Income and Substitution Effects in Aid Allocation

Author

Bas Machielsen

Published

January 23, 2026

Introducton

While working on a model of foreign aid allocation, I encountered a puzzling intuition clash regarding the effects of increased (decreased) aid effectiveness on the total spending on foreign security. Intuitively, one might think that if aid becomes more effective, a country would need to spend less money to achieve the same level of security. However, standard economic theory suggests that when the “price” of a good decreases (in this case, the cost-effectiveness of foreign aid), the quantity demanded should increase, potentially leading to higher total spending.

To resolve this paradox, we need to carefully analyze the income and substitution effects using a more general utility framework, particularly focusing on the case where foreign security is treated as an inferior good. Particularly, we have to allow Foreign Security (\(S\)) to potentially be an inferior good, and apply the Slutsky decomposition to the expenditure function directly.

Setting

We analyze the impact of changes in the effectiveness of foreign aid on the total spending on foreign security. We are a utility-maximizing agent who allocates resources between Domestic Defense (\(D\)) and Foreign Security (\(S\)). The effectiveness of foreign aid is denoted by \(e\), which inversely affects the price of foreign security.

  • \(I\): Fixed Income.
  • \(P_s = 1/e\): Price of Foreign Security (inverse of effectiveness).
  • \(S(P_s, I)\): The Marshallian demand for Foreign Security.
  • \(A = P_s \cdot S\): The quantity of money psent on aid (Expenditure).

We want to find the sign of \(\frac{\partial A}{\partial e}\).

To do this, we first find \(\frac{\partial A}{\partial P_s}\) (change in spending w.r.t price) and recall that Price and Effectiveness move in opposite directions.

Analysis

We start by differentiating the Spending Identity (\(A = P_s \cdot S\)) with respect to Price (\(P_s\)):

\(\frac{\partial A}{\partial P_s} = \underbrace{S}_{\text{Price Effect}} + \underbrace{P_s \cdot \frac{\partial S}{\partial P_s}}_{\text{Quantity Effect}}\)

This equation tells us that when price rises, spending changes for two reasons:

  1. Price Effect (\(S\)): You pay more for the units you are already buying.
  2. Quantity Effect (\(P_s \frac{\partial S}{\partial P_s}\)): You adjust the number of units you buy.

We substitute the term \(\frac{\partial S}{\partial P_s}\) using the Slutsky Equation:

\(\frac{\partial S}{\partial P_s} = \underbrace{\frac{\partial S^h}{\partial P_s}}_{\text{Substitution (SE)}} - \underbrace{S \frac{\partial S}{\partial I}}_{\text{Income (IE)}}\)

Substitute this back into the spending derivative:

\(\frac{\partial A}{\partial P_s} = S + P_s \left[ \frac{\partial S^h}{\partial P_s} - S \frac{\partial S}{\partial I} \right]\)

Rearranging terms to group the effects:

\(\frac{\partial A}{\partial P_s} = \underbrace{S \left( 1 - P_s \frac{\partial S}{\partial I} \right)}_{\text{Modified Income Effect Term}} + \underbrace{P_s \frac{\partial S^h}{\partial P_s}}_{\text{Pure Substitution Effect on Spending}}\)

Inferior vs Normal Goods

Let’s analyze what happens when effectiveness rises (\(e \uparrow\)), which means price fals (\(P_s \downarrow\)).

We look at the components of the change. Note that because we are looking at a price drop, the mathematical signs flip regarding the impact on \(A\).

  1. Pure Substitution Effect

\(\text{Term: } P_s \frac{\partial S^h}{\partial P_s}\)

Since \(\frac{\partial S^h}{\partial P_s}\) is always negative (Law of Demand), as aid becomes more effective (price drops), you substitute toward aid and away from domestic defense. Since you buy more units, this effect pushes you to increase spending (or at least keeps spending high).

  1. Income Effect

\(\text{Term: } - P_s S \frac{\partial S}{\partial I}\)

Here, the sign of \(\frac{\partial S}{\partial I}\) is crucial. If \(S\) is a normal good, then \(\frac{\partial S}{\partial I} > 0\), and the Income Effect would push spending up as well. However, if \(S\) is an inferior good, then \(\frac{\partial S}{\partial I} < 0\), and the Income Effect pushes spending down.

In the latter case, the price of aid falls (\(P_s \downarrow\)), your Real Income increases, and because Foreign Security is inferior, you decide you don’t need as much foreign security now that you are “richer” (perhaps you prefer to rely on domestic defense \(D\), which implies sovereignty/pride). Therefore you reduce your Quantity of \(S\). This effect drives spending down.

Summary

Under the assumption that Foreign Security is an inferior good, the overall effect of an increase in aid effectiveness (\(e \uparrow\)) on total spending (\(A\)) can theoretically be negative. The mechanism is that the Income Effect might dominate the Substitution Effect, leading to a reduction in spending on Foreign Security as aid becomes more effective.