebellion (or secession more in general), as well as the role
played by third countries in this process. These predictions could be tested by taking advantage
of the fact that, close to the point where F switches from being a competitor of C to being
a partner, the cost of rebellion falls discretely. One could test whether episodes in which a
large country switched from exporting to importing a commodity x, were associated with more
secession in regions specialised in the production of x, and the role that the large country’s
diplomacy played in all this. I keep this and other related work for future research.
also temporarily helpless, but this did not trigger colonial rebellion. This difference may, at least in part, be
explained by different trade environments in the two periods.
35
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39
Appendix
Full national preferences over trade outcomes. As pC
A = 0, C is an exporter of x whenever
C
C
C
{C, M, F } C
it trades. If δ < 13 , it is pC
{C,M,.} > p{C,M,F } > p{C,.,F } , implying {C, M, .} C
C
C
{C, ., F } C {., M, F }. If δ ∈ 31 , 12 , it is pC
{C,M,F } > p{C,M,.} > p{C,.,F } , implying {C, M, F } C
C
{C, M, .} C {C, ., F } C {., M, F }. If δ ∈ 21 , 23 , it is pC
{C,M,F } > p{C,.,F } > p{C,M,.} , implying
M
{C, M, F } C {C, ., F } C {C, M, .} C {., M, F }. Turning to M , if δ < 31 , it is pM
A > p{.,M,F } >
M
M
{C, M, .} M
pM
{C,M,.} > p{C,M,F } . Thus, M is an importer whenever it trades, and {C, M, F } M
M
M
{., M, F } M {C, ., F }. If δ ∈ 13 , 21 , it is pM
A > p{.,M,F } > p{C,M,F } > p{C,M,.} . Thus, M is an
importer whenever it trades, and {C, M, .} M {C, M, F } M {., M, F } M {C, ., F }. If
M
M
M
δ ∈ 12 , 23 , it is pM
{.,M,F } > pA > p{C,M,F } > p{C,M,.} . Thus, M is an importer in {C, M, .}, and an
3
M M
exporter in {., M, F }. Simple algebra shows that v M (pM
{C,M,.} ) > v (p{.,M,F } ) for δ ∈ (0, 4 ). It
follows that {C, M, .} M {., M, F }, {C, M, F } M {C, ., F }. Finally, turning to F , if δ ∈ 0, 31 ,
it is pF{.,M,F } > pF{C,M,F } > pFA > pF{C,.,F } . Thus, F is an exporter in {., M, F }, an importer in
{C, ., F }. Simple algebra shows that v F (pF{.,M,F } ) > v F (pF{C,.,F } ) for δ ∈ 0, 14 , v F (pF{.,M,F } ) <
v F (pF{C,.,F } ) for δ ∈ 41 , 1 . It follows that {., M, F } F {C, ., F }, {C, M, F } F {C, M, .} if
δ ∈ 0, 41 , {C, ., F } F {., M, F }, {C, M, F } F {C, M, .} if δ ∈ 41 , 31 . If δ ∈ 13 , 12 , it is
pF{.,M,F } > pFA > pF{C,M,F } > pF{C,.,F } . Thus, F is again an exporter in {., M, F }, an importer
in {C, ., F }, and it is {C, ., F } F {., M, F }, {C, M, F } F {C, M, .}. If δ ∈ 12 , 23 , it is
pFA > pF{.,M,F } > pF{C,M,F } > pF{C,.,F } , F is an importer whenever it trades, and {C, ., F } F
{C, M, F } F {., M, F } F {C, M, .}.
Proof to Lemma ??. If δ < 14 , only {., F, M } may be realised in a CPNE: not {C, M, .}
(since τCM = 0), not {C, ., F } (since {., M, F } F,M {C, ., F }), not {C, M, F } (since {., M, F } F
F
{C, M, F }, and {C, M, F } requires τCF = τM
= 1). But {., M, F } may be realised in a CPNE (if
F
τM
= τFM = 1, τCF = 0) since F is at its first best and M may only deviate to {., ., .}. The case
δ≥
1
4
can be argued symmetrically.
