The fiscal impact of the War of the Pacific

Abstract

In the War of the Pacific (1879–1883), Chile defeated Peru and Bolivia, and acquired territories that contained vast deposits of sodium nitrate, a leading fertilizer. Chile’s export tax on nitrates later accounted for at least one half of all government revenue. We employ a multi-country model of export taxation in order to simulate the potential government revenues that Bolivia, Chile and Peru could have earned under the counterfactual scenario that Chile did not conquer the nitrate-rich provinces of its adversaries. Our results are that Peruvian and Bolivian government revenues could have been at least double their historical levels. We estimate that, over the remainder of the nineteenth century, Chile’s earnings from nitrates would have fallen by 80%.

Appendix A: three-country model of export taxation

Three countries export a product to a world market. None of the countries has a domestic market for the product. We assume linear demand and supply, and a perfectly competitive industry. The relevant demand and supply equations are:

World demand:

$$ Q^{D} = A - BP $$

Country 1 supply:

$$ q_{1}^{s} = a_{1} + b_{1} p_{1} $$

Country 2 supply:

$$ q_{2}^{s} = a_{2} + b_{2} p_{2} $$

Country 3 supply:

$$ q_{3}^{s} = a_{3} + b_{3} p_{3} $$

Each country sets an ad valorem tariff with respect to the world price:

$$ P_{1} = P(1 - t_{1} ) $$

$$ P_{2} = P(1 - t_{2} ) $$

$$ P_{3} = P(1 - t_{3} ) $$

We proceed with the equations for country one, noting that the equations for the other countries are symmetric.

Country one residual demand:

$$ \begin{aligned} Q_{1}^{D} & = Q^{D} - q_{2}^{s} - q_{3}^{s} \\ & = (A - a_{2} - a_{3} ) - (B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} ))P \\ \end{aligned} $$

The marginal revenue function for country one is:

$$ {\text{MR}}_{1} = \frac{{ - 2Q_{1}^{D} + A - a_{2} - a_{3} }}{{B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} )}} $$

The marginal cost function for country one is:

$$ {\text{MC}}_{1} = p_{1} = \frac{1}{{b_{1} }}(q_{1}^{s} - a_{1} ) $$

We now solve for the reaction functions yielding the Nash optimum export taxes.

First, set ${\text{MC}}_{1} = {\text{MR}}_{1}$

and $q_{1}^{s} = Q_{1}^{D} = q_{1}$

$$ \frac{{q_{1} }}{{b_{1} }} - \frac{{a_{1} }}{{b_{1} }} = \frac{{ - 2q_{1} + A - a_{2} - a_{3} }}{{B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} )}} $$

solve for q ₁

$$ q_{1} = \frac{{a_{1} (B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} )) + (A - a_{2} - a_{3} )b_{1} }}{{B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} ) + 2b_{1} }} $$

Now, by substituting the above expression for q ₁ into country one’s demand we get P as a function of the parameters, A, B, a _i , b _i .

$$ P = - \left( {\frac{{q_{1} - A + a_{2} + a_{3} }}{{B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} )}}} \right) $$

And by substituting the expression for q ₁ into country one’s supply we get p ₁ as a function of the parameters.

$$ p_{1} = \frac{1}{{b_{1} }}(q_{1} - a_{1} ) $$

With expressions for P and p ₁, we can derive the reaction function for optimal export tariff:

$$ t_{1} = \frac{{P - p_{1} }}{P} $$

We now solve the reaction functions yielding the Nash maximum revenue taxes for country one. From above, we know that:

$$ T = (P - p_{1} )q_{1} $$

where T is tax revenue.

$$ P = - \left( {\frac{{q_{1} - A + a_{2} + a_{3} }}{{B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} )}}} \right) $$

Solving for T:

$$ \begin{aligned} T & = \left( {\frac{{ - q_{1} + A - a_{2} - a_{3} }}{{B()}} - \frac{{q_{1} - a_{1} }}{{b_{1} }}} \right)q_{1} \\ & = \left[ {\left( {\frac{{A - a_{2} - a_{3} }}{{B()}} + \frac{{a_{1} }}{{b_{1} }}} \right) - \left( {\frac{1}{{B()}} + \frac{1}{{b_{1} }}} \right)q_{1} } \right]q_{1} \\ \end{aligned} $$

Maximize taxing revenue, the first-order condition is:

$$ \frac{{A - a_{2} - a_{3} }}{{B()}} + \frac{{a_{1} }}{{b_{1} }} - 2q_{1} \left( {\frac{1}{{B()}} + \frac{1}{{b_{1} }}} \right) = 0 $$

Solving for q ₁

$$ q_{1} = \frac{{a_{1} (B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} )) + (A - a_{2} - a_{3} )b_{1} }}{{2(B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} ) + b_{1} )}} $$

Substituting, we obtain expressions for P and p ₁ as a function of the parameters.

$$ P = - \left( {\frac{{q_{1} - A + a_{2} + a_{3} }}{{B + b_{2} (1 - t_{2} ) + b_{3} (1 - t_{3} )}}} \right) $$

$$ p_{1} = (1/b_{1} )*(q_{1} - a_{1} ). $$

And from these we get the reaction function for country one:

$$ t_{1} = (P - p_{1} )/P. $$

This is the same form as the reaction function under the optimal tariff, but q ₁ under the maximum revenue tariff is less than q ₁ under the optimal tariff. Note that under the maximum revenue tariff the denominator is multiplied by two. As consequence, when substitutions are made, the reaction functions are algebraically different. We solved the simulations using the mathematical software MAPLE.