Chapter 1 Introduction
3.4 What is Behind Crop Fixed Effects?
3.4.1 Theoretical Framework
Model
There are two agents: Europeans and Africans. The Europeans make profit by buying or producing crops in Africa and reselling them at higher prices in Europe.
They decide whether to leave the production to Africans (peasant agriculture) or to establish plantations. If they decide to establish plantations, they use contract labor to procure the workers they need.8 If the production is left to Africans, the Europeans can choose between two institutional arrangements: free peasant production or com- pulsory production. If the free peasant production is chosen, Africans decide whether to produce cash crops for the colonizers or to stay in the native economy. If compul- sory production is chosen, the Africans have to provide the established quantity at a price fixed by the colonizers.
The total population isL. The only input for production is laborl and production functions arey=E(l) in the European sector (plantation or peasant agriculture) and N(L−l) in the native economy. All production functions are increasing and non- convex. The costs for Europeans of the different institutional arrangements are the following: F(y) increasing with F(0) >0 if they use contract labor; Q(y) increasing withQ(0) >0 if they use compulsory cultivation;pAE(l) if they use , wherepA is the price paid to Africans for production.9
If the Europeans use free peasant production, the price pA is determined in equi- librium. In particular it is such that the marginal utility of labor for Africans is the same in the native economy and in the peasant agriculture sector. That is pA=N0(L−l)/E0(l).
The Europeans choose l to maximize their profit under each of the three institu-
8The level of compulsion under contract labor varies. Formally, Africans work for the colonizers for a wagef ∈[0, w], where 0 is the wage when contract labor is equal to forced labor andwis the free labor wage.
9These costs include the cost of collecting the produce or recruiting the workers, monitoring costs, and enforcement costs.
tions and compare the payoffs. The profit functions are the following:
• ΠM = (p−NE0(L−l)0(l) )E(l) if free peasant production
• ΠQ =pE(l)−Q(E(l)) if compulsory production and peasant agriculture
• ΠP = pE(l)−F(E(l)) if contract labor and plantation where p is the world market price.
To derive an explicit solution, let us assume specific functional forms. In particular:
N(l) = log(l), E(l) = αl, Q(y) = Q+qy, F(y) = F +f y.10 f and q may include respectively the non-market wage and the non-market price of crops paid to Africans.
The market price of crops in equilibrium is pA= 1/α(L−l).
Let us assume that the fixed cost of establishing plantations and using contract labor is higher than the fixed cost of relying on peasant agriculture and using com- pulsory cultivation, that is F > Q.
Consider now the profit functions under each labor arrangement.
• Free peasant agriculture. The profit function is ΠM = (p−α(L−l)1 )αl. This is maximized when l = L −q
L
αp.11 The maximum profit is then Π∗M = (√
αLp−1)2 if p≥1/(αL), Π∗M = 0 otherwise.
• Compulsory cultivation and peasant agriculture. The profit function is ΠQ = pαl −Q−qαl. This is maximized when l = L if p > αLQ +q; l = 0 otherwise. The maximum profit is Π∗Q = pαL −Q −qαL if p > αLQ + q;
Π∗Q = 0 otherwise.12
10The main predictions of the model are not affected by these specific functional forms. See end of section.
11The ‘plus’ solution is not acceptable because it would implyl > L. Sincel must be greater or equal to zero, we must havep≥1/(αL).
12l =L is consistent with the universalistic character of French colonial institutions: no one is excluded from corv´ee labor and quotas obligations.
• Contract labor and plantation. The profit function is ΠP =pαl−F −f αl.
This is maximized when l = L if p > αLF +f; l = 0 otherwise. The maximum profit is then Π∗P =pαL−F −f αL if p > αLF +f; Π∗P = 0 otherwise.
The following proposition defines the choice of the colonizer.
Proposition 1 The colonizer chooses the following institutions:
• free peasant production, if the price is low,1/(αL)< p < minn(1+F+f αL)2
4αL ,(1+Q+qαL)4αL 2o .
• compulsory production, if the price is high and either the marginal cost of compulsory production or the scale of production are low. In particular if p > (1+Q+qαL)4αL 2 and q < f or if p > (1+Q+qαL)4αL 2, q > f, and αL < Fq−f−Q.
• contract labor, if the price is high and both marginal cost of compulsory produc- tion and scale of production are high. In particular if p > (1+F4αL+f αL)2, q > f, and αL > F−Qq−f .
• no production, otherwise.
Proof. Compare the maximized profits for each of the three institutions. The profit from free peasant production is higher than the profit from contract labor if p <
(1+F+f αL)2
4αL and higher than the profit from compulsory production ifp < (1+Q+qαL)4αL 2. The profit of contract labor is higher than the profit of compulsory production when q > f and αL > Fq−f−Q.
The profit of compulsory production is higher than the profit of contract labor whenq < f or whenq > f andαL < F−Qq−f . Moreover, notice that (1+Q+qαL)4αL 2 > αLQ +q:
if the profit from compulsory production is higher than the profit from free peas- ant production , then the profit from compulsory production is positive. Similarly
(1+F+f αL)2
4αL > αLF +f: if the profit from contract labor is higher than the profit from free peasant production, then the profit from contract labor is positive
Notice that the proposition is true in the most general case. As long as the marginal productivity of labor is decreasing in the indigenous sector, free peasant production is chosen only when the price is low. When the price is high:
• compulsory production is chosen whenF(0)> Q(0) and F0(.)> Q0(.), or when Q(0)> F(0), Q0(.)< F0(.), and production is high;
• contract labor is chosen when Q(0) > F(0) and Q0(.)> F0(.), or when F(0)>
Q(0), F0(.)< Q0(.), and production is high.
In the next subsection, I provide the intuition for these results.
IntuitionFigure 3.1 shows the choice of institution for a given crop as a function of the colony’s total productive capacity αL and prices p in each of the two possible cases: f < q or f ≥ q.13 Green regions represent contract labor/plantation, yellow regions represent compulsory cultivation/peasant agriculture, red regions represent free peasant production, white regions represent absence of production.
Effect of price (p). Given the decreasing productivity of labor in the indigenous economy, the marginal cost of labor under free peasant production is increasing.
For this reason, when the colonizers use free peasant production they do not use the whole labor force of the colony. On the other hand, under contract labor and compulsory production all the labor force is employed since the cost of labor is linear.
Thus, an increase in price increases the colonizer’s revenue more under contract labor /compulsory production than under free peasant production .
If the price of the crop is low, then the total revenue from the colony is not enough to pay for the fixed costs of compulsory production or plantation. Free peasant pro- duction is chosen. As the price increases, the revenue from free peasant production increases less than the revenue from the other two institutions. At a certain price,
13The values of parameters F and Q are 10 when low and 20 when high; the values of parameters f and q are 4 when low and 8 when high;α= 1.
Figure 3.1 Institutional regions
the revenue is high enough to compensate the fix cost and contract labor/compulsory production is chosen.
Effect of scale of production (αL). The fixed cost of establishing a planta- tion is higher than that of establishing compulsory cultivation (F > Q). Thus, in places where the cost of an additional unit is higher under contract labor than under compulsory production (f > q), the colonizer will only choose between compulsory production and free peasant production. This is the situation of panel A of figure 3.1.
In places where the cost of an additional unit is higher under compulsory pro- duction than under contract labor (q > f), when the scale of production increases enough contract labor becomes more attractive than compulsory production, as shown in panel B of figure 3.1.