Proof to Lemma ??. Only {C, M, F } may be realised in a CPNE: not {C, M, .} (since
{C, M, F } M,F {C, M, .} if δ < 31 , {C, M, F } C,F {C, M, .} if δ ≥ 31 ), not {C, ., F } (since
{C, M, F } C,M {C, ., F }), not {., M, F } (since {C, M, F } C,M {., M, F } if δ <
1
,
3
and
{C, M, .} C,M {., M, F } if δ ≥ 13 ). But {C, M, F } may be realised in a CPNE, since if δ < 13 , M
is at its first best, and C and F could only either individually deviate to {., M, F } and {C, M, .},
40
or jointly deviate to {C, ., F } ≺C {C, M, F }; if δ ≥ 13 , C is at its first best, M and F could
only either individually deviate to {C, ., F } and {C, M, .}, or jointly deviate to {., M, F } (and
{C, M, F } M {., M, F } if δ ∈ 0, 21 , {C, M, F } F {., M, F } if δ ∈ 21 , 23 .
Proof to Proposition ??. Denote by τbT (E), TbT (E) the policy equilibrium that is realised at
M ’s constrained optimum. If T = 1, it must be: Tb1 (E) = min [ΠC (b
τ1 (E)|δ) + µ − ΠC (R|δ) − B], ΠC (b
τ1 (E)|δ
But then, M ’s payoff at the constrained optimum is V M (b
τ1 (E)|δ) = ΠM (b
τ1 (E)|δ)+ΠC (b
τ1 (E)|δ)+
const. = v M [pM (b
τ1 (E)|δ)] + v C [pC (b
τ1 (E)|δ)] + const. I have already shown (see footnote ??)
that M ’s desired trade outcome must be realised in a CPNE. Thus, the trade outcome that
must be realised at the constrained optimum is the one that maximises v M (pM ) + v C (pC ), and
this is always {C, M, F }. To see the latter point, use the notation CM to denote preference
by C and M joint (by the criterion of maximising v C (pC ) + v M (pM )). It is {C, M, F } CM
{C, M, .}, since v M (p) + v C (p) = 2
p+ 1−δ
√2
2 p
reaches a minimum at
1−δ
=
2
C,M
Furthermore, it is {C, M, F } C,M {C, ., F }; and it is {C, M, F } M
pC
{C,M,.} = p{C,M,.} .
{., M, F } if δ <
1
,
3
{C, M, F } CM {C, M, .} C,M {., M, F } if δ > 31 . Since τb1 (E) is such that the trade outcome
is {C, M, F }, we can write Tb1 (E) = min [ΠC (I) + µ − ΠC (R|δ) − B], ΠC (I) . It immediately
follows that, if µ ≥ µ1 (δ) and µ ∈ [µ(δ), µ1 (δ)), it is respectively Tb1 (E) = ΠC (I) > 0 and
Tb1 (E) = ΠC (I) + µ − ΠC (R|δ) − B > 0. Since S = E leads to the same trade outcome that
is realised when S = I and to a positive transfer, to set S = E must be optimal for M . If
µ < µ(δ), however, it is Tb1 (E) < 0, and it must be optimal for M to set S = I. If T = 0,
it is always Tb0 (E) = 0. To find τb0 (E), suppose µ ≥ µ0 (δ), so that M ’s constrained optimum
coincides with its unconstrained optimum. At the latter, the trade outcome is {C, M, F } if
δ < 13 , {C, M, .} if δ ≥ 13 . This implies that µ0 (δ) = B + ΠC (R|δ) − ΠC (I) = µ(δ) if δ < 13 ,
1
1
µ0 (δ) = B + ΠC (R|δ) − ΠC (pC
{C,M,.} ) > µ(δ) if δ ≥ 3 . Consider first the case where δ < 3 . If
µ > µ0 (δ), M ’s payoff at the constrained optimum is the same as at its unconstrained optimum,
and so it must set S = E. If µ < µ(δ), M must offer to C a better outcome than {C, M, F } at
the constrained optimum. However the only better outcome for C is {C, M, .} ≺M {C, M, F }.
