In many developing regions, including islands like Roatan, a puzzling economic situation exists: despite the long-term cost-effectiveness of water sanitation plants, they are often not built. This article aims to explore this paradox through mathematical analysis, shedding light on the complex factors that contribute to this seemingly irrational economic decision.

## The Basic Economics: A Cost Comparison

Let’s start by comparing the costs of bottled water versus a water sanitation plant:

- Bottled Water Cost (BWC):

- Annual cost per person: $740 (conservative estimate for Roatan)
- Population (P): 75,000 (approximate for Roatan)
- Time horizon (T): 25 years (typical lifespan of a water plant) Total Bottled Water Cost (TBWC) = BWC * P * T

TBWC = $740 * 75,000 * 25 = $1,387,500,000

- Water Sanitation Plant Cost (WSPC):

- Construction cost (CC): $10,000,000
- Annual operating cost (OC): $1,000,000 Total Water Sanitation Plant Cost (TWSPC) = CC + (OC * T)

TWSPC = $10,000,000 + ($1,000,000 * 25) = $35,000,000

The cost difference (CD) over 25 years:

CD = TBWC – TWSPC = $1,387,500,000 – $35,000,000 = $1,352,500,000

This simple calculation suggests that a water sanitation plant could save the community over $1.3 billion over 25 years. So why isn’t it built?

## The Time Value of Money: Net Present Value Analysis

One key factor is the time value of money. Future savings are worth less than immediate costs. Let’s use Net Present Value (NPV) analysis with a discount rate (r) of 10% (common in developing regions due to high risk and inflation):

NPV = -CC + Σ(CFt / (1+r)^t)

Where:

CC = Construction cost

CFt = Cash flow in year t (savings from not buying bottled water minus operating costs)

r = Discount rate

t = Year (1 to 25)

CFt = (BWC * P) – OC = ($740 * 75,000) – $1,000,000 = $54,500,000

Using this formula (which can be calculated precisely using spreadsheet software), we find:

NPV ≈ $395,000,000

While still positive, this number is significantly lower than the simple cost difference calculated earlier. This NPV represents the project’s value in today’s money, accounting for the time value of money.

## The Funding Gap: Initial Capital Requirements

Despite the positive NPV, the initial capital requirement presents a significant barrier. Let’s compare it to the annual government budget:

Assuming Roatan’s government budget is 5% of its GDP:

- Roatan’s GDP (estimated): $200 million
- Annual government budget: $10 million

The water plant’s construction cost ($10 million) represents an entire year’s budget. This highlights the difficulty in allocating funds to a single project, despite its long-term benefits.

## Risk and Uncertainty: Scenario Analysis

Future benefits are also uncertain. Let’s consider three scenarios:

- Best case: 2% annual population growth
- Base case: Stable population
- Worst case: 1% annual population decline

We can calculate the NPV for each scenario:

- Best case NPV ≈ $480 million
- Base case NPV ≈ $395 million (as calculated earlier)
- Worst case NPV ≈ $310 million

While all scenarios show a positive NPV, the uncertainty represented by this range ($170 million difference between best and worst cases) can make decision-makers hesitant.

## Conclusion: The Economic Paradox Explained

The mathematical analysis reveals several key points:

- While a water sanitation plant is cost-effective in the long run, the benefits are spread over many years, while the costs are front-loaded.
- The high discount rate significantly reduces the present value of future savings.
- The initial capital requirement is prohibitively high compared to available resources.
- Uncertainty about future population and economic conditions adds risk to the investment.

These factors combine to create a situation where, despite clear long-term benefits, the short-term economic realities make it challenging to undertake such a project. This paradox highlights the need for innovative financing solutions, international aid focused on infrastructure, and policies that prioritize long-term planning in developing regions.