Thus, M must set S = I in this case. Next, consider the case where δ ∈ 13 , 32 . If µ > µ0 (δ), M ’s
payoff at the constrained optimum is the same as at its unconstrained optimum, and so it must
set S = E. If µ < µ0 (δ), M must offer to C a better outcome than {C, M, .} at the constrained
optimum. The only better outcome for C is {C, M, F }, which gives M the same payoff as under
independence. If µ ∈ µ(δ), µ0 (δ) , to offer {C, M, F } is enough to prevent a revolution, and so
41
M must set S = E. If µ < µ(δ), to offer {C, M, F } is not enough to prevent a revolution, and
so M must set S = I.
Proof to Proposition ??. The first part follows from Lemma ??. From Proposition ??, it is
S = I iff µ < µ(δ) = B − [ΠC (I) − ΠC (R|δ)]. By Lemma ??, ΠC (R|δ) = 0 for δ < 41 , ΠC (R|δ) =
q
pC
{C,.,F }
2
< 14 ,
> 0 if δ ≥ 14 . The latter is increasing in δ, since pC
{C,.,F } is. Thus, µ(δ) is constant for
strictly higher and increasing if δ ≥ 14 . This proves the pattern of decolonisation. If T = 1,
∗
R µ1 (δ) ∗
R N T∗
1 (T ∗ )2
∗ Nµ −[µ(δ)+T ]
expected extraction is µ(δ)
dµ
=
+
T
. The
[T − (µ1 (δ) − µ)]dµ + µ µ(δ) N
Nµ 2
Nµ
µ
δ
1
fact that this is decreasing in µ(δ) implies that expected extraction follows a pattern symmetric
to that of decolonisation.
Proof to Proposition 4. Denote punishment by S = P . If η T (δ, r) ≤ 0, PrT {S ∈ {R, P }} =
0. If η T (δ, r) > 0:57
η 1 (δ,r)
µ(δ) [η 1 (δ, r)]2
+
Nµ
2Nµ Nη
µ(δ)
Z µ0
µ(δ)
µ(δ) η 0 (δ, r) µ0 − µ(δ)
Pr {S ∈ {I, R, P }} =
+
f (µ)G[η 0 (δ, r)]dµ =
+
0
Nµ
Nµ
Nµ Nη
µ(δ)
η 1 (δ, r)µ(δ) [η 1 (δ, r)]2
+
Pr {S = R} = r
1
Nµ Nη
2Nµ Nη
(
)
η 0 (δ, r)µ(δ) η 0 (δ, r) µ0 − µ(δ)
.
Pr {S = R} = r
+
0
Nµ Nη
Nµ Nη
µ(δ)
Pr {S ∈ {I, R, P }} =
+
1
Nµ
Z
f (µ)G[η 1 (δ, r) − (µ − µ(δ))]dµ =
Study η 1 (δ, r) = (1 − r)T ∗ − r ΠM (I|δ) − ΠM (R|δ) . Let B ≡ ΠM (I|δ) − ΠM (R|δ) . It is:
B|δ< 1 =
1
3
4
B|δ> 1 =
4
1
3
√ +1−δ
δ √
q
= const. −
3− 2
2
2 12
√
+1−δ
1 √
q
− vAM = const −
δ 3+2 1−δ ,
2
2 1
+1−δ
q
−
2 13
1
2
3
from which it is easy to show that B is decreasing in δ for δ 6= 41 , increasing for δ = 14 . Fur thermore, B > 0 for δ ∈ 0, 23 , B = 0 for δ = 32 . With T ∗ constant, this implies the following
b η 1 (δ, r) > 0 if
pattern for η 1 (δ, r): there exist r, r0 such that, if r ≥ r, η 1 (δ, r) < 0 if δ < δ,
δ > δb (where δb > 41 ); if r ≤ r0 , η 1 (δ, r) ≥ 0 ∀δ; if r ∈ (r0 , r), η 1 (δ, r) is negative then positive for
57
Notice that it is always η T (δ, r) < µT − µ(δ).
42
1
1
1 2
,
,
turns
negative
at
δ
=
,
negative
then
positive
again
for
δ
∈
,
. The equilibrium
3 4
4
4 3
b it is Pr1 {S ∈ {R, P }} = 0, and we are back to the
then looks as follows: if r > r and δ ≤ δ,
b Pr1 {S ∈ {I, R, P }} and Pr1 {S = R} are increasing in δ, since
baseline. If r > r and δ > δ,
δ∈
1
µ(δ) is non-decreasing and η 1 (δ, r) is increasing. By the same logic, if r < r0 and δ 6= 14 , the two
probabilities are also increasing in δ. Furthermore, there exists r ≤ r0 such that, if r < r, they
are also increasing at δ = 41 . To see this, notice that:
∂Pr1 {S ∈ {I, R, P }} ∂δ
=
δ= 14
∂µ(δ) ∂δ δ= 41
Nµ
2η 1 (δ, r)
+
∂η 1 (δ,r) ∂δ
δ= 14
(19)
2Nµ Nη
"
#
∂µ(δ) r
∂Pr1 {S = R} ∂η 1 (δ, r) 1 = Nµ Nη
1 µ(δ) + η 1 (δ, r) + η 1 (δ, r) ∂δ 1 ,
∂δ
∂δ
δ=
δ=
δ=
4
4
4
(20)
and both derivatives are positive for r small enough.58 Finally, for r ∈ (r, r), Pr1 {S ∈ {I, R, P }}
and Pr1 {S = R} are non-decreasing in δ for δ 6= 41 , can be increasing or decreasing for δ = 14 .
τ0 (E)|δ) − ΠM (I|δ) −rB. Let A ≡ ΠM (e
τ0 (E)|δ) − ΠM (I|δ) .
Next, study η 0 (δ, r) = (1−r) ΠM (e
1 2
M M
)
=
0.
If
δ
∈
If δ < 31 , A = v M (pM
)
−
v
(p
, , it is A = v M (pM
{C,M,F }
{C,M,F }
{C,M,.} ) −
3 3
1
1 2
v M (pM
{C,M,F } ), This is zero for δ = 3 , positive for δ ∈ 3 , 3 . Furthermore, it has a parabola
shape, since:
!
r
1
3
∂A 5 1
5
=
− + +
(1 − δ) > 0 iff δ <
∂δ δ> 1
1−δ
4 2
2
8
3
!
r
1
3 1
1 3
∂ 2 A =
− √
+
< 0.
3
2
∂δ δ> 1
4 1−δ 2 2
2
2(1
−
δ)
3
The pattern of A and B together imply the following pattern for η 0 (δ, r). For r < 1, there
exist δ ∈ 13 , 23 such that, in the range 0, 32 , it is η 0 (δ, r) > 0 iff δ ∈ δ, 23 . Furthermore,
η 0 (δ, r) is parabola shaped in this range.59 There then exists δ ∈ δ, 23 (and δ > 58 ) such
58
The first term on the RHS of (20) is positive, and does not depend on r; the second term is negative, but
converges to 0 as r converges to 0. Whereas for (21), this converges to 0 from above as r converges to 0, since
the term in brackets converges to a positive and finite number.
59
Given (1 − r)A is parabola shaped, and −rB is increasing,
η 0 (δ,r) is either
q
√ parabola
shaped or increasing.
∂η 0 (δ,r) 3
1
1
3
1
However, the fact that
3 − √1−δ < 0 implies that it is
2 = (1 − r) 1−δ − 4 + 2 (1 − δ) + r 2
∂δ
δ= 3
parabola shaped. Finally, since η 0 (δ, r) is negative for δ = 13 , positive for δ = 23 , there exists δ as indicated.
43
that η 0 (δ, r) is increasing for δ < δ, decreasing for δ > δ. The equilibrium then is: if δ < δ,
Pr0 {S ∈ {R, P }} = 0. If δ ∈ δ, δ , Pr0 {S ∈ {I, R, P }} and Pr0 {S = R} are positive and
increasing in δ, since µ(δ), µ0 − µ(δ) and η 0 (δ, r) are. If δ > δ, they are positive but may be
increasing or decreasing, since η 0 (δ, r) is decreasing. Also, δ is increasing in r, since η 0 (δ, r)
∂B is decreasing; and δ is increasing in r, since ∂A
<
0
(since
−
> 0), which implies
∂δ δ=δ
∂δ δ=δ
∂ 2 η 0 (δ,r) > 0. For r = 1, η 0 (δ, r) < 0 ∀δ, and δ = δ = 32 .
∂δ∂r δ=δ
n h
i
h
io
∗
C
F
F
C
F
F
C
Proof to Proposition 5. Define δ (φ ) = argδ v p{.,M,F } (φ ) = v p{C,.,F } (φ ) ,
C
C
where pF{C,.,F } (φC ) =
shown numerically60
δ+ φ3
2−φC
1− φ3
2+φC
. It is δ ∗ (0) = 41 . Furthermore, it can be
that δ ∗ (φC ) is increasing, and δ ∗ 21 = 13 . Define δe = δ ∗ (b) ∈ 41 , 13 (since
and pF{.,M,F } (φC ) =
b is small, it must be δ ∗ (b) < 13 ). Suppose φC
d is the expected decentralised equilibrium, and
C C bC
bC
consider M ’s decision to set a cap φbC ∈ 0, φC
d . Denote by v [p (φ ), φ ] C’s utility when
facing price pC (φbC ). Its first derivative is:
C
∂v C [pC (φbC ), φbC ]
=
∂ φbC
∂
bC )pC (φ
bC )+ φb
(1−φ
3
√
2
bC )
pC (φ
∂ φbC
= −pC (φbC ) +
bC
1 ∂pC (φbC )
pC (φbC ) − pC
A (φ )
+
(1 − φbC )
,
3
∂ φbC
2pC (φbC )
(21)
p
where (1 − φbC )
C (φ
bC )−pC (φ
bC )
A
C
C
b
2p (φ )
1
C
bC
bC
= −mC
x . If δ < 4 , Π (R|δ, φ ) is at a minimum, and π(δ, φ ) at
a maximum, for φbC = 0. This implies that M ’s payoff is at a maximum for φbC = 0. To see
C
C
bC
bC
bC
this, notice that C faces pC
A (φ ) after a revolution, ∀φ ∈ 0, φd . Thus, it is Π (R|δ, φ ) =
v C [pC (φbC ), φbC ], which is increasing in φbC ∀φbC ∈ [0, φC ]. The latter point can be seen by noticing
A
d
1
bC
−pC
A (φ ) + 3
bC
bC
φ
3
that the derivative in (??) simplifies to
= − 1−φbC + 13 , which is positive ∀φbC ≤ 21 . If
e ΠC (R|δ, φbC ) is again at a minimum for φ = 0, implying that M ’s payoff is at a maximum
δ > δ,
bC ) after a revolution, ∀φbC ∈ 0, φC .
for φbC = 0. This is because, in this case, C faces p = pC
(
φ
d
{C,.,F }
C
C
C C
C
C
b
b
b
It is then Π (R|δ, φ ) = v [p{C,.,F } (φ ), φ ], which is increasing in φbC ∀φbC ∈ 0, φC
d . To see
the latter point, notice that the derivative in (??) is positive for φbC small (which is granted by
bC
δ+ φ
1
1
C
3
bC
the fact that b, and thus φC
d , is small). This is because −p{C,.,F } (φ ) + 3 = − 2−φ
bC + 3 , which
∂pC
bC )
(φ
}
A bC
bC
bC small. If
is positive for δ < 23 1 − φbC ; {C,.,F
> 0; and pC
bC
{C,.,F } (φ ) > pC (φ ) with φ
∂φ
h i
h
i
C
C
C
δ ∈ 14 , δe , define φ ≡ argφCd δ ∗ (φC
, it is δ ≥ δ ∗ (φC
d ) = δ . If φd ∈ 0, φ
d ), implying that C
60
Results are available from the author upon request.
44
bC
bC
faces pC
0, φC
. Earlier logic then implies that M ’s payoff is
d
{C,.,F } (φ ) after a revolution,∀φ ∈
i
C
∗
C
C bC
at a maximum for φbC = 0. If φC
d ∈ φ , b , it is δ < δ (φd ). This implies that C faces pA (φ ) if
C
C
C
bC
b
φbC ∈ (φ , φC
d ], p{C,.,F } (φ ) if φ ∈ [0, φ ]. Earlier logic then implies that M ’s payoff is decreasing
C
C
in φbC for φbC ∈ (φ , φC ], but this does not need to be true for φb ∈ [0, φ ]. Thus, while M ’s payoff
d
d
must be at a maximum for some φbC < φC
d , this does not need to be zero. 45